an accurate predictor-corrector-type nonstandard finite
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Research ArticleAn Accurate Predictor-Corrector-Type Nonstandard FiniteDifference Scheme for an SEIR Epidemic Model
Asma Farooqi12 Riaz Ahmad1 Rashada Farooqi3 Sayer O Alharbi4
Dumitru Baleanu 567 Muhammad Rafiq8 Ilyas Khan 4 and M O Ahmad9
1Faculty of Science Yibin University Yibin 644000 China2School of Energy and Power Engineering Xirsquoan Jiaotong University Xirsquoan 710049 China3Wah Medical College POF Hospital Wah Cantt 47040 Pakistan4Department of Mathematics College of Science Al-Zulfi Majmaah University Al-Majmaah 11952 Saudi Arabia5Department of Mathematics Cankaya University Ankara 06790 Turkey6Institute of Space Sciences Magurele 077125 Romania7Department of Medical Research China Medical University Hospital China Medical University Taichung 40447 Taiwan8Department of Mathematics Faculty of Sciences University of Central Punjab Lahore Pakistan9Department of Mathematics and Statistics University of Lahore Lahore 54590 Pakistan
Correspondence should be addressed to Dumitru Baleanu baleanumailcmuhorgtw and Ilyas Khan isaidmuedusa
Received 11 September 2020 Revised 23 September 2020 Accepted 22 November 2020 Published 15 December 2020
Academic Editor Hijaz Ahmad
Copyright copy 2020 Asma Farooqi et al-is is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
-e present work deals with the construction development and analysis of a viable normalized predictor-corrector-typenonstandard finite difference scheme for the SEIR model concerning the transmission dynamics of measles -e proposednumerical scheme double refines the solution and gives realistic results even for large step sizes thus making it economical whenintegrating over long time periods Moreover it is dynamically consistent with a continuous system and unconditionallyconvergent and preserves the positive behavior of the state variables involved in the system Simulations are performed toguarantee the results and its effectiveness is compared with well-known numerical methods such as RungendashKutta (RK) and Eulermethod of a predictor-corrector type
1 Introduction
Mathematicians and biologists have been working for a longtime on the biological process of life science-ey succeededin evaluating some remarkable results from their work Animportant mark in mathematical biology is the mathe-matical modeling of infectious diseases [1] Measles is one ofsuch a highly infectious childhood disease caused by re-spiratory infection by a Morbilli virus measles virus ArthurRansom first observed the irregular cyclic behavior ofmeasles that is considered as a mainly conspicuous facet ofmeasles -e age structure of the population contact im-migration rate and the school seasons were known as acrucial phase for the swell of measles [2ndash4] WilliamHammer in 1906 published a discrete numerical model for
the transmission of measles epidemic Later assumption ofldquoMass Actionrdquo is applied to that model which is the basic ruleto the current theory of deterministic modeling of infectiousdiseases [5]
Society has a keen concern in knowing the major evo-lution for the spread of diseases Analytical results give asolution to these problems but for limited cases and causesmany problems -e homotopy perturbation method andvariational iteration method can be used for the solution ofthe epidemic models [6] However the first choice to solvethese laws of nature is the numerical method based on adifference scheme for good approximations [7ndash9] In gen-eral already developed numerical schemes such as EulerRungendashKutta and others at times stop working by gener-ating nonphysical results -ese unnecessary oscillations
HindawiJournal of MathematicsVolume 2020 Article ID 8830829 18 pageshttpsdoiorg10115520208830829
contrived chaos and false fixed points [10] Moreover somemethods are unsuccessful if we check them on larger stepsizes [11] To avoid such discrepancies the numericalschemes based on the ldquononstandard finite difference method(NSFD)rdquo are established -ese techniques were first de-veloped by R E Mickens [7 12 13] -e created numericalschemes preserve the essential properties such as dynamicalconsistency stability and equilibrium points [14ndash21] Re-searchers have developed competitive NSFD schemes forepidemic diseases Many of these NSFD schemes are con-sistent for small step sizes with the continuousmodel but forlarge step sizes the unwanted oscillations have been ob-served Piyanwong Jansen and Twizel have constructed apositive and unconditionally stable scheme for SIR and SEIRmodels respectively [22 23] Nevertheless the lack of ap-plication of conservation law in their developed schemesexplicitly caused impracticable and unrealistic solutionswhile Abraham and Gilberto have developed NSFD schemesof the SIR epidemic model to obtain the physically realisticsolutions for all step sizes where they apply the conservationlaw in addition to nonlocal approximation [24]
In this paper we have developed a normalized NSFDMof predictor-corrector- (PC-) type inspired by the previouswork discussed to double refine the numerical solution of anonlinear dynamics regarding the transmission of measlesTo keep the method explicit we will use the forward dif-ference approximations for the first derivative terms -enonlocal approximations are used to tackle the nonlinearterms with φ(h) as a nonclassical denominator function Byusing this idea the measles model will converge to equi-librium points even for the larger step sizes
2 The Mathematical Frame of Work
In this work the dynamics of the measles epidemic de-scribed by the SEIRmathematical model suggested by Jansenand E H Twizel is considered [23] In the SEIR model thetotal human population is categorized as susceptible ex-posed infectious and recovered subpopulations denoted byS E I and R respectively Consider the flow of the SEIRmodel for the measles epidemic as shown in Figure 1Here
S susceptible individualsE exposed individualsI infected individualR recovered individualμ birth rate and death rateβ the rate at which susceptible individuals are in-fected by those who are infectiousσ the rate at which exposed individuals becomeinfectedc the rate at which infected individuals recover
Here μ β σ and c are considered as positive parametersFurthermore we assume that (E0 I0 ne 0) Consider N is theconstant size of the population so that the number of re-covered individuals R R(t) at time t is defined byR N minus S minus E minus I A susceptible is a move to the exposed
model where the individual is infected but yet not infec-tious After sometimes the individual becomes infectiousand enters into the infected compartment and in this waythe disease spreads into the population
-e mathematical model is written as
dS
dt μN minus μS minus βIS tgt 0 S(0) S
0
dE
dt βIS minus μE minus σE tgt 0 E(0) E
0
dI
dt σE minus μI minus cI tgt 0 I(0) I
0
dR
dt cI minus μR tgt 0 R(0) R
0
⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭
(1)
where
S + E + I + R N (2)
Equation (2) shows that the total size of the populationremains constant which is connected with the continuoussystem (1) -e following two points give the equilibriumpoints of (1)
-e disease-free equilibrium (DFE) point (N 0 0)
-e endemic equilibrium (EE) point (NR0 μNμ+
σ(1 minus 1R0) μβ(R0 minus 1))
Here R0 σβN(μ + σ)(μ + c) is the basic reproductivenumber associated with the measles model
For the sake of brevity we are not mentioning RK-4 andEuler-PC scheme
3 Numerical Modeling
To the construction of NSFD scheme the continuous system(1) is discretized using the forward difference approximationfor the first-order time derivatives -us if f(t) is differ-entiable the fprime(t) can be approximated by
df(t)
dt
f(t + h) minus f(t)
φ(h)+ O(h) as h⟶ 0 (3)
where φ(h) is a real-valued function satisfying the conditionφ(h)⟶ 0 as h⟶ 0 We have
φ(h) 1 minus eminus h
(4)
μN βSI μE
μS
γI
μI μR
I(t) R(t)
S(t) E(t)
σE
Figure 1 Flowchart of the SEIR Model for measles epidemic
2 Journal of Mathematics
-us the NSFD scheme for system (1) takes the form
Sn+1
minus Sn
φ(h) μN minus μS
n+1minus βS
n+1I
n
En+1
minus En
φ(h) βS
n+1I
nminus μE
n+1minus σE
n+1
In+1
minus In
φ(h) σE
n+1minus μI
n+1minus cI
n+1
Rn+1
minus Rn
φ(h) cI
n+1minus μR
n+1
⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭
(5)
From equation (5) we have
(1 + φ(h)μ) Sn+1
+ En+1
+ In+1
+ Rn+1
1113960 1113961
φ(h)μN minus φ(h)μ Sn
+ En
+ In
+ Rn
( 1113857(6)
-us if Sn + En + In + Rn N for all nge 0 then Sn+1+
En+1 + In+1 + Rn+1 N for all nge 0-us the conservation law property is satisfied System
(5) is written as
Sn+1
S
n+ μφ(h)N
1 + φ(h) μ + βIn
( 1113857
En+1
E
n+ φ(h)βS
n+1I
n
1 + φ(h)(μ + σ)
In+1
I
n+ φ(h)σE
n+1
1 + φ(h)(μ + c)
⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭
(7)
Now the positivity of Sn+1 En+1 In+1 andRn+1 is guar-anteed if 0lt Sn lt 1 0ltEn lt 1 0lt In lt 1 and 0ltRn lt 1 for allnge 0 -us the constructed scheme has the followingproperties
-e conservation law is satisfied-e positivity and boundedness of the solution issatisfied for system (7) we have that if 0lt Sn lt 10ltEn lt 1 0lt In lt 1 and 0ltRn lt 1 then 0lt Sn+1 lt 10ltEn+1 lt 1 0lt In+1 lt 1 and 0ltRn+1 lt 1 for all nge 0
4 NSFD Predictor-Corrector Scheme
-is section is improved by the approach of a predictor-corrector-type NSFD scheme (7) to obtain the benefits ofboth methods For the development of this scheme firstlysystem (7) is taken as a predictor scheme ie
Sn+1p
Sn
+ μφ(h)N
1 + φ(h) μ + βIn
( 1113857
En+1p
En
+ φ(h)βSn+1p I
n
1 + φ(h)(μ + σ)
In+1p
In
+ φ(h)σEn+1p
1 + φ(h)(μ + c)
⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭
(8)
Now we evaluate system (1) at time t + φ(h) and in-troduce the term ϵminus 1Sn+1φ(h) (where ϵ is just like the ac-celerating factor and its range is (0lt ϵlt 1)) -us theexpression will be NSFDCL scheme that preserves conser-vation law represented as
Sn+1
minus Sn
φ(h) μN minus μS
n+1minus βS
n+1I
n+1minusϵminus 1
Sn+1
φ(h)+ϵminus 1
Sn+1
φ(h)
En+1
minus En
φ(h) βS
n+1I
nminus (μ + σ)E
n+1
In+1
minus In
φ(h) σE
n+1minus (μ + c)I
n
Rn+1
minus Rn
φ(h) cI
n+1minus μR
n+1
(9)
-us the corrector scheme is obtained as
Sn+1c
Sn
+ μφ(h)N + ϵminus 1S
n+1p
1 + ϵminus 1 + μφ(h) + βφ(h)In+1p
En+1c
En
+ φ(h)βSn+1c I
n+1p
1 + φ(h)(μ + σ)
In+1c
In
+ φ(h)σEn+1c
1 + φ(h)(μ + c)
⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭
(10)
where Rn+1 N minus Sn+1c minus En+1
c minus In+1c
5 Convergence Analysis
In this section the unconditional convergence of the nu-merical solution is presented by the proposed methodTaking the values as
F1(S E I) S + μNϕ(h)
1 + ϕ(h)[μ + βI]
F2(S E I) E + βϕ(h)IF1
1 + ϕ(h)[μ + σ]
F3(S E I) I + σϕ(h)F2
1 + ϕ(h)[μ + c]
G1(S E I) S + μNϕ(h) + ϵminus 1
F1
1 + ϵminus 1 + μϕ(h) + βϕ(l)F3
G2(S E I) E + βϕ(h)G1F3
1 + ϕ(h)[μ + σ]
G3(S E I) I + σϕ(h)G2
1 + ϕ(h)[μ + c]
(11)
where F1 F2 andF3 are given as in equation (8)
Journal of Mathematics 3
-e convergence and stability analysis of scheme (10) iscarried out by calculating the eigenvalues of the Jacobian ofthe linearized scheme and studying its behavior and
evaluated at the fixed points If (Slowast Elowast Ilowast) is the fixed pointof system (1) then the Jacobian matrix JG is given by
JG Slowast Elowast Ilowast
( 1113857
zG1 Slowast Elowast Ilowast
( 1113857
zS
zG1 Slowast Elowast Ilowast
( 1113857
zE
zG1 Slowast Elowast Ilowast
( 1113857
zI
zG2 Slowast Elowast Ilowast
( 1113857
zS
zG2 Slowast Elowast Ilowast
( 1113857
zE
zG2 Slowast Elowast Ilowast
( 1113857
zI
zG3 Slowast Elowast Ilowast
( 1113857
zS
zG3 Slowast Elowast Ilowast
( 1113857
zE
zG3 Slowast Elowast Ilowast
( 1113857
zI
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
(12)
where
zF1
zS
1[1 + μϕ(h) + βϕ(h)I]
zF1
zE
zF1
zI
minus (S + μNϕ(h))βϕ(h)
[1 + μϕ(h) + βϕ(h)I]2
zF2
zS
βϕ(h)IzF1zS
1 + ϕ(h)[μ + σ]
zF2
zE1 + βϕ(h)IzF1zE
1 + ϕ(h)[μ + σ]
zF2
zIβϕ(h) F1 + IzF1zI1113864 1113865
1 + ϕ(h)[μ + σ]
zF3
zE
σϕ(h)zF2zE
1 + ϕ(h)[μ + c]
zF3
zI1 + σϕ(h)zF2zS
1 + ϕ(h)[μ + c]
zG1
zS
1 + ϵminus 1zF1zS1113872 1113873 1 + ϵminus 1
+ μϕ(h) + βϕ(h)F31113960 1113961 minus S + μNϕ(h) + ϵminus 1F11113872 1113873βϕ(h)zF3zS
1 + ϵminus 1 + μϕ(h) + βϕ(l)F31113960 11139612
zG1
zEϵminus 1
zF1zE1113872 1113873 1 + ϵminus 1+ μϕ(h) + βϕ(h)F31113960 1113961 minus S + μNϕ(h) + ϵminus 1
F11113872 1113873βϕ(h)zF3zE
1 + ϵminus 1 + μϕ(h) + βϕ(h)F31113960 11139612
zG1
zIϵminus 1
zF1zI1113872 1113873 1 + ϵminus 1+ μϕ(h) + βϕ(h)F31113960 1113961 minus S + μNϕ(h) + ϵminus 1
F11113872 1113873βϕ(h)zF3zI
1 + ϵminus 1 + μϕ(h) + βϕ(h)F31113960 11139612
zG2
zS
βϕ(h)
1 + ϕ(h)[μ + σ]
zG1
zSF3 + G1
zF3
zS1113896 1113897
zG2
zE1 + βϕ(h) zG1zEF3 + G1zF3zE1113858 1113859
1 + ϕ(h)[μ + σ]
4 Journal of Mathematics
zG2
zI
βϕ(h)
1 + ϕ(h)[μ + σ]
zG1
zIF3 + G1
zF3
zI1113896 1113897
zG3
zS
σϕ(h)zG2zS
1 + ϕ(h)[μ + c]
zG3
zE
σϕ(h)zG2zE
1 + ϕ(h)[μ + c]
zG3
zI1 + σϕ(h)zG2zI
1 + ϕ(h)[μ + c]
(13)
51 Disease-Free Equilibrium (SoEo Io) (N 0 0) -evalues of functions and its derivatives at DFE point (N 0 0)
are as
F1(N 0 0) N F2(N 0 0) 0 F3(N 0 0) 0
G1(N 0 0) N G2(N 0 0) 0 G3(N 0 0) 0
zF1(N 0 0)
zS
11 + μϕ(h)
zF1(N 0 0)
zE 0
zF1(N 0 0)
zI
minus Nβϕ(h)
[1 + μϕ(h)]
zF2(N 0 0)
zS 0
zF2(N 0 0)
zE
11 + ϕ(h)[μ + σ]
zF2(N 0 0)
zI
βϕ(h)
1 + ϕ(h)[μ + σ]
zF3(N 0 0)
zS 0
zF3(N 0 0)
zE
σϕ(h)
[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]]
zF3(N 0 0)
zI1 +[σϕ(h)βNϕ(h)1 + ϕ(h)(μ + c)]
1 + ϕ(h)[μ + c]
zG1(N 0 0)
zS
11 + μϕ(h)
zG1(N 0 0)
zE
minus Nσϕ(h)βϕ(h)
1 + μϕ(h) + ϵminus 11113960 1113961[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]
Journal of Mathematics 5
zG1(N 0 0)
zI
minus Nβϕ(h)
1 + ϵminus 1+ μϕ(h)1113960 1113961
ϵminus 1
1 + μϕ(h)+
1[1 + ϕ(h)(μ + c)
+σNβϕ2
(h)
[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]1113890 1113891
zG2(N 0 0)
zS 0
zG2(N 0 0)
zE
11 + ϕ(h)(μ + σ)
1 +Nβσϕ2(h)
[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]1113890 1113891
zG2(N 0 0)
zI
βϕ(h)
[1 + ϕ(h)(μ + σ)]
1 + ϕ(h)(μ + σ) + Nσβϕ2(h)
[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]1113896 1113897
zG3(N 0 0)
zS
zG3(N 0 0)
zE
σϕ(h)
[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]1 +
σβNϕ2(h)
[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]1113896 1113897
zG3(N 0 0)
zI
1
[1 + ϕ(h)(μ + σ)]+
Nσβϕ2(h)
[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]2 +
N2β2σ2ϕ4(h)
[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + σ)]
2
(14)
For ease in the calculation we use the followingconventions
α μNϕ(h)gt 0 η1 μϕ(h)gt 0 η2
βϕ(h)gt 0 η3 σϕ(h)gt 0
θ 1 + ϵminus 1+ μϕ(h) 1 + ϵminus 1
+ η1 gt 1
δ1 1 + ϕ(h)[μ + σ]gt 1 δ2
(15)
It is clear that θgt η1 1 + η1 gt δ1 1 + η1 gt δ2 and δ2 θ minus ϵminus 1 + cϕ(h) If R0 σβN(μ + σ)(μ + c)lt 1 the Ja-cobian calculated at disease-free points (Slowast0 Elowast0 Ilowast0 )
(N 0 0) is given as
JG(N 0 0)
zG1(N 0 0)
zS
zG1(N 0 0)
zE
zG1(N 0 0)
zI
zG1(N 0 0)
zS
zG1(N 0 0)
zE
zG1(N 0 0)
zI
zG1(N 0 0)
zS
zG1(N 0 0)
zE
zG1(N 0 0)
zI
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
(16)
Since the partial derivatives using the conventions are
zG1(N 0 0)
zS
11 + η1
zG1(N 0 0)
zE
minus Nη2η3θδ1δ2
zG1(N 0 0)
zI
minus Nη2θϵminus 1
1 + η1+
1δ2
+Nη2η3δ1δ2
1113890 1113891
zG2(N 0 0)
zS 0
zG2(N 0 0)
zE
1δ1
+Nη2η3δ21δ2
1113890 1113891
zG2(N 0 0)
zI
Nη2δ1δ2
+N
2η22η3δ21δ2
1113890 1113891
zG3(N 0 0)
zS 0
zG3(N 0 0)
zE
η3δ1δ2
+Nη2η
23
δ21δ22
1113890 1113891
zG3(N 0 0)
zI
1δ2
+Nη2η3δ1δ
22
+N
2η22η23
δ21δ22
1113890 1113891
(17)
6 Journal of Mathematics
So
JG(N 0 0)
11 + η1
minus Nη2η3θδ1δ2
minus Nη2θϵminus 1
1 + η1+
1δ2
+Nη2η3δ1δ2
1113890 1113891
01δ1
+Nη2η3δ21δ2
1113890 1113891Nη2δ1δ2
+N
2η22η3δ21δ2
1113890 1113891
0η3δ1δ2
+Nη2η
23
δ21δ22
1113890 11138911δ2
+Nη2η3δ1δ
22
+N
2η22η23
δ21δ22
1113890 1113891
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
(18)
To calculate the eigenvalues put det (J minus λI) 0
λ1 11 + η1 11 + μϕ(h)lt 1 And
λ2 minus λδ1δ2 + Nη2η3
δ21δ21113888 1113889 +
δ21δ2 + δ1Nη2η3 + N2η22η
23
δ21δ22
1113888 11138891113890 1113891
+δ1δ2 + Nη2η3
δ21δ21113888 1113889
δ21δ2 + δ1Nη2η3 + N2η22η
23
δ21δ22
1113888 1113889 minusNη2δ1 + N
2η22η3δ21δ2
1113888 1113889η3δ1δ2 + Nη2η
23
δ21δ22
1113888 11138891113890 1113891 0
λ2 minus Aλ + B 0
(19)
where
A δ1δ2 + Nη2η3δ21δ21113872 1113873 + δ21δ2 + δ1Nη2η3 + N
2η22η23δ
21δ
221113872 1113873 trace J
lowast
B δ1δ2 + Nη2η3
δ21δ21113888 1113889
δ21δ2 + δ1Nη2η3 + N2η22η
23
δ21δ22
1113888 1113889 minusNη2δ1 + N
2η22η3δ21δ2
1113888 1113889η3δ1δ2 + Nη2η
23
δ21δ22
1113888 11138891113890 1113891 det Jlowast
Jlowast
1δ1
+Nη2η3δ21δ2
1113890 1113891Nη2δ1δ2
+N
2η22η3δ21δ2
1113890 1113891
η3δ1δ2
+Nη2η
23
δ21δ22
1113890 11138911δ2
+Nη2η3δ1δ
22
+N
2η22η23
δ21δ22
1113890 1113891
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
(20)
To calculate the eigenvalues of Jlowast the following lemma isproved
Lemma 1 For the quadratic equation λ2 minus λA + B 0 bothroots satisfy |λi|lt 1 i 2 3 if and only if the followingconditions are satisfied [25]
(1) 1 minus A + Bgt 0
(2) 1 + A + Bgt 0
(3)Blt 1
(21)
Let us define A TraceJlowast B DetJlowast -erefore
Journal of Mathematics 7
A δ1 + δ2( 1113857 Nη2η3 + δ1δ2( 11138571113858 1113859 + N
2η22η23
δ21δ22
gt 0
B 1
δ1δ2+
Nη2η3δ21δ
22lt 1
f(minus 1) 1 + A + Bgt 0
f(0) B 1
δ1δ2+
Nη2η3δ21δ
22lt
1δ1δ2
+Nη2η3δ1δ2
1 + Nη2η3
δ1δ2
1 + R0(μ + σ)(μ + c)ϕ2(h)
1 + ϕ2(h)(μ + σ)(μ + c) + ϕ(h)[(μ + σ)(μ + c)]
lt 1 sinceR0 lt 1( 1113857
rArrBlt 1
(22)
Now for
f(1) 1 minus A + B
δ21δ
22 + δ1 + δ2( 1113857 Nη2η3 + δ1δ2( 11138571113858 1113859 + N
2η22η23 + Nη2η3 + δ1δ2
δ21δ22
gt 0
(23)
As all the conditions of the lemma hold for R0 lt 1therefore the absolute value of both eigenvalues of Jlowast is less
than one whenever R0 lt 1 -us the numerical scheme inequations (8) and (10) will converge unconditionally todisease-free equilibrium (N 0 0) for any value of time step h
whenever R0 lt 1 which is also computationally verified inFigure 2(a)
52 Endemic Equilibrium (SeEe Ie) (NR0 μN μ + σ(1 minus
1 R0) μβ(R0 minus 1)) -e value of functions and its deriv-atives at an endemic point is as
F1 Se Ee Ie( 1113857 N
R0
F2 Se Ee Ie( 1113857 μN 1 minus 1R0( 1113857
(μ + σ)
F3 Se Ee Ie( 1113857 μβ
R0 minus 1( 1113857
G1 Se Ee Ie( 1113857 N
R0
G2 Se Ee Ie( 1113857 μβ
R0 minus 1( 1113857
G3 Se Ee Ie( 1113857 μN 1 minus 1R0( 1113857
(μ + σ)
(24)
8 Journal of Mathematics
zF1 Se Ee Ie( 1113857
zS
11 + μϕ(h)R0
zF1 Se Ee Ie( 1113857
zE 0
zF1 Se Ee Ie( 1113857
zI
minus Nβϕ(h)
1 + μϕ(h)R0
zF2 Se Ee Ie( 1113857
zS
μϕ(h) R0 minus 1( 1113857
1 + μϕ(h)R01113858 1113859[1 + ϕ(h)(μ + σ)]
zF2 Se Ee Ie( 1113857
zE
1[1 + ϕ(h)(μ + σ)]
zF2 Se Ee Ie( 1113857
zI
minus Nβϕ(h) μϕ(l)R20 minus 2μϕ(h)R0 minus 11113960 1113961
R0[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]
zF3 Se Ee Ie( 1113857
zS
μσϕ2(h) R0 minus 1( 1113857
[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)] 1 + μϕ(h)R01113858 1113859
zF3 Se Ee Ie( 1113857
zE
σϕ(h)
[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]
zF3 Se Ee Ie( 1113857
zI
R0[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859 minus σNβϕ2(h) μϕ(h)R20 minus 2μϕ(h) minus 11113960 1113961
R0[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859[1 + ϕ(h)(μ + c)]
(25)
zG1 Se Ee Ie( 1113857
zS
11 + μϕ(h)R01113858 1113859
minusNμβσϕ3(h) R0 minus 1( 1113857
R0 1 + ϵminus 1 + μϕ(h)R01113960 1113961[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859[1 + ϕ(h)(μ + c)]
zG1 Se Ee Ie( 1113857
zE
minus Nβσϕ2(h)
R0 1 + ϵminus 1 + μϕ(h)R01113960 1113961[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]
Eige
n va
lues
Jacobian at DFE
0492
0494
0496
0498
05
0502
0504
0506
0508
051
1000 2000 3000 4000 5000 6000 7000 8000 9000 100000Step size
(a)
098
0981
0982
0983
0984
0985
0986
0987
0988
0989Jacobian at EE
Eige
n va
lues
1000 2000 3000 4000 5000 6000 7000 8000 9000 100000Step size
(b)
Figure 2 Spectral radius of the Jacobian at equilibrium points
Journal of Mathematics 9
zG1 Se Ee Ie( 1113857
zI
minus Nβϕ(h)
1 + ϵminus 1 + μϕ(h)R01113960 11139611 + μϕ(h)R0
middot ϵminus 1+
R0[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859 minus σNβϕ2(h) μϕ(h)R20 minus 2μϕ(h)R0 minus 11113960 1113961
R20[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859[1 + ϕ(h)(μ + c)]
⎡⎣ ⎤⎦
zG2 Se Ee Ie( 1113857
zS
μβϕ(h) R0 minus 1( 1113857
β[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 11138591113890
minusNμβ2σϕ4(h)μ R0 minus 1( 1113857
2
βR0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)] 1 + ϵminus 1 + μϕ(h)R01113960 1113961 1 + μϕ(h)R01113858 1113859
+Nμβσϕ3(h) R0 minus 1( 1113857
R0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)] 1 + μϕ(h)R01113858 1113859
1113891
zG2 Se Ee Ie( 1113857
zE
1[1 + ϕ(h)(μ + σ)]
+Nβσϕ2(h)
R0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)]
minusNμβ2σϕ3
(h) R0 minus 1( 1113857
βR0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)] 1 + ϵminus 1 + μϕ(h)R01113960 1113961
zG2 Se Ee Ie( 1113857
zI
minus Nμβ2ϕ2(h)ϵminus 1R0 minus 1( 1113857
β[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859 1 + ϵminus 1 + μϕ(h)R01113960 1113961
+Nμβ2ϕ2(h) R0 minus 1( 1113857 R0[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859 minus Nβσϕ2(h) μϕ(h)R
20 minus 2μϕ(h)R0 minus 11113960 11139611113966 1113967
βR20[1 + ϕ(h)(μ + σ)]
2[1 + ϕ(h)(μ + c)] 1 + ϵminus 1 + μϕ(h)R01113960 1113961 1 + μϕ(h)R01113858 1113859
+Nβϕ(h)
R0[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]minus
N2β2σϕ3(h) μϕ(h)R
20 minus 2μϕ(h)R0 minus 11113960 1113961
R20[1 + ϕ(h)(μ + σ)]
2[1 + ϕ(h)(μ + c)] 1 + μϕ(h)R01113858 1113859
zG3 Se Ee Ie( 1113857
zS
σϕ(h)
[1 + ϕ(h)(μ + c)]
μβϕ(h) R0 minus 1( 1113857
β[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 11138591113890
minusNμβ2σϕ4(h)μ R0 minus 1( 1113857
2
βR0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)] 1 + ϵminus 1 + μϕ(h)R01113960 1113961 1 + μϕ(h)R01113858 1113859
+Nμβσϕ3(h) R0 minus 1( 1113857
R0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)] 1 + μϕ(h)R01113858 1113859
1113891
zG3 Se Ee Ie( 1113857
zE
σϕ(h)
[1 + ϕ(h)(μ + c)]
1[1 + ϕ(h)(μ + σ)]
minusNμβ2σϕ3(h) R0 minus 1( 1113857
βR0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)] 1 + ϵminus 1 + μϕ(h)R01113960 1113961
⎡⎢⎣
+Nβσϕ2
(h)
R0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)]
1113891
zG3 Se Ee Ie( 1113857
zI
σϕ(h)
[1 + ϕ(h)(μ + c)]
minus Nμβ2ϕ2(h)ϵminus 1R0 minus 1( 1113857
β[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859 1 + ϵminus 1 + μϕ(h)R01113960 1113961⎡⎢⎣
+Nμβ2ϕ2(h) R0 minus 1( 1113857 R0[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859 minus Nβσϕ2(h) μϕ(h)R
20 minus 2μϕ(h)R0 minus 11113960 11139611113966 1113967
βR20[1 + ϕ(h)(μ + σ)]
2[1 + ϕ(h)(μ + c)] 1 + ϵminus 1 + μϕ(h)R01113960 1113961 1 + μϕ(h)R01113858 1113859
⎤⎥⎦
10 Journal of Mathematics
Table 1 Comparison of NSFD scheme with Euler and RK-4 schemes
l 001 01 100 1000Euler PC Convergence Divergence Divergence DivergenceRK-4 Convergence Divergence Divergence DivergenceNSFDPC Convergence Convergence Convergence Convergence
times107
NSFD-PC
Disease-free equilibrium
1
15
2
25
3
35
4
45
5
Susc
eptib
le fr
actio
n
20 40 60 80 100 120 140 160 180 2000Time (years) h = 001
(a)
Disease-free equilibriumtimes104
NSFD-PC
02 04 06 08 1 12 14 16 18 20Time (years) h = 001
0
05
1
15
2
25
3
35
4
45
5
Expo
sed
fract
ion
(b)
times104
NSFD-PC
Disease-free equilibrium
0
05
1
15
2
25
3
Infe
cted
frac
tion
02 04 06 08 1 12 14 16 18 20Time (years) h = 001
(c)
times107
NSFD-PC
X 5675Y 2435e + 007
Endemic equilibrium
100 200 300 400 500 6000Time (years) h = 001
1
15
2
25
3
35
Susc
eptib
le fr
actio
n
(d)
Figure 3 Continued
Journal of Mathematics 11
+Nβϕ(h)
R0[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]minus
N2β2σϕ3(h) μϕ(h)R
20 minus 2μϕ(h)R0 minus 11113960 1113961
R20[1 + ϕ(h)(μ + σ)]
2[1 + ϕ(h)(μ + c)] 1 + μϕ(h)R01113858 1113859
(26)
-e Jacobian evaluated at an endemic point(Slowaste Elowaste Ilowaste ) (Se Ee Ie) (NR0 μNμ + σ(1 minus 1R0) μβ(R0 minus 1)) is evaluated as
JG Slowaste Elowaste Ilowaste( 1113857
zG1 Slowaste Elowaste Ilowaste( 1113857
zS
zG1 Slowaste Elowaste Ilowaste( 1113857
zE
zG1 Slowaste Elowaste Ilowaste( 1113857
zI
zG1 Slowaste Elowaste Ilowaste( 1113857
zS
zG1 Slowaste Elowaste Ilowaste( 1113857
zE
zG1 Slowaste Elowaste Ilowaste( 1113857
zI
zG1 Slowaste Elowaste Ilowaste( 1113857
zS
zG1 Slowaste Elowaste Ilowaste( 1113857
zE
zG1 Slowaste Elowaste Ilowaste( 1113857
zI
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
(27)
It is tedious to calculate the eigenvalues of the Jacobian ofendemic equilibrium analytically so we have plotted thelargest eigenvalue against each step size and Figure 2(b)shows that for all step sizes the spectral radius of the Ja-cobian of EE remains less than one if R0 gt 1 which impliesthat the numerical scheme in equations (8) and (10) isunconditionally convergent if R0 gt 1 for all step sizes
6 Numerical Results and Discussion
All the methods showed the convergence for small step sizesin Table 1 However for large values of step sizes only thenormalized NSFDPC converge to the correct disease-freepoint for β 01 times 10minus 5 as well as the correct endemic pointfor β 03 times 10minus 5 -is means that for the value of the basic
reproductive number (threshold parameter) R0 lt 1 thenDFE is stable ie solution to the system (S E I R) areconverging to it -is can be seen from Figures 3(a)ndash3(c) forh 001 Also for the value of the basic reproductivenumber (threshold parameter) R0 gt 1 then EE is stable iesolution to the system (S E I R) is converging to it-is canbe seen from Figures 3(d)ndash3(f ) for h 001
Figures 4(a)ndash4(c) show how the NSFDPC method ofDFE converges to the equilibrium point for different stepsizes Figures 4(d)ndash4(f) show how the NSFDPC method ofEE converges to the equilibrium point for different step sizes
For comparison with the well-known RK-4 method tosystem (1) using the parameter values given in Table 2 it isfound that in Figures 5(a)ndash5(f) the numerical solutionconverges for h 001 for both equilibrium points and
times105
NSFD-PC
X 562Y 1125e + 004
Endemic equilibrium
100 200 300 400 500 6000Time (years) h = 001
0
1
2
3
4
5
6
7
8Ex
pose
d fra
ctio
n
(e)
times105
NSFD-PC
X 5523Y 7023
Infe
cted
frac
tion
Endemic equilibrium
100 200 300 400 500 6000Time (years) h = 001
0
05
1
15
2
25
3
35
4
45
5
(f )
Figure 3 Time plots of susceptible exposed and infected population using the NSFDPC method with h 001
12 Journal of Mathematics
times107
NSFD-PC
Disease-free equilibrium
20 40 60 80 100 120 140 160 180 2000Time (years) h = 01
1
15
2
25
3
35
4
45
5Su
scep
tible
frac
tion
(a)
times107
NSFD-PC
Disease-free equilibrium
500 1000 1500 2000 2500 3000 3500 40000Time (years) h = 10
1
15
2
25
3
35
4
45
5
Susc
eptib
le fr
actio
n
(b)
Disease-free equilibriumtimes107
times105
NSFD-PC
05 1 15 2 25 3 35 40Time (years) h = 1000
1
15
2
25
3
35
4
45
5
Susc
eptib
le fr
actio
n
(c)
X 5827Y 7022
Infe
cted
frac
tion
Endemic equilibriumtimes104
NSFD-PC
100 200 300 400 500 6000Time (years) h = 01
0
2
4
6
8
10
12
14
16
18
(d)
Endemic equilibriumtimes104
NSFD-PC
X 9600Y 7022
Infe
cted
frac
tion
1000 2000 3000 4000 5000 6000 7000 8000 9000 100000Time (years) h = 10
0
05
1
15
2
25
3
(e)
Endemic equilibriumtimes104
times105
NSFD-PC
X 861e + 005Y 7022
Infe
cted
frac
tion
1 2 3 4 5 6 7 8 9 100Time (years) h = 1000
0
05
1
15
2
25
3
(f )
Figure 4 Time plots of susceptible and infected population using the NSFDPC method with different step sizes (a) h 01 (b) h 10 and(c) h 1000
Journal of Mathematics 13
Table 2 Parameters value of the measles model
Parameters N μ σ c
5 times 107yearminus 1 002 456 yearminus 1 73 yearminus 1
DFE (S0 E0 I0) (N 0 0) β 01 times 10minus 5
EE (Se Ee Ie) (2435 times 107 1125 times 104 7022671) β 03 times 10minus 5
Disease-free equilibrium
RK4
times107
1
15
2
25
3
35
4
45
5
Susc
eptib
le fr
actio
n
20 40 60 80 100 120 140 160 180 2000Time (years) h = 001
(a)
Expo
sed
fract
ion
Disease-free equilibrium
RK4
times104
0
05
1
15
2
25
3
35
4
45
5
02 04 06 08 1 12 14 16 18 20Time (years) h = 001
(b)
times104
Infe
cted
frac
tion
Disease-free equilibrium
RK4
0
05
1
15
2
25
3
02 04 06 08 1 12 14 16 18 20Time (years) h = 001
(c)
times107
X 5578Y 2435e + 007
Endemic equilibrium
RK4
1
15
2
25
3
35
Susc
eptib
le F
ract
ion
100 200 300 400 500 6000Time (years) h = 001
(d)
Figure 5 Continued
14 Journal of Mathematics
Figure 6 shows overflow for a critical value of step size ieh 01 of DFE and EE respectively
Another popular numerical method is the Eulerpredictor-corrector (E-PC) technique which employs theexplicit Euler method as a predictor and the trapezoidalrule as the corrector -is combination is second-orderaccurate but has similar stability properties in PECEmode to the explicit Euler method alone Figures 7(a)ndash7(f ) show convergence results for h 001 in both casesthat are DFE and EE -e value of h 01 which produces
overflow when solving the system with the parametervalues of Table 2 for the PECE combination can be seen inFigure 8
-us the presented numerical results demonstrated thatNSFDPC has better convergence property following theEuler PC and RK-4 as shown in Figure 9 It is proved thatthe approximations made by other standard numericalmethods experience difficulties in preserving either thestability or the positivity of the solutions or both butNSFDPC is unconditionally convergent
times105
X 5772Y 1125e + 004
Endemic equilibrium
RK4
Expo
sed
fract
ion
0
1
2
3
4
5
6
7
8
9
10
100 200 300 400 500 6000Time (years) h = 001
(e)
times105
X 7659Y 7022
RK4
Endemic equilibrium
Infe
cted
frac
tion
0
1
2
3
4
5
6
100 200 300 400 500 600 700 8000Time (years) h = 001
(f )
Figure 5 Time plots of susceptible exposed and infected population using the RK-4 method with h 001
times1087 RK-4 scheme (disease-free equilibrium)
SusceptibleExposedInfected
01 02 03 04 05 06 07 08 09 10Time (years) h = 01
ndash2
ndash15
ndash1
ndash05
0
05
1
15
2
Popu
latio
n fra
ctio
ns
(a)
RK-4 scheme (endemic equilibrium)
SusceptibleExposedInfected
times10145
01 02 03 04 05 06 07 08 09 10Time (years) h = 01
ndash8
ndash6
ndash4
ndash2
0
2
4
6
8
Popu
latio
n fra
ctio
ns
(b)
Figure 6 Divergence graphs of the population using the RK-4 method with step size h 01
Journal of Mathematics 15
Euler-PC
Disease-free equilibriumtimes107
1
15
2
25
3
35
4
45
5Su
scep
tible
frac
tion
20 40 60 80 100 120 140 160 180 2000Time (years) h = 001
(a)
Euler-PC
Disease-free equilibrium
Expo
sed
fract
ion
times104
0
05
1
15
2
25
3
35
4
45
5
02 04 06 08 1 12 14 16 18 20Time (years) h = 001
(b)
Euler-PC
Disease-free equilibrium
Infe
cted
frac
tion
times104
0
05
1
15
2
25
3
35
02 04 06 08 1 12 14 16 18 20Time (years) h = 001
(c)
Euler-PC
X 5733Y 2435e + 007
Endemic equilibriumtimes107
1
15
2
25
3
35Su
scep
tible
frac
tion
100 200 300 400 500 6000Time (years) h = 001
(d)
Euler-PC
X 5662Y 1118e + 004
Endemic equilibrium
Expo
sed
fract
ion
times105
0
2
4
6
8
10
12
100 200 300 400 500 6000Time (years) h = 001
(e)
Euler-PC
X 5427Y 7025
Endemic equilibrium
Infe
cted
frac
tion
times105
0
1
2
3
4
5
6
7
100 200 300 400 500 6000Time (years) h = 001
(f )
Figure 7 Time plots of population using the Euler-PC method with h 001
16 Journal of Mathematics
7 Conclusions
In this paper a normalized NSFDPC for the SEIR modelconcerning the transmission dynamics of measles is con-structed and analysed -is proposed numerical scheme isvery competitive It is qualitatively stable that is it doublerefines the solution and gives realistic results even for largestep sizes It is dynamically consistent with a continuoussystem and unconditionally convergent and satisfies the
positivity of the state variables involved in the systemSimulations are carried out and its usefulness is comparedwith a well-known numerical method of standard differenceschemes such as RK4 and Euler predictor-corrector method-e standard finite difference schemes are highly dependenton step sizes and initial value problems but NSFDPC isindependent of these two features which make it morepractical Also this method saves computation time andmemory
Euler-PC scheme (disease-free equilibrium)times1041
ndash3
ndash2
ndash1
0
1
2
3
Popu
latio
n fra
ctio
ns
01 02 03 04 05 06 07 08 09 10Time (years) h = 01
SusceptibleExposedInfected
(a)
Euler-PC scheme (endemic equilibrium)times1053
ndash6
ndash4
ndash2
0
2
4
6
Popu
latio
n fra
ctio
ns
01 02 03 04 05 06 07 08 09 10Time (years) h = 01
SusceptibleExposedInfected
(b)
Figure 8 Divergence graphs of the population using the Euler-PC method with step size h 01
NSFD-PCRK-4
times1092
Infe
cted
frac
tion
ndash5
ndash4
ndash3
ndash2
ndash1
0
1
02 04 06 08 1 12 14 16 18 20Time (years) h = 01
(a)
NSFD-PCEuler-PC
times10213
Infe
cted
frac
tion
ndash4
ndash35
ndash3
ndash25
ndash2
ndash15
ndash1
ndash05
0
05
02 04 06 08 1 12 14 16 18 20Time (years) h = 01
(b)
Figure 9 Comparative convergence graphs of the NSFDPC method
Journal of Mathematics 17
Data Availability
-e data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
-e authors declare no conflicts of interest
Acknowledgments
-e authors extend their appreciation to the Deanship ofScientific Research at Majmaah University for funding thiswork under Project no RGP-2019-6
References
[1] E Hunter B Namee and J Kelleher ldquoAn open-data-drivenagent-based model to simulate infectious disease outbreaksrdquoPLoS One vol 14 no 1 pp 1ndash35 2018
[2] E A Bakare Y A Adekunle and K O Kadiri ldquoModeling andsimulation of the dynamics of the transmission of measlesrdquoInternational Journal of Computer Trends and Trends andTechnology vol 3 no 1 pp 174ndash177 2012
[3] L Zhou Y Wang Y Li and Y Michael ldquoGlobal dynamics ofa discrete age-structured SIR epidemic model with applica-tions to measles vaccination strategiesrdquo Mathematical Bio-sciences vol 308 pp 27ndash37 2019
[4] P Widyaningsih D R S Saputro and A W NugroholdquoSusceptible exposed infected recovery (SEIR) model withimmigration equilibria points and its applicationrdquo AmericanInstitute of Physics vol 1 pp 1ndash7 2018
[5] M G Roberts and J A P HeesterbeekMathematical Modelsin Epidemiology Mathematical Models UNESCO-EOLSSAbu Dhabi UAE 2007
[6] J H He F Y Ji and H M Sedighi ldquoDifference equation vsdifferential equation on different scalesrdquo International Journalof Numerical Methods for Heat amp Fluid Flow vol 18pp 0961ndash5539 2020
[7] R E Mickens Nonstandard Finite Difference Models of Dif-ferential Equations World Scientific Singapore 1994
[8] R E Mickens Applications of Nonstandard Finite DifferenceSchemes World Scientific Singapore 2000
[9] R E Mickens ldquoNumerical integration of population modelssatisfying conservation laws NSFD methodsrdquo Journal ofBiological Dynamics vol 1 no 4 pp 427ndash436 2007
[10] J D Lambert Computational Methods in Ordinary Differ-ential Equations Wiley New York NY USA 1973
[11] S M Moghadas M E Alexander B D Corbett andA B Gumel ldquoA positivity-preserving mickens-type dis-cretization of an epidemic modelrdquo Journal of DifferenceEquations and Applications vol 9 no 11 pp 1037ndash10512003
[12] R E Mickens ldquoNonstandard finite difference schemes fordifferential equationsrdquo Journal of Difference Equations andApplications vol 8 no 9 pp 823ndash847 2002
[13] R E Mickens ldquoDynamic consistency a fundamental prin-ciple for constructing nonstandard finite difference schemesfor differential equationsrdquo Journal of Difference Equations andApplications vol 11 no 7 pp 645ndash653 2005
[14] L-I W Roeger and R W Barnard ldquoPreservation of localdynamics when applying central difference methods appli-cation to SIR modelrdquo Journal of Difference Equations andApplications vol 13 no 4 pp 333ndash340 2007
[15] K C Patidar ldquoOn the use of non-standard finite differencemethodsrdquo Journal of Difference Equations and Applicationsvol 11 no 7 pp 735ndash758 2005
[16] R Anguelov and J M-S Lubuma ldquoNonstandard finite dif-ference method by nonlocal approximationrdquo Mathematicsand Computers in Simulation vol 61 no 3-6 pp 465ndash4752003
[17] R Anguelov and J M-S Lubuma ldquoContributions to themathematics of the nonstandard finite difference method andapplicationsrdquo Numerical Methods for Partial DifferentialEquations vol 17 no 5 pp 518ndash543 2001
[18] B M Chen-Charpentier D T Dimitrov andH V Kojouharov ldquoNumerical simulation of multi-speciesbiofilms in porous media for different kineticsrdquo Mathematicsand Computers in Simulation vol 79 no 6 pp 1846ndash18612009
[19] D T Dimitrov and H V Kojouharov ldquoNonstandard finite-difference schemes for general two-dimensional autonomousdynamical systemsrdquo Applied Mathematics Letters vol 18no 7 pp 769ndash774 2005
[20] A J Arenas J A Morantildeo and J C Cortes ldquoNon-standardnumerical method for a mathematical model of RSV epide-miological transmissionrdquo Computers amp Mathematics withApplications vol 56 no 3 pp 670ndash678 2008
[21] L Jodar R J Villanueva A J Arenas and G C GonzalezldquoNonstandard numerical methods for a mathematical modelfor influenza diseaserdquo Mathematics and Computers in Sim-ulation vol 79 no 3 pp 622ndash633 2008
[22] W Piyawong E H Twizell and A B Gumel ldquoAn uncon-ditionally convergent finite-difference scheme for the SIRmodelrdquo Applied Mathematics and Computation vol 146no 2-3 pp 611ndash625 2003
[23] H Jansen and E H Twizell ldquoAn unconditionally convergentdiscretization of the SEIR modelrdquo Mathematics and Com-puters in Simulation vol 58 no 2 pp 147ndash158 2002
[24] A J Arenas G Gonzalez-Parra and B M Chen-CharpentierldquoA nonstandard numerical scheme of predictor-correctortype for epidemic modelsrdquo Computers amp Mathematics withApplications vol 59 no 12 pp 3740ndash3749 2010
[25] F Brauer and C Castillo-Chavez Mathematical Models inPopulation Biology and Epidemiology Springer Berlin Ger-many 2001
18 Journal of Mathematics
contrived chaos and false fixed points [10] Moreover somemethods are unsuccessful if we check them on larger stepsizes [11] To avoid such discrepancies the numericalschemes based on the ldquononstandard finite difference method(NSFD)rdquo are established -ese techniques were first de-veloped by R E Mickens [7 12 13] -e created numericalschemes preserve the essential properties such as dynamicalconsistency stability and equilibrium points [14ndash21] Re-searchers have developed competitive NSFD schemes forepidemic diseases Many of these NSFD schemes are con-sistent for small step sizes with the continuousmodel but forlarge step sizes the unwanted oscillations have been ob-served Piyanwong Jansen and Twizel have constructed apositive and unconditionally stable scheme for SIR and SEIRmodels respectively [22 23] Nevertheless the lack of ap-plication of conservation law in their developed schemesexplicitly caused impracticable and unrealistic solutionswhile Abraham and Gilberto have developed NSFD schemesof the SIR epidemic model to obtain the physically realisticsolutions for all step sizes where they apply the conservationlaw in addition to nonlocal approximation [24]
In this paper we have developed a normalized NSFDMof predictor-corrector- (PC-) type inspired by the previouswork discussed to double refine the numerical solution of anonlinear dynamics regarding the transmission of measlesTo keep the method explicit we will use the forward dif-ference approximations for the first derivative terms -enonlocal approximations are used to tackle the nonlinearterms with φ(h) as a nonclassical denominator function Byusing this idea the measles model will converge to equi-librium points even for the larger step sizes
2 The Mathematical Frame of Work
In this work the dynamics of the measles epidemic de-scribed by the SEIRmathematical model suggested by Jansenand E H Twizel is considered [23] In the SEIR model thetotal human population is categorized as susceptible ex-posed infectious and recovered subpopulations denoted byS E I and R respectively Consider the flow of the SEIRmodel for the measles epidemic as shown in Figure 1Here
S susceptible individualsE exposed individualsI infected individualR recovered individualμ birth rate and death rateβ the rate at which susceptible individuals are in-fected by those who are infectiousσ the rate at which exposed individuals becomeinfectedc the rate at which infected individuals recover
Here μ β σ and c are considered as positive parametersFurthermore we assume that (E0 I0 ne 0) Consider N is theconstant size of the population so that the number of re-covered individuals R R(t) at time t is defined byR N minus S minus E minus I A susceptible is a move to the exposed
model where the individual is infected but yet not infec-tious After sometimes the individual becomes infectiousand enters into the infected compartment and in this waythe disease spreads into the population
-e mathematical model is written as
dS
dt μN minus μS minus βIS tgt 0 S(0) S
0
dE
dt βIS minus μE minus σE tgt 0 E(0) E
0
dI
dt σE minus μI minus cI tgt 0 I(0) I
0
dR
dt cI minus μR tgt 0 R(0) R
0
⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭
(1)
where
S + E + I + R N (2)
Equation (2) shows that the total size of the populationremains constant which is connected with the continuoussystem (1) -e following two points give the equilibriumpoints of (1)
-e disease-free equilibrium (DFE) point (N 0 0)
-e endemic equilibrium (EE) point (NR0 μNμ+
σ(1 minus 1R0) μβ(R0 minus 1))
Here R0 σβN(μ + σ)(μ + c) is the basic reproductivenumber associated with the measles model
For the sake of brevity we are not mentioning RK-4 andEuler-PC scheme
3 Numerical Modeling
To the construction of NSFD scheme the continuous system(1) is discretized using the forward difference approximationfor the first-order time derivatives -us if f(t) is differ-entiable the fprime(t) can be approximated by
df(t)
dt
f(t + h) minus f(t)
φ(h)+ O(h) as h⟶ 0 (3)
where φ(h) is a real-valued function satisfying the conditionφ(h)⟶ 0 as h⟶ 0 We have
φ(h) 1 minus eminus h
(4)
μN βSI μE
μS
γI
μI μR
I(t) R(t)
S(t) E(t)
σE
Figure 1 Flowchart of the SEIR Model for measles epidemic
2 Journal of Mathematics
-us the NSFD scheme for system (1) takes the form
Sn+1
minus Sn
φ(h) μN minus μS
n+1minus βS
n+1I
n
En+1
minus En
φ(h) βS
n+1I
nminus μE
n+1minus σE
n+1
In+1
minus In
φ(h) σE
n+1minus μI
n+1minus cI
n+1
Rn+1
minus Rn
φ(h) cI
n+1minus μR
n+1
⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭
(5)
From equation (5) we have
(1 + φ(h)μ) Sn+1
+ En+1
+ In+1
+ Rn+1
1113960 1113961
φ(h)μN minus φ(h)μ Sn
+ En
+ In
+ Rn
( 1113857(6)
-us if Sn + En + In + Rn N for all nge 0 then Sn+1+
En+1 + In+1 + Rn+1 N for all nge 0-us the conservation law property is satisfied System
(5) is written as
Sn+1
S
n+ μφ(h)N
1 + φ(h) μ + βIn
( 1113857
En+1
E
n+ φ(h)βS
n+1I
n
1 + φ(h)(μ + σ)
In+1
I
n+ φ(h)σE
n+1
1 + φ(h)(μ + c)
⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭
(7)
Now the positivity of Sn+1 En+1 In+1 andRn+1 is guar-anteed if 0lt Sn lt 1 0ltEn lt 1 0lt In lt 1 and 0ltRn lt 1 for allnge 0 -us the constructed scheme has the followingproperties
-e conservation law is satisfied-e positivity and boundedness of the solution issatisfied for system (7) we have that if 0lt Sn lt 10ltEn lt 1 0lt In lt 1 and 0ltRn lt 1 then 0lt Sn+1 lt 10ltEn+1 lt 1 0lt In+1 lt 1 and 0ltRn+1 lt 1 for all nge 0
4 NSFD Predictor-Corrector Scheme
-is section is improved by the approach of a predictor-corrector-type NSFD scheme (7) to obtain the benefits ofboth methods For the development of this scheme firstlysystem (7) is taken as a predictor scheme ie
Sn+1p
Sn
+ μφ(h)N
1 + φ(h) μ + βIn
( 1113857
En+1p
En
+ φ(h)βSn+1p I
n
1 + φ(h)(μ + σ)
In+1p
In
+ φ(h)σEn+1p
1 + φ(h)(μ + c)
⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭
(8)
Now we evaluate system (1) at time t + φ(h) and in-troduce the term ϵminus 1Sn+1φ(h) (where ϵ is just like the ac-celerating factor and its range is (0lt ϵlt 1)) -us theexpression will be NSFDCL scheme that preserves conser-vation law represented as
Sn+1
minus Sn
φ(h) μN minus μS
n+1minus βS
n+1I
n+1minusϵminus 1
Sn+1
φ(h)+ϵminus 1
Sn+1
φ(h)
En+1
minus En
φ(h) βS
n+1I
nminus (μ + σ)E
n+1
In+1
minus In
φ(h) σE
n+1minus (μ + c)I
n
Rn+1
minus Rn
φ(h) cI
n+1minus μR
n+1
(9)
-us the corrector scheme is obtained as
Sn+1c
Sn
+ μφ(h)N + ϵminus 1S
n+1p
1 + ϵminus 1 + μφ(h) + βφ(h)In+1p
En+1c
En
+ φ(h)βSn+1c I
n+1p
1 + φ(h)(μ + σ)
In+1c
In
+ φ(h)σEn+1c
1 + φ(h)(μ + c)
⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭
(10)
where Rn+1 N minus Sn+1c minus En+1
c minus In+1c
5 Convergence Analysis
In this section the unconditional convergence of the nu-merical solution is presented by the proposed methodTaking the values as
F1(S E I) S + μNϕ(h)
1 + ϕ(h)[μ + βI]
F2(S E I) E + βϕ(h)IF1
1 + ϕ(h)[μ + σ]
F3(S E I) I + σϕ(h)F2
1 + ϕ(h)[μ + c]
G1(S E I) S + μNϕ(h) + ϵminus 1
F1
1 + ϵminus 1 + μϕ(h) + βϕ(l)F3
G2(S E I) E + βϕ(h)G1F3
1 + ϕ(h)[μ + σ]
G3(S E I) I + σϕ(h)G2
1 + ϕ(h)[μ + c]
(11)
where F1 F2 andF3 are given as in equation (8)
Journal of Mathematics 3
-e convergence and stability analysis of scheme (10) iscarried out by calculating the eigenvalues of the Jacobian ofthe linearized scheme and studying its behavior and
evaluated at the fixed points If (Slowast Elowast Ilowast) is the fixed pointof system (1) then the Jacobian matrix JG is given by
JG Slowast Elowast Ilowast
( 1113857
zG1 Slowast Elowast Ilowast
( 1113857
zS
zG1 Slowast Elowast Ilowast
( 1113857
zE
zG1 Slowast Elowast Ilowast
( 1113857
zI
zG2 Slowast Elowast Ilowast
( 1113857
zS
zG2 Slowast Elowast Ilowast
( 1113857
zE
zG2 Slowast Elowast Ilowast
( 1113857
zI
zG3 Slowast Elowast Ilowast
( 1113857
zS
zG3 Slowast Elowast Ilowast
( 1113857
zE
zG3 Slowast Elowast Ilowast
( 1113857
zI
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
(12)
where
zF1
zS
1[1 + μϕ(h) + βϕ(h)I]
zF1
zE
zF1
zI
minus (S + μNϕ(h))βϕ(h)
[1 + μϕ(h) + βϕ(h)I]2
zF2
zS
βϕ(h)IzF1zS
1 + ϕ(h)[μ + σ]
zF2
zE1 + βϕ(h)IzF1zE
1 + ϕ(h)[μ + σ]
zF2
zIβϕ(h) F1 + IzF1zI1113864 1113865
1 + ϕ(h)[μ + σ]
zF3
zE
σϕ(h)zF2zE
1 + ϕ(h)[μ + c]
zF3
zI1 + σϕ(h)zF2zS
1 + ϕ(h)[μ + c]
zG1
zS
1 + ϵminus 1zF1zS1113872 1113873 1 + ϵminus 1
+ μϕ(h) + βϕ(h)F31113960 1113961 minus S + μNϕ(h) + ϵminus 1F11113872 1113873βϕ(h)zF3zS
1 + ϵminus 1 + μϕ(h) + βϕ(l)F31113960 11139612
zG1
zEϵminus 1
zF1zE1113872 1113873 1 + ϵminus 1+ μϕ(h) + βϕ(h)F31113960 1113961 minus S + μNϕ(h) + ϵminus 1
F11113872 1113873βϕ(h)zF3zE
1 + ϵminus 1 + μϕ(h) + βϕ(h)F31113960 11139612
zG1
zIϵminus 1
zF1zI1113872 1113873 1 + ϵminus 1+ μϕ(h) + βϕ(h)F31113960 1113961 minus S + μNϕ(h) + ϵminus 1
F11113872 1113873βϕ(h)zF3zI
1 + ϵminus 1 + μϕ(h) + βϕ(h)F31113960 11139612
zG2
zS
βϕ(h)
1 + ϕ(h)[μ + σ]
zG1
zSF3 + G1
zF3
zS1113896 1113897
zG2
zE1 + βϕ(h) zG1zEF3 + G1zF3zE1113858 1113859
1 + ϕ(h)[μ + σ]
4 Journal of Mathematics
zG2
zI
βϕ(h)
1 + ϕ(h)[μ + σ]
zG1
zIF3 + G1
zF3
zI1113896 1113897
zG3
zS
σϕ(h)zG2zS
1 + ϕ(h)[μ + c]
zG3
zE
σϕ(h)zG2zE
1 + ϕ(h)[μ + c]
zG3
zI1 + σϕ(h)zG2zI
1 + ϕ(h)[μ + c]
(13)
51 Disease-Free Equilibrium (SoEo Io) (N 0 0) -evalues of functions and its derivatives at DFE point (N 0 0)
are as
F1(N 0 0) N F2(N 0 0) 0 F3(N 0 0) 0
G1(N 0 0) N G2(N 0 0) 0 G3(N 0 0) 0
zF1(N 0 0)
zS
11 + μϕ(h)
zF1(N 0 0)
zE 0
zF1(N 0 0)
zI
minus Nβϕ(h)
[1 + μϕ(h)]
zF2(N 0 0)
zS 0
zF2(N 0 0)
zE
11 + ϕ(h)[μ + σ]
zF2(N 0 0)
zI
βϕ(h)
1 + ϕ(h)[μ + σ]
zF3(N 0 0)
zS 0
zF3(N 0 0)
zE
σϕ(h)
[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]]
zF3(N 0 0)
zI1 +[σϕ(h)βNϕ(h)1 + ϕ(h)(μ + c)]
1 + ϕ(h)[μ + c]
zG1(N 0 0)
zS
11 + μϕ(h)
zG1(N 0 0)
zE
minus Nσϕ(h)βϕ(h)
1 + μϕ(h) + ϵminus 11113960 1113961[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]
Journal of Mathematics 5
zG1(N 0 0)
zI
minus Nβϕ(h)
1 + ϵminus 1+ μϕ(h)1113960 1113961
ϵminus 1
1 + μϕ(h)+
1[1 + ϕ(h)(μ + c)
+σNβϕ2
(h)
[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]1113890 1113891
zG2(N 0 0)
zS 0
zG2(N 0 0)
zE
11 + ϕ(h)(μ + σ)
1 +Nβσϕ2(h)
[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]1113890 1113891
zG2(N 0 0)
zI
βϕ(h)
[1 + ϕ(h)(μ + σ)]
1 + ϕ(h)(μ + σ) + Nσβϕ2(h)
[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]1113896 1113897
zG3(N 0 0)
zS
zG3(N 0 0)
zE
σϕ(h)
[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]1 +
σβNϕ2(h)
[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]1113896 1113897
zG3(N 0 0)
zI
1
[1 + ϕ(h)(μ + σ)]+
Nσβϕ2(h)
[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]2 +
N2β2σ2ϕ4(h)
[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + σ)]
2
(14)
For ease in the calculation we use the followingconventions
α μNϕ(h)gt 0 η1 μϕ(h)gt 0 η2
βϕ(h)gt 0 η3 σϕ(h)gt 0
θ 1 + ϵminus 1+ μϕ(h) 1 + ϵminus 1
+ η1 gt 1
δ1 1 + ϕ(h)[μ + σ]gt 1 δ2
(15)
It is clear that θgt η1 1 + η1 gt δ1 1 + η1 gt δ2 and δ2 θ minus ϵminus 1 + cϕ(h) If R0 σβN(μ + σ)(μ + c)lt 1 the Ja-cobian calculated at disease-free points (Slowast0 Elowast0 Ilowast0 )
(N 0 0) is given as
JG(N 0 0)
zG1(N 0 0)
zS
zG1(N 0 0)
zE
zG1(N 0 0)
zI
zG1(N 0 0)
zS
zG1(N 0 0)
zE
zG1(N 0 0)
zI
zG1(N 0 0)
zS
zG1(N 0 0)
zE
zG1(N 0 0)
zI
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
(16)
Since the partial derivatives using the conventions are
zG1(N 0 0)
zS
11 + η1
zG1(N 0 0)
zE
minus Nη2η3θδ1δ2
zG1(N 0 0)
zI
minus Nη2θϵminus 1
1 + η1+
1δ2
+Nη2η3δ1δ2
1113890 1113891
zG2(N 0 0)
zS 0
zG2(N 0 0)
zE
1δ1
+Nη2η3δ21δ2
1113890 1113891
zG2(N 0 0)
zI
Nη2δ1δ2
+N
2η22η3δ21δ2
1113890 1113891
zG3(N 0 0)
zS 0
zG3(N 0 0)
zE
η3δ1δ2
+Nη2η
23
δ21δ22
1113890 1113891
zG3(N 0 0)
zI
1δ2
+Nη2η3δ1δ
22
+N
2η22η23
δ21δ22
1113890 1113891
(17)
6 Journal of Mathematics
So
JG(N 0 0)
11 + η1
minus Nη2η3θδ1δ2
minus Nη2θϵminus 1
1 + η1+
1δ2
+Nη2η3δ1δ2
1113890 1113891
01δ1
+Nη2η3δ21δ2
1113890 1113891Nη2δ1δ2
+N
2η22η3δ21δ2
1113890 1113891
0η3δ1δ2
+Nη2η
23
δ21δ22
1113890 11138911δ2
+Nη2η3δ1δ
22
+N
2η22η23
δ21δ22
1113890 1113891
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
(18)
To calculate the eigenvalues put det (J minus λI) 0
λ1 11 + η1 11 + μϕ(h)lt 1 And
λ2 minus λδ1δ2 + Nη2η3
δ21δ21113888 1113889 +
δ21δ2 + δ1Nη2η3 + N2η22η
23
δ21δ22
1113888 11138891113890 1113891
+δ1δ2 + Nη2η3
δ21δ21113888 1113889
δ21δ2 + δ1Nη2η3 + N2η22η
23
δ21δ22
1113888 1113889 minusNη2δ1 + N
2η22η3δ21δ2
1113888 1113889η3δ1δ2 + Nη2η
23
δ21δ22
1113888 11138891113890 1113891 0
λ2 minus Aλ + B 0
(19)
where
A δ1δ2 + Nη2η3δ21δ21113872 1113873 + δ21δ2 + δ1Nη2η3 + N
2η22η23δ
21δ
221113872 1113873 trace J
lowast
B δ1δ2 + Nη2η3
δ21δ21113888 1113889
δ21δ2 + δ1Nη2η3 + N2η22η
23
δ21δ22
1113888 1113889 minusNη2δ1 + N
2η22η3δ21δ2
1113888 1113889η3δ1δ2 + Nη2η
23
δ21δ22
1113888 11138891113890 1113891 det Jlowast
Jlowast
1δ1
+Nη2η3δ21δ2
1113890 1113891Nη2δ1δ2
+N
2η22η3δ21δ2
1113890 1113891
η3δ1δ2
+Nη2η
23
δ21δ22
1113890 11138911δ2
+Nη2η3δ1δ
22
+N
2η22η23
δ21δ22
1113890 1113891
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
(20)
To calculate the eigenvalues of Jlowast the following lemma isproved
Lemma 1 For the quadratic equation λ2 minus λA + B 0 bothroots satisfy |λi|lt 1 i 2 3 if and only if the followingconditions are satisfied [25]
(1) 1 minus A + Bgt 0
(2) 1 + A + Bgt 0
(3)Blt 1
(21)
Let us define A TraceJlowast B DetJlowast -erefore
Journal of Mathematics 7
A δ1 + δ2( 1113857 Nη2η3 + δ1δ2( 11138571113858 1113859 + N
2η22η23
δ21δ22
gt 0
B 1
δ1δ2+
Nη2η3δ21δ
22lt 1
f(minus 1) 1 + A + Bgt 0
f(0) B 1
δ1δ2+
Nη2η3δ21δ
22lt
1δ1δ2
+Nη2η3δ1δ2
1 + Nη2η3
δ1δ2
1 + R0(μ + σ)(μ + c)ϕ2(h)
1 + ϕ2(h)(μ + σ)(μ + c) + ϕ(h)[(μ + σ)(μ + c)]
lt 1 sinceR0 lt 1( 1113857
rArrBlt 1
(22)
Now for
f(1) 1 minus A + B
δ21δ
22 + δ1 + δ2( 1113857 Nη2η3 + δ1δ2( 11138571113858 1113859 + N
2η22η23 + Nη2η3 + δ1δ2
δ21δ22
gt 0
(23)
As all the conditions of the lemma hold for R0 lt 1therefore the absolute value of both eigenvalues of Jlowast is less
than one whenever R0 lt 1 -us the numerical scheme inequations (8) and (10) will converge unconditionally todisease-free equilibrium (N 0 0) for any value of time step h
whenever R0 lt 1 which is also computationally verified inFigure 2(a)
52 Endemic Equilibrium (SeEe Ie) (NR0 μN μ + σ(1 minus
1 R0) μβ(R0 minus 1)) -e value of functions and its deriv-atives at an endemic point is as
F1 Se Ee Ie( 1113857 N
R0
F2 Se Ee Ie( 1113857 μN 1 minus 1R0( 1113857
(μ + σ)
F3 Se Ee Ie( 1113857 μβ
R0 minus 1( 1113857
G1 Se Ee Ie( 1113857 N
R0
G2 Se Ee Ie( 1113857 μβ
R0 minus 1( 1113857
G3 Se Ee Ie( 1113857 μN 1 minus 1R0( 1113857
(μ + σ)
(24)
8 Journal of Mathematics
zF1 Se Ee Ie( 1113857
zS
11 + μϕ(h)R0
zF1 Se Ee Ie( 1113857
zE 0
zF1 Se Ee Ie( 1113857
zI
minus Nβϕ(h)
1 + μϕ(h)R0
zF2 Se Ee Ie( 1113857
zS
μϕ(h) R0 minus 1( 1113857
1 + μϕ(h)R01113858 1113859[1 + ϕ(h)(μ + σ)]
zF2 Se Ee Ie( 1113857
zE
1[1 + ϕ(h)(μ + σ)]
zF2 Se Ee Ie( 1113857
zI
minus Nβϕ(h) μϕ(l)R20 minus 2μϕ(h)R0 minus 11113960 1113961
R0[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]
zF3 Se Ee Ie( 1113857
zS
μσϕ2(h) R0 minus 1( 1113857
[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)] 1 + μϕ(h)R01113858 1113859
zF3 Se Ee Ie( 1113857
zE
σϕ(h)
[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]
zF3 Se Ee Ie( 1113857
zI
R0[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859 minus σNβϕ2(h) μϕ(h)R20 minus 2μϕ(h) minus 11113960 1113961
R0[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859[1 + ϕ(h)(μ + c)]
(25)
zG1 Se Ee Ie( 1113857
zS
11 + μϕ(h)R01113858 1113859
minusNμβσϕ3(h) R0 minus 1( 1113857
R0 1 + ϵminus 1 + μϕ(h)R01113960 1113961[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859[1 + ϕ(h)(μ + c)]
zG1 Se Ee Ie( 1113857
zE
minus Nβσϕ2(h)
R0 1 + ϵminus 1 + μϕ(h)R01113960 1113961[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]
Eige
n va
lues
Jacobian at DFE
0492
0494
0496
0498
05
0502
0504
0506
0508
051
1000 2000 3000 4000 5000 6000 7000 8000 9000 100000Step size
(a)
098
0981
0982
0983
0984
0985
0986
0987
0988
0989Jacobian at EE
Eige
n va
lues
1000 2000 3000 4000 5000 6000 7000 8000 9000 100000Step size
(b)
Figure 2 Spectral radius of the Jacobian at equilibrium points
Journal of Mathematics 9
zG1 Se Ee Ie( 1113857
zI
minus Nβϕ(h)
1 + ϵminus 1 + μϕ(h)R01113960 11139611 + μϕ(h)R0
middot ϵminus 1+
R0[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859 minus σNβϕ2(h) μϕ(h)R20 minus 2μϕ(h)R0 minus 11113960 1113961
R20[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859[1 + ϕ(h)(μ + c)]
⎡⎣ ⎤⎦
zG2 Se Ee Ie( 1113857
zS
μβϕ(h) R0 minus 1( 1113857
β[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 11138591113890
minusNμβ2σϕ4(h)μ R0 minus 1( 1113857
2
βR0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)] 1 + ϵminus 1 + μϕ(h)R01113960 1113961 1 + μϕ(h)R01113858 1113859
+Nμβσϕ3(h) R0 minus 1( 1113857
R0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)] 1 + μϕ(h)R01113858 1113859
1113891
zG2 Se Ee Ie( 1113857
zE
1[1 + ϕ(h)(μ + σ)]
+Nβσϕ2(h)
R0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)]
minusNμβ2σϕ3
(h) R0 minus 1( 1113857
βR0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)] 1 + ϵminus 1 + μϕ(h)R01113960 1113961
zG2 Se Ee Ie( 1113857
zI
minus Nμβ2ϕ2(h)ϵminus 1R0 minus 1( 1113857
β[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859 1 + ϵminus 1 + μϕ(h)R01113960 1113961
+Nμβ2ϕ2(h) R0 minus 1( 1113857 R0[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859 minus Nβσϕ2(h) μϕ(h)R
20 minus 2μϕ(h)R0 minus 11113960 11139611113966 1113967
βR20[1 + ϕ(h)(μ + σ)]
2[1 + ϕ(h)(μ + c)] 1 + ϵminus 1 + μϕ(h)R01113960 1113961 1 + μϕ(h)R01113858 1113859
+Nβϕ(h)
R0[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]minus
N2β2σϕ3(h) μϕ(h)R
20 minus 2μϕ(h)R0 minus 11113960 1113961
R20[1 + ϕ(h)(μ + σ)]
2[1 + ϕ(h)(μ + c)] 1 + μϕ(h)R01113858 1113859
zG3 Se Ee Ie( 1113857
zS
σϕ(h)
[1 + ϕ(h)(μ + c)]
μβϕ(h) R0 minus 1( 1113857
β[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 11138591113890
minusNμβ2σϕ4(h)μ R0 minus 1( 1113857
2
βR0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)] 1 + ϵminus 1 + μϕ(h)R01113960 1113961 1 + μϕ(h)R01113858 1113859
+Nμβσϕ3(h) R0 minus 1( 1113857
R0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)] 1 + μϕ(h)R01113858 1113859
1113891
zG3 Se Ee Ie( 1113857
zE
σϕ(h)
[1 + ϕ(h)(μ + c)]
1[1 + ϕ(h)(μ + σ)]
minusNμβ2σϕ3(h) R0 minus 1( 1113857
βR0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)] 1 + ϵminus 1 + μϕ(h)R01113960 1113961
⎡⎢⎣
+Nβσϕ2
(h)
R0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)]
1113891
zG3 Se Ee Ie( 1113857
zI
σϕ(h)
[1 + ϕ(h)(μ + c)]
minus Nμβ2ϕ2(h)ϵminus 1R0 minus 1( 1113857
β[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859 1 + ϵminus 1 + μϕ(h)R01113960 1113961⎡⎢⎣
+Nμβ2ϕ2(h) R0 minus 1( 1113857 R0[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859 minus Nβσϕ2(h) μϕ(h)R
20 minus 2μϕ(h)R0 minus 11113960 11139611113966 1113967
βR20[1 + ϕ(h)(μ + σ)]
2[1 + ϕ(h)(μ + c)] 1 + ϵminus 1 + μϕ(h)R01113960 1113961 1 + μϕ(h)R01113858 1113859
⎤⎥⎦
10 Journal of Mathematics
Table 1 Comparison of NSFD scheme with Euler and RK-4 schemes
l 001 01 100 1000Euler PC Convergence Divergence Divergence DivergenceRK-4 Convergence Divergence Divergence DivergenceNSFDPC Convergence Convergence Convergence Convergence
times107
NSFD-PC
Disease-free equilibrium
1
15
2
25
3
35
4
45
5
Susc
eptib
le fr
actio
n
20 40 60 80 100 120 140 160 180 2000Time (years) h = 001
(a)
Disease-free equilibriumtimes104
NSFD-PC
02 04 06 08 1 12 14 16 18 20Time (years) h = 001
0
05
1
15
2
25
3
35
4
45
5
Expo
sed
fract
ion
(b)
times104
NSFD-PC
Disease-free equilibrium
0
05
1
15
2
25
3
Infe
cted
frac
tion
02 04 06 08 1 12 14 16 18 20Time (years) h = 001
(c)
times107
NSFD-PC
X 5675Y 2435e + 007
Endemic equilibrium
100 200 300 400 500 6000Time (years) h = 001
1
15
2
25
3
35
Susc
eptib
le fr
actio
n
(d)
Figure 3 Continued
Journal of Mathematics 11
+Nβϕ(h)
R0[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]minus
N2β2σϕ3(h) μϕ(h)R
20 minus 2μϕ(h)R0 minus 11113960 1113961
R20[1 + ϕ(h)(μ + σ)]
2[1 + ϕ(h)(μ + c)] 1 + μϕ(h)R01113858 1113859
(26)
-e Jacobian evaluated at an endemic point(Slowaste Elowaste Ilowaste ) (Se Ee Ie) (NR0 μNμ + σ(1 minus 1R0) μβ(R0 minus 1)) is evaluated as
JG Slowaste Elowaste Ilowaste( 1113857
zG1 Slowaste Elowaste Ilowaste( 1113857
zS
zG1 Slowaste Elowaste Ilowaste( 1113857
zE
zG1 Slowaste Elowaste Ilowaste( 1113857
zI
zG1 Slowaste Elowaste Ilowaste( 1113857
zS
zG1 Slowaste Elowaste Ilowaste( 1113857
zE
zG1 Slowaste Elowaste Ilowaste( 1113857
zI
zG1 Slowaste Elowaste Ilowaste( 1113857
zS
zG1 Slowaste Elowaste Ilowaste( 1113857
zE
zG1 Slowaste Elowaste Ilowaste( 1113857
zI
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
(27)
It is tedious to calculate the eigenvalues of the Jacobian ofendemic equilibrium analytically so we have plotted thelargest eigenvalue against each step size and Figure 2(b)shows that for all step sizes the spectral radius of the Ja-cobian of EE remains less than one if R0 gt 1 which impliesthat the numerical scheme in equations (8) and (10) isunconditionally convergent if R0 gt 1 for all step sizes
6 Numerical Results and Discussion
All the methods showed the convergence for small step sizesin Table 1 However for large values of step sizes only thenormalized NSFDPC converge to the correct disease-freepoint for β 01 times 10minus 5 as well as the correct endemic pointfor β 03 times 10minus 5 -is means that for the value of the basic
reproductive number (threshold parameter) R0 lt 1 thenDFE is stable ie solution to the system (S E I R) areconverging to it -is can be seen from Figures 3(a)ndash3(c) forh 001 Also for the value of the basic reproductivenumber (threshold parameter) R0 gt 1 then EE is stable iesolution to the system (S E I R) is converging to it-is canbe seen from Figures 3(d)ndash3(f ) for h 001
Figures 4(a)ndash4(c) show how the NSFDPC method ofDFE converges to the equilibrium point for different stepsizes Figures 4(d)ndash4(f) show how the NSFDPC method ofEE converges to the equilibrium point for different step sizes
For comparison with the well-known RK-4 method tosystem (1) using the parameter values given in Table 2 it isfound that in Figures 5(a)ndash5(f) the numerical solutionconverges for h 001 for both equilibrium points and
times105
NSFD-PC
X 562Y 1125e + 004
Endemic equilibrium
100 200 300 400 500 6000Time (years) h = 001
0
1
2
3
4
5
6
7
8Ex
pose
d fra
ctio
n
(e)
times105
NSFD-PC
X 5523Y 7023
Infe
cted
frac
tion
Endemic equilibrium
100 200 300 400 500 6000Time (years) h = 001
0
05
1
15
2
25
3
35
4
45
5
(f )
Figure 3 Time plots of susceptible exposed and infected population using the NSFDPC method with h 001
12 Journal of Mathematics
times107
NSFD-PC
Disease-free equilibrium
20 40 60 80 100 120 140 160 180 2000Time (years) h = 01
1
15
2
25
3
35
4
45
5Su
scep
tible
frac
tion
(a)
times107
NSFD-PC
Disease-free equilibrium
500 1000 1500 2000 2500 3000 3500 40000Time (years) h = 10
1
15
2
25
3
35
4
45
5
Susc
eptib
le fr
actio
n
(b)
Disease-free equilibriumtimes107
times105
NSFD-PC
05 1 15 2 25 3 35 40Time (years) h = 1000
1
15
2
25
3
35
4
45
5
Susc
eptib
le fr
actio
n
(c)
X 5827Y 7022
Infe
cted
frac
tion
Endemic equilibriumtimes104
NSFD-PC
100 200 300 400 500 6000Time (years) h = 01
0
2
4
6
8
10
12
14
16
18
(d)
Endemic equilibriumtimes104
NSFD-PC
X 9600Y 7022
Infe
cted
frac
tion
1000 2000 3000 4000 5000 6000 7000 8000 9000 100000Time (years) h = 10
0
05
1
15
2
25
3
(e)
Endemic equilibriumtimes104
times105
NSFD-PC
X 861e + 005Y 7022
Infe
cted
frac
tion
1 2 3 4 5 6 7 8 9 100Time (years) h = 1000
0
05
1
15
2
25
3
(f )
Figure 4 Time plots of susceptible and infected population using the NSFDPC method with different step sizes (a) h 01 (b) h 10 and(c) h 1000
Journal of Mathematics 13
Table 2 Parameters value of the measles model
Parameters N μ σ c
5 times 107yearminus 1 002 456 yearminus 1 73 yearminus 1
DFE (S0 E0 I0) (N 0 0) β 01 times 10minus 5
EE (Se Ee Ie) (2435 times 107 1125 times 104 7022671) β 03 times 10minus 5
Disease-free equilibrium
RK4
times107
1
15
2
25
3
35
4
45
5
Susc
eptib
le fr
actio
n
20 40 60 80 100 120 140 160 180 2000Time (years) h = 001
(a)
Expo
sed
fract
ion
Disease-free equilibrium
RK4
times104
0
05
1
15
2
25
3
35
4
45
5
02 04 06 08 1 12 14 16 18 20Time (years) h = 001
(b)
times104
Infe
cted
frac
tion
Disease-free equilibrium
RK4
0
05
1
15
2
25
3
02 04 06 08 1 12 14 16 18 20Time (years) h = 001
(c)
times107
X 5578Y 2435e + 007
Endemic equilibrium
RK4
1
15
2
25
3
35
Susc
eptib
le F
ract
ion
100 200 300 400 500 6000Time (years) h = 001
(d)
Figure 5 Continued
14 Journal of Mathematics
Figure 6 shows overflow for a critical value of step size ieh 01 of DFE and EE respectively
Another popular numerical method is the Eulerpredictor-corrector (E-PC) technique which employs theexplicit Euler method as a predictor and the trapezoidalrule as the corrector -is combination is second-orderaccurate but has similar stability properties in PECEmode to the explicit Euler method alone Figures 7(a)ndash7(f ) show convergence results for h 001 in both casesthat are DFE and EE -e value of h 01 which produces
overflow when solving the system with the parametervalues of Table 2 for the PECE combination can be seen inFigure 8
-us the presented numerical results demonstrated thatNSFDPC has better convergence property following theEuler PC and RK-4 as shown in Figure 9 It is proved thatthe approximations made by other standard numericalmethods experience difficulties in preserving either thestability or the positivity of the solutions or both butNSFDPC is unconditionally convergent
times105
X 5772Y 1125e + 004
Endemic equilibrium
RK4
Expo
sed
fract
ion
0
1
2
3
4
5
6
7
8
9
10
100 200 300 400 500 6000Time (years) h = 001
(e)
times105
X 7659Y 7022
RK4
Endemic equilibrium
Infe
cted
frac
tion
0
1
2
3
4
5
6
100 200 300 400 500 600 700 8000Time (years) h = 001
(f )
Figure 5 Time plots of susceptible exposed and infected population using the RK-4 method with h 001
times1087 RK-4 scheme (disease-free equilibrium)
SusceptibleExposedInfected
01 02 03 04 05 06 07 08 09 10Time (years) h = 01
ndash2
ndash15
ndash1
ndash05
0
05
1
15
2
Popu
latio
n fra
ctio
ns
(a)
RK-4 scheme (endemic equilibrium)
SusceptibleExposedInfected
times10145
01 02 03 04 05 06 07 08 09 10Time (years) h = 01
ndash8
ndash6
ndash4
ndash2
0
2
4
6
8
Popu
latio
n fra
ctio
ns
(b)
Figure 6 Divergence graphs of the population using the RK-4 method with step size h 01
Journal of Mathematics 15
Euler-PC
Disease-free equilibriumtimes107
1
15
2
25
3
35
4
45
5Su
scep
tible
frac
tion
20 40 60 80 100 120 140 160 180 2000Time (years) h = 001
(a)
Euler-PC
Disease-free equilibrium
Expo
sed
fract
ion
times104
0
05
1
15
2
25
3
35
4
45
5
02 04 06 08 1 12 14 16 18 20Time (years) h = 001
(b)
Euler-PC
Disease-free equilibrium
Infe
cted
frac
tion
times104
0
05
1
15
2
25
3
35
02 04 06 08 1 12 14 16 18 20Time (years) h = 001
(c)
Euler-PC
X 5733Y 2435e + 007
Endemic equilibriumtimes107
1
15
2
25
3
35Su
scep
tible
frac
tion
100 200 300 400 500 6000Time (years) h = 001
(d)
Euler-PC
X 5662Y 1118e + 004
Endemic equilibrium
Expo
sed
fract
ion
times105
0
2
4
6
8
10
12
100 200 300 400 500 6000Time (years) h = 001
(e)
Euler-PC
X 5427Y 7025
Endemic equilibrium
Infe
cted
frac
tion
times105
0
1
2
3
4
5
6
7
100 200 300 400 500 6000Time (years) h = 001
(f )
Figure 7 Time plots of population using the Euler-PC method with h 001
16 Journal of Mathematics
7 Conclusions
In this paper a normalized NSFDPC for the SEIR modelconcerning the transmission dynamics of measles is con-structed and analysed -is proposed numerical scheme isvery competitive It is qualitatively stable that is it doublerefines the solution and gives realistic results even for largestep sizes It is dynamically consistent with a continuoussystem and unconditionally convergent and satisfies the
positivity of the state variables involved in the systemSimulations are carried out and its usefulness is comparedwith a well-known numerical method of standard differenceschemes such as RK4 and Euler predictor-corrector method-e standard finite difference schemes are highly dependenton step sizes and initial value problems but NSFDPC isindependent of these two features which make it morepractical Also this method saves computation time andmemory
Euler-PC scheme (disease-free equilibrium)times1041
ndash3
ndash2
ndash1
0
1
2
3
Popu
latio
n fra
ctio
ns
01 02 03 04 05 06 07 08 09 10Time (years) h = 01
SusceptibleExposedInfected
(a)
Euler-PC scheme (endemic equilibrium)times1053
ndash6
ndash4
ndash2
0
2
4
6
Popu
latio
n fra
ctio
ns
01 02 03 04 05 06 07 08 09 10Time (years) h = 01
SusceptibleExposedInfected
(b)
Figure 8 Divergence graphs of the population using the Euler-PC method with step size h 01
NSFD-PCRK-4
times1092
Infe
cted
frac
tion
ndash5
ndash4
ndash3
ndash2
ndash1
0
1
02 04 06 08 1 12 14 16 18 20Time (years) h = 01
(a)
NSFD-PCEuler-PC
times10213
Infe
cted
frac
tion
ndash4
ndash35
ndash3
ndash25
ndash2
ndash15
ndash1
ndash05
0
05
02 04 06 08 1 12 14 16 18 20Time (years) h = 01
(b)
Figure 9 Comparative convergence graphs of the NSFDPC method
Journal of Mathematics 17
Data Availability
-e data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
-e authors declare no conflicts of interest
Acknowledgments
-e authors extend their appreciation to the Deanship ofScientific Research at Majmaah University for funding thiswork under Project no RGP-2019-6
References
[1] E Hunter B Namee and J Kelleher ldquoAn open-data-drivenagent-based model to simulate infectious disease outbreaksrdquoPLoS One vol 14 no 1 pp 1ndash35 2018
[2] E A Bakare Y A Adekunle and K O Kadiri ldquoModeling andsimulation of the dynamics of the transmission of measlesrdquoInternational Journal of Computer Trends and Trends andTechnology vol 3 no 1 pp 174ndash177 2012
[3] L Zhou Y Wang Y Li and Y Michael ldquoGlobal dynamics ofa discrete age-structured SIR epidemic model with applica-tions to measles vaccination strategiesrdquo Mathematical Bio-sciences vol 308 pp 27ndash37 2019
[4] P Widyaningsih D R S Saputro and A W NugroholdquoSusceptible exposed infected recovery (SEIR) model withimmigration equilibria points and its applicationrdquo AmericanInstitute of Physics vol 1 pp 1ndash7 2018
[5] M G Roberts and J A P HeesterbeekMathematical Modelsin Epidemiology Mathematical Models UNESCO-EOLSSAbu Dhabi UAE 2007
[6] J H He F Y Ji and H M Sedighi ldquoDifference equation vsdifferential equation on different scalesrdquo International Journalof Numerical Methods for Heat amp Fluid Flow vol 18pp 0961ndash5539 2020
[7] R E Mickens Nonstandard Finite Difference Models of Dif-ferential Equations World Scientific Singapore 1994
[8] R E Mickens Applications of Nonstandard Finite DifferenceSchemes World Scientific Singapore 2000
[9] R E Mickens ldquoNumerical integration of population modelssatisfying conservation laws NSFD methodsrdquo Journal ofBiological Dynamics vol 1 no 4 pp 427ndash436 2007
[10] J D Lambert Computational Methods in Ordinary Differ-ential Equations Wiley New York NY USA 1973
[11] S M Moghadas M E Alexander B D Corbett andA B Gumel ldquoA positivity-preserving mickens-type dis-cretization of an epidemic modelrdquo Journal of DifferenceEquations and Applications vol 9 no 11 pp 1037ndash10512003
[12] R E Mickens ldquoNonstandard finite difference schemes fordifferential equationsrdquo Journal of Difference Equations andApplications vol 8 no 9 pp 823ndash847 2002
[13] R E Mickens ldquoDynamic consistency a fundamental prin-ciple for constructing nonstandard finite difference schemesfor differential equationsrdquo Journal of Difference Equations andApplications vol 11 no 7 pp 645ndash653 2005
[14] L-I W Roeger and R W Barnard ldquoPreservation of localdynamics when applying central difference methods appli-cation to SIR modelrdquo Journal of Difference Equations andApplications vol 13 no 4 pp 333ndash340 2007
[15] K C Patidar ldquoOn the use of non-standard finite differencemethodsrdquo Journal of Difference Equations and Applicationsvol 11 no 7 pp 735ndash758 2005
[16] R Anguelov and J M-S Lubuma ldquoNonstandard finite dif-ference method by nonlocal approximationrdquo Mathematicsand Computers in Simulation vol 61 no 3-6 pp 465ndash4752003
[17] R Anguelov and J M-S Lubuma ldquoContributions to themathematics of the nonstandard finite difference method andapplicationsrdquo Numerical Methods for Partial DifferentialEquations vol 17 no 5 pp 518ndash543 2001
[18] B M Chen-Charpentier D T Dimitrov andH V Kojouharov ldquoNumerical simulation of multi-speciesbiofilms in porous media for different kineticsrdquo Mathematicsand Computers in Simulation vol 79 no 6 pp 1846ndash18612009
[19] D T Dimitrov and H V Kojouharov ldquoNonstandard finite-difference schemes for general two-dimensional autonomousdynamical systemsrdquo Applied Mathematics Letters vol 18no 7 pp 769ndash774 2005
[20] A J Arenas J A Morantildeo and J C Cortes ldquoNon-standardnumerical method for a mathematical model of RSV epide-miological transmissionrdquo Computers amp Mathematics withApplications vol 56 no 3 pp 670ndash678 2008
[21] L Jodar R J Villanueva A J Arenas and G C GonzalezldquoNonstandard numerical methods for a mathematical modelfor influenza diseaserdquo Mathematics and Computers in Sim-ulation vol 79 no 3 pp 622ndash633 2008
[22] W Piyawong E H Twizell and A B Gumel ldquoAn uncon-ditionally convergent finite-difference scheme for the SIRmodelrdquo Applied Mathematics and Computation vol 146no 2-3 pp 611ndash625 2003
[23] H Jansen and E H Twizell ldquoAn unconditionally convergentdiscretization of the SEIR modelrdquo Mathematics and Com-puters in Simulation vol 58 no 2 pp 147ndash158 2002
[24] A J Arenas G Gonzalez-Parra and B M Chen-CharpentierldquoA nonstandard numerical scheme of predictor-correctortype for epidemic modelsrdquo Computers amp Mathematics withApplications vol 59 no 12 pp 3740ndash3749 2010
[25] F Brauer and C Castillo-Chavez Mathematical Models inPopulation Biology and Epidemiology Springer Berlin Ger-many 2001
18 Journal of Mathematics
-us the NSFD scheme for system (1) takes the form
Sn+1
minus Sn
φ(h) μN minus μS
n+1minus βS
n+1I
n
En+1
minus En
φ(h) βS
n+1I
nminus μE
n+1minus σE
n+1
In+1
minus In
φ(h) σE
n+1minus μI
n+1minus cI
n+1
Rn+1
minus Rn
φ(h) cI
n+1minus μR
n+1
⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭
(5)
From equation (5) we have
(1 + φ(h)μ) Sn+1
+ En+1
+ In+1
+ Rn+1
1113960 1113961
φ(h)μN minus φ(h)μ Sn
+ En
+ In
+ Rn
( 1113857(6)
-us if Sn + En + In + Rn N for all nge 0 then Sn+1+
En+1 + In+1 + Rn+1 N for all nge 0-us the conservation law property is satisfied System
(5) is written as
Sn+1
S
n+ μφ(h)N
1 + φ(h) μ + βIn
( 1113857
En+1
E
n+ φ(h)βS
n+1I
n
1 + φ(h)(μ + σ)
In+1
I
n+ φ(h)σE
n+1
1 + φ(h)(μ + c)
⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭
(7)
Now the positivity of Sn+1 En+1 In+1 andRn+1 is guar-anteed if 0lt Sn lt 1 0ltEn lt 1 0lt In lt 1 and 0ltRn lt 1 for allnge 0 -us the constructed scheme has the followingproperties
-e conservation law is satisfied-e positivity and boundedness of the solution issatisfied for system (7) we have that if 0lt Sn lt 10ltEn lt 1 0lt In lt 1 and 0ltRn lt 1 then 0lt Sn+1 lt 10ltEn+1 lt 1 0lt In+1 lt 1 and 0ltRn+1 lt 1 for all nge 0
4 NSFD Predictor-Corrector Scheme
-is section is improved by the approach of a predictor-corrector-type NSFD scheme (7) to obtain the benefits ofboth methods For the development of this scheme firstlysystem (7) is taken as a predictor scheme ie
Sn+1p
Sn
+ μφ(h)N
1 + φ(h) μ + βIn
( 1113857
En+1p
En
+ φ(h)βSn+1p I
n
1 + φ(h)(μ + σ)
In+1p
In
+ φ(h)σEn+1p
1 + φ(h)(μ + c)
⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭
(8)
Now we evaluate system (1) at time t + φ(h) and in-troduce the term ϵminus 1Sn+1φ(h) (where ϵ is just like the ac-celerating factor and its range is (0lt ϵlt 1)) -us theexpression will be NSFDCL scheme that preserves conser-vation law represented as
Sn+1
minus Sn
φ(h) μN minus μS
n+1minus βS
n+1I
n+1minusϵminus 1
Sn+1
φ(h)+ϵminus 1
Sn+1
φ(h)
En+1
minus En
φ(h) βS
n+1I
nminus (μ + σ)E
n+1
In+1
minus In
φ(h) σE
n+1minus (μ + c)I
n
Rn+1
minus Rn
φ(h) cI
n+1minus μR
n+1
(9)
-us the corrector scheme is obtained as
Sn+1c
Sn
+ μφ(h)N + ϵminus 1S
n+1p
1 + ϵminus 1 + μφ(h) + βφ(h)In+1p
En+1c
En
+ φ(h)βSn+1c I
n+1p
1 + φ(h)(μ + σ)
In+1c
In
+ φ(h)σEn+1c
1 + φ(h)(μ + c)
⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭
(10)
where Rn+1 N minus Sn+1c minus En+1
c minus In+1c
5 Convergence Analysis
In this section the unconditional convergence of the nu-merical solution is presented by the proposed methodTaking the values as
F1(S E I) S + μNϕ(h)
1 + ϕ(h)[μ + βI]
F2(S E I) E + βϕ(h)IF1
1 + ϕ(h)[μ + σ]
F3(S E I) I + σϕ(h)F2
1 + ϕ(h)[μ + c]
G1(S E I) S + μNϕ(h) + ϵminus 1
F1
1 + ϵminus 1 + μϕ(h) + βϕ(l)F3
G2(S E I) E + βϕ(h)G1F3
1 + ϕ(h)[μ + σ]
G3(S E I) I + σϕ(h)G2
1 + ϕ(h)[μ + c]
(11)
where F1 F2 andF3 are given as in equation (8)
Journal of Mathematics 3
-e convergence and stability analysis of scheme (10) iscarried out by calculating the eigenvalues of the Jacobian ofthe linearized scheme and studying its behavior and
evaluated at the fixed points If (Slowast Elowast Ilowast) is the fixed pointof system (1) then the Jacobian matrix JG is given by
JG Slowast Elowast Ilowast
( 1113857
zG1 Slowast Elowast Ilowast
( 1113857
zS
zG1 Slowast Elowast Ilowast
( 1113857
zE
zG1 Slowast Elowast Ilowast
( 1113857
zI
zG2 Slowast Elowast Ilowast
( 1113857
zS
zG2 Slowast Elowast Ilowast
( 1113857
zE
zG2 Slowast Elowast Ilowast
( 1113857
zI
zG3 Slowast Elowast Ilowast
( 1113857
zS
zG3 Slowast Elowast Ilowast
( 1113857
zE
zG3 Slowast Elowast Ilowast
( 1113857
zI
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
(12)
where
zF1
zS
1[1 + μϕ(h) + βϕ(h)I]
zF1
zE
zF1
zI
minus (S + μNϕ(h))βϕ(h)
[1 + μϕ(h) + βϕ(h)I]2
zF2
zS
βϕ(h)IzF1zS
1 + ϕ(h)[μ + σ]
zF2
zE1 + βϕ(h)IzF1zE
1 + ϕ(h)[μ + σ]
zF2
zIβϕ(h) F1 + IzF1zI1113864 1113865
1 + ϕ(h)[μ + σ]
zF3
zE
σϕ(h)zF2zE
1 + ϕ(h)[μ + c]
zF3
zI1 + σϕ(h)zF2zS
1 + ϕ(h)[μ + c]
zG1
zS
1 + ϵminus 1zF1zS1113872 1113873 1 + ϵminus 1
+ μϕ(h) + βϕ(h)F31113960 1113961 minus S + μNϕ(h) + ϵminus 1F11113872 1113873βϕ(h)zF3zS
1 + ϵminus 1 + μϕ(h) + βϕ(l)F31113960 11139612
zG1
zEϵminus 1
zF1zE1113872 1113873 1 + ϵminus 1+ μϕ(h) + βϕ(h)F31113960 1113961 minus S + μNϕ(h) + ϵminus 1
F11113872 1113873βϕ(h)zF3zE
1 + ϵminus 1 + μϕ(h) + βϕ(h)F31113960 11139612
zG1
zIϵminus 1
zF1zI1113872 1113873 1 + ϵminus 1+ μϕ(h) + βϕ(h)F31113960 1113961 minus S + μNϕ(h) + ϵminus 1
F11113872 1113873βϕ(h)zF3zI
1 + ϵminus 1 + μϕ(h) + βϕ(h)F31113960 11139612
zG2
zS
βϕ(h)
1 + ϕ(h)[μ + σ]
zG1
zSF3 + G1
zF3
zS1113896 1113897
zG2
zE1 + βϕ(h) zG1zEF3 + G1zF3zE1113858 1113859
1 + ϕ(h)[μ + σ]
4 Journal of Mathematics
zG2
zI
βϕ(h)
1 + ϕ(h)[μ + σ]
zG1
zIF3 + G1
zF3
zI1113896 1113897
zG3
zS
σϕ(h)zG2zS
1 + ϕ(h)[μ + c]
zG3
zE
σϕ(h)zG2zE
1 + ϕ(h)[μ + c]
zG3
zI1 + σϕ(h)zG2zI
1 + ϕ(h)[μ + c]
(13)
51 Disease-Free Equilibrium (SoEo Io) (N 0 0) -evalues of functions and its derivatives at DFE point (N 0 0)
are as
F1(N 0 0) N F2(N 0 0) 0 F3(N 0 0) 0
G1(N 0 0) N G2(N 0 0) 0 G3(N 0 0) 0
zF1(N 0 0)
zS
11 + μϕ(h)
zF1(N 0 0)
zE 0
zF1(N 0 0)
zI
minus Nβϕ(h)
[1 + μϕ(h)]
zF2(N 0 0)
zS 0
zF2(N 0 0)
zE
11 + ϕ(h)[μ + σ]
zF2(N 0 0)
zI
βϕ(h)
1 + ϕ(h)[μ + σ]
zF3(N 0 0)
zS 0
zF3(N 0 0)
zE
σϕ(h)
[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]]
zF3(N 0 0)
zI1 +[σϕ(h)βNϕ(h)1 + ϕ(h)(μ + c)]
1 + ϕ(h)[μ + c]
zG1(N 0 0)
zS
11 + μϕ(h)
zG1(N 0 0)
zE
minus Nσϕ(h)βϕ(h)
1 + μϕ(h) + ϵminus 11113960 1113961[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]
Journal of Mathematics 5
zG1(N 0 0)
zI
minus Nβϕ(h)
1 + ϵminus 1+ μϕ(h)1113960 1113961
ϵminus 1
1 + μϕ(h)+
1[1 + ϕ(h)(μ + c)
+σNβϕ2
(h)
[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]1113890 1113891
zG2(N 0 0)
zS 0
zG2(N 0 0)
zE
11 + ϕ(h)(μ + σ)
1 +Nβσϕ2(h)
[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]1113890 1113891
zG2(N 0 0)
zI
βϕ(h)
[1 + ϕ(h)(μ + σ)]
1 + ϕ(h)(μ + σ) + Nσβϕ2(h)
[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]1113896 1113897
zG3(N 0 0)
zS
zG3(N 0 0)
zE
σϕ(h)
[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]1 +
σβNϕ2(h)
[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]1113896 1113897
zG3(N 0 0)
zI
1
[1 + ϕ(h)(μ + σ)]+
Nσβϕ2(h)
[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]2 +
N2β2σ2ϕ4(h)
[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + σ)]
2
(14)
For ease in the calculation we use the followingconventions
α μNϕ(h)gt 0 η1 μϕ(h)gt 0 η2
βϕ(h)gt 0 η3 σϕ(h)gt 0
θ 1 + ϵminus 1+ μϕ(h) 1 + ϵminus 1
+ η1 gt 1
δ1 1 + ϕ(h)[μ + σ]gt 1 δ2
(15)
It is clear that θgt η1 1 + η1 gt δ1 1 + η1 gt δ2 and δ2 θ minus ϵminus 1 + cϕ(h) If R0 σβN(μ + σ)(μ + c)lt 1 the Ja-cobian calculated at disease-free points (Slowast0 Elowast0 Ilowast0 )
(N 0 0) is given as
JG(N 0 0)
zG1(N 0 0)
zS
zG1(N 0 0)
zE
zG1(N 0 0)
zI
zG1(N 0 0)
zS
zG1(N 0 0)
zE
zG1(N 0 0)
zI
zG1(N 0 0)
zS
zG1(N 0 0)
zE
zG1(N 0 0)
zI
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
(16)
Since the partial derivatives using the conventions are
zG1(N 0 0)
zS
11 + η1
zG1(N 0 0)
zE
minus Nη2η3θδ1δ2
zG1(N 0 0)
zI
minus Nη2θϵminus 1
1 + η1+
1δ2
+Nη2η3δ1δ2
1113890 1113891
zG2(N 0 0)
zS 0
zG2(N 0 0)
zE
1δ1
+Nη2η3δ21δ2
1113890 1113891
zG2(N 0 0)
zI
Nη2δ1δ2
+N
2η22η3δ21δ2
1113890 1113891
zG3(N 0 0)
zS 0
zG3(N 0 0)
zE
η3δ1δ2
+Nη2η
23
δ21δ22
1113890 1113891
zG3(N 0 0)
zI
1δ2
+Nη2η3δ1δ
22
+N
2η22η23
δ21δ22
1113890 1113891
(17)
6 Journal of Mathematics
So
JG(N 0 0)
11 + η1
minus Nη2η3θδ1δ2
minus Nη2θϵminus 1
1 + η1+
1δ2
+Nη2η3δ1δ2
1113890 1113891
01δ1
+Nη2η3δ21δ2
1113890 1113891Nη2δ1δ2
+N
2η22η3δ21δ2
1113890 1113891
0η3δ1δ2
+Nη2η
23
δ21δ22
1113890 11138911δ2
+Nη2η3δ1δ
22
+N
2η22η23
δ21δ22
1113890 1113891
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
(18)
To calculate the eigenvalues put det (J minus λI) 0
λ1 11 + η1 11 + μϕ(h)lt 1 And
λ2 minus λδ1δ2 + Nη2η3
δ21δ21113888 1113889 +
δ21δ2 + δ1Nη2η3 + N2η22η
23
δ21δ22
1113888 11138891113890 1113891
+δ1δ2 + Nη2η3
δ21δ21113888 1113889
δ21δ2 + δ1Nη2η3 + N2η22η
23
δ21δ22
1113888 1113889 minusNη2δ1 + N
2η22η3δ21δ2
1113888 1113889η3δ1δ2 + Nη2η
23
δ21δ22
1113888 11138891113890 1113891 0
λ2 minus Aλ + B 0
(19)
where
A δ1δ2 + Nη2η3δ21δ21113872 1113873 + δ21δ2 + δ1Nη2η3 + N
2η22η23δ
21δ
221113872 1113873 trace J
lowast
B δ1δ2 + Nη2η3
δ21δ21113888 1113889
δ21δ2 + δ1Nη2η3 + N2η22η
23
δ21δ22
1113888 1113889 minusNη2δ1 + N
2η22η3δ21δ2
1113888 1113889η3δ1δ2 + Nη2η
23
δ21δ22
1113888 11138891113890 1113891 det Jlowast
Jlowast
1δ1
+Nη2η3δ21δ2
1113890 1113891Nη2δ1δ2
+N
2η22η3δ21δ2
1113890 1113891
η3δ1δ2
+Nη2η
23
δ21δ22
1113890 11138911δ2
+Nη2η3δ1δ
22
+N
2η22η23
δ21δ22
1113890 1113891
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
(20)
To calculate the eigenvalues of Jlowast the following lemma isproved
Lemma 1 For the quadratic equation λ2 minus λA + B 0 bothroots satisfy |λi|lt 1 i 2 3 if and only if the followingconditions are satisfied [25]
(1) 1 minus A + Bgt 0
(2) 1 + A + Bgt 0
(3)Blt 1
(21)
Let us define A TraceJlowast B DetJlowast -erefore
Journal of Mathematics 7
A δ1 + δ2( 1113857 Nη2η3 + δ1δ2( 11138571113858 1113859 + N
2η22η23
δ21δ22
gt 0
B 1
δ1δ2+
Nη2η3δ21δ
22lt 1
f(minus 1) 1 + A + Bgt 0
f(0) B 1
δ1δ2+
Nη2η3δ21δ
22lt
1δ1δ2
+Nη2η3δ1δ2
1 + Nη2η3
δ1δ2
1 + R0(μ + σ)(μ + c)ϕ2(h)
1 + ϕ2(h)(μ + σ)(μ + c) + ϕ(h)[(μ + σ)(μ + c)]
lt 1 sinceR0 lt 1( 1113857
rArrBlt 1
(22)
Now for
f(1) 1 minus A + B
δ21δ
22 + δ1 + δ2( 1113857 Nη2η3 + δ1δ2( 11138571113858 1113859 + N
2η22η23 + Nη2η3 + δ1δ2
δ21δ22
gt 0
(23)
As all the conditions of the lemma hold for R0 lt 1therefore the absolute value of both eigenvalues of Jlowast is less
than one whenever R0 lt 1 -us the numerical scheme inequations (8) and (10) will converge unconditionally todisease-free equilibrium (N 0 0) for any value of time step h
whenever R0 lt 1 which is also computationally verified inFigure 2(a)
52 Endemic Equilibrium (SeEe Ie) (NR0 μN μ + σ(1 minus
1 R0) μβ(R0 minus 1)) -e value of functions and its deriv-atives at an endemic point is as
F1 Se Ee Ie( 1113857 N
R0
F2 Se Ee Ie( 1113857 μN 1 minus 1R0( 1113857
(μ + σ)
F3 Se Ee Ie( 1113857 μβ
R0 minus 1( 1113857
G1 Se Ee Ie( 1113857 N
R0
G2 Se Ee Ie( 1113857 μβ
R0 minus 1( 1113857
G3 Se Ee Ie( 1113857 μN 1 minus 1R0( 1113857
(μ + σ)
(24)
8 Journal of Mathematics
zF1 Se Ee Ie( 1113857
zS
11 + μϕ(h)R0
zF1 Se Ee Ie( 1113857
zE 0
zF1 Se Ee Ie( 1113857
zI
minus Nβϕ(h)
1 + μϕ(h)R0
zF2 Se Ee Ie( 1113857
zS
μϕ(h) R0 minus 1( 1113857
1 + μϕ(h)R01113858 1113859[1 + ϕ(h)(μ + σ)]
zF2 Se Ee Ie( 1113857
zE
1[1 + ϕ(h)(μ + σ)]
zF2 Se Ee Ie( 1113857
zI
minus Nβϕ(h) μϕ(l)R20 minus 2μϕ(h)R0 minus 11113960 1113961
R0[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]
zF3 Se Ee Ie( 1113857
zS
μσϕ2(h) R0 minus 1( 1113857
[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)] 1 + μϕ(h)R01113858 1113859
zF3 Se Ee Ie( 1113857
zE
σϕ(h)
[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]
zF3 Se Ee Ie( 1113857
zI
R0[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859 minus σNβϕ2(h) μϕ(h)R20 minus 2μϕ(h) minus 11113960 1113961
R0[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859[1 + ϕ(h)(μ + c)]
(25)
zG1 Se Ee Ie( 1113857
zS
11 + μϕ(h)R01113858 1113859
minusNμβσϕ3(h) R0 minus 1( 1113857
R0 1 + ϵminus 1 + μϕ(h)R01113960 1113961[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859[1 + ϕ(h)(μ + c)]
zG1 Se Ee Ie( 1113857
zE
minus Nβσϕ2(h)
R0 1 + ϵminus 1 + μϕ(h)R01113960 1113961[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]
Eige
n va
lues
Jacobian at DFE
0492
0494
0496
0498
05
0502
0504
0506
0508
051
1000 2000 3000 4000 5000 6000 7000 8000 9000 100000Step size
(a)
098
0981
0982
0983
0984
0985
0986
0987
0988
0989Jacobian at EE
Eige
n va
lues
1000 2000 3000 4000 5000 6000 7000 8000 9000 100000Step size
(b)
Figure 2 Spectral radius of the Jacobian at equilibrium points
Journal of Mathematics 9
zG1 Se Ee Ie( 1113857
zI
minus Nβϕ(h)
1 + ϵminus 1 + μϕ(h)R01113960 11139611 + μϕ(h)R0
middot ϵminus 1+
R0[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859 minus σNβϕ2(h) μϕ(h)R20 minus 2μϕ(h)R0 minus 11113960 1113961
R20[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859[1 + ϕ(h)(μ + c)]
⎡⎣ ⎤⎦
zG2 Se Ee Ie( 1113857
zS
μβϕ(h) R0 minus 1( 1113857
β[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 11138591113890
minusNμβ2σϕ4(h)μ R0 minus 1( 1113857
2
βR0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)] 1 + ϵminus 1 + μϕ(h)R01113960 1113961 1 + μϕ(h)R01113858 1113859
+Nμβσϕ3(h) R0 minus 1( 1113857
R0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)] 1 + μϕ(h)R01113858 1113859
1113891
zG2 Se Ee Ie( 1113857
zE
1[1 + ϕ(h)(μ + σ)]
+Nβσϕ2(h)
R0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)]
minusNμβ2σϕ3
(h) R0 minus 1( 1113857
βR0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)] 1 + ϵminus 1 + μϕ(h)R01113960 1113961
zG2 Se Ee Ie( 1113857
zI
minus Nμβ2ϕ2(h)ϵminus 1R0 minus 1( 1113857
β[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859 1 + ϵminus 1 + μϕ(h)R01113960 1113961
+Nμβ2ϕ2(h) R0 minus 1( 1113857 R0[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859 minus Nβσϕ2(h) μϕ(h)R
20 minus 2μϕ(h)R0 minus 11113960 11139611113966 1113967
βR20[1 + ϕ(h)(μ + σ)]
2[1 + ϕ(h)(μ + c)] 1 + ϵminus 1 + μϕ(h)R01113960 1113961 1 + μϕ(h)R01113858 1113859
+Nβϕ(h)
R0[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]minus
N2β2σϕ3(h) μϕ(h)R
20 minus 2μϕ(h)R0 minus 11113960 1113961
R20[1 + ϕ(h)(μ + σ)]
2[1 + ϕ(h)(μ + c)] 1 + μϕ(h)R01113858 1113859
zG3 Se Ee Ie( 1113857
zS
σϕ(h)
[1 + ϕ(h)(μ + c)]
μβϕ(h) R0 minus 1( 1113857
β[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 11138591113890
minusNμβ2σϕ4(h)μ R0 minus 1( 1113857
2
βR0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)] 1 + ϵminus 1 + μϕ(h)R01113960 1113961 1 + μϕ(h)R01113858 1113859
+Nμβσϕ3(h) R0 minus 1( 1113857
R0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)] 1 + μϕ(h)R01113858 1113859
1113891
zG3 Se Ee Ie( 1113857
zE
σϕ(h)
[1 + ϕ(h)(μ + c)]
1[1 + ϕ(h)(μ + σ)]
minusNμβ2σϕ3(h) R0 minus 1( 1113857
βR0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)] 1 + ϵminus 1 + μϕ(h)R01113960 1113961
⎡⎢⎣
+Nβσϕ2
(h)
R0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)]
1113891
zG3 Se Ee Ie( 1113857
zI
σϕ(h)
[1 + ϕ(h)(μ + c)]
minus Nμβ2ϕ2(h)ϵminus 1R0 minus 1( 1113857
β[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859 1 + ϵminus 1 + μϕ(h)R01113960 1113961⎡⎢⎣
+Nμβ2ϕ2(h) R0 minus 1( 1113857 R0[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859 minus Nβσϕ2(h) μϕ(h)R
20 minus 2μϕ(h)R0 minus 11113960 11139611113966 1113967
βR20[1 + ϕ(h)(μ + σ)]
2[1 + ϕ(h)(μ + c)] 1 + ϵminus 1 + μϕ(h)R01113960 1113961 1 + μϕ(h)R01113858 1113859
⎤⎥⎦
10 Journal of Mathematics
Table 1 Comparison of NSFD scheme with Euler and RK-4 schemes
l 001 01 100 1000Euler PC Convergence Divergence Divergence DivergenceRK-4 Convergence Divergence Divergence DivergenceNSFDPC Convergence Convergence Convergence Convergence
times107
NSFD-PC
Disease-free equilibrium
1
15
2
25
3
35
4
45
5
Susc
eptib
le fr
actio
n
20 40 60 80 100 120 140 160 180 2000Time (years) h = 001
(a)
Disease-free equilibriumtimes104
NSFD-PC
02 04 06 08 1 12 14 16 18 20Time (years) h = 001
0
05
1
15
2
25
3
35
4
45
5
Expo
sed
fract
ion
(b)
times104
NSFD-PC
Disease-free equilibrium
0
05
1
15
2
25
3
Infe
cted
frac
tion
02 04 06 08 1 12 14 16 18 20Time (years) h = 001
(c)
times107
NSFD-PC
X 5675Y 2435e + 007
Endemic equilibrium
100 200 300 400 500 6000Time (years) h = 001
1
15
2
25
3
35
Susc
eptib
le fr
actio
n
(d)
Figure 3 Continued
Journal of Mathematics 11
+Nβϕ(h)
R0[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]minus
N2β2σϕ3(h) μϕ(h)R
20 minus 2μϕ(h)R0 minus 11113960 1113961
R20[1 + ϕ(h)(μ + σ)]
2[1 + ϕ(h)(μ + c)] 1 + μϕ(h)R01113858 1113859
(26)
-e Jacobian evaluated at an endemic point(Slowaste Elowaste Ilowaste ) (Se Ee Ie) (NR0 μNμ + σ(1 minus 1R0) μβ(R0 minus 1)) is evaluated as
JG Slowaste Elowaste Ilowaste( 1113857
zG1 Slowaste Elowaste Ilowaste( 1113857
zS
zG1 Slowaste Elowaste Ilowaste( 1113857
zE
zG1 Slowaste Elowaste Ilowaste( 1113857
zI
zG1 Slowaste Elowaste Ilowaste( 1113857
zS
zG1 Slowaste Elowaste Ilowaste( 1113857
zE
zG1 Slowaste Elowaste Ilowaste( 1113857
zI
zG1 Slowaste Elowaste Ilowaste( 1113857
zS
zG1 Slowaste Elowaste Ilowaste( 1113857
zE
zG1 Slowaste Elowaste Ilowaste( 1113857
zI
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
(27)
It is tedious to calculate the eigenvalues of the Jacobian ofendemic equilibrium analytically so we have plotted thelargest eigenvalue against each step size and Figure 2(b)shows that for all step sizes the spectral radius of the Ja-cobian of EE remains less than one if R0 gt 1 which impliesthat the numerical scheme in equations (8) and (10) isunconditionally convergent if R0 gt 1 for all step sizes
6 Numerical Results and Discussion
All the methods showed the convergence for small step sizesin Table 1 However for large values of step sizes only thenormalized NSFDPC converge to the correct disease-freepoint for β 01 times 10minus 5 as well as the correct endemic pointfor β 03 times 10minus 5 -is means that for the value of the basic
reproductive number (threshold parameter) R0 lt 1 thenDFE is stable ie solution to the system (S E I R) areconverging to it -is can be seen from Figures 3(a)ndash3(c) forh 001 Also for the value of the basic reproductivenumber (threshold parameter) R0 gt 1 then EE is stable iesolution to the system (S E I R) is converging to it-is canbe seen from Figures 3(d)ndash3(f ) for h 001
Figures 4(a)ndash4(c) show how the NSFDPC method ofDFE converges to the equilibrium point for different stepsizes Figures 4(d)ndash4(f) show how the NSFDPC method ofEE converges to the equilibrium point for different step sizes
For comparison with the well-known RK-4 method tosystem (1) using the parameter values given in Table 2 it isfound that in Figures 5(a)ndash5(f) the numerical solutionconverges for h 001 for both equilibrium points and
times105
NSFD-PC
X 562Y 1125e + 004
Endemic equilibrium
100 200 300 400 500 6000Time (years) h = 001
0
1
2
3
4
5
6
7
8Ex
pose
d fra
ctio
n
(e)
times105
NSFD-PC
X 5523Y 7023
Infe
cted
frac
tion
Endemic equilibrium
100 200 300 400 500 6000Time (years) h = 001
0
05
1
15
2
25
3
35
4
45
5
(f )
Figure 3 Time plots of susceptible exposed and infected population using the NSFDPC method with h 001
12 Journal of Mathematics
times107
NSFD-PC
Disease-free equilibrium
20 40 60 80 100 120 140 160 180 2000Time (years) h = 01
1
15
2
25
3
35
4
45
5Su
scep
tible
frac
tion
(a)
times107
NSFD-PC
Disease-free equilibrium
500 1000 1500 2000 2500 3000 3500 40000Time (years) h = 10
1
15
2
25
3
35
4
45
5
Susc
eptib
le fr
actio
n
(b)
Disease-free equilibriumtimes107
times105
NSFD-PC
05 1 15 2 25 3 35 40Time (years) h = 1000
1
15
2
25
3
35
4
45
5
Susc
eptib
le fr
actio
n
(c)
X 5827Y 7022
Infe
cted
frac
tion
Endemic equilibriumtimes104
NSFD-PC
100 200 300 400 500 6000Time (years) h = 01
0
2
4
6
8
10
12
14
16
18
(d)
Endemic equilibriumtimes104
NSFD-PC
X 9600Y 7022
Infe
cted
frac
tion
1000 2000 3000 4000 5000 6000 7000 8000 9000 100000Time (years) h = 10
0
05
1
15
2
25
3
(e)
Endemic equilibriumtimes104
times105
NSFD-PC
X 861e + 005Y 7022
Infe
cted
frac
tion
1 2 3 4 5 6 7 8 9 100Time (years) h = 1000
0
05
1
15
2
25
3
(f )
Figure 4 Time plots of susceptible and infected population using the NSFDPC method with different step sizes (a) h 01 (b) h 10 and(c) h 1000
Journal of Mathematics 13
Table 2 Parameters value of the measles model
Parameters N μ σ c
5 times 107yearminus 1 002 456 yearminus 1 73 yearminus 1
DFE (S0 E0 I0) (N 0 0) β 01 times 10minus 5
EE (Se Ee Ie) (2435 times 107 1125 times 104 7022671) β 03 times 10minus 5
Disease-free equilibrium
RK4
times107
1
15
2
25
3
35
4
45
5
Susc
eptib
le fr
actio
n
20 40 60 80 100 120 140 160 180 2000Time (years) h = 001
(a)
Expo
sed
fract
ion
Disease-free equilibrium
RK4
times104
0
05
1
15
2
25
3
35
4
45
5
02 04 06 08 1 12 14 16 18 20Time (years) h = 001
(b)
times104
Infe
cted
frac
tion
Disease-free equilibrium
RK4
0
05
1
15
2
25
3
02 04 06 08 1 12 14 16 18 20Time (years) h = 001
(c)
times107
X 5578Y 2435e + 007
Endemic equilibrium
RK4
1
15
2
25
3
35
Susc
eptib
le F
ract
ion
100 200 300 400 500 6000Time (years) h = 001
(d)
Figure 5 Continued
14 Journal of Mathematics
Figure 6 shows overflow for a critical value of step size ieh 01 of DFE and EE respectively
Another popular numerical method is the Eulerpredictor-corrector (E-PC) technique which employs theexplicit Euler method as a predictor and the trapezoidalrule as the corrector -is combination is second-orderaccurate but has similar stability properties in PECEmode to the explicit Euler method alone Figures 7(a)ndash7(f ) show convergence results for h 001 in both casesthat are DFE and EE -e value of h 01 which produces
overflow when solving the system with the parametervalues of Table 2 for the PECE combination can be seen inFigure 8
-us the presented numerical results demonstrated thatNSFDPC has better convergence property following theEuler PC and RK-4 as shown in Figure 9 It is proved thatthe approximations made by other standard numericalmethods experience difficulties in preserving either thestability or the positivity of the solutions or both butNSFDPC is unconditionally convergent
times105
X 5772Y 1125e + 004
Endemic equilibrium
RK4
Expo
sed
fract
ion
0
1
2
3
4
5
6
7
8
9
10
100 200 300 400 500 6000Time (years) h = 001
(e)
times105
X 7659Y 7022
RK4
Endemic equilibrium
Infe
cted
frac
tion
0
1
2
3
4
5
6
100 200 300 400 500 600 700 8000Time (years) h = 001
(f )
Figure 5 Time plots of susceptible exposed and infected population using the RK-4 method with h 001
times1087 RK-4 scheme (disease-free equilibrium)
SusceptibleExposedInfected
01 02 03 04 05 06 07 08 09 10Time (years) h = 01
ndash2
ndash15
ndash1
ndash05
0
05
1
15
2
Popu
latio
n fra
ctio
ns
(a)
RK-4 scheme (endemic equilibrium)
SusceptibleExposedInfected
times10145
01 02 03 04 05 06 07 08 09 10Time (years) h = 01
ndash8
ndash6
ndash4
ndash2
0
2
4
6
8
Popu
latio
n fra
ctio
ns
(b)
Figure 6 Divergence graphs of the population using the RK-4 method with step size h 01
Journal of Mathematics 15
Euler-PC
Disease-free equilibriumtimes107
1
15
2
25
3
35
4
45
5Su
scep
tible
frac
tion
20 40 60 80 100 120 140 160 180 2000Time (years) h = 001
(a)
Euler-PC
Disease-free equilibrium
Expo
sed
fract
ion
times104
0
05
1
15
2
25
3
35
4
45
5
02 04 06 08 1 12 14 16 18 20Time (years) h = 001
(b)
Euler-PC
Disease-free equilibrium
Infe
cted
frac
tion
times104
0
05
1
15
2
25
3
35
02 04 06 08 1 12 14 16 18 20Time (years) h = 001
(c)
Euler-PC
X 5733Y 2435e + 007
Endemic equilibriumtimes107
1
15
2
25
3
35Su
scep
tible
frac
tion
100 200 300 400 500 6000Time (years) h = 001
(d)
Euler-PC
X 5662Y 1118e + 004
Endemic equilibrium
Expo
sed
fract
ion
times105
0
2
4
6
8
10
12
100 200 300 400 500 6000Time (years) h = 001
(e)
Euler-PC
X 5427Y 7025
Endemic equilibrium
Infe
cted
frac
tion
times105
0
1
2
3
4
5
6
7
100 200 300 400 500 6000Time (years) h = 001
(f )
Figure 7 Time plots of population using the Euler-PC method with h 001
16 Journal of Mathematics
7 Conclusions
In this paper a normalized NSFDPC for the SEIR modelconcerning the transmission dynamics of measles is con-structed and analysed -is proposed numerical scheme isvery competitive It is qualitatively stable that is it doublerefines the solution and gives realistic results even for largestep sizes It is dynamically consistent with a continuoussystem and unconditionally convergent and satisfies the
positivity of the state variables involved in the systemSimulations are carried out and its usefulness is comparedwith a well-known numerical method of standard differenceschemes such as RK4 and Euler predictor-corrector method-e standard finite difference schemes are highly dependenton step sizes and initial value problems but NSFDPC isindependent of these two features which make it morepractical Also this method saves computation time andmemory
Euler-PC scheme (disease-free equilibrium)times1041
ndash3
ndash2
ndash1
0
1
2
3
Popu
latio
n fra
ctio
ns
01 02 03 04 05 06 07 08 09 10Time (years) h = 01
SusceptibleExposedInfected
(a)
Euler-PC scheme (endemic equilibrium)times1053
ndash6
ndash4
ndash2
0
2
4
6
Popu
latio
n fra
ctio
ns
01 02 03 04 05 06 07 08 09 10Time (years) h = 01
SusceptibleExposedInfected
(b)
Figure 8 Divergence graphs of the population using the Euler-PC method with step size h 01
NSFD-PCRK-4
times1092
Infe
cted
frac
tion
ndash5
ndash4
ndash3
ndash2
ndash1
0
1
02 04 06 08 1 12 14 16 18 20Time (years) h = 01
(a)
NSFD-PCEuler-PC
times10213
Infe
cted
frac
tion
ndash4
ndash35
ndash3
ndash25
ndash2
ndash15
ndash1
ndash05
0
05
02 04 06 08 1 12 14 16 18 20Time (years) h = 01
(b)
Figure 9 Comparative convergence graphs of the NSFDPC method
Journal of Mathematics 17
Data Availability
-e data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
-e authors declare no conflicts of interest
Acknowledgments
-e authors extend their appreciation to the Deanship ofScientific Research at Majmaah University for funding thiswork under Project no RGP-2019-6
References
[1] E Hunter B Namee and J Kelleher ldquoAn open-data-drivenagent-based model to simulate infectious disease outbreaksrdquoPLoS One vol 14 no 1 pp 1ndash35 2018
[2] E A Bakare Y A Adekunle and K O Kadiri ldquoModeling andsimulation of the dynamics of the transmission of measlesrdquoInternational Journal of Computer Trends and Trends andTechnology vol 3 no 1 pp 174ndash177 2012
[3] L Zhou Y Wang Y Li and Y Michael ldquoGlobal dynamics ofa discrete age-structured SIR epidemic model with applica-tions to measles vaccination strategiesrdquo Mathematical Bio-sciences vol 308 pp 27ndash37 2019
[4] P Widyaningsih D R S Saputro and A W NugroholdquoSusceptible exposed infected recovery (SEIR) model withimmigration equilibria points and its applicationrdquo AmericanInstitute of Physics vol 1 pp 1ndash7 2018
[5] M G Roberts and J A P HeesterbeekMathematical Modelsin Epidemiology Mathematical Models UNESCO-EOLSSAbu Dhabi UAE 2007
[6] J H He F Y Ji and H M Sedighi ldquoDifference equation vsdifferential equation on different scalesrdquo International Journalof Numerical Methods for Heat amp Fluid Flow vol 18pp 0961ndash5539 2020
[7] R E Mickens Nonstandard Finite Difference Models of Dif-ferential Equations World Scientific Singapore 1994
[8] R E Mickens Applications of Nonstandard Finite DifferenceSchemes World Scientific Singapore 2000
[9] R E Mickens ldquoNumerical integration of population modelssatisfying conservation laws NSFD methodsrdquo Journal ofBiological Dynamics vol 1 no 4 pp 427ndash436 2007
[10] J D Lambert Computational Methods in Ordinary Differ-ential Equations Wiley New York NY USA 1973
[11] S M Moghadas M E Alexander B D Corbett andA B Gumel ldquoA positivity-preserving mickens-type dis-cretization of an epidemic modelrdquo Journal of DifferenceEquations and Applications vol 9 no 11 pp 1037ndash10512003
[12] R E Mickens ldquoNonstandard finite difference schemes fordifferential equationsrdquo Journal of Difference Equations andApplications vol 8 no 9 pp 823ndash847 2002
[13] R E Mickens ldquoDynamic consistency a fundamental prin-ciple for constructing nonstandard finite difference schemesfor differential equationsrdquo Journal of Difference Equations andApplications vol 11 no 7 pp 645ndash653 2005
[14] L-I W Roeger and R W Barnard ldquoPreservation of localdynamics when applying central difference methods appli-cation to SIR modelrdquo Journal of Difference Equations andApplications vol 13 no 4 pp 333ndash340 2007
[15] K C Patidar ldquoOn the use of non-standard finite differencemethodsrdquo Journal of Difference Equations and Applicationsvol 11 no 7 pp 735ndash758 2005
[16] R Anguelov and J M-S Lubuma ldquoNonstandard finite dif-ference method by nonlocal approximationrdquo Mathematicsand Computers in Simulation vol 61 no 3-6 pp 465ndash4752003
[17] R Anguelov and J M-S Lubuma ldquoContributions to themathematics of the nonstandard finite difference method andapplicationsrdquo Numerical Methods for Partial DifferentialEquations vol 17 no 5 pp 518ndash543 2001
[18] B M Chen-Charpentier D T Dimitrov andH V Kojouharov ldquoNumerical simulation of multi-speciesbiofilms in porous media for different kineticsrdquo Mathematicsand Computers in Simulation vol 79 no 6 pp 1846ndash18612009
[19] D T Dimitrov and H V Kojouharov ldquoNonstandard finite-difference schemes for general two-dimensional autonomousdynamical systemsrdquo Applied Mathematics Letters vol 18no 7 pp 769ndash774 2005
[20] A J Arenas J A Morantildeo and J C Cortes ldquoNon-standardnumerical method for a mathematical model of RSV epide-miological transmissionrdquo Computers amp Mathematics withApplications vol 56 no 3 pp 670ndash678 2008
[21] L Jodar R J Villanueva A J Arenas and G C GonzalezldquoNonstandard numerical methods for a mathematical modelfor influenza diseaserdquo Mathematics and Computers in Sim-ulation vol 79 no 3 pp 622ndash633 2008
[22] W Piyawong E H Twizell and A B Gumel ldquoAn uncon-ditionally convergent finite-difference scheme for the SIRmodelrdquo Applied Mathematics and Computation vol 146no 2-3 pp 611ndash625 2003
[23] H Jansen and E H Twizell ldquoAn unconditionally convergentdiscretization of the SEIR modelrdquo Mathematics and Com-puters in Simulation vol 58 no 2 pp 147ndash158 2002
[24] A J Arenas G Gonzalez-Parra and B M Chen-CharpentierldquoA nonstandard numerical scheme of predictor-correctortype for epidemic modelsrdquo Computers amp Mathematics withApplications vol 59 no 12 pp 3740ndash3749 2010
[25] F Brauer and C Castillo-Chavez Mathematical Models inPopulation Biology and Epidemiology Springer Berlin Ger-many 2001
18 Journal of Mathematics
-e convergence and stability analysis of scheme (10) iscarried out by calculating the eigenvalues of the Jacobian ofthe linearized scheme and studying its behavior and
evaluated at the fixed points If (Slowast Elowast Ilowast) is the fixed pointof system (1) then the Jacobian matrix JG is given by
JG Slowast Elowast Ilowast
( 1113857
zG1 Slowast Elowast Ilowast
( 1113857
zS
zG1 Slowast Elowast Ilowast
( 1113857
zE
zG1 Slowast Elowast Ilowast
( 1113857
zI
zG2 Slowast Elowast Ilowast
( 1113857
zS
zG2 Slowast Elowast Ilowast
( 1113857
zE
zG2 Slowast Elowast Ilowast
( 1113857
zI
zG3 Slowast Elowast Ilowast
( 1113857
zS
zG3 Slowast Elowast Ilowast
( 1113857
zE
zG3 Slowast Elowast Ilowast
( 1113857
zI
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
(12)
where
zF1
zS
1[1 + μϕ(h) + βϕ(h)I]
zF1
zE
zF1
zI
minus (S + μNϕ(h))βϕ(h)
[1 + μϕ(h) + βϕ(h)I]2
zF2
zS
βϕ(h)IzF1zS
1 + ϕ(h)[μ + σ]
zF2
zE1 + βϕ(h)IzF1zE
1 + ϕ(h)[μ + σ]
zF2
zIβϕ(h) F1 + IzF1zI1113864 1113865
1 + ϕ(h)[μ + σ]
zF3
zE
σϕ(h)zF2zE
1 + ϕ(h)[μ + c]
zF3
zI1 + σϕ(h)zF2zS
1 + ϕ(h)[μ + c]
zG1
zS
1 + ϵminus 1zF1zS1113872 1113873 1 + ϵminus 1
+ μϕ(h) + βϕ(h)F31113960 1113961 minus S + μNϕ(h) + ϵminus 1F11113872 1113873βϕ(h)zF3zS
1 + ϵminus 1 + μϕ(h) + βϕ(l)F31113960 11139612
zG1
zEϵminus 1
zF1zE1113872 1113873 1 + ϵminus 1+ μϕ(h) + βϕ(h)F31113960 1113961 minus S + μNϕ(h) + ϵminus 1
F11113872 1113873βϕ(h)zF3zE
1 + ϵminus 1 + μϕ(h) + βϕ(h)F31113960 11139612
zG1
zIϵminus 1
zF1zI1113872 1113873 1 + ϵminus 1+ μϕ(h) + βϕ(h)F31113960 1113961 minus S + μNϕ(h) + ϵminus 1
F11113872 1113873βϕ(h)zF3zI
1 + ϵminus 1 + μϕ(h) + βϕ(h)F31113960 11139612
zG2
zS
βϕ(h)
1 + ϕ(h)[μ + σ]
zG1
zSF3 + G1
zF3
zS1113896 1113897
zG2
zE1 + βϕ(h) zG1zEF3 + G1zF3zE1113858 1113859
1 + ϕ(h)[μ + σ]
4 Journal of Mathematics
zG2
zI
βϕ(h)
1 + ϕ(h)[μ + σ]
zG1
zIF3 + G1
zF3
zI1113896 1113897
zG3
zS
σϕ(h)zG2zS
1 + ϕ(h)[μ + c]
zG3
zE
σϕ(h)zG2zE
1 + ϕ(h)[μ + c]
zG3
zI1 + σϕ(h)zG2zI
1 + ϕ(h)[μ + c]
(13)
51 Disease-Free Equilibrium (SoEo Io) (N 0 0) -evalues of functions and its derivatives at DFE point (N 0 0)
are as
F1(N 0 0) N F2(N 0 0) 0 F3(N 0 0) 0
G1(N 0 0) N G2(N 0 0) 0 G3(N 0 0) 0
zF1(N 0 0)
zS
11 + μϕ(h)
zF1(N 0 0)
zE 0
zF1(N 0 0)
zI
minus Nβϕ(h)
[1 + μϕ(h)]
zF2(N 0 0)
zS 0
zF2(N 0 0)
zE
11 + ϕ(h)[μ + σ]
zF2(N 0 0)
zI
βϕ(h)
1 + ϕ(h)[μ + σ]
zF3(N 0 0)
zS 0
zF3(N 0 0)
zE
σϕ(h)
[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]]
zF3(N 0 0)
zI1 +[σϕ(h)βNϕ(h)1 + ϕ(h)(μ + c)]
1 + ϕ(h)[μ + c]
zG1(N 0 0)
zS
11 + μϕ(h)
zG1(N 0 0)
zE
minus Nσϕ(h)βϕ(h)
1 + μϕ(h) + ϵminus 11113960 1113961[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]
Journal of Mathematics 5
zG1(N 0 0)
zI
minus Nβϕ(h)
1 + ϵminus 1+ μϕ(h)1113960 1113961
ϵminus 1
1 + μϕ(h)+
1[1 + ϕ(h)(μ + c)
+σNβϕ2
(h)
[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]1113890 1113891
zG2(N 0 0)
zS 0
zG2(N 0 0)
zE
11 + ϕ(h)(μ + σ)
1 +Nβσϕ2(h)
[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]1113890 1113891
zG2(N 0 0)
zI
βϕ(h)
[1 + ϕ(h)(μ + σ)]
1 + ϕ(h)(μ + σ) + Nσβϕ2(h)
[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]1113896 1113897
zG3(N 0 0)
zS
zG3(N 0 0)
zE
σϕ(h)
[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]1 +
σβNϕ2(h)
[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]1113896 1113897
zG3(N 0 0)
zI
1
[1 + ϕ(h)(μ + σ)]+
Nσβϕ2(h)
[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]2 +
N2β2σ2ϕ4(h)
[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + σ)]
2
(14)
For ease in the calculation we use the followingconventions
α μNϕ(h)gt 0 η1 μϕ(h)gt 0 η2
βϕ(h)gt 0 η3 σϕ(h)gt 0
θ 1 + ϵminus 1+ μϕ(h) 1 + ϵminus 1
+ η1 gt 1
δ1 1 + ϕ(h)[μ + σ]gt 1 δ2
(15)
It is clear that θgt η1 1 + η1 gt δ1 1 + η1 gt δ2 and δ2 θ minus ϵminus 1 + cϕ(h) If R0 σβN(μ + σ)(μ + c)lt 1 the Ja-cobian calculated at disease-free points (Slowast0 Elowast0 Ilowast0 )
(N 0 0) is given as
JG(N 0 0)
zG1(N 0 0)
zS
zG1(N 0 0)
zE
zG1(N 0 0)
zI
zG1(N 0 0)
zS
zG1(N 0 0)
zE
zG1(N 0 0)
zI
zG1(N 0 0)
zS
zG1(N 0 0)
zE
zG1(N 0 0)
zI
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
(16)
Since the partial derivatives using the conventions are
zG1(N 0 0)
zS
11 + η1
zG1(N 0 0)
zE
minus Nη2η3θδ1δ2
zG1(N 0 0)
zI
minus Nη2θϵminus 1
1 + η1+
1δ2
+Nη2η3δ1δ2
1113890 1113891
zG2(N 0 0)
zS 0
zG2(N 0 0)
zE
1δ1
+Nη2η3δ21δ2
1113890 1113891
zG2(N 0 0)
zI
Nη2δ1δ2
+N
2η22η3δ21δ2
1113890 1113891
zG3(N 0 0)
zS 0
zG3(N 0 0)
zE
η3δ1δ2
+Nη2η
23
δ21δ22
1113890 1113891
zG3(N 0 0)
zI
1δ2
+Nη2η3δ1δ
22
+N
2η22η23
δ21δ22
1113890 1113891
(17)
6 Journal of Mathematics
So
JG(N 0 0)
11 + η1
minus Nη2η3θδ1δ2
minus Nη2θϵminus 1
1 + η1+
1δ2
+Nη2η3δ1δ2
1113890 1113891
01δ1
+Nη2η3δ21δ2
1113890 1113891Nη2δ1δ2
+N
2η22η3δ21δ2
1113890 1113891
0η3δ1δ2
+Nη2η
23
δ21δ22
1113890 11138911δ2
+Nη2η3δ1δ
22
+N
2η22η23
δ21δ22
1113890 1113891
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
(18)
To calculate the eigenvalues put det (J minus λI) 0
λ1 11 + η1 11 + μϕ(h)lt 1 And
λ2 minus λδ1δ2 + Nη2η3
δ21δ21113888 1113889 +
δ21δ2 + δ1Nη2η3 + N2η22η
23
δ21δ22
1113888 11138891113890 1113891
+δ1δ2 + Nη2η3
δ21δ21113888 1113889
δ21δ2 + δ1Nη2η3 + N2η22η
23
δ21δ22
1113888 1113889 minusNη2δ1 + N
2η22η3δ21δ2
1113888 1113889η3δ1δ2 + Nη2η
23
δ21δ22
1113888 11138891113890 1113891 0
λ2 minus Aλ + B 0
(19)
where
A δ1δ2 + Nη2η3δ21δ21113872 1113873 + δ21δ2 + δ1Nη2η3 + N
2η22η23δ
21δ
221113872 1113873 trace J
lowast
B δ1δ2 + Nη2η3
δ21δ21113888 1113889
δ21δ2 + δ1Nη2η3 + N2η22η
23
δ21δ22
1113888 1113889 minusNη2δ1 + N
2η22η3δ21δ2
1113888 1113889η3δ1δ2 + Nη2η
23
δ21δ22
1113888 11138891113890 1113891 det Jlowast
Jlowast
1δ1
+Nη2η3δ21δ2
1113890 1113891Nη2δ1δ2
+N
2η22η3δ21δ2
1113890 1113891
η3δ1δ2
+Nη2η
23
δ21δ22
1113890 11138911δ2
+Nη2η3δ1δ
22
+N
2η22η23
δ21δ22
1113890 1113891
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
(20)
To calculate the eigenvalues of Jlowast the following lemma isproved
Lemma 1 For the quadratic equation λ2 minus λA + B 0 bothroots satisfy |λi|lt 1 i 2 3 if and only if the followingconditions are satisfied [25]
(1) 1 minus A + Bgt 0
(2) 1 + A + Bgt 0
(3)Blt 1
(21)
Let us define A TraceJlowast B DetJlowast -erefore
Journal of Mathematics 7
A δ1 + δ2( 1113857 Nη2η3 + δ1δ2( 11138571113858 1113859 + N
2η22η23
δ21δ22
gt 0
B 1
δ1δ2+
Nη2η3δ21δ
22lt 1
f(minus 1) 1 + A + Bgt 0
f(0) B 1
δ1δ2+
Nη2η3δ21δ
22lt
1δ1δ2
+Nη2η3δ1δ2
1 + Nη2η3
δ1δ2
1 + R0(μ + σ)(μ + c)ϕ2(h)
1 + ϕ2(h)(μ + σ)(μ + c) + ϕ(h)[(μ + σ)(μ + c)]
lt 1 sinceR0 lt 1( 1113857
rArrBlt 1
(22)
Now for
f(1) 1 minus A + B
δ21δ
22 + δ1 + δ2( 1113857 Nη2η3 + δ1δ2( 11138571113858 1113859 + N
2η22η23 + Nη2η3 + δ1δ2
δ21δ22
gt 0
(23)
As all the conditions of the lemma hold for R0 lt 1therefore the absolute value of both eigenvalues of Jlowast is less
than one whenever R0 lt 1 -us the numerical scheme inequations (8) and (10) will converge unconditionally todisease-free equilibrium (N 0 0) for any value of time step h
whenever R0 lt 1 which is also computationally verified inFigure 2(a)
52 Endemic Equilibrium (SeEe Ie) (NR0 μN μ + σ(1 minus
1 R0) μβ(R0 minus 1)) -e value of functions and its deriv-atives at an endemic point is as
F1 Se Ee Ie( 1113857 N
R0
F2 Se Ee Ie( 1113857 μN 1 minus 1R0( 1113857
(μ + σ)
F3 Se Ee Ie( 1113857 μβ
R0 minus 1( 1113857
G1 Se Ee Ie( 1113857 N
R0
G2 Se Ee Ie( 1113857 μβ
R0 minus 1( 1113857
G3 Se Ee Ie( 1113857 μN 1 minus 1R0( 1113857
(μ + σ)
(24)
8 Journal of Mathematics
zF1 Se Ee Ie( 1113857
zS
11 + μϕ(h)R0
zF1 Se Ee Ie( 1113857
zE 0
zF1 Se Ee Ie( 1113857
zI
minus Nβϕ(h)
1 + μϕ(h)R0
zF2 Se Ee Ie( 1113857
zS
μϕ(h) R0 minus 1( 1113857
1 + μϕ(h)R01113858 1113859[1 + ϕ(h)(μ + σ)]
zF2 Se Ee Ie( 1113857
zE
1[1 + ϕ(h)(μ + σ)]
zF2 Se Ee Ie( 1113857
zI
minus Nβϕ(h) μϕ(l)R20 minus 2μϕ(h)R0 minus 11113960 1113961
R0[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]
zF3 Se Ee Ie( 1113857
zS
μσϕ2(h) R0 minus 1( 1113857
[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)] 1 + μϕ(h)R01113858 1113859
zF3 Se Ee Ie( 1113857
zE
σϕ(h)
[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]
zF3 Se Ee Ie( 1113857
zI
R0[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859 minus σNβϕ2(h) μϕ(h)R20 minus 2μϕ(h) minus 11113960 1113961
R0[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859[1 + ϕ(h)(μ + c)]
(25)
zG1 Se Ee Ie( 1113857
zS
11 + μϕ(h)R01113858 1113859
minusNμβσϕ3(h) R0 minus 1( 1113857
R0 1 + ϵminus 1 + μϕ(h)R01113960 1113961[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859[1 + ϕ(h)(μ + c)]
zG1 Se Ee Ie( 1113857
zE
minus Nβσϕ2(h)
R0 1 + ϵminus 1 + μϕ(h)R01113960 1113961[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]
Eige
n va
lues
Jacobian at DFE
0492
0494
0496
0498
05
0502
0504
0506
0508
051
1000 2000 3000 4000 5000 6000 7000 8000 9000 100000Step size
(a)
098
0981
0982
0983
0984
0985
0986
0987
0988
0989Jacobian at EE
Eige
n va
lues
1000 2000 3000 4000 5000 6000 7000 8000 9000 100000Step size
(b)
Figure 2 Spectral radius of the Jacobian at equilibrium points
Journal of Mathematics 9
zG1 Se Ee Ie( 1113857
zI
minus Nβϕ(h)
1 + ϵminus 1 + μϕ(h)R01113960 11139611 + μϕ(h)R0
middot ϵminus 1+
R0[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859 minus σNβϕ2(h) μϕ(h)R20 minus 2μϕ(h)R0 minus 11113960 1113961
R20[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859[1 + ϕ(h)(μ + c)]
⎡⎣ ⎤⎦
zG2 Se Ee Ie( 1113857
zS
μβϕ(h) R0 minus 1( 1113857
β[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 11138591113890
minusNμβ2σϕ4(h)μ R0 minus 1( 1113857
2
βR0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)] 1 + ϵminus 1 + μϕ(h)R01113960 1113961 1 + μϕ(h)R01113858 1113859
+Nμβσϕ3(h) R0 minus 1( 1113857
R0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)] 1 + μϕ(h)R01113858 1113859
1113891
zG2 Se Ee Ie( 1113857
zE
1[1 + ϕ(h)(μ + σ)]
+Nβσϕ2(h)
R0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)]
minusNμβ2σϕ3
(h) R0 minus 1( 1113857
βR0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)] 1 + ϵminus 1 + μϕ(h)R01113960 1113961
zG2 Se Ee Ie( 1113857
zI
minus Nμβ2ϕ2(h)ϵminus 1R0 minus 1( 1113857
β[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859 1 + ϵminus 1 + μϕ(h)R01113960 1113961
+Nμβ2ϕ2(h) R0 minus 1( 1113857 R0[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859 minus Nβσϕ2(h) μϕ(h)R
20 minus 2μϕ(h)R0 minus 11113960 11139611113966 1113967
βR20[1 + ϕ(h)(μ + σ)]
2[1 + ϕ(h)(μ + c)] 1 + ϵminus 1 + μϕ(h)R01113960 1113961 1 + μϕ(h)R01113858 1113859
+Nβϕ(h)
R0[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]minus
N2β2σϕ3(h) μϕ(h)R
20 minus 2μϕ(h)R0 minus 11113960 1113961
R20[1 + ϕ(h)(μ + σ)]
2[1 + ϕ(h)(μ + c)] 1 + μϕ(h)R01113858 1113859
zG3 Se Ee Ie( 1113857
zS
σϕ(h)
[1 + ϕ(h)(μ + c)]
μβϕ(h) R0 minus 1( 1113857
β[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 11138591113890
minusNμβ2σϕ4(h)μ R0 minus 1( 1113857
2
βR0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)] 1 + ϵminus 1 + μϕ(h)R01113960 1113961 1 + μϕ(h)R01113858 1113859
+Nμβσϕ3(h) R0 minus 1( 1113857
R0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)] 1 + μϕ(h)R01113858 1113859
1113891
zG3 Se Ee Ie( 1113857
zE
σϕ(h)
[1 + ϕ(h)(μ + c)]
1[1 + ϕ(h)(μ + σ)]
minusNμβ2σϕ3(h) R0 minus 1( 1113857
βR0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)] 1 + ϵminus 1 + μϕ(h)R01113960 1113961
⎡⎢⎣
+Nβσϕ2
(h)
R0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)]
1113891
zG3 Se Ee Ie( 1113857
zI
σϕ(h)
[1 + ϕ(h)(μ + c)]
minus Nμβ2ϕ2(h)ϵminus 1R0 minus 1( 1113857
β[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859 1 + ϵminus 1 + μϕ(h)R01113960 1113961⎡⎢⎣
+Nμβ2ϕ2(h) R0 minus 1( 1113857 R0[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859 minus Nβσϕ2(h) μϕ(h)R
20 minus 2μϕ(h)R0 minus 11113960 11139611113966 1113967
βR20[1 + ϕ(h)(μ + σ)]
2[1 + ϕ(h)(μ + c)] 1 + ϵminus 1 + μϕ(h)R01113960 1113961 1 + μϕ(h)R01113858 1113859
⎤⎥⎦
10 Journal of Mathematics
Table 1 Comparison of NSFD scheme with Euler and RK-4 schemes
l 001 01 100 1000Euler PC Convergence Divergence Divergence DivergenceRK-4 Convergence Divergence Divergence DivergenceNSFDPC Convergence Convergence Convergence Convergence
times107
NSFD-PC
Disease-free equilibrium
1
15
2
25
3
35
4
45
5
Susc
eptib
le fr
actio
n
20 40 60 80 100 120 140 160 180 2000Time (years) h = 001
(a)
Disease-free equilibriumtimes104
NSFD-PC
02 04 06 08 1 12 14 16 18 20Time (years) h = 001
0
05
1
15
2
25
3
35
4
45
5
Expo
sed
fract
ion
(b)
times104
NSFD-PC
Disease-free equilibrium
0
05
1
15
2
25
3
Infe
cted
frac
tion
02 04 06 08 1 12 14 16 18 20Time (years) h = 001
(c)
times107
NSFD-PC
X 5675Y 2435e + 007
Endemic equilibrium
100 200 300 400 500 6000Time (years) h = 001
1
15
2
25
3
35
Susc
eptib
le fr
actio
n
(d)
Figure 3 Continued
Journal of Mathematics 11
+Nβϕ(h)
R0[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]minus
N2β2σϕ3(h) μϕ(h)R
20 minus 2μϕ(h)R0 minus 11113960 1113961
R20[1 + ϕ(h)(μ + σ)]
2[1 + ϕ(h)(μ + c)] 1 + μϕ(h)R01113858 1113859
(26)
-e Jacobian evaluated at an endemic point(Slowaste Elowaste Ilowaste ) (Se Ee Ie) (NR0 μNμ + σ(1 minus 1R0) μβ(R0 minus 1)) is evaluated as
JG Slowaste Elowaste Ilowaste( 1113857
zG1 Slowaste Elowaste Ilowaste( 1113857
zS
zG1 Slowaste Elowaste Ilowaste( 1113857
zE
zG1 Slowaste Elowaste Ilowaste( 1113857
zI
zG1 Slowaste Elowaste Ilowaste( 1113857
zS
zG1 Slowaste Elowaste Ilowaste( 1113857
zE
zG1 Slowaste Elowaste Ilowaste( 1113857
zI
zG1 Slowaste Elowaste Ilowaste( 1113857
zS
zG1 Slowaste Elowaste Ilowaste( 1113857
zE
zG1 Slowaste Elowaste Ilowaste( 1113857
zI
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
(27)
It is tedious to calculate the eigenvalues of the Jacobian ofendemic equilibrium analytically so we have plotted thelargest eigenvalue against each step size and Figure 2(b)shows that for all step sizes the spectral radius of the Ja-cobian of EE remains less than one if R0 gt 1 which impliesthat the numerical scheme in equations (8) and (10) isunconditionally convergent if R0 gt 1 for all step sizes
6 Numerical Results and Discussion
All the methods showed the convergence for small step sizesin Table 1 However for large values of step sizes only thenormalized NSFDPC converge to the correct disease-freepoint for β 01 times 10minus 5 as well as the correct endemic pointfor β 03 times 10minus 5 -is means that for the value of the basic
reproductive number (threshold parameter) R0 lt 1 thenDFE is stable ie solution to the system (S E I R) areconverging to it -is can be seen from Figures 3(a)ndash3(c) forh 001 Also for the value of the basic reproductivenumber (threshold parameter) R0 gt 1 then EE is stable iesolution to the system (S E I R) is converging to it-is canbe seen from Figures 3(d)ndash3(f ) for h 001
Figures 4(a)ndash4(c) show how the NSFDPC method ofDFE converges to the equilibrium point for different stepsizes Figures 4(d)ndash4(f) show how the NSFDPC method ofEE converges to the equilibrium point for different step sizes
For comparison with the well-known RK-4 method tosystem (1) using the parameter values given in Table 2 it isfound that in Figures 5(a)ndash5(f) the numerical solutionconverges for h 001 for both equilibrium points and
times105
NSFD-PC
X 562Y 1125e + 004
Endemic equilibrium
100 200 300 400 500 6000Time (years) h = 001
0
1
2
3
4
5
6
7
8Ex
pose
d fra
ctio
n
(e)
times105
NSFD-PC
X 5523Y 7023
Infe
cted
frac
tion
Endemic equilibrium
100 200 300 400 500 6000Time (years) h = 001
0
05
1
15
2
25
3
35
4
45
5
(f )
Figure 3 Time plots of susceptible exposed and infected population using the NSFDPC method with h 001
12 Journal of Mathematics
times107
NSFD-PC
Disease-free equilibrium
20 40 60 80 100 120 140 160 180 2000Time (years) h = 01
1
15
2
25
3
35
4
45
5Su
scep
tible
frac
tion
(a)
times107
NSFD-PC
Disease-free equilibrium
500 1000 1500 2000 2500 3000 3500 40000Time (years) h = 10
1
15
2
25
3
35
4
45
5
Susc
eptib
le fr
actio
n
(b)
Disease-free equilibriumtimes107
times105
NSFD-PC
05 1 15 2 25 3 35 40Time (years) h = 1000
1
15
2
25
3
35
4
45
5
Susc
eptib
le fr
actio
n
(c)
X 5827Y 7022
Infe
cted
frac
tion
Endemic equilibriumtimes104
NSFD-PC
100 200 300 400 500 6000Time (years) h = 01
0
2
4
6
8
10
12
14
16
18
(d)
Endemic equilibriumtimes104
NSFD-PC
X 9600Y 7022
Infe
cted
frac
tion
1000 2000 3000 4000 5000 6000 7000 8000 9000 100000Time (years) h = 10
0
05
1
15
2
25
3
(e)
Endemic equilibriumtimes104
times105
NSFD-PC
X 861e + 005Y 7022
Infe
cted
frac
tion
1 2 3 4 5 6 7 8 9 100Time (years) h = 1000
0
05
1
15
2
25
3
(f )
Figure 4 Time plots of susceptible and infected population using the NSFDPC method with different step sizes (a) h 01 (b) h 10 and(c) h 1000
Journal of Mathematics 13
Table 2 Parameters value of the measles model
Parameters N μ σ c
5 times 107yearminus 1 002 456 yearminus 1 73 yearminus 1
DFE (S0 E0 I0) (N 0 0) β 01 times 10minus 5
EE (Se Ee Ie) (2435 times 107 1125 times 104 7022671) β 03 times 10minus 5
Disease-free equilibrium
RK4
times107
1
15
2
25
3
35
4
45
5
Susc
eptib
le fr
actio
n
20 40 60 80 100 120 140 160 180 2000Time (years) h = 001
(a)
Expo
sed
fract
ion
Disease-free equilibrium
RK4
times104
0
05
1
15
2
25
3
35
4
45
5
02 04 06 08 1 12 14 16 18 20Time (years) h = 001
(b)
times104
Infe
cted
frac
tion
Disease-free equilibrium
RK4
0
05
1
15
2
25
3
02 04 06 08 1 12 14 16 18 20Time (years) h = 001
(c)
times107
X 5578Y 2435e + 007
Endemic equilibrium
RK4
1
15
2
25
3
35
Susc
eptib
le F
ract
ion
100 200 300 400 500 6000Time (years) h = 001
(d)
Figure 5 Continued
14 Journal of Mathematics
Figure 6 shows overflow for a critical value of step size ieh 01 of DFE and EE respectively
Another popular numerical method is the Eulerpredictor-corrector (E-PC) technique which employs theexplicit Euler method as a predictor and the trapezoidalrule as the corrector -is combination is second-orderaccurate but has similar stability properties in PECEmode to the explicit Euler method alone Figures 7(a)ndash7(f ) show convergence results for h 001 in both casesthat are DFE and EE -e value of h 01 which produces
overflow when solving the system with the parametervalues of Table 2 for the PECE combination can be seen inFigure 8
-us the presented numerical results demonstrated thatNSFDPC has better convergence property following theEuler PC and RK-4 as shown in Figure 9 It is proved thatthe approximations made by other standard numericalmethods experience difficulties in preserving either thestability or the positivity of the solutions or both butNSFDPC is unconditionally convergent
times105
X 5772Y 1125e + 004
Endemic equilibrium
RK4
Expo
sed
fract
ion
0
1
2
3
4
5
6
7
8
9
10
100 200 300 400 500 6000Time (years) h = 001
(e)
times105
X 7659Y 7022
RK4
Endemic equilibrium
Infe
cted
frac
tion
0
1
2
3
4
5
6
100 200 300 400 500 600 700 8000Time (years) h = 001
(f )
Figure 5 Time plots of susceptible exposed and infected population using the RK-4 method with h 001
times1087 RK-4 scheme (disease-free equilibrium)
SusceptibleExposedInfected
01 02 03 04 05 06 07 08 09 10Time (years) h = 01
ndash2
ndash15
ndash1
ndash05
0
05
1
15
2
Popu
latio
n fra
ctio
ns
(a)
RK-4 scheme (endemic equilibrium)
SusceptibleExposedInfected
times10145
01 02 03 04 05 06 07 08 09 10Time (years) h = 01
ndash8
ndash6
ndash4
ndash2
0
2
4
6
8
Popu
latio
n fra
ctio
ns
(b)
Figure 6 Divergence graphs of the population using the RK-4 method with step size h 01
Journal of Mathematics 15
Euler-PC
Disease-free equilibriumtimes107
1
15
2
25
3
35
4
45
5Su
scep
tible
frac
tion
20 40 60 80 100 120 140 160 180 2000Time (years) h = 001
(a)
Euler-PC
Disease-free equilibrium
Expo
sed
fract
ion
times104
0
05
1
15
2
25
3
35
4
45
5
02 04 06 08 1 12 14 16 18 20Time (years) h = 001
(b)
Euler-PC
Disease-free equilibrium
Infe
cted
frac
tion
times104
0
05
1
15
2
25
3
35
02 04 06 08 1 12 14 16 18 20Time (years) h = 001
(c)
Euler-PC
X 5733Y 2435e + 007
Endemic equilibriumtimes107
1
15
2
25
3
35Su
scep
tible
frac
tion
100 200 300 400 500 6000Time (years) h = 001
(d)
Euler-PC
X 5662Y 1118e + 004
Endemic equilibrium
Expo
sed
fract
ion
times105
0
2
4
6
8
10
12
100 200 300 400 500 6000Time (years) h = 001
(e)
Euler-PC
X 5427Y 7025
Endemic equilibrium
Infe
cted
frac
tion
times105
0
1
2
3
4
5
6
7
100 200 300 400 500 6000Time (years) h = 001
(f )
Figure 7 Time plots of population using the Euler-PC method with h 001
16 Journal of Mathematics
7 Conclusions
In this paper a normalized NSFDPC for the SEIR modelconcerning the transmission dynamics of measles is con-structed and analysed -is proposed numerical scheme isvery competitive It is qualitatively stable that is it doublerefines the solution and gives realistic results even for largestep sizes It is dynamically consistent with a continuoussystem and unconditionally convergent and satisfies the
positivity of the state variables involved in the systemSimulations are carried out and its usefulness is comparedwith a well-known numerical method of standard differenceschemes such as RK4 and Euler predictor-corrector method-e standard finite difference schemes are highly dependenton step sizes and initial value problems but NSFDPC isindependent of these two features which make it morepractical Also this method saves computation time andmemory
Euler-PC scheme (disease-free equilibrium)times1041
ndash3
ndash2
ndash1
0
1
2
3
Popu
latio
n fra
ctio
ns
01 02 03 04 05 06 07 08 09 10Time (years) h = 01
SusceptibleExposedInfected
(a)
Euler-PC scheme (endemic equilibrium)times1053
ndash6
ndash4
ndash2
0
2
4
6
Popu
latio
n fra
ctio
ns
01 02 03 04 05 06 07 08 09 10Time (years) h = 01
SusceptibleExposedInfected
(b)
Figure 8 Divergence graphs of the population using the Euler-PC method with step size h 01
NSFD-PCRK-4
times1092
Infe
cted
frac
tion
ndash5
ndash4
ndash3
ndash2
ndash1
0
1
02 04 06 08 1 12 14 16 18 20Time (years) h = 01
(a)
NSFD-PCEuler-PC
times10213
Infe
cted
frac
tion
ndash4
ndash35
ndash3
ndash25
ndash2
ndash15
ndash1
ndash05
0
05
02 04 06 08 1 12 14 16 18 20Time (years) h = 01
(b)
Figure 9 Comparative convergence graphs of the NSFDPC method
Journal of Mathematics 17
Data Availability
-e data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
-e authors declare no conflicts of interest
Acknowledgments
-e authors extend their appreciation to the Deanship ofScientific Research at Majmaah University for funding thiswork under Project no RGP-2019-6
References
[1] E Hunter B Namee and J Kelleher ldquoAn open-data-drivenagent-based model to simulate infectious disease outbreaksrdquoPLoS One vol 14 no 1 pp 1ndash35 2018
[2] E A Bakare Y A Adekunle and K O Kadiri ldquoModeling andsimulation of the dynamics of the transmission of measlesrdquoInternational Journal of Computer Trends and Trends andTechnology vol 3 no 1 pp 174ndash177 2012
[3] L Zhou Y Wang Y Li and Y Michael ldquoGlobal dynamics ofa discrete age-structured SIR epidemic model with applica-tions to measles vaccination strategiesrdquo Mathematical Bio-sciences vol 308 pp 27ndash37 2019
[4] P Widyaningsih D R S Saputro and A W NugroholdquoSusceptible exposed infected recovery (SEIR) model withimmigration equilibria points and its applicationrdquo AmericanInstitute of Physics vol 1 pp 1ndash7 2018
[5] M G Roberts and J A P HeesterbeekMathematical Modelsin Epidemiology Mathematical Models UNESCO-EOLSSAbu Dhabi UAE 2007
[6] J H He F Y Ji and H M Sedighi ldquoDifference equation vsdifferential equation on different scalesrdquo International Journalof Numerical Methods for Heat amp Fluid Flow vol 18pp 0961ndash5539 2020
[7] R E Mickens Nonstandard Finite Difference Models of Dif-ferential Equations World Scientific Singapore 1994
[8] R E Mickens Applications of Nonstandard Finite DifferenceSchemes World Scientific Singapore 2000
[9] R E Mickens ldquoNumerical integration of population modelssatisfying conservation laws NSFD methodsrdquo Journal ofBiological Dynamics vol 1 no 4 pp 427ndash436 2007
[10] J D Lambert Computational Methods in Ordinary Differ-ential Equations Wiley New York NY USA 1973
[11] S M Moghadas M E Alexander B D Corbett andA B Gumel ldquoA positivity-preserving mickens-type dis-cretization of an epidemic modelrdquo Journal of DifferenceEquations and Applications vol 9 no 11 pp 1037ndash10512003
[12] R E Mickens ldquoNonstandard finite difference schemes fordifferential equationsrdquo Journal of Difference Equations andApplications vol 8 no 9 pp 823ndash847 2002
[13] R E Mickens ldquoDynamic consistency a fundamental prin-ciple for constructing nonstandard finite difference schemesfor differential equationsrdquo Journal of Difference Equations andApplications vol 11 no 7 pp 645ndash653 2005
[14] L-I W Roeger and R W Barnard ldquoPreservation of localdynamics when applying central difference methods appli-cation to SIR modelrdquo Journal of Difference Equations andApplications vol 13 no 4 pp 333ndash340 2007
[15] K C Patidar ldquoOn the use of non-standard finite differencemethodsrdquo Journal of Difference Equations and Applicationsvol 11 no 7 pp 735ndash758 2005
[16] R Anguelov and J M-S Lubuma ldquoNonstandard finite dif-ference method by nonlocal approximationrdquo Mathematicsand Computers in Simulation vol 61 no 3-6 pp 465ndash4752003
[17] R Anguelov and J M-S Lubuma ldquoContributions to themathematics of the nonstandard finite difference method andapplicationsrdquo Numerical Methods for Partial DifferentialEquations vol 17 no 5 pp 518ndash543 2001
[18] B M Chen-Charpentier D T Dimitrov andH V Kojouharov ldquoNumerical simulation of multi-speciesbiofilms in porous media for different kineticsrdquo Mathematicsand Computers in Simulation vol 79 no 6 pp 1846ndash18612009
[19] D T Dimitrov and H V Kojouharov ldquoNonstandard finite-difference schemes for general two-dimensional autonomousdynamical systemsrdquo Applied Mathematics Letters vol 18no 7 pp 769ndash774 2005
[20] A J Arenas J A Morantildeo and J C Cortes ldquoNon-standardnumerical method for a mathematical model of RSV epide-miological transmissionrdquo Computers amp Mathematics withApplications vol 56 no 3 pp 670ndash678 2008
[21] L Jodar R J Villanueva A J Arenas and G C GonzalezldquoNonstandard numerical methods for a mathematical modelfor influenza diseaserdquo Mathematics and Computers in Sim-ulation vol 79 no 3 pp 622ndash633 2008
[22] W Piyawong E H Twizell and A B Gumel ldquoAn uncon-ditionally convergent finite-difference scheme for the SIRmodelrdquo Applied Mathematics and Computation vol 146no 2-3 pp 611ndash625 2003
[23] H Jansen and E H Twizell ldquoAn unconditionally convergentdiscretization of the SEIR modelrdquo Mathematics and Com-puters in Simulation vol 58 no 2 pp 147ndash158 2002
[24] A J Arenas G Gonzalez-Parra and B M Chen-CharpentierldquoA nonstandard numerical scheme of predictor-correctortype for epidemic modelsrdquo Computers amp Mathematics withApplications vol 59 no 12 pp 3740ndash3749 2010
[25] F Brauer and C Castillo-Chavez Mathematical Models inPopulation Biology and Epidemiology Springer Berlin Ger-many 2001
18 Journal of Mathematics
zG2
zI
βϕ(h)
1 + ϕ(h)[μ + σ]
zG1
zIF3 + G1
zF3
zI1113896 1113897
zG3
zS
σϕ(h)zG2zS
1 + ϕ(h)[μ + c]
zG3
zE
σϕ(h)zG2zE
1 + ϕ(h)[μ + c]
zG3
zI1 + σϕ(h)zG2zI
1 + ϕ(h)[μ + c]
(13)
51 Disease-Free Equilibrium (SoEo Io) (N 0 0) -evalues of functions and its derivatives at DFE point (N 0 0)
are as
F1(N 0 0) N F2(N 0 0) 0 F3(N 0 0) 0
G1(N 0 0) N G2(N 0 0) 0 G3(N 0 0) 0
zF1(N 0 0)
zS
11 + μϕ(h)
zF1(N 0 0)
zE 0
zF1(N 0 0)
zI
minus Nβϕ(h)
[1 + μϕ(h)]
zF2(N 0 0)
zS 0
zF2(N 0 0)
zE
11 + ϕ(h)[μ + σ]
zF2(N 0 0)
zI
βϕ(h)
1 + ϕ(h)[μ + σ]
zF3(N 0 0)
zS 0
zF3(N 0 0)
zE
σϕ(h)
[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]]
zF3(N 0 0)
zI1 +[σϕ(h)βNϕ(h)1 + ϕ(h)(μ + c)]
1 + ϕ(h)[μ + c]
zG1(N 0 0)
zS
11 + μϕ(h)
zG1(N 0 0)
zE
minus Nσϕ(h)βϕ(h)
1 + μϕ(h) + ϵminus 11113960 1113961[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]
Journal of Mathematics 5
zG1(N 0 0)
zI
minus Nβϕ(h)
1 + ϵminus 1+ μϕ(h)1113960 1113961
ϵminus 1
1 + μϕ(h)+
1[1 + ϕ(h)(μ + c)
+σNβϕ2
(h)
[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]1113890 1113891
zG2(N 0 0)
zS 0
zG2(N 0 0)
zE
11 + ϕ(h)(μ + σ)
1 +Nβσϕ2(h)
[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]1113890 1113891
zG2(N 0 0)
zI
βϕ(h)
[1 + ϕ(h)(μ + σ)]
1 + ϕ(h)(μ + σ) + Nσβϕ2(h)
[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]1113896 1113897
zG3(N 0 0)
zS
zG3(N 0 0)
zE
σϕ(h)
[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]1 +
σβNϕ2(h)
[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]1113896 1113897
zG3(N 0 0)
zI
1
[1 + ϕ(h)(μ + σ)]+
Nσβϕ2(h)
[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]2 +
N2β2σ2ϕ4(h)
[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + σ)]
2
(14)
For ease in the calculation we use the followingconventions
α μNϕ(h)gt 0 η1 μϕ(h)gt 0 η2
βϕ(h)gt 0 η3 σϕ(h)gt 0
θ 1 + ϵminus 1+ μϕ(h) 1 + ϵminus 1
+ η1 gt 1
δ1 1 + ϕ(h)[μ + σ]gt 1 δ2
(15)
It is clear that θgt η1 1 + η1 gt δ1 1 + η1 gt δ2 and δ2 θ minus ϵminus 1 + cϕ(h) If R0 σβN(μ + σ)(μ + c)lt 1 the Ja-cobian calculated at disease-free points (Slowast0 Elowast0 Ilowast0 )
(N 0 0) is given as
JG(N 0 0)
zG1(N 0 0)
zS
zG1(N 0 0)
zE
zG1(N 0 0)
zI
zG1(N 0 0)
zS
zG1(N 0 0)
zE
zG1(N 0 0)
zI
zG1(N 0 0)
zS
zG1(N 0 0)
zE
zG1(N 0 0)
zI
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
(16)
Since the partial derivatives using the conventions are
zG1(N 0 0)
zS
11 + η1
zG1(N 0 0)
zE
minus Nη2η3θδ1δ2
zG1(N 0 0)
zI
minus Nη2θϵminus 1
1 + η1+
1δ2
+Nη2η3δ1δ2
1113890 1113891
zG2(N 0 0)
zS 0
zG2(N 0 0)
zE
1δ1
+Nη2η3δ21δ2
1113890 1113891
zG2(N 0 0)
zI
Nη2δ1δ2
+N
2η22η3δ21δ2
1113890 1113891
zG3(N 0 0)
zS 0
zG3(N 0 0)
zE
η3δ1δ2
+Nη2η
23
δ21δ22
1113890 1113891
zG3(N 0 0)
zI
1δ2
+Nη2η3δ1δ
22
+N
2η22η23
δ21δ22
1113890 1113891
(17)
6 Journal of Mathematics
So
JG(N 0 0)
11 + η1
minus Nη2η3θδ1δ2
minus Nη2θϵminus 1
1 + η1+
1δ2
+Nη2η3δ1δ2
1113890 1113891
01δ1
+Nη2η3δ21δ2
1113890 1113891Nη2δ1δ2
+N
2η22η3δ21δ2
1113890 1113891
0η3δ1δ2
+Nη2η
23
δ21δ22
1113890 11138911δ2
+Nη2η3δ1δ
22
+N
2η22η23
δ21δ22
1113890 1113891
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
(18)
To calculate the eigenvalues put det (J minus λI) 0
λ1 11 + η1 11 + μϕ(h)lt 1 And
λ2 minus λδ1δ2 + Nη2η3
δ21δ21113888 1113889 +
δ21δ2 + δ1Nη2η3 + N2η22η
23
δ21δ22
1113888 11138891113890 1113891
+δ1δ2 + Nη2η3
δ21δ21113888 1113889
δ21δ2 + δ1Nη2η3 + N2η22η
23
δ21δ22
1113888 1113889 minusNη2δ1 + N
2η22η3δ21δ2
1113888 1113889η3δ1δ2 + Nη2η
23
δ21δ22
1113888 11138891113890 1113891 0
λ2 minus Aλ + B 0
(19)
where
A δ1δ2 + Nη2η3δ21δ21113872 1113873 + δ21δ2 + δ1Nη2η3 + N
2η22η23δ
21δ
221113872 1113873 trace J
lowast
B δ1δ2 + Nη2η3
δ21δ21113888 1113889
δ21δ2 + δ1Nη2η3 + N2η22η
23
δ21δ22
1113888 1113889 minusNη2δ1 + N
2η22η3δ21δ2
1113888 1113889η3δ1δ2 + Nη2η
23
δ21δ22
1113888 11138891113890 1113891 det Jlowast
Jlowast
1δ1
+Nη2η3δ21δ2
1113890 1113891Nη2δ1δ2
+N
2η22η3δ21δ2
1113890 1113891
η3δ1δ2
+Nη2η
23
δ21δ22
1113890 11138911δ2
+Nη2η3δ1δ
22
+N
2η22η23
δ21δ22
1113890 1113891
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
(20)
To calculate the eigenvalues of Jlowast the following lemma isproved
Lemma 1 For the quadratic equation λ2 minus λA + B 0 bothroots satisfy |λi|lt 1 i 2 3 if and only if the followingconditions are satisfied [25]
(1) 1 minus A + Bgt 0
(2) 1 + A + Bgt 0
(3)Blt 1
(21)
Let us define A TraceJlowast B DetJlowast -erefore
Journal of Mathematics 7
A δ1 + δ2( 1113857 Nη2η3 + δ1δ2( 11138571113858 1113859 + N
2η22η23
δ21δ22
gt 0
B 1
δ1δ2+
Nη2η3δ21δ
22lt 1
f(minus 1) 1 + A + Bgt 0
f(0) B 1
δ1δ2+
Nη2η3δ21δ
22lt
1δ1δ2
+Nη2η3δ1δ2
1 + Nη2η3
δ1δ2
1 + R0(μ + σ)(μ + c)ϕ2(h)
1 + ϕ2(h)(μ + σ)(μ + c) + ϕ(h)[(μ + σ)(μ + c)]
lt 1 sinceR0 lt 1( 1113857
rArrBlt 1
(22)
Now for
f(1) 1 minus A + B
δ21δ
22 + δ1 + δ2( 1113857 Nη2η3 + δ1δ2( 11138571113858 1113859 + N
2η22η23 + Nη2η3 + δ1δ2
δ21δ22
gt 0
(23)
As all the conditions of the lemma hold for R0 lt 1therefore the absolute value of both eigenvalues of Jlowast is less
than one whenever R0 lt 1 -us the numerical scheme inequations (8) and (10) will converge unconditionally todisease-free equilibrium (N 0 0) for any value of time step h
whenever R0 lt 1 which is also computationally verified inFigure 2(a)
52 Endemic Equilibrium (SeEe Ie) (NR0 μN μ + σ(1 minus
1 R0) μβ(R0 minus 1)) -e value of functions and its deriv-atives at an endemic point is as
F1 Se Ee Ie( 1113857 N
R0
F2 Se Ee Ie( 1113857 μN 1 minus 1R0( 1113857
(μ + σ)
F3 Se Ee Ie( 1113857 μβ
R0 minus 1( 1113857
G1 Se Ee Ie( 1113857 N
R0
G2 Se Ee Ie( 1113857 μβ
R0 minus 1( 1113857
G3 Se Ee Ie( 1113857 μN 1 minus 1R0( 1113857
(μ + σ)
(24)
8 Journal of Mathematics
zF1 Se Ee Ie( 1113857
zS
11 + μϕ(h)R0
zF1 Se Ee Ie( 1113857
zE 0
zF1 Se Ee Ie( 1113857
zI
minus Nβϕ(h)
1 + μϕ(h)R0
zF2 Se Ee Ie( 1113857
zS
μϕ(h) R0 minus 1( 1113857
1 + μϕ(h)R01113858 1113859[1 + ϕ(h)(μ + σ)]
zF2 Se Ee Ie( 1113857
zE
1[1 + ϕ(h)(μ + σ)]
zF2 Se Ee Ie( 1113857
zI
minus Nβϕ(h) μϕ(l)R20 minus 2μϕ(h)R0 minus 11113960 1113961
R0[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]
zF3 Se Ee Ie( 1113857
zS
μσϕ2(h) R0 minus 1( 1113857
[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)] 1 + μϕ(h)R01113858 1113859
zF3 Se Ee Ie( 1113857
zE
σϕ(h)
[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]
zF3 Se Ee Ie( 1113857
zI
R0[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859 minus σNβϕ2(h) μϕ(h)R20 minus 2μϕ(h) minus 11113960 1113961
R0[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859[1 + ϕ(h)(μ + c)]
(25)
zG1 Se Ee Ie( 1113857
zS
11 + μϕ(h)R01113858 1113859
minusNμβσϕ3(h) R0 minus 1( 1113857
R0 1 + ϵminus 1 + μϕ(h)R01113960 1113961[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859[1 + ϕ(h)(μ + c)]
zG1 Se Ee Ie( 1113857
zE
minus Nβσϕ2(h)
R0 1 + ϵminus 1 + μϕ(h)R01113960 1113961[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]
Eige
n va
lues
Jacobian at DFE
0492
0494
0496
0498
05
0502
0504
0506
0508
051
1000 2000 3000 4000 5000 6000 7000 8000 9000 100000Step size
(a)
098
0981
0982
0983
0984
0985
0986
0987
0988
0989Jacobian at EE
Eige
n va
lues
1000 2000 3000 4000 5000 6000 7000 8000 9000 100000Step size
(b)
Figure 2 Spectral radius of the Jacobian at equilibrium points
Journal of Mathematics 9
zG1 Se Ee Ie( 1113857
zI
minus Nβϕ(h)
1 + ϵminus 1 + μϕ(h)R01113960 11139611 + μϕ(h)R0
middot ϵminus 1+
R0[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859 minus σNβϕ2(h) μϕ(h)R20 minus 2μϕ(h)R0 minus 11113960 1113961
R20[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859[1 + ϕ(h)(μ + c)]
⎡⎣ ⎤⎦
zG2 Se Ee Ie( 1113857
zS
μβϕ(h) R0 minus 1( 1113857
β[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 11138591113890
minusNμβ2σϕ4(h)μ R0 minus 1( 1113857
2
βR0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)] 1 + ϵminus 1 + μϕ(h)R01113960 1113961 1 + μϕ(h)R01113858 1113859
+Nμβσϕ3(h) R0 minus 1( 1113857
R0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)] 1 + μϕ(h)R01113858 1113859
1113891
zG2 Se Ee Ie( 1113857
zE
1[1 + ϕ(h)(μ + σ)]
+Nβσϕ2(h)
R0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)]
minusNμβ2σϕ3
(h) R0 minus 1( 1113857
βR0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)] 1 + ϵminus 1 + μϕ(h)R01113960 1113961
zG2 Se Ee Ie( 1113857
zI
minus Nμβ2ϕ2(h)ϵminus 1R0 minus 1( 1113857
β[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859 1 + ϵminus 1 + μϕ(h)R01113960 1113961
+Nμβ2ϕ2(h) R0 minus 1( 1113857 R0[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859 minus Nβσϕ2(h) μϕ(h)R
20 minus 2μϕ(h)R0 minus 11113960 11139611113966 1113967
βR20[1 + ϕ(h)(μ + σ)]
2[1 + ϕ(h)(μ + c)] 1 + ϵminus 1 + μϕ(h)R01113960 1113961 1 + μϕ(h)R01113858 1113859
+Nβϕ(h)
R0[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]minus
N2β2σϕ3(h) μϕ(h)R
20 minus 2μϕ(h)R0 minus 11113960 1113961
R20[1 + ϕ(h)(μ + σ)]
2[1 + ϕ(h)(μ + c)] 1 + μϕ(h)R01113858 1113859
zG3 Se Ee Ie( 1113857
zS
σϕ(h)
[1 + ϕ(h)(μ + c)]
μβϕ(h) R0 minus 1( 1113857
β[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 11138591113890
minusNμβ2σϕ4(h)μ R0 minus 1( 1113857
2
βR0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)] 1 + ϵminus 1 + μϕ(h)R01113960 1113961 1 + μϕ(h)R01113858 1113859
+Nμβσϕ3(h) R0 minus 1( 1113857
R0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)] 1 + μϕ(h)R01113858 1113859
1113891
zG3 Se Ee Ie( 1113857
zE
σϕ(h)
[1 + ϕ(h)(μ + c)]
1[1 + ϕ(h)(μ + σ)]
minusNμβ2σϕ3(h) R0 minus 1( 1113857
βR0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)] 1 + ϵminus 1 + μϕ(h)R01113960 1113961
⎡⎢⎣
+Nβσϕ2
(h)
R0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)]
1113891
zG3 Se Ee Ie( 1113857
zI
σϕ(h)
[1 + ϕ(h)(μ + c)]
minus Nμβ2ϕ2(h)ϵminus 1R0 minus 1( 1113857
β[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859 1 + ϵminus 1 + μϕ(h)R01113960 1113961⎡⎢⎣
+Nμβ2ϕ2(h) R0 minus 1( 1113857 R0[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859 minus Nβσϕ2(h) μϕ(h)R
20 minus 2μϕ(h)R0 minus 11113960 11139611113966 1113967
βR20[1 + ϕ(h)(μ + σ)]
2[1 + ϕ(h)(μ + c)] 1 + ϵminus 1 + μϕ(h)R01113960 1113961 1 + μϕ(h)R01113858 1113859
⎤⎥⎦
10 Journal of Mathematics
Table 1 Comparison of NSFD scheme with Euler and RK-4 schemes
l 001 01 100 1000Euler PC Convergence Divergence Divergence DivergenceRK-4 Convergence Divergence Divergence DivergenceNSFDPC Convergence Convergence Convergence Convergence
times107
NSFD-PC
Disease-free equilibrium
1
15
2
25
3
35
4
45
5
Susc
eptib
le fr
actio
n
20 40 60 80 100 120 140 160 180 2000Time (years) h = 001
(a)
Disease-free equilibriumtimes104
NSFD-PC
02 04 06 08 1 12 14 16 18 20Time (years) h = 001
0
05
1
15
2
25
3
35
4
45
5
Expo
sed
fract
ion
(b)
times104
NSFD-PC
Disease-free equilibrium
0
05
1
15
2
25
3
Infe
cted
frac
tion
02 04 06 08 1 12 14 16 18 20Time (years) h = 001
(c)
times107
NSFD-PC
X 5675Y 2435e + 007
Endemic equilibrium
100 200 300 400 500 6000Time (years) h = 001
1
15
2
25
3
35
Susc
eptib
le fr
actio
n
(d)
Figure 3 Continued
Journal of Mathematics 11
+Nβϕ(h)
R0[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]minus
N2β2σϕ3(h) μϕ(h)R
20 minus 2μϕ(h)R0 minus 11113960 1113961
R20[1 + ϕ(h)(μ + σ)]
2[1 + ϕ(h)(μ + c)] 1 + μϕ(h)R01113858 1113859
(26)
-e Jacobian evaluated at an endemic point(Slowaste Elowaste Ilowaste ) (Se Ee Ie) (NR0 μNμ + σ(1 minus 1R0) μβ(R0 minus 1)) is evaluated as
JG Slowaste Elowaste Ilowaste( 1113857
zG1 Slowaste Elowaste Ilowaste( 1113857
zS
zG1 Slowaste Elowaste Ilowaste( 1113857
zE
zG1 Slowaste Elowaste Ilowaste( 1113857
zI
zG1 Slowaste Elowaste Ilowaste( 1113857
zS
zG1 Slowaste Elowaste Ilowaste( 1113857
zE
zG1 Slowaste Elowaste Ilowaste( 1113857
zI
zG1 Slowaste Elowaste Ilowaste( 1113857
zS
zG1 Slowaste Elowaste Ilowaste( 1113857
zE
zG1 Slowaste Elowaste Ilowaste( 1113857
zI
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
(27)
It is tedious to calculate the eigenvalues of the Jacobian ofendemic equilibrium analytically so we have plotted thelargest eigenvalue against each step size and Figure 2(b)shows that for all step sizes the spectral radius of the Ja-cobian of EE remains less than one if R0 gt 1 which impliesthat the numerical scheme in equations (8) and (10) isunconditionally convergent if R0 gt 1 for all step sizes
6 Numerical Results and Discussion
All the methods showed the convergence for small step sizesin Table 1 However for large values of step sizes only thenormalized NSFDPC converge to the correct disease-freepoint for β 01 times 10minus 5 as well as the correct endemic pointfor β 03 times 10minus 5 -is means that for the value of the basic
reproductive number (threshold parameter) R0 lt 1 thenDFE is stable ie solution to the system (S E I R) areconverging to it -is can be seen from Figures 3(a)ndash3(c) forh 001 Also for the value of the basic reproductivenumber (threshold parameter) R0 gt 1 then EE is stable iesolution to the system (S E I R) is converging to it-is canbe seen from Figures 3(d)ndash3(f ) for h 001
Figures 4(a)ndash4(c) show how the NSFDPC method ofDFE converges to the equilibrium point for different stepsizes Figures 4(d)ndash4(f) show how the NSFDPC method ofEE converges to the equilibrium point for different step sizes
For comparison with the well-known RK-4 method tosystem (1) using the parameter values given in Table 2 it isfound that in Figures 5(a)ndash5(f) the numerical solutionconverges for h 001 for both equilibrium points and
times105
NSFD-PC
X 562Y 1125e + 004
Endemic equilibrium
100 200 300 400 500 6000Time (years) h = 001
0
1
2
3
4
5
6
7
8Ex
pose
d fra
ctio
n
(e)
times105
NSFD-PC
X 5523Y 7023
Infe
cted
frac
tion
Endemic equilibrium
100 200 300 400 500 6000Time (years) h = 001
0
05
1
15
2
25
3
35
4
45
5
(f )
Figure 3 Time plots of susceptible exposed and infected population using the NSFDPC method with h 001
12 Journal of Mathematics
times107
NSFD-PC
Disease-free equilibrium
20 40 60 80 100 120 140 160 180 2000Time (years) h = 01
1
15
2
25
3
35
4
45
5Su
scep
tible
frac
tion
(a)
times107
NSFD-PC
Disease-free equilibrium
500 1000 1500 2000 2500 3000 3500 40000Time (years) h = 10
1
15
2
25
3
35
4
45
5
Susc
eptib
le fr
actio
n
(b)
Disease-free equilibriumtimes107
times105
NSFD-PC
05 1 15 2 25 3 35 40Time (years) h = 1000
1
15
2
25
3
35
4
45
5
Susc
eptib
le fr
actio
n
(c)
X 5827Y 7022
Infe
cted
frac
tion
Endemic equilibriumtimes104
NSFD-PC
100 200 300 400 500 6000Time (years) h = 01
0
2
4
6
8
10
12
14
16
18
(d)
Endemic equilibriumtimes104
NSFD-PC
X 9600Y 7022
Infe
cted
frac
tion
1000 2000 3000 4000 5000 6000 7000 8000 9000 100000Time (years) h = 10
0
05
1
15
2
25
3
(e)
Endemic equilibriumtimes104
times105
NSFD-PC
X 861e + 005Y 7022
Infe
cted
frac
tion
1 2 3 4 5 6 7 8 9 100Time (years) h = 1000
0
05
1
15
2
25
3
(f )
Figure 4 Time plots of susceptible and infected population using the NSFDPC method with different step sizes (a) h 01 (b) h 10 and(c) h 1000
Journal of Mathematics 13
Table 2 Parameters value of the measles model
Parameters N μ σ c
5 times 107yearminus 1 002 456 yearminus 1 73 yearminus 1
DFE (S0 E0 I0) (N 0 0) β 01 times 10minus 5
EE (Se Ee Ie) (2435 times 107 1125 times 104 7022671) β 03 times 10minus 5
Disease-free equilibrium
RK4
times107
1
15
2
25
3
35
4
45
5
Susc
eptib
le fr
actio
n
20 40 60 80 100 120 140 160 180 2000Time (years) h = 001
(a)
Expo
sed
fract
ion
Disease-free equilibrium
RK4
times104
0
05
1
15
2
25
3
35
4
45
5
02 04 06 08 1 12 14 16 18 20Time (years) h = 001
(b)
times104
Infe
cted
frac
tion
Disease-free equilibrium
RK4
0
05
1
15
2
25
3
02 04 06 08 1 12 14 16 18 20Time (years) h = 001
(c)
times107
X 5578Y 2435e + 007
Endemic equilibrium
RK4
1
15
2
25
3
35
Susc
eptib
le F
ract
ion
100 200 300 400 500 6000Time (years) h = 001
(d)
Figure 5 Continued
14 Journal of Mathematics
Figure 6 shows overflow for a critical value of step size ieh 01 of DFE and EE respectively
Another popular numerical method is the Eulerpredictor-corrector (E-PC) technique which employs theexplicit Euler method as a predictor and the trapezoidalrule as the corrector -is combination is second-orderaccurate but has similar stability properties in PECEmode to the explicit Euler method alone Figures 7(a)ndash7(f ) show convergence results for h 001 in both casesthat are DFE and EE -e value of h 01 which produces
overflow when solving the system with the parametervalues of Table 2 for the PECE combination can be seen inFigure 8
-us the presented numerical results demonstrated thatNSFDPC has better convergence property following theEuler PC and RK-4 as shown in Figure 9 It is proved thatthe approximations made by other standard numericalmethods experience difficulties in preserving either thestability or the positivity of the solutions or both butNSFDPC is unconditionally convergent
times105
X 5772Y 1125e + 004
Endemic equilibrium
RK4
Expo
sed
fract
ion
0
1
2
3
4
5
6
7
8
9
10
100 200 300 400 500 6000Time (years) h = 001
(e)
times105
X 7659Y 7022
RK4
Endemic equilibrium
Infe
cted
frac
tion
0
1
2
3
4
5
6
100 200 300 400 500 600 700 8000Time (years) h = 001
(f )
Figure 5 Time plots of susceptible exposed and infected population using the RK-4 method with h 001
times1087 RK-4 scheme (disease-free equilibrium)
SusceptibleExposedInfected
01 02 03 04 05 06 07 08 09 10Time (years) h = 01
ndash2
ndash15
ndash1
ndash05
0
05
1
15
2
Popu
latio
n fra
ctio
ns
(a)
RK-4 scheme (endemic equilibrium)
SusceptibleExposedInfected
times10145
01 02 03 04 05 06 07 08 09 10Time (years) h = 01
ndash8
ndash6
ndash4
ndash2
0
2
4
6
8
Popu
latio
n fra
ctio
ns
(b)
Figure 6 Divergence graphs of the population using the RK-4 method with step size h 01
Journal of Mathematics 15
Euler-PC
Disease-free equilibriumtimes107
1
15
2
25
3
35
4
45
5Su
scep
tible
frac
tion
20 40 60 80 100 120 140 160 180 2000Time (years) h = 001
(a)
Euler-PC
Disease-free equilibrium
Expo
sed
fract
ion
times104
0
05
1
15
2
25
3
35
4
45
5
02 04 06 08 1 12 14 16 18 20Time (years) h = 001
(b)
Euler-PC
Disease-free equilibrium
Infe
cted
frac
tion
times104
0
05
1
15
2
25
3
35
02 04 06 08 1 12 14 16 18 20Time (years) h = 001
(c)
Euler-PC
X 5733Y 2435e + 007
Endemic equilibriumtimes107
1
15
2
25
3
35Su
scep
tible
frac
tion
100 200 300 400 500 6000Time (years) h = 001
(d)
Euler-PC
X 5662Y 1118e + 004
Endemic equilibrium
Expo
sed
fract
ion
times105
0
2
4
6
8
10
12
100 200 300 400 500 6000Time (years) h = 001
(e)
Euler-PC
X 5427Y 7025
Endemic equilibrium
Infe
cted
frac
tion
times105
0
1
2
3
4
5
6
7
100 200 300 400 500 6000Time (years) h = 001
(f )
Figure 7 Time plots of population using the Euler-PC method with h 001
16 Journal of Mathematics
7 Conclusions
In this paper a normalized NSFDPC for the SEIR modelconcerning the transmission dynamics of measles is con-structed and analysed -is proposed numerical scheme isvery competitive It is qualitatively stable that is it doublerefines the solution and gives realistic results even for largestep sizes It is dynamically consistent with a continuoussystem and unconditionally convergent and satisfies the
positivity of the state variables involved in the systemSimulations are carried out and its usefulness is comparedwith a well-known numerical method of standard differenceschemes such as RK4 and Euler predictor-corrector method-e standard finite difference schemes are highly dependenton step sizes and initial value problems but NSFDPC isindependent of these two features which make it morepractical Also this method saves computation time andmemory
Euler-PC scheme (disease-free equilibrium)times1041
ndash3
ndash2
ndash1
0
1
2
3
Popu
latio
n fra
ctio
ns
01 02 03 04 05 06 07 08 09 10Time (years) h = 01
SusceptibleExposedInfected
(a)
Euler-PC scheme (endemic equilibrium)times1053
ndash6
ndash4
ndash2
0
2
4
6
Popu
latio
n fra
ctio
ns
01 02 03 04 05 06 07 08 09 10Time (years) h = 01
SusceptibleExposedInfected
(b)
Figure 8 Divergence graphs of the population using the Euler-PC method with step size h 01
NSFD-PCRK-4
times1092
Infe
cted
frac
tion
ndash5
ndash4
ndash3
ndash2
ndash1
0
1
02 04 06 08 1 12 14 16 18 20Time (years) h = 01
(a)
NSFD-PCEuler-PC
times10213
Infe
cted
frac
tion
ndash4
ndash35
ndash3
ndash25
ndash2
ndash15
ndash1
ndash05
0
05
02 04 06 08 1 12 14 16 18 20Time (years) h = 01
(b)
Figure 9 Comparative convergence graphs of the NSFDPC method
Journal of Mathematics 17
Data Availability
-e data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
-e authors declare no conflicts of interest
Acknowledgments
-e authors extend their appreciation to the Deanship ofScientific Research at Majmaah University for funding thiswork under Project no RGP-2019-6
References
[1] E Hunter B Namee and J Kelleher ldquoAn open-data-drivenagent-based model to simulate infectious disease outbreaksrdquoPLoS One vol 14 no 1 pp 1ndash35 2018
[2] E A Bakare Y A Adekunle and K O Kadiri ldquoModeling andsimulation of the dynamics of the transmission of measlesrdquoInternational Journal of Computer Trends and Trends andTechnology vol 3 no 1 pp 174ndash177 2012
[3] L Zhou Y Wang Y Li and Y Michael ldquoGlobal dynamics ofa discrete age-structured SIR epidemic model with applica-tions to measles vaccination strategiesrdquo Mathematical Bio-sciences vol 308 pp 27ndash37 2019
[4] P Widyaningsih D R S Saputro and A W NugroholdquoSusceptible exposed infected recovery (SEIR) model withimmigration equilibria points and its applicationrdquo AmericanInstitute of Physics vol 1 pp 1ndash7 2018
[5] M G Roberts and J A P HeesterbeekMathematical Modelsin Epidemiology Mathematical Models UNESCO-EOLSSAbu Dhabi UAE 2007
[6] J H He F Y Ji and H M Sedighi ldquoDifference equation vsdifferential equation on different scalesrdquo International Journalof Numerical Methods for Heat amp Fluid Flow vol 18pp 0961ndash5539 2020
[7] R E Mickens Nonstandard Finite Difference Models of Dif-ferential Equations World Scientific Singapore 1994
[8] R E Mickens Applications of Nonstandard Finite DifferenceSchemes World Scientific Singapore 2000
[9] R E Mickens ldquoNumerical integration of population modelssatisfying conservation laws NSFD methodsrdquo Journal ofBiological Dynamics vol 1 no 4 pp 427ndash436 2007
[10] J D Lambert Computational Methods in Ordinary Differ-ential Equations Wiley New York NY USA 1973
[11] S M Moghadas M E Alexander B D Corbett andA B Gumel ldquoA positivity-preserving mickens-type dis-cretization of an epidemic modelrdquo Journal of DifferenceEquations and Applications vol 9 no 11 pp 1037ndash10512003
[12] R E Mickens ldquoNonstandard finite difference schemes fordifferential equationsrdquo Journal of Difference Equations andApplications vol 8 no 9 pp 823ndash847 2002
[13] R E Mickens ldquoDynamic consistency a fundamental prin-ciple for constructing nonstandard finite difference schemesfor differential equationsrdquo Journal of Difference Equations andApplications vol 11 no 7 pp 645ndash653 2005
[14] L-I W Roeger and R W Barnard ldquoPreservation of localdynamics when applying central difference methods appli-cation to SIR modelrdquo Journal of Difference Equations andApplications vol 13 no 4 pp 333ndash340 2007
[15] K C Patidar ldquoOn the use of non-standard finite differencemethodsrdquo Journal of Difference Equations and Applicationsvol 11 no 7 pp 735ndash758 2005
[16] R Anguelov and J M-S Lubuma ldquoNonstandard finite dif-ference method by nonlocal approximationrdquo Mathematicsand Computers in Simulation vol 61 no 3-6 pp 465ndash4752003
[17] R Anguelov and J M-S Lubuma ldquoContributions to themathematics of the nonstandard finite difference method andapplicationsrdquo Numerical Methods for Partial DifferentialEquations vol 17 no 5 pp 518ndash543 2001
[18] B M Chen-Charpentier D T Dimitrov andH V Kojouharov ldquoNumerical simulation of multi-speciesbiofilms in porous media for different kineticsrdquo Mathematicsand Computers in Simulation vol 79 no 6 pp 1846ndash18612009
[19] D T Dimitrov and H V Kojouharov ldquoNonstandard finite-difference schemes for general two-dimensional autonomousdynamical systemsrdquo Applied Mathematics Letters vol 18no 7 pp 769ndash774 2005
[20] A J Arenas J A Morantildeo and J C Cortes ldquoNon-standardnumerical method for a mathematical model of RSV epide-miological transmissionrdquo Computers amp Mathematics withApplications vol 56 no 3 pp 670ndash678 2008
[21] L Jodar R J Villanueva A J Arenas and G C GonzalezldquoNonstandard numerical methods for a mathematical modelfor influenza diseaserdquo Mathematics and Computers in Sim-ulation vol 79 no 3 pp 622ndash633 2008
[22] W Piyawong E H Twizell and A B Gumel ldquoAn uncon-ditionally convergent finite-difference scheme for the SIRmodelrdquo Applied Mathematics and Computation vol 146no 2-3 pp 611ndash625 2003
[23] H Jansen and E H Twizell ldquoAn unconditionally convergentdiscretization of the SEIR modelrdquo Mathematics and Com-puters in Simulation vol 58 no 2 pp 147ndash158 2002
[24] A J Arenas G Gonzalez-Parra and B M Chen-CharpentierldquoA nonstandard numerical scheme of predictor-correctortype for epidemic modelsrdquo Computers amp Mathematics withApplications vol 59 no 12 pp 3740ndash3749 2010
[25] F Brauer and C Castillo-Chavez Mathematical Models inPopulation Biology and Epidemiology Springer Berlin Ger-many 2001
18 Journal of Mathematics
zG1(N 0 0)
zI
minus Nβϕ(h)
1 + ϵminus 1+ μϕ(h)1113960 1113961
ϵminus 1
1 + μϕ(h)+
1[1 + ϕ(h)(μ + c)
+σNβϕ2
(h)
[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]1113890 1113891
zG2(N 0 0)
zS 0
zG2(N 0 0)
zE
11 + ϕ(h)(μ + σ)
1 +Nβσϕ2(h)
[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]1113890 1113891
zG2(N 0 0)
zI
βϕ(h)
[1 + ϕ(h)(μ + σ)]
1 + ϕ(h)(μ + σ) + Nσβϕ2(h)
[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]1113896 1113897
zG3(N 0 0)
zS
zG3(N 0 0)
zE
σϕ(h)
[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]1 +
σβNϕ2(h)
[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]1113896 1113897
zG3(N 0 0)
zI
1
[1 + ϕ(h)(μ + σ)]+
Nσβϕ2(h)
[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]2 +
N2β2σ2ϕ4(h)
[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + σ)]
2
(14)
For ease in the calculation we use the followingconventions
α μNϕ(h)gt 0 η1 μϕ(h)gt 0 η2
βϕ(h)gt 0 η3 σϕ(h)gt 0
θ 1 + ϵminus 1+ μϕ(h) 1 + ϵminus 1
+ η1 gt 1
δ1 1 + ϕ(h)[μ + σ]gt 1 δ2
(15)
It is clear that θgt η1 1 + η1 gt δ1 1 + η1 gt δ2 and δ2 θ minus ϵminus 1 + cϕ(h) If R0 σβN(μ + σ)(μ + c)lt 1 the Ja-cobian calculated at disease-free points (Slowast0 Elowast0 Ilowast0 )
(N 0 0) is given as
JG(N 0 0)
zG1(N 0 0)
zS
zG1(N 0 0)
zE
zG1(N 0 0)
zI
zG1(N 0 0)
zS
zG1(N 0 0)
zE
zG1(N 0 0)
zI
zG1(N 0 0)
zS
zG1(N 0 0)
zE
zG1(N 0 0)
zI
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
(16)
Since the partial derivatives using the conventions are
zG1(N 0 0)
zS
11 + η1
zG1(N 0 0)
zE
minus Nη2η3θδ1δ2
zG1(N 0 0)
zI
minus Nη2θϵminus 1
1 + η1+
1δ2
+Nη2η3δ1δ2
1113890 1113891
zG2(N 0 0)
zS 0
zG2(N 0 0)
zE
1δ1
+Nη2η3δ21δ2
1113890 1113891
zG2(N 0 0)
zI
Nη2δ1δ2
+N
2η22η3δ21δ2
1113890 1113891
zG3(N 0 0)
zS 0
zG3(N 0 0)
zE
η3δ1δ2
+Nη2η
23
δ21δ22
1113890 1113891
zG3(N 0 0)
zI
1δ2
+Nη2η3δ1δ
22
+N
2η22η23
δ21δ22
1113890 1113891
(17)
6 Journal of Mathematics
So
JG(N 0 0)
11 + η1
minus Nη2η3θδ1δ2
minus Nη2θϵminus 1
1 + η1+
1δ2
+Nη2η3δ1δ2
1113890 1113891
01δ1
+Nη2η3δ21δ2
1113890 1113891Nη2δ1δ2
+N
2η22η3δ21δ2
1113890 1113891
0η3δ1δ2
+Nη2η
23
δ21δ22
1113890 11138911δ2
+Nη2η3δ1δ
22
+N
2η22η23
δ21δ22
1113890 1113891
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
(18)
To calculate the eigenvalues put det (J minus λI) 0
λ1 11 + η1 11 + μϕ(h)lt 1 And
λ2 minus λδ1δ2 + Nη2η3
δ21δ21113888 1113889 +
δ21δ2 + δ1Nη2η3 + N2η22η
23
δ21δ22
1113888 11138891113890 1113891
+δ1δ2 + Nη2η3
δ21δ21113888 1113889
δ21δ2 + δ1Nη2η3 + N2η22η
23
δ21δ22
1113888 1113889 minusNη2δ1 + N
2η22η3δ21δ2
1113888 1113889η3δ1δ2 + Nη2η
23
δ21δ22
1113888 11138891113890 1113891 0
λ2 minus Aλ + B 0
(19)
where
A δ1δ2 + Nη2η3δ21δ21113872 1113873 + δ21δ2 + δ1Nη2η3 + N
2η22η23δ
21δ
221113872 1113873 trace J
lowast
B δ1δ2 + Nη2η3
δ21δ21113888 1113889
δ21δ2 + δ1Nη2η3 + N2η22η
23
δ21δ22
1113888 1113889 minusNη2δ1 + N
2η22η3δ21δ2
1113888 1113889η3δ1δ2 + Nη2η
23
δ21δ22
1113888 11138891113890 1113891 det Jlowast
Jlowast
1δ1
+Nη2η3δ21δ2
1113890 1113891Nη2δ1δ2
+N
2η22η3δ21δ2
1113890 1113891
η3δ1δ2
+Nη2η
23
δ21δ22
1113890 11138911δ2
+Nη2η3δ1δ
22
+N
2η22η23
δ21δ22
1113890 1113891
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
(20)
To calculate the eigenvalues of Jlowast the following lemma isproved
Lemma 1 For the quadratic equation λ2 minus λA + B 0 bothroots satisfy |λi|lt 1 i 2 3 if and only if the followingconditions are satisfied [25]
(1) 1 minus A + Bgt 0
(2) 1 + A + Bgt 0
(3)Blt 1
(21)
Let us define A TraceJlowast B DetJlowast -erefore
Journal of Mathematics 7
A δ1 + δ2( 1113857 Nη2η3 + δ1δ2( 11138571113858 1113859 + N
2η22η23
δ21δ22
gt 0
B 1
δ1δ2+
Nη2η3δ21δ
22lt 1
f(minus 1) 1 + A + Bgt 0
f(0) B 1
δ1δ2+
Nη2η3δ21δ
22lt
1δ1δ2
+Nη2η3δ1δ2
1 + Nη2η3
δ1δ2
1 + R0(μ + σ)(μ + c)ϕ2(h)
1 + ϕ2(h)(μ + σ)(μ + c) + ϕ(h)[(μ + σ)(μ + c)]
lt 1 sinceR0 lt 1( 1113857
rArrBlt 1
(22)
Now for
f(1) 1 minus A + B
δ21δ
22 + δ1 + δ2( 1113857 Nη2η3 + δ1δ2( 11138571113858 1113859 + N
2η22η23 + Nη2η3 + δ1δ2
δ21δ22
gt 0
(23)
As all the conditions of the lemma hold for R0 lt 1therefore the absolute value of both eigenvalues of Jlowast is less
than one whenever R0 lt 1 -us the numerical scheme inequations (8) and (10) will converge unconditionally todisease-free equilibrium (N 0 0) for any value of time step h
whenever R0 lt 1 which is also computationally verified inFigure 2(a)
52 Endemic Equilibrium (SeEe Ie) (NR0 μN μ + σ(1 minus
1 R0) μβ(R0 minus 1)) -e value of functions and its deriv-atives at an endemic point is as
F1 Se Ee Ie( 1113857 N
R0
F2 Se Ee Ie( 1113857 μN 1 minus 1R0( 1113857
(μ + σ)
F3 Se Ee Ie( 1113857 μβ
R0 minus 1( 1113857
G1 Se Ee Ie( 1113857 N
R0
G2 Se Ee Ie( 1113857 μβ
R0 minus 1( 1113857
G3 Se Ee Ie( 1113857 μN 1 minus 1R0( 1113857
(μ + σ)
(24)
8 Journal of Mathematics
zF1 Se Ee Ie( 1113857
zS
11 + μϕ(h)R0
zF1 Se Ee Ie( 1113857
zE 0
zF1 Se Ee Ie( 1113857
zI
minus Nβϕ(h)
1 + μϕ(h)R0
zF2 Se Ee Ie( 1113857
zS
μϕ(h) R0 minus 1( 1113857
1 + μϕ(h)R01113858 1113859[1 + ϕ(h)(μ + σ)]
zF2 Se Ee Ie( 1113857
zE
1[1 + ϕ(h)(μ + σ)]
zF2 Se Ee Ie( 1113857
zI
minus Nβϕ(h) μϕ(l)R20 minus 2μϕ(h)R0 minus 11113960 1113961
R0[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]
zF3 Se Ee Ie( 1113857
zS
μσϕ2(h) R0 minus 1( 1113857
[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)] 1 + μϕ(h)R01113858 1113859
zF3 Se Ee Ie( 1113857
zE
σϕ(h)
[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]
zF3 Se Ee Ie( 1113857
zI
R0[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859 minus σNβϕ2(h) μϕ(h)R20 minus 2μϕ(h) minus 11113960 1113961
R0[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859[1 + ϕ(h)(μ + c)]
(25)
zG1 Se Ee Ie( 1113857
zS
11 + μϕ(h)R01113858 1113859
minusNμβσϕ3(h) R0 minus 1( 1113857
R0 1 + ϵminus 1 + μϕ(h)R01113960 1113961[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859[1 + ϕ(h)(μ + c)]
zG1 Se Ee Ie( 1113857
zE
minus Nβσϕ2(h)
R0 1 + ϵminus 1 + μϕ(h)R01113960 1113961[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]
Eige
n va
lues
Jacobian at DFE
0492
0494
0496
0498
05
0502
0504
0506
0508
051
1000 2000 3000 4000 5000 6000 7000 8000 9000 100000Step size
(a)
098
0981
0982
0983
0984
0985
0986
0987
0988
0989Jacobian at EE
Eige
n va
lues
1000 2000 3000 4000 5000 6000 7000 8000 9000 100000Step size
(b)
Figure 2 Spectral radius of the Jacobian at equilibrium points
Journal of Mathematics 9
zG1 Se Ee Ie( 1113857
zI
minus Nβϕ(h)
1 + ϵminus 1 + μϕ(h)R01113960 11139611 + μϕ(h)R0
middot ϵminus 1+
R0[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859 minus σNβϕ2(h) μϕ(h)R20 minus 2μϕ(h)R0 minus 11113960 1113961
R20[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859[1 + ϕ(h)(μ + c)]
⎡⎣ ⎤⎦
zG2 Se Ee Ie( 1113857
zS
μβϕ(h) R0 minus 1( 1113857
β[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 11138591113890
minusNμβ2σϕ4(h)μ R0 minus 1( 1113857
2
βR0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)] 1 + ϵminus 1 + μϕ(h)R01113960 1113961 1 + μϕ(h)R01113858 1113859
+Nμβσϕ3(h) R0 minus 1( 1113857
R0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)] 1 + μϕ(h)R01113858 1113859
1113891
zG2 Se Ee Ie( 1113857
zE
1[1 + ϕ(h)(μ + σ)]
+Nβσϕ2(h)
R0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)]
minusNμβ2σϕ3
(h) R0 minus 1( 1113857
βR0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)] 1 + ϵminus 1 + μϕ(h)R01113960 1113961
zG2 Se Ee Ie( 1113857
zI
minus Nμβ2ϕ2(h)ϵminus 1R0 minus 1( 1113857
β[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859 1 + ϵminus 1 + μϕ(h)R01113960 1113961
+Nμβ2ϕ2(h) R0 minus 1( 1113857 R0[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859 minus Nβσϕ2(h) μϕ(h)R
20 minus 2μϕ(h)R0 minus 11113960 11139611113966 1113967
βR20[1 + ϕ(h)(μ + σ)]
2[1 + ϕ(h)(μ + c)] 1 + ϵminus 1 + μϕ(h)R01113960 1113961 1 + μϕ(h)R01113858 1113859
+Nβϕ(h)
R0[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]minus
N2β2σϕ3(h) μϕ(h)R
20 minus 2μϕ(h)R0 minus 11113960 1113961
R20[1 + ϕ(h)(μ + σ)]
2[1 + ϕ(h)(μ + c)] 1 + μϕ(h)R01113858 1113859
zG3 Se Ee Ie( 1113857
zS
σϕ(h)
[1 + ϕ(h)(μ + c)]
μβϕ(h) R0 minus 1( 1113857
β[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 11138591113890
minusNμβ2σϕ4(h)μ R0 minus 1( 1113857
2
βR0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)] 1 + ϵminus 1 + μϕ(h)R01113960 1113961 1 + μϕ(h)R01113858 1113859
+Nμβσϕ3(h) R0 minus 1( 1113857
R0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)] 1 + μϕ(h)R01113858 1113859
1113891
zG3 Se Ee Ie( 1113857
zE
σϕ(h)
[1 + ϕ(h)(μ + c)]
1[1 + ϕ(h)(μ + σ)]
minusNμβ2σϕ3(h) R0 minus 1( 1113857
βR0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)] 1 + ϵminus 1 + μϕ(h)R01113960 1113961
⎡⎢⎣
+Nβσϕ2
(h)
R0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)]
1113891
zG3 Se Ee Ie( 1113857
zI
σϕ(h)
[1 + ϕ(h)(μ + c)]
minus Nμβ2ϕ2(h)ϵminus 1R0 minus 1( 1113857
β[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859 1 + ϵminus 1 + μϕ(h)R01113960 1113961⎡⎢⎣
+Nμβ2ϕ2(h) R0 minus 1( 1113857 R0[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859 minus Nβσϕ2(h) μϕ(h)R
20 minus 2μϕ(h)R0 minus 11113960 11139611113966 1113967
βR20[1 + ϕ(h)(μ + σ)]
2[1 + ϕ(h)(μ + c)] 1 + ϵminus 1 + μϕ(h)R01113960 1113961 1 + μϕ(h)R01113858 1113859
⎤⎥⎦
10 Journal of Mathematics
Table 1 Comparison of NSFD scheme with Euler and RK-4 schemes
l 001 01 100 1000Euler PC Convergence Divergence Divergence DivergenceRK-4 Convergence Divergence Divergence DivergenceNSFDPC Convergence Convergence Convergence Convergence
times107
NSFD-PC
Disease-free equilibrium
1
15
2
25
3
35
4
45
5
Susc
eptib
le fr
actio
n
20 40 60 80 100 120 140 160 180 2000Time (years) h = 001
(a)
Disease-free equilibriumtimes104
NSFD-PC
02 04 06 08 1 12 14 16 18 20Time (years) h = 001
0
05
1
15
2
25
3
35
4
45
5
Expo
sed
fract
ion
(b)
times104
NSFD-PC
Disease-free equilibrium
0
05
1
15
2
25
3
Infe
cted
frac
tion
02 04 06 08 1 12 14 16 18 20Time (years) h = 001
(c)
times107
NSFD-PC
X 5675Y 2435e + 007
Endemic equilibrium
100 200 300 400 500 6000Time (years) h = 001
1
15
2
25
3
35
Susc
eptib
le fr
actio
n
(d)
Figure 3 Continued
Journal of Mathematics 11
+Nβϕ(h)
R0[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]minus
N2β2σϕ3(h) μϕ(h)R
20 minus 2μϕ(h)R0 minus 11113960 1113961
R20[1 + ϕ(h)(μ + σ)]
2[1 + ϕ(h)(μ + c)] 1 + μϕ(h)R01113858 1113859
(26)
-e Jacobian evaluated at an endemic point(Slowaste Elowaste Ilowaste ) (Se Ee Ie) (NR0 μNμ + σ(1 minus 1R0) μβ(R0 minus 1)) is evaluated as
JG Slowaste Elowaste Ilowaste( 1113857
zG1 Slowaste Elowaste Ilowaste( 1113857
zS
zG1 Slowaste Elowaste Ilowaste( 1113857
zE
zG1 Slowaste Elowaste Ilowaste( 1113857
zI
zG1 Slowaste Elowaste Ilowaste( 1113857
zS
zG1 Slowaste Elowaste Ilowaste( 1113857
zE
zG1 Slowaste Elowaste Ilowaste( 1113857
zI
zG1 Slowaste Elowaste Ilowaste( 1113857
zS
zG1 Slowaste Elowaste Ilowaste( 1113857
zE
zG1 Slowaste Elowaste Ilowaste( 1113857
zI
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
(27)
It is tedious to calculate the eigenvalues of the Jacobian ofendemic equilibrium analytically so we have plotted thelargest eigenvalue against each step size and Figure 2(b)shows that for all step sizes the spectral radius of the Ja-cobian of EE remains less than one if R0 gt 1 which impliesthat the numerical scheme in equations (8) and (10) isunconditionally convergent if R0 gt 1 for all step sizes
6 Numerical Results and Discussion
All the methods showed the convergence for small step sizesin Table 1 However for large values of step sizes only thenormalized NSFDPC converge to the correct disease-freepoint for β 01 times 10minus 5 as well as the correct endemic pointfor β 03 times 10minus 5 -is means that for the value of the basic
reproductive number (threshold parameter) R0 lt 1 thenDFE is stable ie solution to the system (S E I R) areconverging to it -is can be seen from Figures 3(a)ndash3(c) forh 001 Also for the value of the basic reproductivenumber (threshold parameter) R0 gt 1 then EE is stable iesolution to the system (S E I R) is converging to it-is canbe seen from Figures 3(d)ndash3(f ) for h 001
Figures 4(a)ndash4(c) show how the NSFDPC method ofDFE converges to the equilibrium point for different stepsizes Figures 4(d)ndash4(f) show how the NSFDPC method ofEE converges to the equilibrium point for different step sizes
For comparison with the well-known RK-4 method tosystem (1) using the parameter values given in Table 2 it isfound that in Figures 5(a)ndash5(f) the numerical solutionconverges for h 001 for both equilibrium points and
times105
NSFD-PC
X 562Y 1125e + 004
Endemic equilibrium
100 200 300 400 500 6000Time (years) h = 001
0
1
2
3
4
5
6
7
8Ex
pose
d fra
ctio
n
(e)
times105
NSFD-PC
X 5523Y 7023
Infe
cted
frac
tion
Endemic equilibrium
100 200 300 400 500 6000Time (years) h = 001
0
05
1
15
2
25
3
35
4
45
5
(f )
Figure 3 Time plots of susceptible exposed and infected population using the NSFDPC method with h 001
12 Journal of Mathematics
times107
NSFD-PC
Disease-free equilibrium
20 40 60 80 100 120 140 160 180 2000Time (years) h = 01
1
15
2
25
3
35
4
45
5Su
scep
tible
frac
tion
(a)
times107
NSFD-PC
Disease-free equilibrium
500 1000 1500 2000 2500 3000 3500 40000Time (years) h = 10
1
15
2
25
3
35
4
45
5
Susc
eptib
le fr
actio
n
(b)
Disease-free equilibriumtimes107
times105
NSFD-PC
05 1 15 2 25 3 35 40Time (years) h = 1000
1
15
2
25
3
35
4
45
5
Susc
eptib
le fr
actio
n
(c)
X 5827Y 7022
Infe
cted
frac
tion
Endemic equilibriumtimes104
NSFD-PC
100 200 300 400 500 6000Time (years) h = 01
0
2
4
6
8
10
12
14
16
18
(d)
Endemic equilibriumtimes104
NSFD-PC
X 9600Y 7022
Infe
cted
frac
tion
1000 2000 3000 4000 5000 6000 7000 8000 9000 100000Time (years) h = 10
0
05
1
15
2
25
3
(e)
Endemic equilibriumtimes104
times105
NSFD-PC
X 861e + 005Y 7022
Infe
cted
frac
tion
1 2 3 4 5 6 7 8 9 100Time (years) h = 1000
0
05
1
15
2
25
3
(f )
Figure 4 Time plots of susceptible and infected population using the NSFDPC method with different step sizes (a) h 01 (b) h 10 and(c) h 1000
Journal of Mathematics 13
Table 2 Parameters value of the measles model
Parameters N μ σ c
5 times 107yearminus 1 002 456 yearminus 1 73 yearminus 1
DFE (S0 E0 I0) (N 0 0) β 01 times 10minus 5
EE (Se Ee Ie) (2435 times 107 1125 times 104 7022671) β 03 times 10minus 5
Disease-free equilibrium
RK4
times107
1
15
2
25
3
35
4
45
5
Susc
eptib
le fr
actio
n
20 40 60 80 100 120 140 160 180 2000Time (years) h = 001
(a)
Expo
sed
fract
ion
Disease-free equilibrium
RK4
times104
0
05
1
15
2
25
3
35
4
45
5
02 04 06 08 1 12 14 16 18 20Time (years) h = 001
(b)
times104
Infe
cted
frac
tion
Disease-free equilibrium
RK4
0
05
1
15
2
25
3
02 04 06 08 1 12 14 16 18 20Time (years) h = 001
(c)
times107
X 5578Y 2435e + 007
Endemic equilibrium
RK4
1
15
2
25
3
35
Susc
eptib
le F
ract
ion
100 200 300 400 500 6000Time (years) h = 001
(d)
Figure 5 Continued
14 Journal of Mathematics
Figure 6 shows overflow for a critical value of step size ieh 01 of DFE and EE respectively
Another popular numerical method is the Eulerpredictor-corrector (E-PC) technique which employs theexplicit Euler method as a predictor and the trapezoidalrule as the corrector -is combination is second-orderaccurate but has similar stability properties in PECEmode to the explicit Euler method alone Figures 7(a)ndash7(f ) show convergence results for h 001 in both casesthat are DFE and EE -e value of h 01 which produces
overflow when solving the system with the parametervalues of Table 2 for the PECE combination can be seen inFigure 8
-us the presented numerical results demonstrated thatNSFDPC has better convergence property following theEuler PC and RK-4 as shown in Figure 9 It is proved thatthe approximations made by other standard numericalmethods experience difficulties in preserving either thestability or the positivity of the solutions or both butNSFDPC is unconditionally convergent
times105
X 5772Y 1125e + 004
Endemic equilibrium
RK4
Expo
sed
fract
ion
0
1
2
3
4
5
6
7
8
9
10
100 200 300 400 500 6000Time (years) h = 001
(e)
times105
X 7659Y 7022
RK4
Endemic equilibrium
Infe
cted
frac
tion
0
1
2
3
4
5
6
100 200 300 400 500 600 700 8000Time (years) h = 001
(f )
Figure 5 Time plots of susceptible exposed and infected population using the RK-4 method with h 001
times1087 RK-4 scheme (disease-free equilibrium)
SusceptibleExposedInfected
01 02 03 04 05 06 07 08 09 10Time (years) h = 01
ndash2
ndash15
ndash1
ndash05
0
05
1
15
2
Popu
latio
n fra
ctio
ns
(a)
RK-4 scheme (endemic equilibrium)
SusceptibleExposedInfected
times10145
01 02 03 04 05 06 07 08 09 10Time (years) h = 01
ndash8
ndash6
ndash4
ndash2
0
2
4
6
8
Popu
latio
n fra
ctio
ns
(b)
Figure 6 Divergence graphs of the population using the RK-4 method with step size h 01
Journal of Mathematics 15
Euler-PC
Disease-free equilibriumtimes107
1
15
2
25
3
35
4
45
5Su
scep
tible
frac
tion
20 40 60 80 100 120 140 160 180 2000Time (years) h = 001
(a)
Euler-PC
Disease-free equilibrium
Expo
sed
fract
ion
times104
0
05
1
15
2
25
3
35
4
45
5
02 04 06 08 1 12 14 16 18 20Time (years) h = 001
(b)
Euler-PC
Disease-free equilibrium
Infe
cted
frac
tion
times104
0
05
1
15
2
25
3
35
02 04 06 08 1 12 14 16 18 20Time (years) h = 001
(c)
Euler-PC
X 5733Y 2435e + 007
Endemic equilibriumtimes107
1
15
2
25
3
35Su
scep
tible
frac
tion
100 200 300 400 500 6000Time (years) h = 001
(d)
Euler-PC
X 5662Y 1118e + 004
Endemic equilibrium
Expo
sed
fract
ion
times105
0
2
4
6
8
10
12
100 200 300 400 500 6000Time (years) h = 001
(e)
Euler-PC
X 5427Y 7025
Endemic equilibrium
Infe
cted
frac
tion
times105
0
1
2
3
4
5
6
7
100 200 300 400 500 6000Time (years) h = 001
(f )
Figure 7 Time plots of population using the Euler-PC method with h 001
16 Journal of Mathematics
7 Conclusions
In this paper a normalized NSFDPC for the SEIR modelconcerning the transmission dynamics of measles is con-structed and analysed -is proposed numerical scheme isvery competitive It is qualitatively stable that is it doublerefines the solution and gives realistic results even for largestep sizes It is dynamically consistent with a continuoussystem and unconditionally convergent and satisfies the
positivity of the state variables involved in the systemSimulations are carried out and its usefulness is comparedwith a well-known numerical method of standard differenceschemes such as RK4 and Euler predictor-corrector method-e standard finite difference schemes are highly dependenton step sizes and initial value problems but NSFDPC isindependent of these two features which make it morepractical Also this method saves computation time andmemory
Euler-PC scheme (disease-free equilibrium)times1041
ndash3
ndash2
ndash1
0
1
2
3
Popu
latio
n fra
ctio
ns
01 02 03 04 05 06 07 08 09 10Time (years) h = 01
SusceptibleExposedInfected
(a)
Euler-PC scheme (endemic equilibrium)times1053
ndash6
ndash4
ndash2
0
2
4
6
Popu
latio
n fra
ctio
ns
01 02 03 04 05 06 07 08 09 10Time (years) h = 01
SusceptibleExposedInfected
(b)
Figure 8 Divergence graphs of the population using the Euler-PC method with step size h 01
NSFD-PCRK-4
times1092
Infe
cted
frac
tion
ndash5
ndash4
ndash3
ndash2
ndash1
0
1
02 04 06 08 1 12 14 16 18 20Time (years) h = 01
(a)
NSFD-PCEuler-PC
times10213
Infe
cted
frac
tion
ndash4
ndash35
ndash3
ndash25
ndash2
ndash15
ndash1
ndash05
0
05
02 04 06 08 1 12 14 16 18 20Time (years) h = 01
(b)
Figure 9 Comparative convergence graphs of the NSFDPC method
Journal of Mathematics 17
Data Availability
-e data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
-e authors declare no conflicts of interest
Acknowledgments
-e authors extend their appreciation to the Deanship ofScientific Research at Majmaah University for funding thiswork under Project no RGP-2019-6
References
[1] E Hunter B Namee and J Kelleher ldquoAn open-data-drivenagent-based model to simulate infectious disease outbreaksrdquoPLoS One vol 14 no 1 pp 1ndash35 2018
[2] E A Bakare Y A Adekunle and K O Kadiri ldquoModeling andsimulation of the dynamics of the transmission of measlesrdquoInternational Journal of Computer Trends and Trends andTechnology vol 3 no 1 pp 174ndash177 2012
[3] L Zhou Y Wang Y Li and Y Michael ldquoGlobal dynamics ofa discrete age-structured SIR epidemic model with applica-tions to measles vaccination strategiesrdquo Mathematical Bio-sciences vol 308 pp 27ndash37 2019
[4] P Widyaningsih D R S Saputro and A W NugroholdquoSusceptible exposed infected recovery (SEIR) model withimmigration equilibria points and its applicationrdquo AmericanInstitute of Physics vol 1 pp 1ndash7 2018
[5] M G Roberts and J A P HeesterbeekMathematical Modelsin Epidemiology Mathematical Models UNESCO-EOLSSAbu Dhabi UAE 2007
[6] J H He F Y Ji and H M Sedighi ldquoDifference equation vsdifferential equation on different scalesrdquo International Journalof Numerical Methods for Heat amp Fluid Flow vol 18pp 0961ndash5539 2020
[7] R E Mickens Nonstandard Finite Difference Models of Dif-ferential Equations World Scientific Singapore 1994
[8] R E Mickens Applications of Nonstandard Finite DifferenceSchemes World Scientific Singapore 2000
[9] R E Mickens ldquoNumerical integration of population modelssatisfying conservation laws NSFD methodsrdquo Journal ofBiological Dynamics vol 1 no 4 pp 427ndash436 2007
[10] J D Lambert Computational Methods in Ordinary Differ-ential Equations Wiley New York NY USA 1973
[11] S M Moghadas M E Alexander B D Corbett andA B Gumel ldquoA positivity-preserving mickens-type dis-cretization of an epidemic modelrdquo Journal of DifferenceEquations and Applications vol 9 no 11 pp 1037ndash10512003
[12] R E Mickens ldquoNonstandard finite difference schemes fordifferential equationsrdquo Journal of Difference Equations andApplications vol 8 no 9 pp 823ndash847 2002
[13] R E Mickens ldquoDynamic consistency a fundamental prin-ciple for constructing nonstandard finite difference schemesfor differential equationsrdquo Journal of Difference Equations andApplications vol 11 no 7 pp 645ndash653 2005
[14] L-I W Roeger and R W Barnard ldquoPreservation of localdynamics when applying central difference methods appli-cation to SIR modelrdquo Journal of Difference Equations andApplications vol 13 no 4 pp 333ndash340 2007
[15] K C Patidar ldquoOn the use of non-standard finite differencemethodsrdquo Journal of Difference Equations and Applicationsvol 11 no 7 pp 735ndash758 2005
[16] R Anguelov and J M-S Lubuma ldquoNonstandard finite dif-ference method by nonlocal approximationrdquo Mathematicsand Computers in Simulation vol 61 no 3-6 pp 465ndash4752003
[17] R Anguelov and J M-S Lubuma ldquoContributions to themathematics of the nonstandard finite difference method andapplicationsrdquo Numerical Methods for Partial DifferentialEquations vol 17 no 5 pp 518ndash543 2001
[18] B M Chen-Charpentier D T Dimitrov andH V Kojouharov ldquoNumerical simulation of multi-speciesbiofilms in porous media for different kineticsrdquo Mathematicsand Computers in Simulation vol 79 no 6 pp 1846ndash18612009
[19] D T Dimitrov and H V Kojouharov ldquoNonstandard finite-difference schemes for general two-dimensional autonomousdynamical systemsrdquo Applied Mathematics Letters vol 18no 7 pp 769ndash774 2005
[20] A J Arenas J A Morantildeo and J C Cortes ldquoNon-standardnumerical method for a mathematical model of RSV epide-miological transmissionrdquo Computers amp Mathematics withApplications vol 56 no 3 pp 670ndash678 2008
[21] L Jodar R J Villanueva A J Arenas and G C GonzalezldquoNonstandard numerical methods for a mathematical modelfor influenza diseaserdquo Mathematics and Computers in Sim-ulation vol 79 no 3 pp 622ndash633 2008
[22] W Piyawong E H Twizell and A B Gumel ldquoAn uncon-ditionally convergent finite-difference scheme for the SIRmodelrdquo Applied Mathematics and Computation vol 146no 2-3 pp 611ndash625 2003
[23] H Jansen and E H Twizell ldquoAn unconditionally convergentdiscretization of the SEIR modelrdquo Mathematics and Com-puters in Simulation vol 58 no 2 pp 147ndash158 2002
[24] A J Arenas G Gonzalez-Parra and B M Chen-CharpentierldquoA nonstandard numerical scheme of predictor-correctortype for epidemic modelsrdquo Computers amp Mathematics withApplications vol 59 no 12 pp 3740ndash3749 2010
[25] F Brauer and C Castillo-Chavez Mathematical Models inPopulation Biology and Epidemiology Springer Berlin Ger-many 2001
18 Journal of Mathematics
So
JG(N 0 0)
11 + η1
minus Nη2η3θδ1δ2
minus Nη2θϵminus 1
1 + η1+
1δ2
+Nη2η3δ1δ2
1113890 1113891
01δ1
+Nη2η3δ21δ2
1113890 1113891Nη2δ1δ2
+N
2η22η3δ21δ2
1113890 1113891
0η3δ1δ2
+Nη2η
23
δ21δ22
1113890 11138911δ2
+Nη2η3δ1δ
22
+N
2η22η23
δ21δ22
1113890 1113891
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
(18)
To calculate the eigenvalues put det (J minus λI) 0
λ1 11 + η1 11 + μϕ(h)lt 1 And
λ2 minus λδ1δ2 + Nη2η3
δ21δ21113888 1113889 +
δ21δ2 + δ1Nη2η3 + N2η22η
23
δ21δ22
1113888 11138891113890 1113891
+δ1δ2 + Nη2η3
δ21δ21113888 1113889
δ21δ2 + δ1Nη2η3 + N2η22η
23
δ21δ22
1113888 1113889 minusNη2δ1 + N
2η22η3δ21δ2
1113888 1113889η3δ1δ2 + Nη2η
23
δ21δ22
1113888 11138891113890 1113891 0
λ2 minus Aλ + B 0
(19)
where
A δ1δ2 + Nη2η3δ21δ21113872 1113873 + δ21δ2 + δ1Nη2η3 + N
2η22η23δ
21δ
221113872 1113873 trace J
lowast
B δ1δ2 + Nη2η3
δ21δ21113888 1113889
δ21δ2 + δ1Nη2η3 + N2η22η
23
δ21δ22
1113888 1113889 minusNη2δ1 + N
2η22η3δ21δ2
1113888 1113889η3δ1δ2 + Nη2η
23
δ21δ22
1113888 11138891113890 1113891 det Jlowast
Jlowast
1δ1
+Nη2η3δ21δ2
1113890 1113891Nη2δ1δ2
+N
2η22η3δ21δ2
1113890 1113891
η3δ1δ2
+Nη2η
23
δ21δ22
1113890 11138911δ2
+Nη2η3δ1δ
22
+N
2η22η23
δ21δ22
1113890 1113891
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
(20)
To calculate the eigenvalues of Jlowast the following lemma isproved
Lemma 1 For the quadratic equation λ2 minus λA + B 0 bothroots satisfy |λi|lt 1 i 2 3 if and only if the followingconditions are satisfied [25]
(1) 1 minus A + Bgt 0
(2) 1 + A + Bgt 0
(3)Blt 1
(21)
Let us define A TraceJlowast B DetJlowast -erefore
Journal of Mathematics 7
A δ1 + δ2( 1113857 Nη2η3 + δ1δ2( 11138571113858 1113859 + N
2η22η23
δ21δ22
gt 0
B 1
δ1δ2+
Nη2η3δ21δ
22lt 1
f(minus 1) 1 + A + Bgt 0
f(0) B 1
δ1δ2+
Nη2η3δ21δ
22lt
1δ1δ2
+Nη2η3δ1δ2
1 + Nη2η3
δ1δ2
1 + R0(μ + σ)(μ + c)ϕ2(h)
1 + ϕ2(h)(μ + σ)(μ + c) + ϕ(h)[(μ + σ)(μ + c)]
lt 1 sinceR0 lt 1( 1113857
rArrBlt 1
(22)
Now for
f(1) 1 minus A + B
δ21δ
22 + δ1 + δ2( 1113857 Nη2η3 + δ1δ2( 11138571113858 1113859 + N
2η22η23 + Nη2η3 + δ1δ2
δ21δ22
gt 0
(23)
As all the conditions of the lemma hold for R0 lt 1therefore the absolute value of both eigenvalues of Jlowast is less
than one whenever R0 lt 1 -us the numerical scheme inequations (8) and (10) will converge unconditionally todisease-free equilibrium (N 0 0) for any value of time step h
whenever R0 lt 1 which is also computationally verified inFigure 2(a)
52 Endemic Equilibrium (SeEe Ie) (NR0 μN μ + σ(1 minus
1 R0) μβ(R0 minus 1)) -e value of functions and its deriv-atives at an endemic point is as
F1 Se Ee Ie( 1113857 N
R0
F2 Se Ee Ie( 1113857 μN 1 minus 1R0( 1113857
(μ + σ)
F3 Se Ee Ie( 1113857 μβ
R0 minus 1( 1113857
G1 Se Ee Ie( 1113857 N
R0
G2 Se Ee Ie( 1113857 μβ
R0 minus 1( 1113857
G3 Se Ee Ie( 1113857 μN 1 minus 1R0( 1113857
(μ + σ)
(24)
8 Journal of Mathematics
zF1 Se Ee Ie( 1113857
zS
11 + μϕ(h)R0
zF1 Se Ee Ie( 1113857
zE 0
zF1 Se Ee Ie( 1113857
zI
minus Nβϕ(h)
1 + μϕ(h)R0
zF2 Se Ee Ie( 1113857
zS
μϕ(h) R0 minus 1( 1113857
1 + μϕ(h)R01113858 1113859[1 + ϕ(h)(μ + σ)]
zF2 Se Ee Ie( 1113857
zE
1[1 + ϕ(h)(μ + σ)]
zF2 Se Ee Ie( 1113857
zI
minus Nβϕ(h) μϕ(l)R20 minus 2μϕ(h)R0 minus 11113960 1113961
R0[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]
zF3 Se Ee Ie( 1113857
zS
μσϕ2(h) R0 minus 1( 1113857
[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)] 1 + μϕ(h)R01113858 1113859
zF3 Se Ee Ie( 1113857
zE
σϕ(h)
[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]
zF3 Se Ee Ie( 1113857
zI
R0[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859 minus σNβϕ2(h) μϕ(h)R20 minus 2μϕ(h) minus 11113960 1113961
R0[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859[1 + ϕ(h)(μ + c)]
(25)
zG1 Se Ee Ie( 1113857
zS
11 + μϕ(h)R01113858 1113859
minusNμβσϕ3(h) R0 minus 1( 1113857
R0 1 + ϵminus 1 + μϕ(h)R01113960 1113961[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859[1 + ϕ(h)(μ + c)]
zG1 Se Ee Ie( 1113857
zE
minus Nβσϕ2(h)
R0 1 + ϵminus 1 + μϕ(h)R01113960 1113961[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]
Eige
n va
lues
Jacobian at DFE
0492
0494
0496
0498
05
0502
0504
0506
0508
051
1000 2000 3000 4000 5000 6000 7000 8000 9000 100000Step size
(a)
098
0981
0982
0983
0984
0985
0986
0987
0988
0989Jacobian at EE
Eige
n va
lues
1000 2000 3000 4000 5000 6000 7000 8000 9000 100000Step size
(b)
Figure 2 Spectral radius of the Jacobian at equilibrium points
Journal of Mathematics 9
zG1 Se Ee Ie( 1113857
zI
minus Nβϕ(h)
1 + ϵminus 1 + μϕ(h)R01113960 11139611 + μϕ(h)R0
middot ϵminus 1+
R0[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859 minus σNβϕ2(h) μϕ(h)R20 minus 2μϕ(h)R0 minus 11113960 1113961
R20[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859[1 + ϕ(h)(μ + c)]
⎡⎣ ⎤⎦
zG2 Se Ee Ie( 1113857
zS
μβϕ(h) R0 minus 1( 1113857
β[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 11138591113890
minusNμβ2σϕ4(h)μ R0 minus 1( 1113857
2
βR0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)] 1 + ϵminus 1 + μϕ(h)R01113960 1113961 1 + μϕ(h)R01113858 1113859
+Nμβσϕ3(h) R0 minus 1( 1113857
R0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)] 1 + μϕ(h)R01113858 1113859
1113891
zG2 Se Ee Ie( 1113857
zE
1[1 + ϕ(h)(μ + σ)]
+Nβσϕ2(h)
R0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)]
minusNμβ2σϕ3
(h) R0 minus 1( 1113857
βR0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)] 1 + ϵminus 1 + μϕ(h)R01113960 1113961
zG2 Se Ee Ie( 1113857
zI
minus Nμβ2ϕ2(h)ϵminus 1R0 minus 1( 1113857
β[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859 1 + ϵminus 1 + μϕ(h)R01113960 1113961
+Nμβ2ϕ2(h) R0 minus 1( 1113857 R0[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859 minus Nβσϕ2(h) μϕ(h)R
20 minus 2μϕ(h)R0 minus 11113960 11139611113966 1113967
βR20[1 + ϕ(h)(μ + σ)]
2[1 + ϕ(h)(μ + c)] 1 + ϵminus 1 + μϕ(h)R01113960 1113961 1 + μϕ(h)R01113858 1113859
+Nβϕ(h)
R0[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]minus
N2β2σϕ3(h) μϕ(h)R
20 minus 2μϕ(h)R0 minus 11113960 1113961
R20[1 + ϕ(h)(μ + σ)]
2[1 + ϕ(h)(μ + c)] 1 + μϕ(h)R01113858 1113859
zG3 Se Ee Ie( 1113857
zS
σϕ(h)
[1 + ϕ(h)(μ + c)]
μβϕ(h) R0 minus 1( 1113857
β[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 11138591113890
minusNμβ2σϕ4(h)μ R0 minus 1( 1113857
2
βR0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)] 1 + ϵminus 1 + μϕ(h)R01113960 1113961 1 + μϕ(h)R01113858 1113859
+Nμβσϕ3(h) R0 minus 1( 1113857
R0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)] 1 + μϕ(h)R01113858 1113859
1113891
zG3 Se Ee Ie( 1113857
zE
σϕ(h)
[1 + ϕ(h)(μ + c)]
1[1 + ϕ(h)(μ + σ)]
minusNμβ2σϕ3(h) R0 minus 1( 1113857
βR0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)] 1 + ϵminus 1 + μϕ(h)R01113960 1113961
⎡⎢⎣
+Nβσϕ2
(h)
R0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)]
1113891
zG3 Se Ee Ie( 1113857
zI
σϕ(h)
[1 + ϕ(h)(μ + c)]
minus Nμβ2ϕ2(h)ϵminus 1R0 minus 1( 1113857
β[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859 1 + ϵminus 1 + μϕ(h)R01113960 1113961⎡⎢⎣
+Nμβ2ϕ2(h) R0 minus 1( 1113857 R0[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859 minus Nβσϕ2(h) μϕ(h)R
20 minus 2μϕ(h)R0 minus 11113960 11139611113966 1113967
βR20[1 + ϕ(h)(μ + σ)]
2[1 + ϕ(h)(μ + c)] 1 + ϵminus 1 + μϕ(h)R01113960 1113961 1 + μϕ(h)R01113858 1113859
⎤⎥⎦
10 Journal of Mathematics
Table 1 Comparison of NSFD scheme with Euler and RK-4 schemes
l 001 01 100 1000Euler PC Convergence Divergence Divergence DivergenceRK-4 Convergence Divergence Divergence DivergenceNSFDPC Convergence Convergence Convergence Convergence
times107
NSFD-PC
Disease-free equilibrium
1
15
2
25
3
35
4
45
5
Susc
eptib
le fr
actio
n
20 40 60 80 100 120 140 160 180 2000Time (years) h = 001
(a)
Disease-free equilibriumtimes104
NSFD-PC
02 04 06 08 1 12 14 16 18 20Time (years) h = 001
0
05
1
15
2
25
3
35
4
45
5
Expo
sed
fract
ion
(b)
times104
NSFD-PC
Disease-free equilibrium
0
05
1
15
2
25
3
Infe
cted
frac
tion
02 04 06 08 1 12 14 16 18 20Time (years) h = 001
(c)
times107
NSFD-PC
X 5675Y 2435e + 007
Endemic equilibrium
100 200 300 400 500 6000Time (years) h = 001
1
15
2
25
3
35
Susc
eptib
le fr
actio
n
(d)
Figure 3 Continued
Journal of Mathematics 11
+Nβϕ(h)
R0[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]minus
N2β2σϕ3(h) μϕ(h)R
20 minus 2μϕ(h)R0 minus 11113960 1113961
R20[1 + ϕ(h)(μ + σ)]
2[1 + ϕ(h)(μ + c)] 1 + μϕ(h)R01113858 1113859
(26)
-e Jacobian evaluated at an endemic point(Slowaste Elowaste Ilowaste ) (Se Ee Ie) (NR0 μNμ + σ(1 minus 1R0) μβ(R0 minus 1)) is evaluated as
JG Slowaste Elowaste Ilowaste( 1113857
zG1 Slowaste Elowaste Ilowaste( 1113857
zS
zG1 Slowaste Elowaste Ilowaste( 1113857
zE
zG1 Slowaste Elowaste Ilowaste( 1113857
zI
zG1 Slowaste Elowaste Ilowaste( 1113857
zS
zG1 Slowaste Elowaste Ilowaste( 1113857
zE
zG1 Slowaste Elowaste Ilowaste( 1113857
zI
zG1 Slowaste Elowaste Ilowaste( 1113857
zS
zG1 Slowaste Elowaste Ilowaste( 1113857
zE
zG1 Slowaste Elowaste Ilowaste( 1113857
zI
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
(27)
It is tedious to calculate the eigenvalues of the Jacobian ofendemic equilibrium analytically so we have plotted thelargest eigenvalue against each step size and Figure 2(b)shows that for all step sizes the spectral radius of the Ja-cobian of EE remains less than one if R0 gt 1 which impliesthat the numerical scheme in equations (8) and (10) isunconditionally convergent if R0 gt 1 for all step sizes
6 Numerical Results and Discussion
All the methods showed the convergence for small step sizesin Table 1 However for large values of step sizes only thenormalized NSFDPC converge to the correct disease-freepoint for β 01 times 10minus 5 as well as the correct endemic pointfor β 03 times 10minus 5 -is means that for the value of the basic
reproductive number (threshold parameter) R0 lt 1 thenDFE is stable ie solution to the system (S E I R) areconverging to it -is can be seen from Figures 3(a)ndash3(c) forh 001 Also for the value of the basic reproductivenumber (threshold parameter) R0 gt 1 then EE is stable iesolution to the system (S E I R) is converging to it-is canbe seen from Figures 3(d)ndash3(f ) for h 001
Figures 4(a)ndash4(c) show how the NSFDPC method ofDFE converges to the equilibrium point for different stepsizes Figures 4(d)ndash4(f) show how the NSFDPC method ofEE converges to the equilibrium point for different step sizes
For comparison with the well-known RK-4 method tosystem (1) using the parameter values given in Table 2 it isfound that in Figures 5(a)ndash5(f) the numerical solutionconverges for h 001 for both equilibrium points and
times105
NSFD-PC
X 562Y 1125e + 004
Endemic equilibrium
100 200 300 400 500 6000Time (years) h = 001
0
1
2
3
4
5
6
7
8Ex
pose
d fra
ctio
n
(e)
times105
NSFD-PC
X 5523Y 7023
Infe
cted
frac
tion
Endemic equilibrium
100 200 300 400 500 6000Time (years) h = 001
0
05
1
15
2
25
3
35
4
45
5
(f )
Figure 3 Time plots of susceptible exposed and infected population using the NSFDPC method with h 001
12 Journal of Mathematics
times107
NSFD-PC
Disease-free equilibrium
20 40 60 80 100 120 140 160 180 2000Time (years) h = 01
1
15
2
25
3
35
4
45
5Su
scep
tible
frac
tion
(a)
times107
NSFD-PC
Disease-free equilibrium
500 1000 1500 2000 2500 3000 3500 40000Time (years) h = 10
1
15
2
25
3
35
4
45
5
Susc
eptib
le fr
actio
n
(b)
Disease-free equilibriumtimes107
times105
NSFD-PC
05 1 15 2 25 3 35 40Time (years) h = 1000
1
15
2
25
3
35
4
45
5
Susc
eptib
le fr
actio
n
(c)
X 5827Y 7022
Infe
cted
frac
tion
Endemic equilibriumtimes104
NSFD-PC
100 200 300 400 500 6000Time (years) h = 01
0
2
4
6
8
10
12
14
16
18
(d)
Endemic equilibriumtimes104
NSFD-PC
X 9600Y 7022
Infe
cted
frac
tion
1000 2000 3000 4000 5000 6000 7000 8000 9000 100000Time (years) h = 10
0
05
1
15
2
25
3
(e)
Endemic equilibriumtimes104
times105
NSFD-PC
X 861e + 005Y 7022
Infe
cted
frac
tion
1 2 3 4 5 6 7 8 9 100Time (years) h = 1000
0
05
1
15
2
25
3
(f )
Figure 4 Time plots of susceptible and infected population using the NSFDPC method with different step sizes (a) h 01 (b) h 10 and(c) h 1000
Journal of Mathematics 13
Table 2 Parameters value of the measles model
Parameters N μ σ c
5 times 107yearminus 1 002 456 yearminus 1 73 yearminus 1
DFE (S0 E0 I0) (N 0 0) β 01 times 10minus 5
EE (Se Ee Ie) (2435 times 107 1125 times 104 7022671) β 03 times 10minus 5
Disease-free equilibrium
RK4
times107
1
15
2
25
3
35
4
45
5
Susc
eptib
le fr
actio
n
20 40 60 80 100 120 140 160 180 2000Time (years) h = 001
(a)
Expo
sed
fract
ion
Disease-free equilibrium
RK4
times104
0
05
1
15
2
25
3
35
4
45
5
02 04 06 08 1 12 14 16 18 20Time (years) h = 001
(b)
times104
Infe
cted
frac
tion
Disease-free equilibrium
RK4
0
05
1
15
2
25
3
02 04 06 08 1 12 14 16 18 20Time (years) h = 001
(c)
times107
X 5578Y 2435e + 007
Endemic equilibrium
RK4
1
15
2
25
3
35
Susc
eptib
le F
ract
ion
100 200 300 400 500 6000Time (years) h = 001
(d)
Figure 5 Continued
14 Journal of Mathematics
Figure 6 shows overflow for a critical value of step size ieh 01 of DFE and EE respectively
Another popular numerical method is the Eulerpredictor-corrector (E-PC) technique which employs theexplicit Euler method as a predictor and the trapezoidalrule as the corrector -is combination is second-orderaccurate but has similar stability properties in PECEmode to the explicit Euler method alone Figures 7(a)ndash7(f ) show convergence results for h 001 in both casesthat are DFE and EE -e value of h 01 which produces
overflow when solving the system with the parametervalues of Table 2 for the PECE combination can be seen inFigure 8
-us the presented numerical results demonstrated thatNSFDPC has better convergence property following theEuler PC and RK-4 as shown in Figure 9 It is proved thatthe approximations made by other standard numericalmethods experience difficulties in preserving either thestability or the positivity of the solutions or both butNSFDPC is unconditionally convergent
times105
X 5772Y 1125e + 004
Endemic equilibrium
RK4
Expo
sed
fract
ion
0
1
2
3
4
5
6
7
8
9
10
100 200 300 400 500 6000Time (years) h = 001
(e)
times105
X 7659Y 7022
RK4
Endemic equilibrium
Infe
cted
frac
tion
0
1
2
3
4
5
6
100 200 300 400 500 600 700 8000Time (years) h = 001
(f )
Figure 5 Time plots of susceptible exposed and infected population using the RK-4 method with h 001
times1087 RK-4 scheme (disease-free equilibrium)
SusceptibleExposedInfected
01 02 03 04 05 06 07 08 09 10Time (years) h = 01
ndash2
ndash15
ndash1
ndash05
0
05
1
15
2
Popu
latio
n fra
ctio
ns
(a)
RK-4 scheme (endemic equilibrium)
SusceptibleExposedInfected
times10145
01 02 03 04 05 06 07 08 09 10Time (years) h = 01
ndash8
ndash6
ndash4
ndash2
0
2
4
6
8
Popu
latio
n fra
ctio
ns
(b)
Figure 6 Divergence graphs of the population using the RK-4 method with step size h 01
Journal of Mathematics 15
Euler-PC
Disease-free equilibriumtimes107
1
15
2
25
3
35
4
45
5Su
scep
tible
frac
tion
20 40 60 80 100 120 140 160 180 2000Time (years) h = 001
(a)
Euler-PC
Disease-free equilibrium
Expo
sed
fract
ion
times104
0
05
1
15
2
25
3
35
4
45
5
02 04 06 08 1 12 14 16 18 20Time (years) h = 001
(b)
Euler-PC
Disease-free equilibrium
Infe
cted
frac
tion
times104
0
05
1
15
2
25
3
35
02 04 06 08 1 12 14 16 18 20Time (years) h = 001
(c)
Euler-PC
X 5733Y 2435e + 007
Endemic equilibriumtimes107
1
15
2
25
3
35Su
scep
tible
frac
tion
100 200 300 400 500 6000Time (years) h = 001
(d)
Euler-PC
X 5662Y 1118e + 004
Endemic equilibrium
Expo
sed
fract
ion
times105
0
2
4
6
8
10
12
100 200 300 400 500 6000Time (years) h = 001
(e)
Euler-PC
X 5427Y 7025
Endemic equilibrium
Infe
cted
frac
tion
times105
0
1
2
3
4
5
6
7
100 200 300 400 500 6000Time (years) h = 001
(f )
Figure 7 Time plots of population using the Euler-PC method with h 001
16 Journal of Mathematics
7 Conclusions
In this paper a normalized NSFDPC for the SEIR modelconcerning the transmission dynamics of measles is con-structed and analysed -is proposed numerical scheme isvery competitive It is qualitatively stable that is it doublerefines the solution and gives realistic results even for largestep sizes It is dynamically consistent with a continuoussystem and unconditionally convergent and satisfies the
positivity of the state variables involved in the systemSimulations are carried out and its usefulness is comparedwith a well-known numerical method of standard differenceschemes such as RK4 and Euler predictor-corrector method-e standard finite difference schemes are highly dependenton step sizes and initial value problems but NSFDPC isindependent of these two features which make it morepractical Also this method saves computation time andmemory
Euler-PC scheme (disease-free equilibrium)times1041
ndash3
ndash2
ndash1
0
1
2
3
Popu
latio
n fra
ctio
ns
01 02 03 04 05 06 07 08 09 10Time (years) h = 01
SusceptibleExposedInfected
(a)
Euler-PC scheme (endemic equilibrium)times1053
ndash6
ndash4
ndash2
0
2
4
6
Popu
latio
n fra
ctio
ns
01 02 03 04 05 06 07 08 09 10Time (years) h = 01
SusceptibleExposedInfected
(b)
Figure 8 Divergence graphs of the population using the Euler-PC method with step size h 01
NSFD-PCRK-4
times1092
Infe
cted
frac
tion
ndash5
ndash4
ndash3
ndash2
ndash1
0
1
02 04 06 08 1 12 14 16 18 20Time (years) h = 01
(a)
NSFD-PCEuler-PC
times10213
Infe
cted
frac
tion
ndash4
ndash35
ndash3
ndash25
ndash2
ndash15
ndash1
ndash05
0
05
02 04 06 08 1 12 14 16 18 20Time (years) h = 01
(b)
Figure 9 Comparative convergence graphs of the NSFDPC method
Journal of Mathematics 17
Data Availability
-e data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
-e authors declare no conflicts of interest
Acknowledgments
-e authors extend their appreciation to the Deanship ofScientific Research at Majmaah University for funding thiswork under Project no RGP-2019-6
References
[1] E Hunter B Namee and J Kelleher ldquoAn open-data-drivenagent-based model to simulate infectious disease outbreaksrdquoPLoS One vol 14 no 1 pp 1ndash35 2018
[2] E A Bakare Y A Adekunle and K O Kadiri ldquoModeling andsimulation of the dynamics of the transmission of measlesrdquoInternational Journal of Computer Trends and Trends andTechnology vol 3 no 1 pp 174ndash177 2012
[3] L Zhou Y Wang Y Li and Y Michael ldquoGlobal dynamics ofa discrete age-structured SIR epidemic model with applica-tions to measles vaccination strategiesrdquo Mathematical Bio-sciences vol 308 pp 27ndash37 2019
[4] P Widyaningsih D R S Saputro and A W NugroholdquoSusceptible exposed infected recovery (SEIR) model withimmigration equilibria points and its applicationrdquo AmericanInstitute of Physics vol 1 pp 1ndash7 2018
[5] M G Roberts and J A P HeesterbeekMathematical Modelsin Epidemiology Mathematical Models UNESCO-EOLSSAbu Dhabi UAE 2007
[6] J H He F Y Ji and H M Sedighi ldquoDifference equation vsdifferential equation on different scalesrdquo International Journalof Numerical Methods for Heat amp Fluid Flow vol 18pp 0961ndash5539 2020
[7] R E Mickens Nonstandard Finite Difference Models of Dif-ferential Equations World Scientific Singapore 1994
[8] R E Mickens Applications of Nonstandard Finite DifferenceSchemes World Scientific Singapore 2000
[9] R E Mickens ldquoNumerical integration of population modelssatisfying conservation laws NSFD methodsrdquo Journal ofBiological Dynamics vol 1 no 4 pp 427ndash436 2007
[10] J D Lambert Computational Methods in Ordinary Differ-ential Equations Wiley New York NY USA 1973
[11] S M Moghadas M E Alexander B D Corbett andA B Gumel ldquoA positivity-preserving mickens-type dis-cretization of an epidemic modelrdquo Journal of DifferenceEquations and Applications vol 9 no 11 pp 1037ndash10512003
[12] R E Mickens ldquoNonstandard finite difference schemes fordifferential equationsrdquo Journal of Difference Equations andApplications vol 8 no 9 pp 823ndash847 2002
[13] R E Mickens ldquoDynamic consistency a fundamental prin-ciple for constructing nonstandard finite difference schemesfor differential equationsrdquo Journal of Difference Equations andApplications vol 11 no 7 pp 645ndash653 2005
[14] L-I W Roeger and R W Barnard ldquoPreservation of localdynamics when applying central difference methods appli-cation to SIR modelrdquo Journal of Difference Equations andApplications vol 13 no 4 pp 333ndash340 2007
[15] K C Patidar ldquoOn the use of non-standard finite differencemethodsrdquo Journal of Difference Equations and Applicationsvol 11 no 7 pp 735ndash758 2005
[16] R Anguelov and J M-S Lubuma ldquoNonstandard finite dif-ference method by nonlocal approximationrdquo Mathematicsand Computers in Simulation vol 61 no 3-6 pp 465ndash4752003
[17] R Anguelov and J M-S Lubuma ldquoContributions to themathematics of the nonstandard finite difference method andapplicationsrdquo Numerical Methods for Partial DifferentialEquations vol 17 no 5 pp 518ndash543 2001
[18] B M Chen-Charpentier D T Dimitrov andH V Kojouharov ldquoNumerical simulation of multi-speciesbiofilms in porous media for different kineticsrdquo Mathematicsand Computers in Simulation vol 79 no 6 pp 1846ndash18612009
[19] D T Dimitrov and H V Kojouharov ldquoNonstandard finite-difference schemes for general two-dimensional autonomousdynamical systemsrdquo Applied Mathematics Letters vol 18no 7 pp 769ndash774 2005
[20] A J Arenas J A Morantildeo and J C Cortes ldquoNon-standardnumerical method for a mathematical model of RSV epide-miological transmissionrdquo Computers amp Mathematics withApplications vol 56 no 3 pp 670ndash678 2008
[21] L Jodar R J Villanueva A J Arenas and G C GonzalezldquoNonstandard numerical methods for a mathematical modelfor influenza diseaserdquo Mathematics and Computers in Sim-ulation vol 79 no 3 pp 622ndash633 2008
[22] W Piyawong E H Twizell and A B Gumel ldquoAn uncon-ditionally convergent finite-difference scheme for the SIRmodelrdquo Applied Mathematics and Computation vol 146no 2-3 pp 611ndash625 2003
[23] H Jansen and E H Twizell ldquoAn unconditionally convergentdiscretization of the SEIR modelrdquo Mathematics and Com-puters in Simulation vol 58 no 2 pp 147ndash158 2002
[24] A J Arenas G Gonzalez-Parra and B M Chen-CharpentierldquoA nonstandard numerical scheme of predictor-correctortype for epidemic modelsrdquo Computers amp Mathematics withApplications vol 59 no 12 pp 3740ndash3749 2010
[25] F Brauer and C Castillo-Chavez Mathematical Models inPopulation Biology and Epidemiology Springer Berlin Ger-many 2001
18 Journal of Mathematics
A δ1 + δ2( 1113857 Nη2η3 + δ1δ2( 11138571113858 1113859 + N
2η22η23
δ21δ22
gt 0
B 1
δ1δ2+
Nη2η3δ21δ
22lt 1
f(minus 1) 1 + A + Bgt 0
f(0) B 1
δ1δ2+
Nη2η3δ21δ
22lt
1δ1δ2
+Nη2η3δ1δ2
1 + Nη2η3
δ1δ2
1 + R0(μ + σ)(μ + c)ϕ2(h)
1 + ϕ2(h)(μ + σ)(μ + c) + ϕ(h)[(μ + σ)(μ + c)]
lt 1 sinceR0 lt 1( 1113857
rArrBlt 1
(22)
Now for
f(1) 1 minus A + B
δ21δ
22 + δ1 + δ2( 1113857 Nη2η3 + δ1δ2( 11138571113858 1113859 + N
2η22η23 + Nη2η3 + δ1δ2
δ21δ22
gt 0
(23)
As all the conditions of the lemma hold for R0 lt 1therefore the absolute value of both eigenvalues of Jlowast is less
than one whenever R0 lt 1 -us the numerical scheme inequations (8) and (10) will converge unconditionally todisease-free equilibrium (N 0 0) for any value of time step h
whenever R0 lt 1 which is also computationally verified inFigure 2(a)
52 Endemic Equilibrium (SeEe Ie) (NR0 μN μ + σ(1 minus
1 R0) μβ(R0 minus 1)) -e value of functions and its deriv-atives at an endemic point is as
F1 Se Ee Ie( 1113857 N
R0
F2 Se Ee Ie( 1113857 μN 1 minus 1R0( 1113857
(μ + σ)
F3 Se Ee Ie( 1113857 μβ
R0 minus 1( 1113857
G1 Se Ee Ie( 1113857 N
R0
G2 Se Ee Ie( 1113857 μβ
R0 minus 1( 1113857
G3 Se Ee Ie( 1113857 μN 1 minus 1R0( 1113857
(μ + σ)
(24)
8 Journal of Mathematics
zF1 Se Ee Ie( 1113857
zS
11 + μϕ(h)R0
zF1 Se Ee Ie( 1113857
zE 0
zF1 Se Ee Ie( 1113857
zI
minus Nβϕ(h)
1 + μϕ(h)R0
zF2 Se Ee Ie( 1113857
zS
μϕ(h) R0 minus 1( 1113857
1 + μϕ(h)R01113858 1113859[1 + ϕ(h)(μ + σ)]
zF2 Se Ee Ie( 1113857
zE
1[1 + ϕ(h)(μ + σ)]
zF2 Se Ee Ie( 1113857
zI
minus Nβϕ(h) μϕ(l)R20 minus 2μϕ(h)R0 minus 11113960 1113961
R0[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]
zF3 Se Ee Ie( 1113857
zS
μσϕ2(h) R0 minus 1( 1113857
[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)] 1 + μϕ(h)R01113858 1113859
zF3 Se Ee Ie( 1113857
zE
σϕ(h)
[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]
zF3 Se Ee Ie( 1113857
zI
R0[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859 minus σNβϕ2(h) μϕ(h)R20 minus 2μϕ(h) minus 11113960 1113961
R0[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859[1 + ϕ(h)(μ + c)]
(25)
zG1 Se Ee Ie( 1113857
zS
11 + μϕ(h)R01113858 1113859
minusNμβσϕ3(h) R0 minus 1( 1113857
R0 1 + ϵminus 1 + μϕ(h)R01113960 1113961[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859[1 + ϕ(h)(μ + c)]
zG1 Se Ee Ie( 1113857
zE
minus Nβσϕ2(h)
R0 1 + ϵminus 1 + μϕ(h)R01113960 1113961[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]
Eige
n va
lues
Jacobian at DFE
0492
0494
0496
0498
05
0502
0504
0506
0508
051
1000 2000 3000 4000 5000 6000 7000 8000 9000 100000Step size
(a)
098
0981
0982
0983
0984
0985
0986
0987
0988
0989Jacobian at EE
Eige
n va
lues
1000 2000 3000 4000 5000 6000 7000 8000 9000 100000Step size
(b)
Figure 2 Spectral radius of the Jacobian at equilibrium points
Journal of Mathematics 9
zG1 Se Ee Ie( 1113857
zI
minus Nβϕ(h)
1 + ϵminus 1 + μϕ(h)R01113960 11139611 + μϕ(h)R0
middot ϵminus 1+
R0[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859 minus σNβϕ2(h) μϕ(h)R20 minus 2μϕ(h)R0 minus 11113960 1113961
R20[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859[1 + ϕ(h)(μ + c)]
⎡⎣ ⎤⎦
zG2 Se Ee Ie( 1113857
zS
μβϕ(h) R0 minus 1( 1113857
β[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 11138591113890
minusNμβ2σϕ4(h)μ R0 minus 1( 1113857
2
βR0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)] 1 + ϵminus 1 + μϕ(h)R01113960 1113961 1 + μϕ(h)R01113858 1113859
+Nμβσϕ3(h) R0 minus 1( 1113857
R0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)] 1 + μϕ(h)R01113858 1113859
1113891
zG2 Se Ee Ie( 1113857
zE
1[1 + ϕ(h)(μ + σ)]
+Nβσϕ2(h)
R0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)]
minusNμβ2σϕ3
(h) R0 minus 1( 1113857
βR0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)] 1 + ϵminus 1 + μϕ(h)R01113960 1113961
zG2 Se Ee Ie( 1113857
zI
minus Nμβ2ϕ2(h)ϵminus 1R0 minus 1( 1113857
β[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859 1 + ϵminus 1 + μϕ(h)R01113960 1113961
+Nμβ2ϕ2(h) R0 minus 1( 1113857 R0[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859 minus Nβσϕ2(h) μϕ(h)R
20 minus 2μϕ(h)R0 minus 11113960 11139611113966 1113967
βR20[1 + ϕ(h)(μ + σ)]
2[1 + ϕ(h)(μ + c)] 1 + ϵminus 1 + μϕ(h)R01113960 1113961 1 + μϕ(h)R01113858 1113859
+Nβϕ(h)
R0[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]minus
N2β2σϕ3(h) μϕ(h)R
20 minus 2μϕ(h)R0 minus 11113960 1113961
R20[1 + ϕ(h)(μ + σ)]
2[1 + ϕ(h)(μ + c)] 1 + μϕ(h)R01113858 1113859
zG3 Se Ee Ie( 1113857
zS
σϕ(h)
[1 + ϕ(h)(μ + c)]
μβϕ(h) R0 minus 1( 1113857
β[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 11138591113890
minusNμβ2σϕ4(h)μ R0 minus 1( 1113857
2
βR0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)] 1 + ϵminus 1 + μϕ(h)R01113960 1113961 1 + μϕ(h)R01113858 1113859
+Nμβσϕ3(h) R0 minus 1( 1113857
R0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)] 1 + μϕ(h)R01113858 1113859
1113891
zG3 Se Ee Ie( 1113857
zE
σϕ(h)
[1 + ϕ(h)(μ + c)]
1[1 + ϕ(h)(μ + σ)]
minusNμβ2σϕ3(h) R0 minus 1( 1113857
βR0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)] 1 + ϵminus 1 + μϕ(h)R01113960 1113961
⎡⎢⎣
+Nβσϕ2
(h)
R0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)]
1113891
zG3 Se Ee Ie( 1113857
zI
σϕ(h)
[1 + ϕ(h)(μ + c)]
minus Nμβ2ϕ2(h)ϵminus 1R0 minus 1( 1113857
β[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859 1 + ϵminus 1 + μϕ(h)R01113960 1113961⎡⎢⎣
+Nμβ2ϕ2(h) R0 minus 1( 1113857 R0[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859 minus Nβσϕ2(h) μϕ(h)R
20 minus 2μϕ(h)R0 minus 11113960 11139611113966 1113967
βR20[1 + ϕ(h)(μ + σ)]
2[1 + ϕ(h)(μ + c)] 1 + ϵminus 1 + μϕ(h)R01113960 1113961 1 + μϕ(h)R01113858 1113859
⎤⎥⎦
10 Journal of Mathematics
Table 1 Comparison of NSFD scheme with Euler and RK-4 schemes
l 001 01 100 1000Euler PC Convergence Divergence Divergence DivergenceRK-4 Convergence Divergence Divergence DivergenceNSFDPC Convergence Convergence Convergence Convergence
times107
NSFD-PC
Disease-free equilibrium
1
15
2
25
3
35
4
45
5
Susc
eptib
le fr
actio
n
20 40 60 80 100 120 140 160 180 2000Time (years) h = 001
(a)
Disease-free equilibriumtimes104
NSFD-PC
02 04 06 08 1 12 14 16 18 20Time (years) h = 001
0
05
1
15
2
25
3
35
4
45
5
Expo
sed
fract
ion
(b)
times104
NSFD-PC
Disease-free equilibrium
0
05
1
15
2
25
3
Infe
cted
frac
tion
02 04 06 08 1 12 14 16 18 20Time (years) h = 001
(c)
times107
NSFD-PC
X 5675Y 2435e + 007
Endemic equilibrium
100 200 300 400 500 6000Time (years) h = 001
1
15
2
25
3
35
Susc
eptib
le fr
actio
n
(d)
Figure 3 Continued
Journal of Mathematics 11
+Nβϕ(h)
R0[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]minus
N2β2σϕ3(h) μϕ(h)R
20 minus 2μϕ(h)R0 minus 11113960 1113961
R20[1 + ϕ(h)(μ + σ)]
2[1 + ϕ(h)(μ + c)] 1 + μϕ(h)R01113858 1113859
(26)
-e Jacobian evaluated at an endemic point(Slowaste Elowaste Ilowaste ) (Se Ee Ie) (NR0 μNμ + σ(1 minus 1R0) μβ(R0 minus 1)) is evaluated as
JG Slowaste Elowaste Ilowaste( 1113857
zG1 Slowaste Elowaste Ilowaste( 1113857
zS
zG1 Slowaste Elowaste Ilowaste( 1113857
zE
zG1 Slowaste Elowaste Ilowaste( 1113857
zI
zG1 Slowaste Elowaste Ilowaste( 1113857
zS
zG1 Slowaste Elowaste Ilowaste( 1113857
zE
zG1 Slowaste Elowaste Ilowaste( 1113857
zI
zG1 Slowaste Elowaste Ilowaste( 1113857
zS
zG1 Slowaste Elowaste Ilowaste( 1113857
zE
zG1 Slowaste Elowaste Ilowaste( 1113857
zI
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
(27)
It is tedious to calculate the eigenvalues of the Jacobian ofendemic equilibrium analytically so we have plotted thelargest eigenvalue against each step size and Figure 2(b)shows that for all step sizes the spectral radius of the Ja-cobian of EE remains less than one if R0 gt 1 which impliesthat the numerical scheme in equations (8) and (10) isunconditionally convergent if R0 gt 1 for all step sizes
6 Numerical Results and Discussion
All the methods showed the convergence for small step sizesin Table 1 However for large values of step sizes only thenormalized NSFDPC converge to the correct disease-freepoint for β 01 times 10minus 5 as well as the correct endemic pointfor β 03 times 10minus 5 -is means that for the value of the basic
reproductive number (threshold parameter) R0 lt 1 thenDFE is stable ie solution to the system (S E I R) areconverging to it -is can be seen from Figures 3(a)ndash3(c) forh 001 Also for the value of the basic reproductivenumber (threshold parameter) R0 gt 1 then EE is stable iesolution to the system (S E I R) is converging to it-is canbe seen from Figures 3(d)ndash3(f ) for h 001
Figures 4(a)ndash4(c) show how the NSFDPC method ofDFE converges to the equilibrium point for different stepsizes Figures 4(d)ndash4(f) show how the NSFDPC method ofEE converges to the equilibrium point for different step sizes
For comparison with the well-known RK-4 method tosystem (1) using the parameter values given in Table 2 it isfound that in Figures 5(a)ndash5(f) the numerical solutionconverges for h 001 for both equilibrium points and
times105
NSFD-PC
X 562Y 1125e + 004
Endemic equilibrium
100 200 300 400 500 6000Time (years) h = 001
0
1
2
3
4
5
6
7
8Ex
pose
d fra
ctio
n
(e)
times105
NSFD-PC
X 5523Y 7023
Infe
cted
frac
tion
Endemic equilibrium
100 200 300 400 500 6000Time (years) h = 001
0
05
1
15
2
25
3
35
4
45
5
(f )
Figure 3 Time plots of susceptible exposed and infected population using the NSFDPC method with h 001
12 Journal of Mathematics
times107
NSFD-PC
Disease-free equilibrium
20 40 60 80 100 120 140 160 180 2000Time (years) h = 01
1
15
2
25
3
35
4
45
5Su
scep
tible
frac
tion
(a)
times107
NSFD-PC
Disease-free equilibrium
500 1000 1500 2000 2500 3000 3500 40000Time (years) h = 10
1
15
2
25
3
35
4
45
5
Susc
eptib
le fr
actio
n
(b)
Disease-free equilibriumtimes107
times105
NSFD-PC
05 1 15 2 25 3 35 40Time (years) h = 1000
1
15
2
25
3
35
4
45
5
Susc
eptib
le fr
actio
n
(c)
X 5827Y 7022
Infe
cted
frac
tion
Endemic equilibriumtimes104
NSFD-PC
100 200 300 400 500 6000Time (years) h = 01
0
2
4
6
8
10
12
14
16
18
(d)
Endemic equilibriumtimes104
NSFD-PC
X 9600Y 7022
Infe
cted
frac
tion
1000 2000 3000 4000 5000 6000 7000 8000 9000 100000Time (years) h = 10
0
05
1
15
2
25
3
(e)
Endemic equilibriumtimes104
times105
NSFD-PC
X 861e + 005Y 7022
Infe
cted
frac
tion
1 2 3 4 5 6 7 8 9 100Time (years) h = 1000
0
05
1
15
2
25
3
(f )
Figure 4 Time plots of susceptible and infected population using the NSFDPC method with different step sizes (a) h 01 (b) h 10 and(c) h 1000
Journal of Mathematics 13
Table 2 Parameters value of the measles model
Parameters N μ σ c
5 times 107yearminus 1 002 456 yearminus 1 73 yearminus 1
DFE (S0 E0 I0) (N 0 0) β 01 times 10minus 5
EE (Se Ee Ie) (2435 times 107 1125 times 104 7022671) β 03 times 10minus 5
Disease-free equilibrium
RK4
times107
1
15
2
25
3
35
4
45
5
Susc
eptib
le fr
actio
n
20 40 60 80 100 120 140 160 180 2000Time (years) h = 001
(a)
Expo
sed
fract
ion
Disease-free equilibrium
RK4
times104
0
05
1
15
2
25
3
35
4
45
5
02 04 06 08 1 12 14 16 18 20Time (years) h = 001
(b)
times104
Infe
cted
frac
tion
Disease-free equilibrium
RK4
0
05
1
15
2
25
3
02 04 06 08 1 12 14 16 18 20Time (years) h = 001
(c)
times107
X 5578Y 2435e + 007
Endemic equilibrium
RK4
1
15
2
25
3
35
Susc
eptib
le F
ract
ion
100 200 300 400 500 6000Time (years) h = 001
(d)
Figure 5 Continued
14 Journal of Mathematics
Figure 6 shows overflow for a critical value of step size ieh 01 of DFE and EE respectively
Another popular numerical method is the Eulerpredictor-corrector (E-PC) technique which employs theexplicit Euler method as a predictor and the trapezoidalrule as the corrector -is combination is second-orderaccurate but has similar stability properties in PECEmode to the explicit Euler method alone Figures 7(a)ndash7(f ) show convergence results for h 001 in both casesthat are DFE and EE -e value of h 01 which produces
overflow when solving the system with the parametervalues of Table 2 for the PECE combination can be seen inFigure 8
-us the presented numerical results demonstrated thatNSFDPC has better convergence property following theEuler PC and RK-4 as shown in Figure 9 It is proved thatthe approximations made by other standard numericalmethods experience difficulties in preserving either thestability or the positivity of the solutions or both butNSFDPC is unconditionally convergent
times105
X 5772Y 1125e + 004
Endemic equilibrium
RK4
Expo
sed
fract
ion
0
1
2
3
4
5
6
7
8
9
10
100 200 300 400 500 6000Time (years) h = 001
(e)
times105
X 7659Y 7022
RK4
Endemic equilibrium
Infe
cted
frac
tion
0
1
2
3
4
5
6
100 200 300 400 500 600 700 8000Time (years) h = 001
(f )
Figure 5 Time plots of susceptible exposed and infected population using the RK-4 method with h 001
times1087 RK-4 scheme (disease-free equilibrium)
SusceptibleExposedInfected
01 02 03 04 05 06 07 08 09 10Time (years) h = 01
ndash2
ndash15
ndash1
ndash05
0
05
1
15
2
Popu
latio
n fra
ctio
ns
(a)
RK-4 scheme (endemic equilibrium)
SusceptibleExposedInfected
times10145
01 02 03 04 05 06 07 08 09 10Time (years) h = 01
ndash8
ndash6
ndash4
ndash2
0
2
4
6
8
Popu
latio
n fra
ctio
ns
(b)
Figure 6 Divergence graphs of the population using the RK-4 method with step size h 01
Journal of Mathematics 15
Euler-PC
Disease-free equilibriumtimes107
1
15
2
25
3
35
4
45
5Su
scep
tible
frac
tion
20 40 60 80 100 120 140 160 180 2000Time (years) h = 001
(a)
Euler-PC
Disease-free equilibrium
Expo
sed
fract
ion
times104
0
05
1
15
2
25
3
35
4
45
5
02 04 06 08 1 12 14 16 18 20Time (years) h = 001
(b)
Euler-PC
Disease-free equilibrium
Infe
cted
frac
tion
times104
0
05
1
15
2
25
3
35
02 04 06 08 1 12 14 16 18 20Time (years) h = 001
(c)
Euler-PC
X 5733Y 2435e + 007
Endemic equilibriumtimes107
1
15
2
25
3
35Su
scep
tible
frac
tion
100 200 300 400 500 6000Time (years) h = 001
(d)
Euler-PC
X 5662Y 1118e + 004
Endemic equilibrium
Expo
sed
fract
ion
times105
0
2
4
6
8
10
12
100 200 300 400 500 6000Time (years) h = 001
(e)
Euler-PC
X 5427Y 7025
Endemic equilibrium
Infe
cted
frac
tion
times105
0
1
2
3
4
5
6
7
100 200 300 400 500 6000Time (years) h = 001
(f )
Figure 7 Time plots of population using the Euler-PC method with h 001
16 Journal of Mathematics
7 Conclusions
In this paper a normalized NSFDPC for the SEIR modelconcerning the transmission dynamics of measles is con-structed and analysed -is proposed numerical scheme isvery competitive It is qualitatively stable that is it doublerefines the solution and gives realistic results even for largestep sizes It is dynamically consistent with a continuoussystem and unconditionally convergent and satisfies the
positivity of the state variables involved in the systemSimulations are carried out and its usefulness is comparedwith a well-known numerical method of standard differenceschemes such as RK4 and Euler predictor-corrector method-e standard finite difference schemes are highly dependenton step sizes and initial value problems but NSFDPC isindependent of these two features which make it morepractical Also this method saves computation time andmemory
Euler-PC scheme (disease-free equilibrium)times1041
ndash3
ndash2
ndash1
0
1
2
3
Popu
latio
n fra
ctio
ns
01 02 03 04 05 06 07 08 09 10Time (years) h = 01
SusceptibleExposedInfected
(a)
Euler-PC scheme (endemic equilibrium)times1053
ndash6
ndash4
ndash2
0
2
4
6
Popu
latio
n fra
ctio
ns
01 02 03 04 05 06 07 08 09 10Time (years) h = 01
SusceptibleExposedInfected
(b)
Figure 8 Divergence graphs of the population using the Euler-PC method with step size h 01
NSFD-PCRK-4
times1092
Infe
cted
frac
tion
ndash5
ndash4
ndash3
ndash2
ndash1
0
1
02 04 06 08 1 12 14 16 18 20Time (years) h = 01
(a)
NSFD-PCEuler-PC
times10213
Infe
cted
frac
tion
ndash4
ndash35
ndash3
ndash25
ndash2
ndash15
ndash1
ndash05
0
05
02 04 06 08 1 12 14 16 18 20Time (years) h = 01
(b)
Figure 9 Comparative convergence graphs of the NSFDPC method
Journal of Mathematics 17
Data Availability
-e data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
-e authors declare no conflicts of interest
Acknowledgments
-e authors extend their appreciation to the Deanship ofScientific Research at Majmaah University for funding thiswork under Project no RGP-2019-6
References
[1] E Hunter B Namee and J Kelleher ldquoAn open-data-drivenagent-based model to simulate infectious disease outbreaksrdquoPLoS One vol 14 no 1 pp 1ndash35 2018
[2] E A Bakare Y A Adekunle and K O Kadiri ldquoModeling andsimulation of the dynamics of the transmission of measlesrdquoInternational Journal of Computer Trends and Trends andTechnology vol 3 no 1 pp 174ndash177 2012
[3] L Zhou Y Wang Y Li and Y Michael ldquoGlobal dynamics ofa discrete age-structured SIR epidemic model with applica-tions to measles vaccination strategiesrdquo Mathematical Bio-sciences vol 308 pp 27ndash37 2019
[4] P Widyaningsih D R S Saputro and A W NugroholdquoSusceptible exposed infected recovery (SEIR) model withimmigration equilibria points and its applicationrdquo AmericanInstitute of Physics vol 1 pp 1ndash7 2018
[5] M G Roberts and J A P HeesterbeekMathematical Modelsin Epidemiology Mathematical Models UNESCO-EOLSSAbu Dhabi UAE 2007
[6] J H He F Y Ji and H M Sedighi ldquoDifference equation vsdifferential equation on different scalesrdquo International Journalof Numerical Methods for Heat amp Fluid Flow vol 18pp 0961ndash5539 2020
[7] R E Mickens Nonstandard Finite Difference Models of Dif-ferential Equations World Scientific Singapore 1994
[8] R E Mickens Applications of Nonstandard Finite DifferenceSchemes World Scientific Singapore 2000
[9] R E Mickens ldquoNumerical integration of population modelssatisfying conservation laws NSFD methodsrdquo Journal ofBiological Dynamics vol 1 no 4 pp 427ndash436 2007
[10] J D Lambert Computational Methods in Ordinary Differ-ential Equations Wiley New York NY USA 1973
[11] S M Moghadas M E Alexander B D Corbett andA B Gumel ldquoA positivity-preserving mickens-type dis-cretization of an epidemic modelrdquo Journal of DifferenceEquations and Applications vol 9 no 11 pp 1037ndash10512003
[12] R E Mickens ldquoNonstandard finite difference schemes fordifferential equationsrdquo Journal of Difference Equations andApplications vol 8 no 9 pp 823ndash847 2002
[13] R E Mickens ldquoDynamic consistency a fundamental prin-ciple for constructing nonstandard finite difference schemesfor differential equationsrdquo Journal of Difference Equations andApplications vol 11 no 7 pp 645ndash653 2005
[14] L-I W Roeger and R W Barnard ldquoPreservation of localdynamics when applying central difference methods appli-cation to SIR modelrdquo Journal of Difference Equations andApplications vol 13 no 4 pp 333ndash340 2007
[15] K C Patidar ldquoOn the use of non-standard finite differencemethodsrdquo Journal of Difference Equations and Applicationsvol 11 no 7 pp 735ndash758 2005
[16] R Anguelov and J M-S Lubuma ldquoNonstandard finite dif-ference method by nonlocal approximationrdquo Mathematicsand Computers in Simulation vol 61 no 3-6 pp 465ndash4752003
[17] R Anguelov and J M-S Lubuma ldquoContributions to themathematics of the nonstandard finite difference method andapplicationsrdquo Numerical Methods for Partial DifferentialEquations vol 17 no 5 pp 518ndash543 2001
[18] B M Chen-Charpentier D T Dimitrov andH V Kojouharov ldquoNumerical simulation of multi-speciesbiofilms in porous media for different kineticsrdquo Mathematicsand Computers in Simulation vol 79 no 6 pp 1846ndash18612009
[19] D T Dimitrov and H V Kojouharov ldquoNonstandard finite-difference schemes for general two-dimensional autonomousdynamical systemsrdquo Applied Mathematics Letters vol 18no 7 pp 769ndash774 2005
[20] A J Arenas J A Morantildeo and J C Cortes ldquoNon-standardnumerical method for a mathematical model of RSV epide-miological transmissionrdquo Computers amp Mathematics withApplications vol 56 no 3 pp 670ndash678 2008
[21] L Jodar R J Villanueva A J Arenas and G C GonzalezldquoNonstandard numerical methods for a mathematical modelfor influenza diseaserdquo Mathematics and Computers in Sim-ulation vol 79 no 3 pp 622ndash633 2008
[22] W Piyawong E H Twizell and A B Gumel ldquoAn uncon-ditionally convergent finite-difference scheme for the SIRmodelrdquo Applied Mathematics and Computation vol 146no 2-3 pp 611ndash625 2003
[23] H Jansen and E H Twizell ldquoAn unconditionally convergentdiscretization of the SEIR modelrdquo Mathematics and Com-puters in Simulation vol 58 no 2 pp 147ndash158 2002
[24] A J Arenas G Gonzalez-Parra and B M Chen-CharpentierldquoA nonstandard numerical scheme of predictor-correctortype for epidemic modelsrdquo Computers amp Mathematics withApplications vol 59 no 12 pp 3740ndash3749 2010
[25] F Brauer and C Castillo-Chavez Mathematical Models inPopulation Biology and Epidemiology Springer Berlin Ger-many 2001
18 Journal of Mathematics
zF1 Se Ee Ie( 1113857
zS
11 + μϕ(h)R0
zF1 Se Ee Ie( 1113857
zE 0
zF1 Se Ee Ie( 1113857
zI
minus Nβϕ(h)
1 + μϕ(h)R0
zF2 Se Ee Ie( 1113857
zS
μϕ(h) R0 minus 1( 1113857
1 + μϕ(h)R01113858 1113859[1 + ϕ(h)(μ + σ)]
zF2 Se Ee Ie( 1113857
zE
1[1 + ϕ(h)(μ + σ)]
zF2 Se Ee Ie( 1113857
zI
minus Nβϕ(h) μϕ(l)R20 minus 2μϕ(h)R0 minus 11113960 1113961
R0[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]
zF3 Se Ee Ie( 1113857
zS
μσϕ2(h) R0 minus 1( 1113857
[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)] 1 + μϕ(h)R01113858 1113859
zF3 Se Ee Ie( 1113857
zE
σϕ(h)
[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]
zF3 Se Ee Ie( 1113857
zI
R0[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859 minus σNβϕ2(h) μϕ(h)R20 minus 2μϕ(h) minus 11113960 1113961
R0[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859[1 + ϕ(h)(μ + c)]
(25)
zG1 Se Ee Ie( 1113857
zS
11 + μϕ(h)R01113858 1113859
minusNμβσϕ3(h) R0 minus 1( 1113857
R0 1 + ϵminus 1 + μϕ(h)R01113960 1113961[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859[1 + ϕ(h)(μ + c)]
zG1 Se Ee Ie( 1113857
zE
minus Nβσϕ2(h)
R0 1 + ϵminus 1 + μϕ(h)R01113960 1113961[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]
Eige
n va
lues
Jacobian at DFE
0492
0494
0496
0498
05
0502
0504
0506
0508
051
1000 2000 3000 4000 5000 6000 7000 8000 9000 100000Step size
(a)
098
0981
0982
0983
0984
0985
0986
0987
0988
0989Jacobian at EE
Eige
n va
lues
1000 2000 3000 4000 5000 6000 7000 8000 9000 100000Step size
(b)
Figure 2 Spectral radius of the Jacobian at equilibrium points
Journal of Mathematics 9
zG1 Se Ee Ie( 1113857
zI
minus Nβϕ(h)
1 + ϵminus 1 + μϕ(h)R01113960 11139611 + μϕ(h)R0
middot ϵminus 1+
R0[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859 minus σNβϕ2(h) μϕ(h)R20 minus 2μϕ(h)R0 minus 11113960 1113961
R20[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859[1 + ϕ(h)(μ + c)]
⎡⎣ ⎤⎦
zG2 Se Ee Ie( 1113857
zS
μβϕ(h) R0 minus 1( 1113857
β[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 11138591113890
minusNμβ2σϕ4(h)μ R0 minus 1( 1113857
2
βR0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)] 1 + ϵminus 1 + μϕ(h)R01113960 1113961 1 + μϕ(h)R01113858 1113859
+Nμβσϕ3(h) R0 minus 1( 1113857
R0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)] 1 + μϕ(h)R01113858 1113859
1113891
zG2 Se Ee Ie( 1113857
zE
1[1 + ϕ(h)(μ + σ)]
+Nβσϕ2(h)
R0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)]
minusNμβ2σϕ3
(h) R0 minus 1( 1113857
βR0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)] 1 + ϵminus 1 + μϕ(h)R01113960 1113961
zG2 Se Ee Ie( 1113857
zI
minus Nμβ2ϕ2(h)ϵminus 1R0 minus 1( 1113857
β[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859 1 + ϵminus 1 + μϕ(h)R01113960 1113961
+Nμβ2ϕ2(h) R0 minus 1( 1113857 R0[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859 minus Nβσϕ2(h) μϕ(h)R
20 minus 2μϕ(h)R0 minus 11113960 11139611113966 1113967
βR20[1 + ϕ(h)(μ + σ)]
2[1 + ϕ(h)(μ + c)] 1 + ϵminus 1 + μϕ(h)R01113960 1113961 1 + μϕ(h)R01113858 1113859
+Nβϕ(h)
R0[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]minus
N2β2σϕ3(h) μϕ(h)R
20 minus 2μϕ(h)R0 minus 11113960 1113961
R20[1 + ϕ(h)(μ + σ)]
2[1 + ϕ(h)(μ + c)] 1 + μϕ(h)R01113858 1113859
zG3 Se Ee Ie( 1113857
zS
σϕ(h)
[1 + ϕ(h)(μ + c)]
μβϕ(h) R0 minus 1( 1113857
β[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 11138591113890
minusNμβ2σϕ4(h)μ R0 minus 1( 1113857
2
βR0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)] 1 + ϵminus 1 + μϕ(h)R01113960 1113961 1 + μϕ(h)R01113858 1113859
+Nμβσϕ3(h) R0 minus 1( 1113857
R0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)] 1 + μϕ(h)R01113858 1113859
1113891
zG3 Se Ee Ie( 1113857
zE
σϕ(h)
[1 + ϕ(h)(μ + c)]
1[1 + ϕ(h)(μ + σ)]
minusNμβ2σϕ3(h) R0 minus 1( 1113857
βR0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)] 1 + ϵminus 1 + μϕ(h)R01113960 1113961
⎡⎢⎣
+Nβσϕ2
(h)
R0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)]
1113891
zG3 Se Ee Ie( 1113857
zI
σϕ(h)
[1 + ϕ(h)(μ + c)]
minus Nμβ2ϕ2(h)ϵminus 1R0 minus 1( 1113857
β[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859 1 + ϵminus 1 + μϕ(h)R01113960 1113961⎡⎢⎣
+Nμβ2ϕ2(h) R0 minus 1( 1113857 R0[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859 minus Nβσϕ2(h) μϕ(h)R
20 minus 2μϕ(h)R0 minus 11113960 11139611113966 1113967
βR20[1 + ϕ(h)(μ + σ)]
2[1 + ϕ(h)(μ + c)] 1 + ϵminus 1 + μϕ(h)R01113960 1113961 1 + μϕ(h)R01113858 1113859
⎤⎥⎦
10 Journal of Mathematics
Table 1 Comparison of NSFD scheme with Euler and RK-4 schemes
l 001 01 100 1000Euler PC Convergence Divergence Divergence DivergenceRK-4 Convergence Divergence Divergence DivergenceNSFDPC Convergence Convergence Convergence Convergence
times107
NSFD-PC
Disease-free equilibrium
1
15
2
25
3
35
4
45
5
Susc
eptib
le fr
actio
n
20 40 60 80 100 120 140 160 180 2000Time (years) h = 001
(a)
Disease-free equilibriumtimes104
NSFD-PC
02 04 06 08 1 12 14 16 18 20Time (years) h = 001
0
05
1
15
2
25
3
35
4
45
5
Expo
sed
fract
ion
(b)
times104
NSFD-PC
Disease-free equilibrium
0
05
1
15
2
25
3
Infe
cted
frac
tion
02 04 06 08 1 12 14 16 18 20Time (years) h = 001
(c)
times107
NSFD-PC
X 5675Y 2435e + 007
Endemic equilibrium
100 200 300 400 500 6000Time (years) h = 001
1
15
2
25
3
35
Susc
eptib
le fr
actio
n
(d)
Figure 3 Continued
Journal of Mathematics 11
+Nβϕ(h)
R0[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]minus
N2β2σϕ3(h) μϕ(h)R
20 minus 2μϕ(h)R0 minus 11113960 1113961
R20[1 + ϕ(h)(μ + σ)]
2[1 + ϕ(h)(μ + c)] 1 + μϕ(h)R01113858 1113859
(26)
-e Jacobian evaluated at an endemic point(Slowaste Elowaste Ilowaste ) (Se Ee Ie) (NR0 μNμ + σ(1 minus 1R0) μβ(R0 minus 1)) is evaluated as
JG Slowaste Elowaste Ilowaste( 1113857
zG1 Slowaste Elowaste Ilowaste( 1113857
zS
zG1 Slowaste Elowaste Ilowaste( 1113857
zE
zG1 Slowaste Elowaste Ilowaste( 1113857
zI
zG1 Slowaste Elowaste Ilowaste( 1113857
zS
zG1 Slowaste Elowaste Ilowaste( 1113857
zE
zG1 Slowaste Elowaste Ilowaste( 1113857
zI
zG1 Slowaste Elowaste Ilowaste( 1113857
zS
zG1 Slowaste Elowaste Ilowaste( 1113857
zE
zG1 Slowaste Elowaste Ilowaste( 1113857
zI
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
(27)
It is tedious to calculate the eigenvalues of the Jacobian ofendemic equilibrium analytically so we have plotted thelargest eigenvalue against each step size and Figure 2(b)shows that for all step sizes the spectral radius of the Ja-cobian of EE remains less than one if R0 gt 1 which impliesthat the numerical scheme in equations (8) and (10) isunconditionally convergent if R0 gt 1 for all step sizes
6 Numerical Results and Discussion
All the methods showed the convergence for small step sizesin Table 1 However for large values of step sizes only thenormalized NSFDPC converge to the correct disease-freepoint for β 01 times 10minus 5 as well as the correct endemic pointfor β 03 times 10minus 5 -is means that for the value of the basic
reproductive number (threshold parameter) R0 lt 1 thenDFE is stable ie solution to the system (S E I R) areconverging to it -is can be seen from Figures 3(a)ndash3(c) forh 001 Also for the value of the basic reproductivenumber (threshold parameter) R0 gt 1 then EE is stable iesolution to the system (S E I R) is converging to it-is canbe seen from Figures 3(d)ndash3(f ) for h 001
Figures 4(a)ndash4(c) show how the NSFDPC method ofDFE converges to the equilibrium point for different stepsizes Figures 4(d)ndash4(f) show how the NSFDPC method ofEE converges to the equilibrium point for different step sizes
For comparison with the well-known RK-4 method tosystem (1) using the parameter values given in Table 2 it isfound that in Figures 5(a)ndash5(f) the numerical solutionconverges for h 001 for both equilibrium points and
times105
NSFD-PC
X 562Y 1125e + 004
Endemic equilibrium
100 200 300 400 500 6000Time (years) h = 001
0
1
2
3
4
5
6
7
8Ex
pose
d fra
ctio
n
(e)
times105
NSFD-PC
X 5523Y 7023
Infe
cted
frac
tion
Endemic equilibrium
100 200 300 400 500 6000Time (years) h = 001
0
05
1
15
2
25
3
35
4
45
5
(f )
Figure 3 Time plots of susceptible exposed and infected population using the NSFDPC method with h 001
12 Journal of Mathematics
times107
NSFD-PC
Disease-free equilibrium
20 40 60 80 100 120 140 160 180 2000Time (years) h = 01
1
15
2
25
3
35
4
45
5Su
scep
tible
frac
tion
(a)
times107
NSFD-PC
Disease-free equilibrium
500 1000 1500 2000 2500 3000 3500 40000Time (years) h = 10
1
15
2
25
3
35
4
45
5
Susc
eptib
le fr
actio
n
(b)
Disease-free equilibriumtimes107
times105
NSFD-PC
05 1 15 2 25 3 35 40Time (years) h = 1000
1
15
2
25
3
35
4
45
5
Susc
eptib
le fr
actio
n
(c)
X 5827Y 7022
Infe
cted
frac
tion
Endemic equilibriumtimes104
NSFD-PC
100 200 300 400 500 6000Time (years) h = 01
0
2
4
6
8
10
12
14
16
18
(d)
Endemic equilibriumtimes104
NSFD-PC
X 9600Y 7022
Infe
cted
frac
tion
1000 2000 3000 4000 5000 6000 7000 8000 9000 100000Time (years) h = 10
0
05
1
15
2
25
3
(e)
Endemic equilibriumtimes104
times105
NSFD-PC
X 861e + 005Y 7022
Infe
cted
frac
tion
1 2 3 4 5 6 7 8 9 100Time (years) h = 1000
0
05
1
15
2
25
3
(f )
Figure 4 Time plots of susceptible and infected population using the NSFDPC method with different step sizes (a) h 01 (b) h 10 and(c) h 1000
Journal of Mathematics 13
Table 2 Parameters value of the measles model
Parameters N μ σ c
5 times 107yearminus 1 002 456 yearminus 1 73 yearminus 1
DFE (S0 E0 I0) (N 0 0) β 01 times 10minus 5
EE (Se Ee Ie) (2435 times 107 1125 times 104 7022671) β 03 times 10minus 5
Disease-free equilibrium
RK4
times107
1
15
2
25
3
35
4
45
5
Susc
eptib
le fr
actio
n
20 40 60 80 100 120 140 160 180 2000Time (years) h = 001
(a)
Expo
sed
fract
ion
Disease-free equilibrium
RK4
times104
0
05
1
15
2
25
3
35
4
45
5
02 04 06 08 1 12 14 16 18 20Time (years) h = 001
(b)
times104
Infe
cted
frac
tion
Disease-free equilibrium
RK4
0
05
1
15
2
25
3
02 04 06 08 1 12 14 16 18 20Time (years) h = 001
(c)
times107
X 5578Y 2435e + 007
Endemic equilibrium
RK4
1
15
2
25
3
35
Susc
eptib
le F
ract
ion
100 200 300 400 500 6000Time (years) h = 001
(d)
Figure 5 Continued
14 Journal of Mathematics
Figure 6 shows overflow for a critical value of step size ieh 01 of DFE and EE respectively
Another popular numerical method is the Eulerpredictor-corrector (E-PC) technique which employs theexplicit Euler method as a predictor and the trapezoidalrule as the corrector -is combination is second-orderaccurate but has similar stability properties in PECEmode to the explicit Euler method alone Figures 7(a)ndash7(f ) show convergence results for h 001 in both casesthat are DFE and EE -e value of h 01 which produces
overflow when solving the system with the parametervalues of Table 2 for the PECE combination can be seen inFigure 8
-us the presented numerical results demonstrated thatNSFDPC has better convergence property following theEuler PC and RK-4 as shown in Figure 9 It is proved thatthe approximations made by other standard numericalmethods experience difficulties in preserving either thestability or the positivity of the solutions or both butNSFDPC is unconditionally convergent
times105
X 5772Y 1125e + 004
Endemic equilibrium
RK4
Expo
sed
fract
ion
0
1
2
3
4
5
6
7
8
9
10
100 200 300 400 500 6000Time (years) h = 001
(e)
times105
X 7659Y 7022
RK4
Endemic equilibrium
Infe
cted
frac
tion
0
1
2
3
4
5
6
100 200 300 400 500 600 700 8000Time (years) h = 001
(f )
Figure 5 Time plots of susceptible exposed and infected population using the RK-4 method with h 001
times1087 RK-4 scheme (disease-free equilibrium)
SusceptibleExposedInfected
01 02 03 04 05 06 07 08 09 10Time (years) h = 01
ndash2
ndash15
ndash1
ndash05
0
05
1
15
2
Popu
latio
n fra
ctio
ns
(a)
RK-4 scheme (endemic equilibrium)
SusceptibleExposedInfected
times10145
01 02 03 04 05 06 07 08 09 10Time (years) h = 01
ndash8
ndash6
ndash4
ndash2
0
2
4
6
8
Popu
latio
n fra
ctio
ns
(b)
Figure 6 Divergence graphs of the population using the RK-4 method with step size h 01
Journal of Mathematics 15
Euler-PC
Disease-free equilibriumtimes107
1
15
2
25
3
35
4
45
5Su
scep
tible
frac
tion
20 40 60 80 100 120 140 160 180 2000Time (years) h = 001
(a)
Euler-PC
Disease-free equilibrium
Expo
sed
fract
ion
times104
0
05
1
15
2
25
3
35
4
45
5
02 04 06 08 1 12 14 16 18 20Time (years) h = 001
(b)
Euler-PC
Disease-free equilibrium
Infe
cted
frac
tion
times104
0
05
1
15
2
25
3
35
02 04 06 08 1 12 14 16 18 20Time (years) h = 001
(c)
Euler-PC
X 5733Y 2435e + 007
Endemic equilibriumtimes107
1
15
2
25
3
35Su
scep
tible
frac
tion
100 200 300 400 500 6000Time (years) h = 001
(d)
Euler-PC
X 5662Y 1118e + 004
Endemic equilibrium
Expo
sed
fract
ion
times105
0
2
4
6
8
10
12
100 200 300 400 500 6000Time (years) h = 001
(e)
Euler-PC
X 5427Y 7025
Endemic equilibrium
Infe
cted
frac
tion
times105
0
1
2
3
4
5
6
7
100 200 300 400 500 6000Time (years) h = 001
(f )
Figure 7 Time plots of population using the Euler-PC method with h 001
16 Journal of Mathematics
7 Conclusions
In this paper a normalized NSFDPC for the SEIR modelconcerning the transmission dynamics of measles is con-structed and analysed -is proposed numerical scheme isvery competitive It is qualitatively stable that is it doublerefines the solution and gives realistic results even for largestep sizes It is dynamically consistent with a continuoussystem and unconditionally convergent and satisfies the
positivity of the state variables involved in the systemSimulations are carried out and its usefulness is comparedwith a well-known numerical method of standard differenceschemes such as RK4 and Euler predictor-corrector method-e standard finite difference schemes are highly dependenton step sizes and initial value problems but NSFDPC isindependent of these two features which make it morepractical Also this method saves computation time andmemory
Euler-PC scheme (disease-free equilibrium)times1041
ndash3
ndash2
ndash1
0
1
2
3
Popu
latio
n fra
ctio
ns
01 02 03 04 05 06 07 08 09 10Time (years) h = 01
SusceptibleExposedInfected
(a)
Euler-PC scheme (endemic equilibrium)times1053
ndash6
ndash4
ndash2
0
2
4
6
Popu
latio
n fra
ctio
ns
01 02 03 04 05 06 07 08 09 10Time (years) h = 01
SusceptibleExposedInfected
(b)
Figure 8 Divergence graphs of the population using the Euler-PC method with step size h 01
NSFD-PCRK-4
times1092
Infe
cted
frac
tion
ndash5
ndash4
ndash3
ndash2
ndash1
0
1
02 04 06 08 1 12 14 16 18 20Time (years) h = 01
(a)
NSFD-PCEuler-PC
times10213
Infe
cted
frac
tion
ndash4
ndash35
ndash3
ndash25
ndash2
ndash15
ndash1
ndash05
0
05
02 04 06 08 1 12 14 16 18 20Time (years) h = 01
(b)
Figure 9 Comparative convergence graphs of the NSFDPC method
Journal of Mathematics 17
Data Availability
-e data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
-e authors declare no conflicts of interest
Acknowledgments
-e authors extend their appreciation to the Deanship ofScientific Research at Majmaah University for funding thiswork under Project no RGP-2019-6
References
[1] E Hunter B Namee and J Kelleher ldquoAn open-data-drivenagent-based model to simulate infectious disease outbreaksrdquoPLoS One vol 14 no 1 pp 1ndash35 2018
[2] E A Bakare Y A Adekunle and K O Kadiri ldquoModeling andsimulation of the dynamics of the transmission of measlesrdquoInternational Journal of Computer Trends and Trends andTechnology vol 3 no 1 pp 174ndash177 2012
[3] L Zhou Y Wang Y Li and Y Michael ldquoGlobal dynamics ofa discrete age-structured SIR epidemic model with applica-tions to measles vaccination strategiesrdquo Mathematical Bio-sciences vol 308 pp 27ndash37 2019
[4] P Widyaningsih D R S Saputro and A W NugroholdquoSusceptible exposed infected recovery (SEIR) model withimmigration equilibria points and its applicationrdquo AmericanInstitute of Physics vol 1 pp 1ndash7 2018
[5] M G Roberts and J A P HeesterbeekMathematical Modelsin Epidemiology Mathematical Models UNESCO-EOLSSAbu Dhabi UAE 2007
[6] J H He F Y Ji and H M Sedighi ldquoDifference equation vsdifferential equation on different scalesrdquo International Journalof Numerical Methods for Heat amp Fluid Flow vol 18pp 0961ndash5539 2020
[7] R E Mickens Nonstandard Finite Difference Models of Dif-ferential Equations World Scientific Singapore 1994
[8] R E Mickens Applications of Nonstandard Finite DifferenceSchemes World Scientific Singapore 2000
[9] R E Mickens ldquoNumerical integration of population modelssatisfying conservation laws NSFD methodsrdquo Journal ofBiological Dynamics vol 1 no 4 pp 427ndash436 2007
[10] J D Lambert Computational Methods in Ordinary Differ-ential Equations Wiley New York NY USA 1973
[11] S M Moghadas M E Alexander B D Corbett andA B Gumel ldquoA positivity-preserving mickens-type dis-cretization of an epidemic modelrdquo Journal of DifferenceEquations and Applications vol 9 no 11 pp 1037ndash10512003
[12] R E Mickens ldquoNonstandard finite difference schemes fordifferential equationsrdquo Journal of Difference Equations andApplications vol 8 no 9 pp 823ndash847 2002
[13] R E Mickens ldquoDynamic consistency a fundamental prin-ciple for constructing nonstandard finite difference schemesfor differential equationsrdquo Journal of Difference Equations andApplications vol 11 no 7 pp 645ndash653 2005
[14] L-I W Roeger and R W Barnard ldquoPreservation of localdynamics when applying central difference methods appli-cation to SIR modelrdquo Journal of Difference Equations andApplications vol 13 no 4 pp 333ndash340 2007
[15] K C Patidar ldquoOn the use of non-standard finite differencemethodsrdquo Journal of Difference Equations and Applicationsvol 11 no 7 pp 735ndash758 2005
[16] R Anguelov and J M-S Lubuma ldquoNonstandard finite dif-ference method by nonlocal approximationrdquo Mathematicsand Computers in Simulation vol 61 no 3-6 pp 465ndash4752003
[17] R Anguelov and J M-S Lubuma ldquoContributions to themathematics of the nonstandard finite difference method andapplicationsrdquo Numerical Methods for Partial DifferentialEquations vol 17 no 5 pp 518ndash543 2001
[18] B M Chen-Charpentier D T Dimitrov andH V Kojouharov ldquoNumerical simulation of multi-speciesbiofilms in porous media for different kineticsrdquo Mathematicsand Computers in Simulation vol 79 no 6 pp 1846ndash18612009
[19] D T Dimitrov and H V Kojouharov ldquoNonstandard finite-difference schemes for general two-dimensional autonomousdynamical systemsrdquo Applied Mathematics Letters vol 18no 7 pp 769ndash774 2005
[20] A J Arenas J A Morantildeo and J C Cortes ldquoNon-standardnumerical method for a mathematical model of RSV epide-miological transmissionrdquo Computers amp Mathematics withApplications vol 56 no 3 pp 670ndash678 2008
[21] L Jodar R J Villanueva A J Arenas and G C GonzalezldquoNonstandard numerical methods for a mathematical modelfor influenza diseaserdquo Mathematics and Computers in Sim-ulation vol 79 no 3 pp 622ndash633 2008
[22] W Piyawong E H Twizell and A B Gumel ldquoAn uncon-ditionally convergent finite-difference scheme for the SIRmodelrdquo Applied Mathematics and Computation vol 146no 2-3 pp 611ndash625 2003
[23] H Jansen and E H Twizell ldquoAn unconditionally convergentdiscretization of the SEIR modelrdquo Mathematics and Com-puters in Simulation vol 58 no 2 pp 147ndash158 2002
[24] A J Arenas G Gonzalez-Parra and B M Chen-CharpentierldquoA nonstandard numerical scheme of predictor-correctortype for epidemic modelsrdquo Computers amp Mathematics withApplications vol 59 no 12 pp 3740ndash3749 2010
[25] F Brauer and C Castillo-Chavez Mathematical Models inPopulation Biology and Epidemiology Springer Berlin Ger-many 2001
18 Journal of Mathematics
zG1 Se Ee Ie( 1113857
zI
minus Nβϕ(h)
1 + ϵminus 1 + μϕ(h)R01113960 11139611 + μϕ(h)R0
middot ϵminus 1+
R0[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859 minus σNβϕ2(h) μϕ(h)R20 minus 2μϕ(h)R0 minus 11113960 1113961
R20[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859[1 + ϕ(h)(μ + c)]
⎡⎣ ⎤⎦
zG2 Se Ee Ie( 1113857
zS
μβϕ(h) R0 minus 1( 1113857
β[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 11138591113890
minusNμβ2σϕ4(h)μ R0 minus 1( 1113857
2
βR0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)] 1 + ϵminus 1 + μϕ(h)R01113960 1113961 1 + μϕ(h)R01113858 1113859
+Nμβσϕ3(h) R0 minus 1( 1113857
R0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)] 1 + μϕ(h)R01113858 1113859
1113891
zG2 Se Ee Ie( 1113857
zE
1[1 + ϕ(h)(μ + σ)]
+Nβσϕ2(h)
R0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)]
minusNμβ2σϕ3
(h) R0 minus 1( 1113857
βR0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)] 1 + ϵminus 1 + μϕ(h)R01113960 1113961
zG2 Se Ee Ie( 1113857
zI
minus Nμβ2ϕ2(h)ϵminus 1R0 minus 1( 1113857
β[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859 1 + ϵminus 1 + μϕ(h)R01113960 1113961
+Nμβ2ϕ2(h) R0 minus 1( 1113857 R0[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859 minus Nβσϕ2(h) μϕ(h)R
20 minus 2μϕ(h)R0 minus 11113960 11139611113966 1113967
βR20[1 + ϕ(h)(μ + σ)]
2[1 + ϕ(h)(μ + c)] 1 + ϵminus 1 + μϕ(h)R01113960 1113961 1 + μϕ(h)R01113858 1113859
+Nβϕ(h)
R0[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]minus
N2β2σϕ3(h) μϕ(h)R
20 minus 2μϕ(h)R0 minus 11113960 1113961
R20[1 + ϕ(h)(μ + σ)]
2[1 + ϕ(h)(μ + c)] 1 + μϕ(h)R01113858 1113859
zG3 Se Ee Ie( 1113857
zS
σϕ(h)
[1 + ϕ(h)(μ + c)]
μβϕ(h) R0 minus 1( 1113857
β[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 11138591113890
minusNμβ2σϕ4(h)μ R0 minus 1( 1113857
2
βR0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)] 1 + ϵminus 1 + μϕ(h)R01113960 1113961 1 + μϕ(h)R01113858 1113859
+Nμβσϕ3(h) R0 minus 1( 1113857
R0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)] 1 + μϕ(h)R01113858 1113859
1113891
zG3 Se Ee Ie( 1113857
zE
σϕ(h)
[1 + ϕ(h)(μ + c)]
1[1 + ϕ(h)(μ + σ)]
minusNμβ2σϕ3(h) R0 minus 1( 1113857
βR0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)] 1 + ϵminus 1 + μϕ(h)R01113960 1113961
⎡⎢⎣
+Nβσϕ2
(h)
R0[1 + ϕ(h)(μ + σ)]2[1 + ϕ(h)(μ + c)]
1113891
zG3 Se Ee Ie( 1113857
zI
σϕ(h)
[1 + ϕ(h)(μ + c)]
minus Nμβ2ϕ2(h)ϵminus 1R0 minus 1( 1113857
β[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859 1 + ϵminus 1 + μϕ(h)R01113960 1113961⎡⎢⎣
+Nμβ2ϕ2(h) R0 minus 1( 1113857 R0[1 + ϕ(h)(μ + σ)] 1 + μϕ(h)R01113858 1113859 minus Nβσϕ2(h) μϕ(h)R
20 minus 2μϕ(h)R0 minus 11113960 11139611113966 1113967
βR20[1 + ϕ(h)(μ + σ)]
2[1 + ϕ(h)(μ + c)] 1 + ϵminus 1 + μϕ(h)R01113960 1113961 1 + μϕ(h)R01113858 1113859
⎤⎥⎦
10 Journal of Mathematics
Table 1 Comparison of NSFD scheme with Euler and RK-4 schemes
l 001 01 100 1000Euler PC Convergence Divergence Divergence DivergenceRK-4 Convergence Divergence Divergence DivergenceNSFDPC Convergence Convergence Convergence Convergence
times107
NSFD-PC
Disease-free equilibrium
1
15
2
25
3
35
4
45
5
Susc
eptib
le fr
actio
n
20 40 60 80 100 120 140 160 180 2000Time (years) h = 001
(a)
Disease-free equilibriumtimes104
NSFD-PC
02 04 06 08 1 12 14 16 18 20Time (years) h = 001
0
05
1
15
2
25
3
35
4
45
5
Expo
sed
fract
ion
(b)
times104
NSFD-PC
Disease-free equilibrium
0
05
1
15
2
25
3
Infe
cted
frac
tion
02 04 06 08 1 12 14 16 18 20Time (years) h = 001
(c)
times107
NSFD-PC
X 5675Y 2435e + 007
Endemic equilibrium
100 200 300 400 500 6000Time (years) h = 001
1
15
2
25
3
35
Susc
eptib
le fr
actio
n
(d)
Figure 3 Continued
Journal of Mathematics 11
+Nβϕ(h)
R0[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]minus
N2β2σϕ3(h) μϕ(h)R
20 minus 2μϕ(h)R0 minus 11113960 1113961
R20[1 + ϕ(h)(μ + σ)]
2[1 + ϕ(h)(μ + c)] 1 + μϕ(h)R01113858 1113859
(26)
-e Jacobian evaluated at an endemic point(Slowaste Elowaste Ilowaste ) (Se Ee Ie) (NR0 μNμ + σ(1 minus 1R0) μβ(R0 minus 1)) is evaluated as
JG Slowaste Elowaste Ilowaste( 1113857
zG1 Slowaste Elowaste Ilowaste( 1113857
zS
zG1 Slowaste Elowaste Ilowaste( 1113857
zE
zG1 Slowaste Elowaste Ilowaste( 1113857
zI
zG1 Slowaste Elowaste Ilowaste( 1113857
zS
zG1 Slowaste Elowaste Ilowaste( 1113857
zE
zG1 Slowaste Elowaste Ilowaste( 1113857
zI
zG1 Slowaste Elowaste Ilowaste( 1113857
zS
zG1 Slowaste Elowaste Ilowaste( 1113857
zE
zG1 Slowaste Elowaste Ilowaste( 1113857
zI
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
(27)
It is tedious to calculate the eigenvalues of the Jacobian ofendemic equilibrium analytically so we have plotted thelargest eigenvalue against each step size and Figure 2(b)shows that for all step sizes the spectral radius of the Ja-cobian of EE remains less than one if R0 gt 1 which impliesthat the numerical scheme in equations (8) and (10) isunconditionally convergent if R0 gt 1 for all step sizes
6 Numerical Results and Discussion
All the methods showed the convergence for small step sizesin Table 1 However for large values of step sizes only thenormalized NSFDPC converge to the correct disease-freepoint for β 01 times 10minus 5 as well as the correct endemic pointfor β 03 times 10minus 5 -is means that for the value of the basic
reproductive number (threshold parameter) R0 lt 1 thenDFE is stable ie solution to the system (S E I R) areconverging to it -is can be seen from Figures 3(a)ndash3(c) forh 001 Also for the value of the basic reproductivenumber (threshold parameter) R0 gt 1 then EE is stable iesolution to the system (S E I R) is converging to it-is canbe seen from Figures 3(d)ndash3(f ) for h 001
Figures 4(a)ndash4(c) show how the NSFDPC method ofDFE converges to the equilibrium point for different stepsizes Figures 4(d)ndash4(f) show how the NSFDPC method ofEE converges to the equilibrium point for different step sizes
For comparison with the well-known RK-4 method tosystem (1) using the parameter values given in Table 2 it isfound that in Figures 5(a)ndash5(f) the numerical solutionconverges for h 001 for both equilibrium points and
times105
NSFD-PC
X 562Y 1125e + 004
Endemic equilibrium
100 200 300 400 500 6000Time (years) h = 001
0
1
2
3
4
5
6
7
8Ex
pose
d fra
ctio
n
(e)
times105
NSFD-PC
X 5523Y 7023
Infe
cted
frac
tion
Endemic equilibrium
100 200 300 400 500 6000Time (years) h = 001
0
05
1
15
2
25
3
35
4
45
5
(f )
Figure 3 Time plots of susceptible exposed and infected population using the NSFDPC method with h 001
12 Journal of Mathematics
times107
NSFD-PC
Disease-free equilibrium
20 40 60 80 100 120 140 160 180 2000Time (years) h = 01
1
15
2
25
3
35
4
45
5Su
scep
tible
frac
tion
(a)
times107
NSFD-PC
Disease-free equilibrium
500 1000 1500 2000 2500 3000 3500 40000Time (years) h = 10
1
15
2
25
3
35
4
45
5
Susc
eptib
le fr
actio
n
(b)
Disease-free equilibriumtimes107
times105
NSFD-PC
05 1 15 2 25 3 35 40Time (years) h = 1000
1
15
2
25
3
35
4
45
5
Susc
eptib
le fr
actio
n
(c)
X 5827Y 7022
Infe
cted
frac
tion
Endemic equilibriumtimes104
NSFD-PC
100 200 300 400 500 6000Time (years) h = 01
0
2
4
6
8
10
12
14
16
18
(d)
Endemic equilibriumtimes104
NSFD-PC
X 9600Y 7022
Infe
cted
frac
tion
1000 2000 3000 4000 5000 6000 7000 8000 9000 100000Time (years) h = 10
0
05
1
15
2
25
3
(e)
Endemic equilibriumtimes104
times105
NSFD-PC
X 861e + 005Y 7022
Infe
cted
frac
tion
1 2 3 4 5 6 7 8 9 100Time (years) h = 1000
0
05
1
15
2
25
3
(f )
Figure 4 Time plots of susceptible and infected population using the NSFDPC method with different step sizes (a) h 01 (b) h 10 and(c) h 1000
Journal of Mathematics 13
Table 2 Parameters value of the measles model
Parameters N μ σ c
5 times 107yearminus 1 002 456 yearminus 1 73 yearminus 1
DFE (S0 E0 I0) (N 0 0) β 01 times 10minus 5
EE (Se Ee Ie) (2435 times 107 1125 times 104 7022671) β 03 times 10minus 5
Disease-free equilibrium
RK4
times107
1
15
2
25
3
35
4
45
5
Susc
eptib
le fr
actio
n
20 40 60 80 100 120 140 160 180 2000Time (years) h = 001
(a)
Expo
sed
fract
ion
Disease-free equilibrium
RK4
times104
0
05
1
15
2
25
3
35
4
45
5
02 04 06 08 1 12 14 16 18 20Time (years) h = 001
(b)
times104
Infe
cted
frac
tion
Disease-free equilibrium
RK4
0
05
1
15
2
25
3
02 04 06 08 1 12 14 16 18 20Time (years) h = 001
(c)
times107
X 5578Y 2435e + 007
Endemic equilibrium
RK4
1
15
2
25
3
35
Susc
eptib
le F
ract
ion
100 200 300 400 500 6000Time (years) h = 001
(d)
Figure 5 Continued
14 Journal of Mathematics
Figure 6 shows overflow for a critical value of step size ieh 01 of DFE and EE respectively
Another popular numerical method is the Eulerpredictor-corrector (E-PC) technique which employs theexplicit Euler method as a predictor and the trapezoidalrule as the corrector -is combination is second-orderaccurate but has similar stability properties in PECEmode to the explicit Euler method alone Figures 7(a)ndash7(f ) show convergence results for h 001 in both casesthat are DFE and EE -e value of h 01 which produces
overflow when solving the system with the parametervalues of Table 2 for the PECE combination can be seen inFigure 8
-us the presented numerical results demonstrated thatNSFDPC has better convergence property following theEuler PC and RK-4 as shown in Figure 9 It is proved thatthe approximations made by other standard numericalmethods experience difficulties in preserving either thestability or the positivity of the solutions or both butNSFDPC is unconditionally convergent
times105
X 5772Y 1125e + 004
Endemic equilibrium
RK4
Expo
sed
fract
ion
0
1
2
3
4
5
6
7
8
9
10
100 200 300 400 500 6000Time (years) h = 001
(e)
times105
X 7659Y 7022
RK4
Endemic equilibrium
Infe
cted
frac
tion
0
1
2
3
4
5
6
100 200 300 400 500 600 700 8000Time (years) h = 001
(f )
Figure 5 Time plots of susceptible exposed and infected population using the RK-4 method with h 001
times1087 RK-4 scheme (disease-free equilibrium)
SusceptibleExposedInfected
01 02 03 04 05 06 07 08 09 10Time (years) h = 01
ndash2
ndash15
ndash1
ndash05
0
05
1
15
2
Popu
latio
n fra
ctio
ns
(a)
RK-4 scheme (endemic equilibrium)
SusceptibleExposedInfected
times10145
01 02 03 04 05 06 07 08 09 10Time (years) h = 01
ndash8
ndash6
ndash4
ndash2
0
2
4
6
8
Popu
latio
n fra
ctio
ns
(b)
Figure 6 Divergence graphs of the population using the RK-4 method with step size h 01
Journal of Mathematics 15
Euler-PC
Disease-free equilibriumtimes107
1
15
2
25
3
35
4
45
5Su
scep
tible
frac
tion
20 40 60 80 100 120 140 160 180 2000Time (years) h = 001
(a)
Euler-PC
Disease-free equilibrium
Expo
sed
fract
ion
times104
0
05
1
15
2
25
3
35
4
45
5
02 04 06 08 1 12 14 16 18 20Time (years) h = 001
(b)
Euler-PC
Disease-free equilibrium
Infe
cted
frac
tion
times104
0
05
1
15
2
25
3
35
02 04 06 08 1 12 14 16 18 20Time (years) h = 001
(c)
Euler-PC
X 5733Y 2435e + 007
Endemic equilibriumtimes107
1
15
2
25
3
35Su
scep
tible
frac
tion
100 200 300 400 500 6000Time (years) h = 001
(d)
Euler-PC
X 5662Y 1118e + 004
Endemic equilibrium
Expo
sed
fract
ion
times105
0
2
4
6
8
10
12
100 200 300 400 500 6000Time (years) h = 001
(e)
Euler-PC
X 5427Y 7025
Endemic equilibrium
Infe
cted
frac
tion
times105
0
1
2
3
4
5
6
7
100 200 300 400 500 6000Time (years) h = 001
(f )
Figure 7 Time plots of population using the Euler-PC method with h 001
16 Journal of Mathematics
7 Conclusions
In this paper a normalized NSFDPC for the SEIR modelconcerning the transmission dynamics of measles is con-structed and analysed -is proposed numerical scheme isvery competitive It is qualitatively stable that is it doublerefines the solution and gives realistic results even for largestep sizes It is dynamically consistent with a continuoussystem and unconditionally convergent and satisfies the
positivity of the state variables involved in the systemSimulations are carried out and its usefulness is comparedwith a well-known numerical method of standard differenceschemes such as RK4 and Euler predictor-corrector method-e standard finite difference schemes are highly dependenton step sizes and initial value problems but NSFDPC isindependent of these two features which make it morepractical Also this method saves computation time andmemory
Euler-PC scheme (disease-free equilibrium)times1041
ndash3
ndash2
ndash1
0
1
2
3
Popu
latio
n fra
ctio
ns
01 02 03 04 05 06 07 08 09 10Time (years) h = 01
SusceptibleExposedInfected
(a)
Euler-PC scheme (endemic equilibrium)times1053
ndash6
ndash4
ndash2
0
2
4
6
Popu
latio
n fra
ctio
ns
01 02 03 04 05 06 07 08 09 10Time (years) h = 01
SusceptibleExposedInfected
(b)
Figure 8 Divergence graphs of the population using the Euler-PC method with step size h 01
NSFD-PCRK-4
times1092
Infe
cted
frac
tion
ndash5
ndash4
ndash3
ndash2
ndash1
0
1
02 04 06 08 1 12 14 16 18 20Time (years) h = 01
(a)
NSFD-PCEuler-PC
times10213
Infe
cted
frac
tion
ndash4
ndash35
ndash3
ndash25
ndash2
ndash15
ndash1
ndash05
0
05
02 04 06 08 1 12 14 16 18 20Time (years) h = 01
(b)
Figure 9 Comparative convergence graphs of the NSFDPC method
Journal of Mathematics 17
Data Availability
-e data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
-e authors declare no conflicts of interest
Acknowledgments
-e authors extend their appreciation to the Deanship ofScientific Research at Majmaah University for funding thiswork under Project no RGP-2019-6
References
[1] E Hunter B Namee and J Kelleher ldquoAn open-data-drivenagent-based model to simulate infectious disease outbreaksrdquoPLoS One vol 14 no 1 pp 1ndash35 2018
[2] E A Bakare Y A Adekunle and K O Kadiri ldquoModeling andsimulation of the dynamics of the transmission of measlesrdquoInternational Journal of Computer Trends and Trends andTechnology vol 3 no 1 pp 174ndash177 2012
[3] L Zhou Y Wang Y Li and Y Michael ldquoGlobal dynamics ofa discrete age-structured SIR epidemic model with applica-tions to measles vaccination strategiesrdquo Mathematical Bio-sciences vol 308 pp 27ndash37 2019
[4] P Widyaningsih D R S Saputro and A W NugroholdquoSusceptible exposed infected recovery (SEIR) model withimmigration equilibria points and its applicationrdquo AmericanInstitute of Physics vol 1 pp 1ndash7 2018
[5] M G Roberts and J A P HeesterbeekMathematical Modelsin Epidemiology Mathematical Models UNESCO-EOLSSAbu Dhabi UAE 2007
[6] J H He F Y Ji and H M Sedighi ldquoDifference equation vsdifferential equation on different scalesrdquo International Journalof Numerical Methods for Heat amp Fluid Flow vol 18pp 0961ndash5539 2020
[7] R E Mickens Nonstandard Finite Difference Models of Dif-ferential Equations World Scientific Singapore 1994
[8] R E Mickens Applications of Nonstandard Finite DifferenceSchemes World Scientific Singapore 2000
[9] R E Mickens ldquoNumerical integration of population modelssatisfying conservation laws NSFD methodsrdquo Journal ofBiological Dynamics vol 1 no 4 pp 427ndash436 2007
[10] J D Lambert Computational Methods in Ordinary Differ-ential Equations Wiley New York NY USA 1973
[11] S M Moghadas M E Alexander B D Corbett andA B Gumel ldquoA positivity-preserving mickens-type dis-cretization of an epidemic modelrdquo Journal of DifferenceEquations and Applications vol 9 no 11 pp 1037ndash10512003
[12] R E Mickens ldquoNonstandard finite difference schemes fordifferential equationsrdquo Journal of Difference Equations andApplications vol 8 no 9 pp 823ndash847 2002
[13] R E Mickens ldquoDynamic consistency a fundamental prin-ciple for constructing nonstandard finite difference schemesfor differential equationsrdquo Journal of Difference Equations andApplications vol 11 no 7 pp 645ndash653 2005
[14] L-I W Roeger and R W Barnard ldquoPreservation of localdynamics when applying central difference methods appli-cation to SIR modelrdquo Journal of Difference Equations andApplications vol 13 no 4 pp 333ndash340 2007
[15] K C Patidar ldquoOn the use of non-standard finite differencemethodsrdquo Journal of Difference Equations and Applicationsvol 11 no 7 pp 735ndash758 2005
[16] R Anguelov and J M-S Lubuma ldquoNonstandard finite dif-ference method by nonlocal approximationrdquo Mathematicsand Computers in Simulation vol 61 no 3-6 pp 465ndash4752003
[17] R Anguelov and J M-S Lubuma ldquoContributions to themathematics of the nonstandard finite difference method andapplicationsrdquo Numerical Methods for Partial DifferentialEquations vol 17 no 5 pp 518ndash543 2001
[18] B M Chen-Charpentier D T Dimitrov andH V Kojouharov ldquoNumerical simulation of multi-speciesbiofilms in porous media for different kineticsrdquo Mathematicsand Computers in Simulation vol 79 no 6 pp 1846ndash18612009
[19] D T Dimitrov and H V Kojouharov ldquoNonstandard finite-difference schemes for general two-dimensional autonomousdynamical systemsrdquo Applied Mathematics Letters vol 18no 7 pp 769ndash774 2005
[20] A J Arenas J A Morantildeo and J C Cortes ldquoNon-standardnumerical method for a mathematical model of RSV epide-miological transmissionrdquo Computers amp Mathematics withApplications vol 56 no 3 pp 670ndash678 2008
[21] L Jodar R J Villanueva A J Arenas and G C GonzalezldquoNonstandard numerical methods for a mathematical modelfor influenza diseaserdquo Mathematics and Computers in Sim-ulation vol 79 no 3 pp 622ndash633 2008
[22] W Piyawong E H Twizell and A B Gumel ldquoAn uncon-ditionally convergent finite-difference scheme for the SIRmodelrdquo Applied Mathematics and Computation vol 146no 2-3 pp 611ndash625 2003
[23] H Jansen and E H Twizell ldquoAn unconditionally convergentdiscretization of the SEIR modelrdquo Mathematics and Com-puters in Simulation vol 58 no 2 pp 147ndash158 2002
[24] A J Arenas G Gonzalez-Parra and B M Chen-CharpentierldquoA nonstandard numerical scheme of predictor-correctortype for epidemic modelsrdquo Computers amp Mathematics withApplications vol 59 no 12 pp 3740ndash3749 2010
[25] F Brauer and C Castillo-Chavez Mathematical Models inPopulation Biology and Epidemiology Springer Berlin Ger-many 2001
18 Journal of Mathematics
Table 1 Comparison of NSFD scheme with Euler and RK-4 schemes
l 001 01 100 1000Euler PC Convergence Divergence Divergence DivergenceRK-4 Convergence Divergence Divergence DivergenceNSFDPC Convergence Convergence Convergence Convergence
times107
NSFD-PC
Disease-free equilibrium
1
15
2
25
3
35
4
45
5
Susc
eptib
le fr
actio
n
20 40 60 80 100 120 140 160 180 2000Time (years) h = 001
(a)
Disease-free equilibriumtimes104
NSFD-PC
02 04 06 08 1 12 14 16 18 20Time (years) h = 001
0
05
1
15
2
25
3
35
4
45
5
Expo
sed
fract
ion
(b)
times104
NSFD-PC
Disease-free equilibrium
0
05
1
15
2
25
3
Infe
cted
frac
tion
02 04 06 08 1 12 14 16 18 20Time (years) h = 001
(c)
times107
NSFD-PC
X 5675Y 2435e + 007
Endemic equilibrium
100 200 300 400 500 6000Time (years) h = 001
1
15
2
25
3
35
Susc
eptib
le fr
actio
n
(d)
Figure 3 Continued
Journal of Mathematics 11
+Nβϕ(h)
R0[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]minus
N2β2σϕ3(h) μϕ(h)R
20 minus 2μϕ(h)R0 minus 11113960 1113961
R20[1 + ϕ(h)(μ + σ)]
2[1 + ϕ(h)(μ + c)] 1 + μϕ(h)R01113858 1113859
(26)
-e Jacobian evaluated at an endemic point(Slowaste Elowaste Ilowaste ) (Se Ee Ie) (NR0 μNμ + σ(1 minus 1R0) μβ(R0 minus 1)) is evaluated as
JG Slowaste Elowaste Ilowaste( 1113857
zG1 Slowaste Elowaste Ilowaste( 1113857
zS
zG1 Slowaste Elowaste Ilowaste( 1113857
zE
zG1 Slowaste Elowaste Ilowaste( 1113857
zI
zG1 Slowaste Elowaste Ilowaste( 1113857
zS
zG1 Slowaste Elowaste Ilowaste( 1113857
zE
zG1 Slowaste Elowaste Ilowaste( 1113857
zI
zG1 Slowaste Elowaste Ilowaste( 1113857
zS
zG1 Slowaste Elowaste Ilowaste( 1113857
zE
zG1 Slowaste Elowaste Ilowaste( 1113857
zI
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
(27)
It is tedious to calculate the eigenvalues of the Jacobian ofendemic equilibrium analytically so we have plotted thelargest eigenvalue against each step size and Figure 2(b)shows that for all step sizes the spectral radius of the Ja-cobian of EE remains less than one if R0 gt 1 which impliesthat the numerical scheme in equations (8) and (10) isunconditionally convergent if R0 gt 1 for all step sizes
6 Numerical Results and Discussion
All the methods showed the convergence for small step sizesin Table 1 However for large values of step sizes only thenormalized NSFDPC converge to the correct disease-freepoint for β 01 times 10minus 5 as well as the correct endemic pointfor β 03 times 10minus 5 -is means that for the value of the basic
reproductive number (threshold parameter) R0 lt 1 thenDFE is stable ie solution to the system (S E I R) areconverging to it -is can be seen from Figures 3(a)ndash3(c) forh 001 Also for the value of the basic reproductivenumber (threshold parameter) R0 gt 1 then EE is stable iesolution to the system (S E I R) is converging to it-is canbe seen from Figures 3(d)ndash3(f ) for h 001
Figures 4(a)ndash4(c) show how the NSFDPC method ofDFE converges to the equilibrium point for different stepsizes Figures 4(d)ndash4(f) show how the NSFDPC method ofEE converges to the equilibrium point for different step sizes
For comparison with the well-known RK-4 method tosystem (1) using the parameter values given in Table 2 it isfound that in Figures 5(a)ndash5(f) the numerical solutionconverges for h 001 for both equilibrium points and
times105
NSFD-PC
X 562Y 1125e + 004
Endemic equilibrium
100 200 300 400 500 6000Time (years) h = 001
0
1
2
3
4
5
6
7
8Ex
pose
d fra
ctio
n
(e)
times105
NSFD-PC
X 5523Y 7023
Infe
cted
frac
tion
Endemic equilibrium
100 200 300 400 500 6000Time (years) h = 001
0
05
1
15
2
25
3
35
4
45
5
(f )
Figure 3 Time plots of susceptible exposed and infected population using the NSFDPC method with h 001
12 Journal of Mathematics
times107
NSFD-PC
Disease-free equilibrium
20 40 60 80 100 120 140 160 180 2000Time (years) h = 01
1
15
2
25
3
35
4
45
5Su
scep
tible
frac
tion
(a)
times107
NSFD-PC
Disease-free equilibrium
500 1000 1500 2000 2500 3000 3500 40000Time (years) h = 10
1
15
2
25
3
35
4
45
5
Susc
eptib
le fr
actio
n
(b)
Disease-free equilibriumtimes107
times105
NSFD-PC
05 1 15 2 25 3 35 40Time (years) h = 1000
1
15
2
25
3
35
4
45
5
Susc
eptib
le fr
actio
n
(c)
X 5827Y 7022
Infe
cted
frac
tion
Endemic equilibriumtimes104
NSFD-PC
100 200 300 400 500 6000Time (years) h = 01
0
2
4
6
8
10
12
14
16
18
(d)
Endemic equilibriumtimes104
NSFD-PC
X 9600Y 7022
Infe
cted
frac
tion
1000 2000 3000 4000 5000 6000 7000 8000 9000 100000Time (years) h = 10
0
05
1
15
2
25
3
(e)
Endemic equilibriumtimes104
times105
NSFD-PC
X 861e + 005Y 7022
Infe
cted
frac
tion
1 2 3 4 5 6 7 8 9 100Time (years) h = 1000
0
05
1
15
2
25
3
(f )
Figure 4 Time plots of susceptible and infected population using the NSFDPC method with different step sizes (a) h 01 (b) h 10 and(c) h 1000
Journal of Mathematics 13
Table 2 Parameters value of the measles model
Parameters N μ σ c
5 times 107yearminus 1 002 456 yearminus 1 73 yearminus 1
DFE (S0 E0 I0) (N 0 0) β 01 times 10minus 5
EE (Se Ee Ie) (2435 times 107 1125 times 104 7022671) β 03 times 10minus 5
Disease-free equilibrium
RK4
times107
1
15
2
25
3
35
4
45
5
Susc
eptib
le fr
actio
n
20 40 60 80 100 120 140 160 180 2000Time (years) h = 001
(a)
Expo
sed
fract
ion
Disease-free equilibrium
RK4
times104
0
05
1
15
2
25
3
35
4
45
5
02 04 06 08 1 12 14 16 18 20Time (years) h = 001
(b)
times104
Infe
cted
frac
tion
Disease-free equilibrium
RK4
0
05
1
15
2
25
3
02 04 06 08 1 12 14 16 18 20Time (years) h = 001
(c)
times107
X 5578Y 2435e + 007
Endemic equilibrium
RK4
1
15
2
25
3
35
Susc
eptib
le F
ract
ion
100 200 300 400 500 6000Time (years) h = 001
(d)
Figure 5 Continued
14 Journal of Mathematics
Figure 6 shows overflow for a critical value of step size ieh 01 of DFE and EE respectively
Another popular numerical method is the Eulerpredictor-corrector (E-PC) technique which employs theexplicit Euler method as a predictor and the trapezoidalrule as the corrector -is combination is second-orderaccurate but has similar stability properties in PECEmode to the explicit Euler method alone Figures 7(a)ndash7(f ) show convergence results for h 001 in both casesthat are DFE and EE -e value of h 01 which produces
overflow when solving the system with the parametervalues of Table 2 for the PECE combination can be seen inFigure 8
-us the presented numerical results demonstrated thatNSFDPC has better convergence property following theEuler PC and RK-4 as shown in Figure 9 It is proved thatthe approximations made by other standard numericalmethods experience difficulties in preserving either thestability or the positivity of the solutions or both butNSFDPC is unconditionally convergent
times105
X 5772Y 1125e + 004
Endemic equilibrium
RK4
Expo
sed
fract
ion
0
1
2
3
4
5
6
7
8
9
10
100 200 300 400 500 6000Time (years) h = 001
(e)
times105
X 7659Y 7022
RK4
Endemic equilibrium
Infe
cted
frac
tion
0
1
2
3
4
5
6
100 200 300 400 500 600 700 8000Time (years) h = 001
(f )
Figure 5 Time plots of susceptible exposed and infected population using the RK-4 method with h 001
times1087 RK-4 scheme (disease-free equilibrium)
SusceptibleExposedInfected
01 02 03 04 05 06 07 08 09 10Time (years) h = 01
ndash2
ndash15
ndash1
ndash05
0
05
1
15
2
Popu
latio
n fra
ctio
ns
(a)
RK-4 scheme (endemic equilibrium)
SusceptibleExposedInfected
times10145
01 02 03 04 05 06 07 08 09 10Time (years) h = 01
ndash8
ndash6
ndash4
ndash2
0
2
4
6
8
Popu
latio
n fra
ctio
ns
(b)
Figure 6 Divergence graphs of the population using the RK-4 method with step size h 01
Journal of Mathematics 15
Euler-PC
Disease-free equilibriumtimes107
1
15
2
25
3
35
4
45
5Su
scep
tible
frac
tion
20 40 60 80 100 120 140 160 180 2000Time (years) h = 001
(a)
Euler-PC
Disease-free equilibrium
Expo
sed
fract
ion
times104
0
05
1
15
2
25
3
35
4
45
5
02 04 06 08 1 12 14 16 18 20Time (years) h = 001
(b)
Euler-PC
Disease-free equilibrium
Infe
cted
frac
tion
times104
0
05
1
15
2
25
3
35
02 04 06 08 1 12 14 16 18 20Time (years) h = 001
(c)
Euler-PC
X 5733Y 2435e + 007
Endemic equilibriumtimes107
1
15
2
25
3
35Su
scep
tible
frac
tion
100 200 300 400 500 6000Time (years) h = 001
(d)
Euler-PC
X 5662Y 1118e + 004
Endemic equilibrium
Expo
sed
fract
ion
times105
0
2
4
6
8
10
12
100 200 300 400 500 6000Time (years) h = 001
(e)
Euler-PC
X 5427Y 7025
Endemic equilibrium
Infe
cted
frac
tion
times105
0
1
2
3
4
5
6
7
100 200 300 400 500 6000Time (years) h = 001
(f )
Figure 7 Time plots of population using the Euler-PC method with h 001
16 Journal of Mathematics
7 Conclusions
In this paper a normalized NSFDPC for the SEIR modelconcerning the transmission dynamics of measles is con-structed and analysed -is proposed numerical scheme isvery competitive It is qualitatively stable that is it doublerefines the solution and gives realistic results even for largestep sizes It is dynamically consistent with a continuoussystem and unconditionally convergent and satisfies the
positivity of the state variables involved in the systemSimulations are carried out and its usefulness is comparedwith a well-known numerical method of standard differenceschemes such as RK4 and Euler predictor-corrector method-e standard finite difference schemes are highly dependenton step sizes and initial value problems but NSFDPC isindependent of these two features which make it morepractical Also this method saves computation time andmemory
Euler-PC scheme (disease-free equilibrium)times1041
ndash3
ndash2
ndash1
0
1
2
3
Popu
latio
n fra
ctio
ns
01 02 03 04 05 06 07 08 09 10Time (years) h = 01
SusceptibleExposedInfected
(a)
Euler-PC scheme (endemic equilibrium)times1053
ndash6
ndash4
ndash2
0
2
4
6
Popu
latio
n fra
ctio
ns
01 02 03 04 05 06 07 08 09 10Time (years) h = 01
SusceptibleExposedInfected
(b)
Figure 8 Divergence graphs of the population using the Euler-PC method with step size h 01
NSFD-PCRK-4
times1092
Infe
cted
frac
tion
ndash5
ndash4
ndash3
ndash2
ndash1
0
1
02 04 06 08 1 12 14 16 18 20Time (years) h = 01
(a)
NSFD-PCEuler-PC
times10213
Infe
cted
frac
tion
ndash4
ndash35
ndash3
ndash25
ndash2
ndash15
ndash1
ndash05
0
05
02 04 06 08 1 12 14 16 18 20Time (years) h = 01
(b)
Figure 9 Comparative convergence graphs of the NSFDPC method
Journal of Mathematics 17
Data Availability
-e data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
-e authors declare no conflicts of interest
Acknowledgments
-e authors extend their appreciation to the Deanship ofScientific Research at Majmaah University for funding thiswork under Project no RGP-2019-6
References
[1] E Hunter B Namee and J Kelleher ldquoAn open-data-drivenagent-based model to simulate infectious disease outbreaksrdquoPLoS One vol 14 no 1 pp 1ndash35 2018
[2] E A Bakare Y A Adekunle and K O Kadiri ldquoModeling andsimulation of the dynamics of the transmission of measlesrdquoInternational Journal of Computer Trends and Trends andTechnology vol 3 no 1 pp 174ndash177 2012
[3] L Zhou Y Wang Y Li and Y Michael ldquoGlobal dynamics ofa discrete age-structured SIR epidemic model with applica-tions to measles vaccination strategiesrdquo Mathematical Bio-sciences vol 308 pp 27ndash37 2019
[4] P Widyaningsih D R S Saputro and A W NugroholdquoSusceptible exposed infected recovery (SEIR) model withimmigration equilibria points and its applicationrdquo AmericanInstitute of Physics vol 1 pp 1ndash7 2018
[5] M G Roberts and J A P HeesterbeekMathematical Modelsin Epidemiology Mathematical Models UNESCO-EOLSSAbu Dhabi UAE 2007
[6] J H He F Y Ji and H M Sedighi ldquoDifference equation vsdifferential equation on different scalesrdquo International Journalof Numerical Methods for Heat amp Fluid Flow vol 18pp 0961ndash5539 2020
[7] R E Mickens Nonstandard Finite Difference Models of Dif-ferential Equations World Scientific Singapore 1994
[8] R E Mickens Applications of Nonstandard Finite DifferenceSchemes World Scientific Singapore 2000
[9] R E Mickens ldquoNumerical integration of population modelssatisfying conservation laws NSFD methodsrdquo Journal ofBiological Dynamics vol 1 no 4 pp 427ndash436 2007
[10] J D Lambert Computational Methods in Ordinary Differ-ential Equations Wiley New York NY USA 1973
[11] S M Moghadas M E Alexander B D Corbett andA B Gumel ldquoA positivity-preserving mickens-type dis-cretization of an epidemic modelrdquo Journal of DifferenceEquations and Applications vol 9 no 11 pp 1037ndash10512003
[12] R E Mickens ldquoNonstandard finite difference schemes fordifferential equationsrdquo Journal of Difference Equations andApplications vol 8 no 9 pp 823ndash847 2002
[13] R E Mickens ldquoDynamic consistency a fundamental prin-ciple for constructing nonstandard finite difference schemesfor differential equationsrdquo Journal of Difference Equations andApplications vol 11 no 7 pp 645ndash653 2005
[14] L-I W Roeger and R W Barnard ldquoPreservation of localdynamics when applying central difference methods appli-cation to SIR modelrdquo Journal of Difference Equations andApplications vol 13 no 4 pp 333ndash340 2007
[15] K C Patidar ldquoOn the use of non-standard finite differencemethodsrdquo Journal of Difference Equations and Applicationsvol 11 no 7 pp 735ndash758 2005
[16] R Anguelov and J M-S Lubuma ldquoNonstandard finite dif-ference method by nonlocal approximationrdquo Mathematicsand Computers in Simulation vol 61 no 3-6 pp 465ndash4752003
[17] R Anguelov and J M-S Lubuma ldquoContributions to themathematics of the nonstandard finite difference method andapplicationsrdquo Numerical Methods for Partial DifferentialEquations vol 17 no 5 pp 518ndash543 2001
[18] B M Chen-Charpentier D T Dimitrov andH V Kojouharov ldquoNumerical simulation of multi-speciesbiofilms in porous media for different kineticsrdquo Mathematicsand Computers in Simulation vol 79 no 6 pp 1846ndash18612009
[19] D T Dimitrov and H V Kojouharov ldquoNonstandard finite-difference schemes for general two-dimensional autonomousdynamical systemsrdquo Applied Mathematics Letters vol 18no 7 pp 769ndash774 2005
[20] A J Arenas J A Morantildeo and J C Cortes ldquoNon-standardnumerical method for a mathematical model of RSV epide-miological transmissionrdquo Computers amp Mathematics withApplications vol 56 no 3 pp 670ndash678 2008
[21] L Jodar R J Villanueva A J Arenas and G C GonzalezldquoNonstandard numerical methods for a mathematical modelfor influenza diseaserdquo Mathematics and Computers in Sim-ulation vol 79 no 3 pp 622ndash633 2008
[22] W Piyawong E H Twizell and A B Gumel ldquoAn uncon-ditionally convergent finite-difference scheme for the SIRmodelrdquo Applied Mathematics and Computation vol 146no 2-3 pp 611ndash625 2003
[23] H Jansen and E H Twizell ldquoAn unconditionally convergentdiscretization of the SEIR modelrdquo Mathematics and Com-puters in Simulation vol 58 no 2 pp 147ndash158 2002
[24] A J Arenas G Gonzalez-Parra and B M Chen-CharpentierldquoA nonstandard numerical scheme of predictor-correctortype for epidemic modelsrdquo Computers amp Mathematics withApplications vol 59 no 12 pp 3740ndash3749 2010
[25] F Brauer and C Castillo-Chavez Mathematical Models inPopulation Biology and Epidemiology Springer Berlin Ger-many 2001
18 Journal of Mathematics
+Nβϕ(h)
R0[1 + ϕ(h)(μ + σ)][1 + ϕ(h)(μ + c)]minus
N2β2σϕ3(h) μϕ(h)R
20 minus 2μϕ(h)R0 minus 11113960 1113961
R20[1 + ϕ(h)(μ + σ)]
2[1 + ϕ(h)(μ + c)] 1 + μϕ(h)R01113858 1113859
(26)
-e Jacobian evaluated at an endemic point(Slowaste Elowaste Ilowaste ) (Se Ee Ie) (NR0 μNμ + σ(1 minus 1R0) μβ(R0 minus 1)) is evaluated as
JG Slowaste Elowaste Ilowaste( 1113857
zG1 Slowaste Elowaste Ilowaste( 1113857
zS
zG1 Slowaste Elowaste Ilowaste( 1113857
zE
zG1 Slowaste Elowaste Ilowaste( 1113857
zI
zG1 Slowaste Elowaste Ilowaste( 1113857
zS
zG1 Slowaste Elowaste Ilowaste( 1113857
zE
zG1 Slowaste Elowaste Ilowaste( 1113857
zI
zG1 Slowaste Elowaste Ilowaste( 1113857
zS
zG1 Slowaste Elowaste Ilowaste( 1113857
zE
zG1 Slowaste Elowaste Ilowaste( 1113857
zI
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
(27)
It is tedious to calculate the eigenvalues of the Jacobian ofendemic equilibrium analytically so we have plotted thelargest eigenvalue against each step size and Figure 2(b)shows that for all step sizes the spectral radius of the Ja-cobian of EE remains less than one if R0 gt 1 which impliesthat the numerical scheme in equations (8) and (10) isunconditionally convergent if R0 gt 1 for all step sizes
6 Numerical Results and Discussion
All the methods showed the convergence for small step sizesin Table 1 However for large values of step sizes only thenormalized NSFDPC converge to the correct disease-freepoint for β 01 times 10minus 5 as well as the correct endemic pointfor β 03 times 10minus 5 -is means that for the value of the basic
reproductive number (threshold parameter) R0 lt 1 thenDFE is stable ie solution to the system (S E I R) areconverging to it -is can be seen from Figures 3(a)ndash3(c) forh 001 Also for the value of the basic reproductivenumber (threshold parameter) R0 gt 1 then EE is stable iesolution to the system (S E I R) is converging to it-is canbe seen from Figures 3(d)ndash3(f ) for h 001
Figures 4(a)ndash4(c) show how the NSFDPC method ofDFE converges to the equilibrium point for different stepsizes Figures 4(d)ndash4(f) show how the NSFDPC method ofEE converges to the equilibrium point for different step sizes
For comparison with the well-known RK-4 method tosystem (1) using the parameter values given in Table 2 it isfound that in Figures 5(a)ndash5(f) the numerical solutionconverges for h 001 for both equilibrium points and
times105
NSFD-PC
X 562Y 1125e + 004
Endemic equilibrium
100 200 300 400 500 6000Time (years) h = 001
0
1
2
3
4
5
6
7
8Ex
pose
d fra
ctio
n
(e)
times105
NSFD-PC
X 5523Y 7023
Infe
cted
frac
tion
Endemic equilibrium
100 200 300 400 500 6000Time (years) h = 001
0
05
1
15
2
25
3
35
4
45
5
(f )
Figure 3 Time plots of susceptible exposed and infected population using the NSFDPC method with h 001
12 Journal of Mathematics
times107
NSFD-PC
Disease-free equilibrium
20 40 60 80 100 120 140 160 180 2000Time (years) h = 01
1
15
2
25
3
35
4
45
5Su
scep
tible
frac
tion
(a)
times107
NSFD-PC
Disease-free equilibrium
500 1000 1500 2000 2500 3000 3500 40000Time (years) h = 10
1
15
2
25
3
35
4
45
5
Susc
eptib
le fr
actio
n
(b)
Disease-free equilibriumtimes107
times105
NSFD-PC
05 1 15 2 25 3 35 40Time (years) h = 1000
1
15
2
25
3
35
4
45
5
Susc
eptib
le fr
actio
n
(c)
X 5827Y 7022
Infe
cted
frac
tion
Endemic equilibriumtimes104
NSFD-PC
100 200 300 400 500 6000Time (years) h = 01
0
2
4
6
8
10
12
14
16
18
(d)
Endemic equilibriumtimes104
NSFD-PC
X 9600Y 7022
Infe
cted
frac
tion
1000 2000 3000 4000 5000 6000 7000 8000 9000 100000Time (years) h = 10
0
05
1
15
2
25
3
(e)
Endemic equilibriumtimes104
times105
NSFD-PC
X 861e + 005Y 7022
Infe
cted
frac
tion
1 2 3 4 5 6 7 8 9 100Time (years) h = 1000
0
05
1
15
2
25
3
(f )
Figure 4 Time plots of susceptible and infected population using the NSFDPC method with different step sizes (a) h 01 (b) h 10 and(c) h 1000
Journal of Mathematics 13
Table 2 Parameters value of the measles model
Parameters N μ σ c
5 times 107yearminus 1 002 456 yearminus 1 73 yearminus 1
DFE (S0 E0 I0) (N 0 0) β 01 times 10minus 5
EE (Se Ee Ie) (2435 times 107 1125 times 104 7022671) β 03 times 10minus 5
Disease-free equilibrium
RK4
times107
1
15
2
25
3
35
4
45
5
Susc
eptib
le fr
actio
n
20 40 60 80 100 120 140 160 180 2000Time (years) h = 001
(a)
Expo
sed
fract
ion
Disease-free equilibrium
RK4
times104
0
05
1
15
2
25
3
35
4
45
5
02 04 06 08 1 12 14 16 18 20Time (years) h = 001
(b)
times104
Infe
cted
frac
tion
Disease-free equilibrium
RK4
0
05
1
15
2
25
3
02 04 06 08 1 12 14 16 18 20Time (years) h = 001
(c)
times107
X 5578Y 2435e + 007
Endemic equilibrium
RK4
1
15
2
25
3
35
Susc
eptib
le F
ract
ion
100 200 300 400 500 6000Time (years) h = 001
(d)
Figure 5 Continued
14 Journal of Mathematics
Figure 6 shows overflow for a critical value of step size ieh 01 of DFE and EE respectively
Another popular numerical method is the Eulerpredictor-corrector (E-PC) technique which employs theexplicit Euler method as a predictor and the trapezoidalrule as the corrector -is combination is second-orderaccurate but has similar stability properties in PECEmode to the explicit Euler method alone Figures 7(a)ndash7(f ) show convergence results for h 001 in both casesthat are DFE and EE -e value of h 01 which produces
overflow when solving the system with the parametervalues of Table 2 for the PECE combination can be seen inFigure 8
-us the presented numerical results demonstrated thatNSFDPC has better convergence property following theEuler PC and RK-4 as shown in Figure 9 It is proved thatthe approximations made by other standard numericalmethods experience difficulties in preserving either thestability or the positivity of the solutions or both butNSFDPC is unconditionally convergent
times105
X 5772Y 1125e + 004
Endemic equilibrium
RK4
Expo
sed
fract
ion
0
1
2
3
4
5
6
7
8
9
10
100 200 300 400 500 6000Time (years) h = 001
(e)
times105
X 7659Y 7022
RK4
Endemic equilibrium
Infe
cted
frac
tion
0
1
2
3
4
5
6
100 200 300 400 500 600 700 8000Time (years) h = 001
(f )
Figure 5 Time plots of susceptible exposed and infected population using the RK-4 method with h 001
times1087 RK-4 scheme (disease-free equilibrium)
SusceptibleExposedInfected
01 02 03 04 05 06 07 08 09 10Time (years) h = 01
ndash2
ndash15
ndash1
ndash05
0
05
1
15
2
Popu
latio
n fra
ctio
ns
(a)
RK-4 scheme (endemic equilibrium)
SusceptibleExposedInfected
times10145
01 02 03 04 05 06 07 08 09 10Time (years) h = 01
ndash8
ndash6
ndash4
ndash2
0
2
4
6
8
Popu
latio
n fra
ctio
ns
(b)
Figure 6 Divergence graphs of the population using the RK-4 method with step size h 01
Journal of Mathematics 15
Euler-PC
Disease-free equilibriumtimes107
1
15
2
25
3
35
4
45
5Su
scep
tible
frac
tion
20 40 60 80 100 120 140 160 180 2000Time (years) h = 001
(a)
Euler-PC
Disease-free equilibrium
Expo
sed
fract
ion
times104
0
05
1
15
2
25
3
35
4
45
5
02 04 06 08 1 12 14 16 18 20Time (years) h = 001
(b)
Euler-PC
Disease-free equilibrium
Infe
cted
frac
tion
times104
0
05
1
15
2
25
3
35
02 04 06 08 1 12 14 16 18 20Time (years) h = 001
(c)
Euler-PC
X 5733Y 2435e + 007
Endemic equilibriumtimes107
1
15
2
25
3
35Su
scep
tible
frac
tion
100 200 300 400 500 6000Time (years) h = 001
(d)
Euler-PC
X 5662Y 1118e + 004
Endemic equilibrium
Expo
sed
fract
ion
times105
0
2
4
6
8
10
12
100 200 300 400 500 6000Time (years) h = 001
(e)
Euler-PC
X 5427Y 7025
Endemic equilibrium
Infe
cted
frac
tion
times105
0
1
2
3
4
5
6
7
100 200 300 400 500 6000Time (years) h = 001
(f )
Figure 7 Time plots of population using the Euler-PC method with h 001
16 Journal of Mathematics
7 Conclusions
In this paper a normalized NSFDPC for the SEIR modelconcerning the transmission dynamics of measles is con-structed and analysed -is proposed numerical scheme isvery competitive It is qualitatively stable that is it doublerefines the solution and gives realistic results even for largestep sizes It is dynamically consistent with a continuoussystem and unconditionally convergent and satisfies the
positivity of the state variables involved in the systemSimulations are carried out and its usefulness is comparedwith a well-known numerical method of standard differenceschemes such as RK4 and Euler predictor-corrector method-e standard finite difference schemes are highly dependenton step sizes and initial value problems but NSFDPC isindependent of these two features which make it morepractical Also this method saves computation time andmemory
Euler-PC scheme (disease-free equilibrium)times1041
ndash3
ndash2
ndash1
0
1
2
3
Popu
latio
n fra
ctio
ns
01 02 03 04 05 06 07 08 09 10Time (years) h = 01
SusceptibleExposedInfected
(a)
Euler-PC scheme (endemic equilibrium)times1053
ndash6
ndash4
ndash2
0
2
4
6
Popu
latio
n fra
ctio
ns
01 02 03 04 05 06 07 08 09 10Time (years) h = 01
SusceptibleExposedInfected
(b)
Figure 8 Divergence graphs of the population using the Euler-PC method with step size h 01
NSFD-PCRK-4
times1092
Infe
cted
frac
tion
ndash5
ndash4
ndash3
ndash2
ndash1
0
1
02 04 06 08 1 12 14 16 18 20Time (years) h = 01
(a)
NSFD-PCEuler-PC
times10213
Infe
cted
frac
tion
ndash4
ndash35
ndash3
ndash25
ndash2
ndash15
ndash1
ndash05
0
05
02 04 06 08 1 12 14 16 18 20Time (years) h = 01
(b)
Figure 9 Comparative convergence graphs of the NSFDPC method
Journal of Mathematics 17
Data Availability
-e data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
-e authors declare no conflicts of interest
Acknowledgments
-e authors extend their appreciation to the Deanship ofScientific Research at Majmaah University for funding thiswork under Project no RGP-2019-6
References
[1] E Hunter B Namee and J Kelleher ldquoAn open-data-drivenagent-based model to simulate infectious disease outbreaksrdquoPLoS One vol 14 no 1 pp 1ndash35 2018
[2] E A Bakare Y A Adekunle and K O Kadiri ldquoModeling andsimulation of the dynamics of the transmission of measlesrdquoInternational Journal of Computer Trends and Trends andTechnology vol 3 no 1 pp 174ndash177 2012
[3] L Zhou Y Wang Y Li and Y Michael ldquoGlobal dynamics ofa discrete age-structured SIR epidemic model with applica-tions to measles vaccination strategiesrdquo Mathematical Bio-sciences vol 308 pp 27ndash37 2019
[4] P Widyaningsih D R S Saputro and A W NugroholdquoSusceptible exposed infected recovery (SEIR) model withimmigration equilibria points and its applicationrdquo AmericanInstitute of Physics vol 1 pp 1ndash7 2018
[5] M G Roberts and J A P HeesterbeekMathematical Modelsin Epidemiology Mathematical Models UNESCO-EOLSSAbu Dhabi UAE 2007
[6] J H He F Y Ji and H M Sedighi ldquoDifference equation vsdifferential equation on different scalesrdquo International Journalof Numerical Methods for Heat amp Fluid Flow vol 18pp 0961ndash5539 2020
[7] R E Mickens Nonstandard Finite Difference Models of Dif-ferential Equations World Scientific Singapore 1994
[8] R E Mickens Applications of Nonstandard Finite DifferenceSchemes World Scientific Singapore 2000
[9] R E Mickens ldquoNumerical integration of population modelssatisfying conservation laws NSFD methodsrdquo Journal ofBiological Dynamics vol 1 no 4 pp 427ndash436 2007
[10] J D Lambert Computational Methods in Ordinary Differ-ential Equations Wiley New York NY USA 1973
[11] S M Moghadas M E Alexander B D Corbett andA B Gumel ldquoA positivity-preserving mickens-type dis-cretization of an epidemic modelrdquo Journal of DifferenceEquations and Applications vol 9 no 11 pp 1037ndash10512003
[12] R E Mickens ldquoNonstandard finite difference schemes fordifferential equationsrdquo Journal of Difference Equations andApplications vol 8 no 9 pp 823ndash847 2002
[13] R E Mickens ldquoDynamic consistency a fundamental prin-ciple for constructing nonstandard finite difference schemesfor differential equationsrdquo Journal of Difference Equations andApplications vol 11 no 7 pp 645ndash653 2005
[14] L-I W Roeger and R W Barnard ldquoPreservation of localdynamics when applying central difference methods appli-cation to SIR modelrdquo Journal of Difference Equations andApplications vol 13 no 4 pp 333ndash340 2007
[15] K C Patidar ldquoOn the use of non-standard finite differencemethodsrdquo Journal of Difference Equations and Applicationsvol 11 no 7 pp 735ndash758 2005
[16] R Anguelov and J M-S Lubuma ldquoNonstandard finite dif-ference method by nonlocal approximationrdquo Mathematicsand Computers in Simulation vol 61 no 3-6 pp 465ndash4752003
[17] R Anguelov and J M-S Lubuma ldquoContributions to themathematics of the nonstandard finite difference method andapplicationsrdquo Numerical Methods for Partial DifferentialEquations vol 17 no 5 pp 518ndash543 2001
[18] B M Chen-Charpentier D T Dimitrov andH V Kojouharov ldquoNumerical simulation of multi-speciesbiofilms in porous media for different kineticsrdquo Mathematicsand Computers in Simulation vol 79 no 6 pp 1846ndash18612009
[19] D T Dimitrov and H V Kojouharov ldquoNonstandard finite-difference schemes for general two-dimensional autonomousdynamical systemsrdquo Applied Mathematics Letters vol 18no 7 pp 769ndash774 2005
[20] A J Arenas J A Morantildeo and J C Cortes ldquoNon-standardnumerical method for a mathematical model of RSV epide-miological transmissionrdquo Computers amp Mathematics withApplications vol 56 no 3 pp 670ndash678 2008
[21] L Jodar R J Villanueva A J Arenas and G C GonzalezldquoNonstandard numerical methods for a mathematical modelfor influenza diseaserdquo Mathematics and Computers in Sim-ulation vol 79 no 3 pp 622ndash633 2008
[22] W Piyawong E H Twizell and A B Gumel ldquoAn uncon-ditionally convergent finite-difference scheme for the SIRmodelrdquo Applied Mathematics and Computation vol 146no 2-3 pp 611ndash625 2003
[23] H Jansen and E H Twizell ldquoAn unconditionally convergentdiscretization of the SEIR modelrdquo Mathematics and Com-puters in Simulation vol 58 no 2 pp 147ndash158 2002
[24] A J Arenas G Gonzalez-Parra and B M Chen-CharpentierldquoA nonstandard numerical scheme of predictor-correctortype for epidemic modelsrdquo Computers amp Mathematics withApplications vol 59 no 12 pp 3740ndash3749 2010
[25] F Brauer and C Castillo-Chavez Mathematical Models inPopulation Biology and Epidemiology Springer Berlin Ger-many 2001
18 Journal of Mathematics
times107
NSFD-PC
Disease-free equilibrium
20 40 60 80 100 120 140 160 180 2000Time (years) h = 01
1
15
2
25
3
35
4
45
5Su
scep
tible
frac
tion
(a)
times107
NSFD-PC
Disease-free equilibrium
500 1000 1500 2000 2500 3000 3500 40000Time (years) h = 10
1
15
2
25
3
35
4
45
5
Susc
eptib
le fr
actio
n
(b)
Disease-free equilibriumtimes107
times105
NSFD-PC
05 1 15 2 25 3 35 40Time (years) h = 1000
1
15
2
25
3
35
4
45
5
Susc
eptib
le fr
actio
n
(c)
X 5827Y 7022
Infe
cted
frac
tion
Endemic equilibriumtimes104
NSFD-PC
100 200 300 400 500 6000Time (years) h = 01
0
2
4
6
8
10
12
14
16
18
(d)
Endemic equilibriumtimes104
NSFD-PC
X 9600Y 7022
Infe
cted
frac
tion
1000 2000 3000 4000 5000 6000 7000 8000 9000 100000Time (years) h = 10
0
05
1
15
2
25
3
(e)
Endemic equilibriumtimes104
times105
NSFD-PC
X 861e + 005Y 7022
Infe
cted
frac
tion
1 2 3 4 5 6 7 8 9 100Time (years) h = 1000
0
05
1
15
2
25
3
(f )
Figure 4 Time plots of susceptible and infected population using the NSFDPC method with different step sizes (a) h 01 (b) h 10 and(c) h 1000
Journal of Mathematics 13
Table 2 Parameters value of the measles model
Parameters N μ σ c
5 times 107yearminus 1 002 456 yearminus 1 73 yearminus 1
DFE (S0 E0 I0) (N 0 0) β 01 times 10minus 5
EE (Se Ee Ie) (2435 times 107 1125 times 104 7022671) β 03 times 10minus 5
Disease-free equilibrium
RK4
times107
1
15
2
25
3
35
4
45
5
Susc
eptib
le fr
actio
n
20 40 60 80 100 120 140 160 180 2000Time (years) h = 001
(a)
Expo
sed
fract
ion
Disease-free equilibrium
RK4
times104
0
05
1
15
2
25
3
35
4
45
5
02 04 06 08 1 12 14 16 18 20Time (years) h = 001
(b)
times104
Infe
cted
frac
tion
Disease-free equilibrium
RK4
0
05
1
15
2
25
3
02 04 06 08 1 12 14 16 18 20Time (years) h = 001
(c)
times107
X 5578Y 2435e + 007
Endemic equilibrium
RK4
1
15
2
25
3
35
Susc
eptib
le F
ract
ion
100 200 300 400 500 6000Time (years) h = 001
(d)
Figure 5 Continued
14 Journal of Mathematics
Figure 6 shows overflow for a critical value of step size ieh 01 of DFE and EE respectively
Another popular numerical method is the Eulerpredictor-corrector (E-PC) technique which employs theexplicit Euler method as a predictor and the trapezoidalrule as the corrector -is combination is second-orderaccurate but has similar stability properties in PECEmode to the explicit Euler method alone Figures 7(a)ndash7(f ) show convergence results for h 001 in both casesthat are DFE and EE -e value of h 01 which produces
overflow when solving the system with the parametervalues of Table 2 for the PECE combination can be seen inFigure 8
-us the presented numerical results demonstrated thatNSFDPC has better convergence property following theEuler PC and RK-4 as shown in Figure 9 It is proved thatthe approximations made by other standard numericalmethods experience difficulties in preserving either thestability or the positivity of the solutions or both butNSFDPC is unconditionally convergent
times105
X 5772Y 1125e + 004
Endemic equilibrium
RK4
Expo
sed
fract
ion
0
1
2
3
4
5
6
7
8
9
10
100 200 300 400 500 6000Time (years) h = 001
(e)
times105
X 7659Y 7022
RK4
Endemic equilibrium
Infe
cted
frac
tion
0
1
2
3
4
5
6
100 200 300 400 500 600 700 8000Time (years) h = 001
(f )
Figure 5 Time plots of susceptible exposed and infected population using the RK-4 method with h 001
times1087 RK-4 scheme (disease-free equilibrium)
SusceptibleExposedInfected
01 02 03 04 05 06 07 08 09 10Time (years) h = 01
ndash2
ndash15
ndash1
ndash05
0
05
1
15
2
Popu
latio
n fra
ctio
ns
(a)
RK-4 scheme (endemic equilibrium)
SusceptibleExposedInfected
times10145
01 02 03 04 05 06 07 08 09 10Time (years) h = 01
ndash8
ndash6
ndash4
ndash2
0
2
4
6
8
Popu
latio
n fra
ctio
ns
(b)
Figure 6 Divergence graphs of the population using the RK-4 method with step size h 01
Journal of Mathematics 15
Euler-PC
Disease-free equilibriumtimes107
1
15
2
25
3
35
4
45
5Su
scep
tible
frac
tion
20 40 60 80 100 120 140 160 180 2000Time (years) h = 001
(a)
Euler-PC
Disease-free equilibrium
Expo
sed
fract
ion
times104
0
05
1
15
2
25
3
35
4
45
5
02 04 06 08 1 12 14 16 18 20Time (years) h = 001
(b)
Euler-PC
Disease-free equilibrium
Infe
cted
frac
tion
times104
0
05
1
15
2
25
3
35
02 04 06 08 1 12 14 16 18 20Time (years) h = 001
(c)
Euler-PC
X 5733Y 2435e + 007
Endemic equilibriumtimes107
1
15
2
25
3
35Su
scep
tible
frac
tion
100 200 300 400 500 6000Time (years) h = 001
(d)
Euler-PC
X 5662Y 1118e + 004
Endemic equilibrium
Expo
sed
fract
ion
times105
0
2
4
6
8
10
12
100 200 300 400 500 6000Time (years) h = 001
(e)
Euler-PC
X 5427Y 7025
Endemic equilibrium
Infe
cted
frac
tion
times105
0
1
2
3
4
5
6
7
100 200 300 400 500 6000Time (years) h = 001
(f )
Figure 7 Time plots of population using the Euler-PC method with h 001
16 Journal of Mathematics
7 Conclusions
In this paper a normalized NSFDPC for the SEIR modelconcerning the transmission dynamics of measles is con-structed and analysed -is proposed numerical scheme isvery competitive It is qualitatively stable that is it doublerefines the solution and gives realistic results even for largestep sizes It is dynamically consistent with a continuoussystem and unconditionally convergent and satisfies the
positivity of the state variables involved in the systemSimulations are carried out and its usefulness is comparedwith a well-known numerical method of standard differenceschemes such as RK4 and Euler predictor-corrector method-e standard finite difference schemes are highly dependenton step sizes and initial value problems but NSFDPC isindependent of these two features which make it morepractical Also this method saves computation time andmemory
Euler-PC scheme (disease-free equilibrium)times1041
ndash3
ndash2
ndash1
0
1
2
3
Popu
latio
n fra
ctio
ns
01 02 03 04 05 06 07 08 09 10Time (years) h = 01
SusceptibleExposedInfected
(a)
Euler-PC scheme (endemic equilibrium)times1053
ndash6
ndash4
ndash2
0
2
4
6
Popu
latio
n fra
ctio
ns
01 02 03 04 05 06 07 08 09 10Time (years) h = 01
SusceptibleExposedInfected
(b)
Figure 8 Divergence graphs of the population using the Euler-PC method with step size h 01
NSFD-PCRK-4
times1092
Infe
cted
frac
tion
ndash5
ndash4
ndash3
ndash2
ndash1
0
1
02 04 06 08 1 12 14 16 18 20Time (years) h = 01
(a)
NSFD-PCEuler-PC
times10213
Infe
cted
frac
tion
ndash4
ndash35
ndash3
ndash25
ndash2
ndash15
ndash1
ndash05
0
05
02 04 06 08 1 12 14 16 18 20Time (years) h = 01
(b)
Figure 9 Comparative convergence graphs of the NSFDPC method
Journal of Mathematics 17
Data Availability
-e data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
-e authors declare no conflicts of interest
Acknowledgments
-e authors extend their appreciation to the Deanship ofScientific Research at Majmaah University for funding thiswork under Project no RGP-2019-6
References
[1] E Hunter B Namee and J Kelleher ldquoAn open-data-drivenagent-based model to simulate infectious disease outbreaksrdquoPLoS One vol 14 no 1 pp 1ndash35 2018
[2] E A Bakare Y A Adekunle and K O Kadiri ldquoModeling andsimulation of the dynamics of the transmission of measlesrdquoInternational Journal of Computer Trends and Trends andTechnology vol 3 no 1 pp 174ndash177 2012
[3] L Zhou Y Wang Y Li and Y Michael ldquoGlobal dynamics ofa discrete age-structured SIR epidemic model with applica-tions to measles vaccination strategiesrdquo Mathematical Bio-sciences vol 308 pp 27ndash37 2019
[4] P Widyaningsih D R S Saputro and A W NugroholdquoSusceptible exposed infected recovery (SEIR) model withimmigration equilibria points and its applicationrdquo AmericanInstitute of Physics vol 1 pp 1ndash7 2018
[5] M G Roberts and J A P HeesterbeekMathematical Modelsin Epidemiology Mathematical Models UNESCO-EOLSSAbu Dhabi UAE 2007
[6] J H He F Y Ji and H M Sedighi ldquoDifference equation vsdifferential equation on different scalesrdquo International Journalof Numerical Methods for Heat amp Fluid Flow vol 18pp 0961ndash5539 2020
[7] R E Mickens Nonstandard Finite Difference Models of Dif-ferential Equations World Scientific Singapore 1994
[8] R E Mickens Applications of Nonstandard Finite DifferenceSchemes World Scientific Singapore 2000
[9] R E Mickens ldquoNumerical integration of population modelssatisfying conservation laws NSFD methodsrdquo Journal ofBiological Dynamics vol 1 no 4 pp 427ndash436 2007
[10] J D Lambert Computational Methods in Ordinary Differ-ential Equations Wiley New York NY USA 1973
[11] S M Moghadas M E Alexander B D Corbett andA B Gumel ldquoA positivity-preserving mickens-type dis-cretization of an epidemic modelrdquo Journal of DifferenceEquations and Applications vol 9 no 11 pp 1037ndash10512003
[12] R E Mickens ldquoNonstandard finite difference schemes fordifferential equationsrdquo Journal of Difference Equations andApplications vol 8 no 9 pp 823ndash847 2002
[13] R E Mickens ldquoDynamic consistency a fundamental prin-ciple for constructing nonstandard finite difference schemesfor differential equationsrdquo Journal of Difference Equations andApplications vol 11 no 7 pp 645ndash653 2005
[14] L-I W Roeger and R W Barnard ldquoPreservation of localdynamics when applying central difference methods appli-cation to SIR modelrdquo Journal of Difference Equations andApplications vol 13 no 4 pp 333ndash340 2007
[15] K C Patidar ldquoOn the use of non-standard finite differencemethodsrdquo Journal of Difference Equations and Applicationsvol 11 no 7 pp 735ndash758 2005
[16] R Anguelov and J M-S Lubuma ldquoNonstandard finite dif-ference method by nonlocal approximationrdquo Mathematicsand Computers in Simulation vol 61 no 3-6 pp 465ndash4752003
[17] R Anguelov and J M-S Lubuma ldquoContributions to themathematics of the nonstandard finite difference method andapplicationsrdquo Numerical Methods for Partial DifferentialEquations vol 17 no 5 pp 518ndash543 2001
[18] B M Chen-Charpentier D T Dimitrov andH V Kojouharov ldquoNumerical simulation of multi-speciesbiofilms in porous media for different kineticsrdquo Mathematicsand Computers in Simulation vol 79 no 6 pp 1846ndash18612009
[19] D T Dimitrov and H V Kojouharov ldquoNonstandard finite-difference schemes for general two-dimensional autonomousdynamical systemsrdquo Applied Mathematics Letters vol 18no 7 pp 769ndash774 2005
[20] A J Arenas J A Morantildeo and J C Cortes ldquoNon-standardnumerical method for a mathematical model of RSV epide-miological transmissionrdquo Computers amp Mathematics withApplications vol 56 no 3 pp 670ndash678 2008
[21] L Jodar R J Villanueva A J Arenas and G C GonzalezldquoNonstandard numerical methods for a mathematical modelfor influenza diseaserdquo Mathematics and Computers in Sim-ulation vol 79 no 3 pp 622ndash633 2008
[22] W Piyawong E H Twizell and A B Gumel ldquoAn uncon-ditionally convergent finite-difference scheme for the SIRmodelrdquo Applied Mathematics and Computation vol 146no 2-3 pp 611ndash625 2003
[23] H Jansen and E H Twizell ldquoAn unconditionally convergentdiscretization of the SEIR modelrdquo Mathematics and Com-puters in Simulation vol 58 no 2 pp 147ndash158 2002
[24] A J Arenas G Gonzalez-Parra and B M Chen-CharpentierldquoA nonstandard numerical scheme of predictor-correctortype for epidemic modelsrdquo Computers amp Mathematics withApplications vol 59 no 12 pp 3740ndash3749 2010
[25] F Brauer and C Castillo-Chavez Mathematical Models inPopulation Biology and Epidemiology Springer Berlin Ger-many 2001
18 Journal of Mathematics
Table 2 Parameters value of the measles model
Parameters N μ σ c
5 times 107yearminus 1 002 456 yearminus 1 73 yearminus 1
DFE (S0 E0 I0) (N 0 0) β 01 times 10minus 5
EE (Se Ee Ie) (2435 times 107 1125 times 104 7022671) β 03 times 10minus 5
Disease-free equilibrium
RK4
times107
1
15
2
25
3
35
4
45
5
Susc
eptib
le fr
actio
n
20 40 60 80 100 120 140 160 180 2000Time (years) h = 001
(a)
Expo
sed
fract
ion
Disease-free equilibrium
RK4
times104
0
05
1
15
2
25
3
35
4
45
5
02 04 06 08 1 12 14 16 18 20Time (years) h = 001
(b)
times104
Infe
cted
frac
tion
Disease-free equilibrium
RK4
0
05
1
15
2
25
3
02 04 06 08 1 12 14 16 18 20Time (years) h = 001
(c)
times107
X 5578Y 2435e + 007
Endemic equilibrium
RK4
1
15
2
25
3
35
Susc
eptib
le F
ract
ion
100 200 300 400 500 6000Time (years) h = 001
(d)
Figure 5 Continued
14 Journal of Mathematics
Figure 6 shows overflow for a critical value of step size ieh 01 of DFE and EE respectively
Another popular numerical method is the Eulerpredictor-corrector (E-PC) technique which employs theexplicit Euler method as a predictor and the trapezoidalrule as the corrector -is combination is second-orderaccurate but has similar stability properties in PECEmode to the explicit Euler method alone Figures 7(a)ndash7(f ) show convergence results for h 001 in both casesthat are DFE and EE -e value of h 01 which produces
overflow when solving the system with the parametervalues of Table 2 for the PECE combination can be seen inFigure 8
-us the presented numerical results demonstrated thatNSFDPC has better convergence property following theEuler PC and RK-4 as shown in Figure 9 It is proved thatthe approximations made by other standard numericalmethods experience difficulties in preserving either thestability or the positivity of the solutions or both butNSFDPC is unconditionally convergent
times105
X 5772Y 1125e + 004
Endemic equilibrium
RK4
Expo
sed
fract
ion
0
1
2
3
4
5
6
7
8
9
10
100 200 300 400 500 6000Time (years) h = 001
(e)
times105
X 7659Y 7022
RK4
Endemic equilibrium
Infe
cted
frac
tion
0
1
2
3
4
5
6
100 200 300 400 500 600 700 8000Time (years) h = 001
(f )
Figure 5 Time plots of susceptible exposed and infected population using the RK-4 method with h 001
times1087 RK-4 scheme (disease-free equilibrium)
SusceptibleExposedInfected
01 02 03 04 05 06 07 08 09 10Time (years) h = 01
ndash2
ndash15
ndash1
ndash05
0
05
1
15
2
Popu
latio
n fra
ctio
ns
(a)
RK-4 scheme (endemic equilibrium)
SusceptibleExposedInfected
times10145
01 02 03 04 05 06 07 08 09 10Time (years) h = 01
ndash8
ndash6
ndash4
ndash2
0
2
4
6
8
Popu
latio
n fra
ctio
ns
(b)
Figure 6 Divergence graphs of the population using the RK-4 method with step size h 01
Journal of Mathematics 15
Euler-PC
Disease-free equilibriumtimes107
1
15
2
25
3
35
4
45
5Su
scep
tible
frac
tion
20 40 60 80 100 120 140 160 180 2000Time (years) h = 001
(a)
Euler-PC
Disease-free equilibrium
Expo
sed
fract
ion
times104
0
05
1
15
2
25
3
35
4
45
5
02 04 06 08 1 12 14 16 18 20Time (years) h = 001
(b)
Euler-PC
Disease-free equilibrium
Infe
cted
frac
tion
times104
0
05
1
15
2
25
3
35
02 04 06 08 1 12 14 16 18 20Time (years) h = 001
(c)
Euler-PC
X 5733Y 2435e + 007
Endemic equilibriumtimes107
1
15
2
25
3
35Su
scep
tible
frac
tion
100 200 300 400 500 6000Time (years) h = 001
(d)
Euler-PC
X 5662Y 1118e + 004
Endemic equilibrium
Expo
sed
fract
ion
times105
0
2
4
6
8
10
12
100 200 300 400 500 6000Time (years) h = 001
(e)
Euler-PC
X 5427Y 7025
Endemic equilibrium
Infe
cted
frac
tion
times105
0
1
2
3
4
5
6
7
100 200 300 400 500 6000Time (years) h = 001
(f )
Figure 7 Time plots of population using the Euler-PC method with h 001
16 Journal of Mathematics
7 Conclusions
In this paper a normalized NSFDPC for the SEIR modelconcerning the transmission dynamics of measles is con-structed and analysed -is proposed numerical scheme isvery competitive It is qualitatively stable that is it doublerefines the solution and gives realistic results even for largestep sizes It is dynamically consistent with a continuoussystem and unconditionally convergent and satisfies the
positivity of the state variables involved in the systemSimulations are carried out and its usefulness is comparedwith a well-known numerical method of standard differenceschemes such as RK4 and Euler predictor-corrector method-e standard finite difference schemes are highly dependenton step sizes and initial value problems but NSFDPC isindependent of these two features which make it morepractical Also this method saves computation time andmemory
Euler-PC scheme (disease-free equilibrium)times1041
ndash3
ndash2
ndash1
0
1
2
3
Popu
latio
n fra
ctio
ns
01 02 03 04 05 06 07 08 09 10Time (years) h = 01
SusceptibleExposedInfected
(a)
Euler-PC scheme (endemic equilibrium)times1053
ndash6
ndash4
ndash2
0
2
4
6
Popu
latio
n fra
ctio
ns
01 02 03 04 05 06 07 08 09 10Time (years) h = 01
SusceptibleExposedInfected
(b)
Figure 8 Divergence graphs of the population using the Euler-PC method with step size h 01
NSFD-PCRK-4
times1092
Infe
cted
frac
tion
ndash5
ndash4
ndash3
ndash2
ndash1
0
1
02 04 06 08 1 12 14 16 18 20Time (years) h = 01
(a)
NSFD-PCEuler-PC
times10213
Infe
cted
frac
tion
ndash4
ndash35
ndash3
ndash25
ndash2
ndash15
ndash1
ndash05
0
05
02 04 06 08 1 12 14 16 18 20Time (years) h = 01
(b)
Figure 9 Comparative convergence graphs of the NSFDPC method
Journal of Mathematics 17
Data Availability
-e data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
-e authors declare no conflicts of interest
Acknowledgments
-e authors extend their appreciation to the Deanship ofScientific Research at Majmaah University for funding thiswork under Project no RGP-2019-6
References
[1] E Hunter B Namee and J Kelleher ldquoAn open-data-drivenagent-based model to simulate infectious disease outbreaksrdquoPLoS One vol 14 no 1 pp 1ndash35 2018
[2] E A Bakare Y A Adekunle and K O Kadiri ldquoModeling andsimulation of the dynamics of the transmission of measlesrdquoInternational Journal of Computer Trends and Trends andTechnology vol 3 no 1 pp 174ndash177 2012
[3] L Zhou Y Wang Y Li and Y Michael ldquoGlobal dynamics ofa discrete age-structured SIR epidemic model with applica-tions to measles vaccination strategiesrdquo Mathematical Bio-sciences vol 308 pp 27ndash37 2019
[4] P Widyaningsih D R S Saputro and A W NugroholdquoSusceptible exposed infected recovery (SEIR) model withimmigration equilibria points and its applicationrdquo AmericanInstitute of Physics vol 1 pp 1ndash7 2018
[5] M G Roberts and J A P HeesterbeekMathematical Modelsin Epidemiology Mathematical Models UNESCO-EOLSSAbu Dhabi UAE 2007
[6] J H He F Y Ji and H M Sedighi ldquoDifference equation vsdifferential equation on different scalesrdquo International Journalof Numerical Methods for Heat amp Fluid Flow vol 18pp 0961ndash5539 2020
[7] R E Mickens Nonstandard Finite Difference Models of Dif-ferential Equations World Scientific Singapore 1994
[8] R E Mickens Applications of Nonstandard Finite DifferenceSchemes World Scientific Singapore 2000
[9] R E Mickens ldquoNumerical integration of population modelssatisfying conservation laws NSFD methodsrdquo Journal ofBiological Dynamics vol 1 no 4 pp 427ndash436 2007
[10] J D Lambert Computational Methods in Ordinary Differ-ential Equations Wiley New York NY USA 1973
[11] S M Moghadas M E Alexander B D Corbett andA B Gumel ldquoA positivity-preserving mickens-type dis-cretization of an epidemic modelrdquo Journal of DifferenceEquations and Applications vol 9 no 11 pp 1037ndash10512003
[12] R E Mickens ldquoNonstandard finite difference schemes fordifferential equationsrdquo Journal of Difference Equations andApplications vol 8 no 9 pp 823ndash847 2002
[13] R E Mickens ldquoDynamic consistency a fundamental prin-ciple for constructing nonstandard finite difference schemesfor differential equationsrdquo Journal of Difference Equations andApplications vol 11 no 7 pp 645ndash653 2005
[14] L-I W Roeger and R W Barnard ldquoPreservation of localdynamics when applying central difference methods appli-cation to SIR modelrdquo Journal of Difference Equations andApplications vol 13 no 4 pp 333ndash340 2007
[15] K C Patidar ldquoOn the use of non-standard finite differencemethodsrdquo Journal of Difference Equations and Applicationsvol 11 no 7 pp 735ndash758 2005
[16] R Anguelov and J M-S Lubuma ldquoNonstandard finite dif-ference method by nonlocal approximationrdquo Mathematicsand Computers in Simulation vol 61 no 3-6 pp 465ndash4752003
[17] R Anguelov and J M-S Lubuma ldquoContributions to themathematics of the nonstandard finite difference method andapplicationsrdquo Numerical Methods for Partial DifferentialEquations vol 17 no 5 pp 518ndash543 2001
[18] B M Chen-Charpentier D T Dimitrov andH V Kojouharov ldquoNumerical simulation of multi-speciesbiofilms in porous media for different kineticsrdquo Mathematicsand Computers in Simulation vol 79 no 6 pp 1846ndash18612009
[19] D T Dimitrov and H V Kojouharov ldquoNonstandard finite-difference schemes for general two-dimensional autonomousdynamical systemsrdquo Applied Mathematics Letters vol 18no 7 pp 769ndash774 2005
[20] A J Arenas J A Morantildeo and J C Cortes ldquoNon-standardnumerical method for a mathematical model of RSV epide-miological transmissionrdquo Computers amp Mathematics withApplications vol 56 no 3 pp 670ndash678 2008
[21] L Jodar R J Villanueva A J Arenas and G C GonzalezldquoNonstandard numerical methods for a mathematical modelfor influenza diseaserdquo Mathematics and Computers in Sim-ulation vol 79 no 3 pp 622ndash633 2008
[22] W Piyawong E H Twizell and A B Gumel ldquoAn uncon-ditionally convergent finite-difference scheme for the SIRmodelrdquo Applied Mathematics and Computation vol 146no 2-3 pp 611ndash625 2003
[23] H Jansen and E H Twizell ldquoAn unconditionally convergentdiscretization of the SEIR modelrdquo Mathematics and Com-puters in Simulation vol 58 no 2 pp 147ndash158 2002
[24] A J Arenas G Gonzalez-Parra and B M Chen-CharpentierldquoA nonstandard numerical scheme of predictor-correctortype for epidemic modelsrdquo Computers amp Mathematics withApplications vol 59 no 12 pp 3740ndash3749 2010
[25] F Brauer and C Castillo-Chavez Mathematical Models inPopulation Biology and Epidemiology Springer Berlin Ger-many 2001
18 Journal of Mathematics
Figure 6 shows overflow for a critical value of step size ieh 01 of DFE and EE respectively
Another popular numerical method is the Eulerpredictor-corrector (E-PC) technique which employs theexplicit Euler method as a predictor and the trapezoidalrule as the corrector -is combination is second-orderaccurate but has similar stability properties in PECEmode to the explicit Euler method alone Figures 7(a)ndash7(f ) show convergence results for h 001 in both casesthat are DFE and EE -e value of h 01 which produces
overflow when solving the system with the parametervalues of Table 2 for the PECE combination can be seen inFigure 8
-us the presented numerical results demonstrated thatNSFDPC has better convergence property following theEuler PC and RK-4 as shown in Figure 9 It is proved thatthe approximations made by other standard numericalmethods experience difficulties in preserving either thestability or the positivity of the solutions or both butNSFDPC is unconditionally convergent
times105
X 5772Y 1125e + 004
Endemic equilibrium
RK4
Expo
sed
fract
ion
0
1
2
3
4
5
6
7
8
9
10
100 200 300 400 500 6000Time (years) h = 001
(e)
times105
X 7659Y 7022
RK4
Endemic equilibrium
Infe
cted
frac
tion
0
1
2
3
4
5
6
100 200 300 400 500 600 700 8000Time (years) h = 001
(f )
Figure 5 Time plots of susceptible exposed and infected population using the RK-4 method with h 001
times1087 RK-4 scheme (disease-free equilibrium)
SusceptibleExposedInfected
01 02 03 04 05 06 07 08 09 10Time (years) h = 01
ndash2
ndash15
ndash1
ndash05
0
05
1
15
2
Popu
latio
n fra
ctio
ns
(a)
RK-4 scheme (endemic equilibrium)
SusceptibleExposedInfected
times10145
01 02 03 04 05 06 07 08 09 10Time (years) h = 01
ndash8
ndash6
ndash4
ndash2
0
2
4
6
8
Popu
latio
n fra
ctio
ns
(b)
Figure 6 Divergence graphs of the population using the RK-4 method with step size h 01
Journal of Mathematics 15
Euler-PC
Disease-free equilibriumtimes107
1
15
2
25
3
35
4
45
5Su
scep
tible
frac
tion
20 40 60 80 100 120 140 160 180 2000Time (years) h = 001
(a)
Euler-PC
Disease-free equilibrium
Expo
sed
fract
ion
times104
0
05
1
15
2
25
3
35
4
45
5
02 04 06 08 1 12 14 16 18 20Time (years) h = 001
(b)
Euler-PC
Disease-free equilibrium
Infe
cted
frac
tion
times104
0
05
1
15
2
25
3
35
02 04 06 08 1 12 14 16 18 20Time (years) h = 001
(c)
Euler-PC
X 5733Y 2435e + 007
Endemic equilibriumtimes107
1
15
2
25
3
35Su
scep
tible
frac
tion
100 200 300 400 500 6000Time (years) h = 001
(d)
Euler-PC
X 5662Y 1118e + 004
Endemic equilibrium
Expo
sed
fract
ion
times105
0
2
4
6
8
10
12
100 200 300 400 500 6000Time (years) h = 001
(e)
Euler-PC
X 5427Y 7025
Endemic equilibrium
Infe
cted
frac
tion
times105
0
1
2
3
4
5
6
7
100 200 300 400 500 6000Time (years) h = 001
(f )
Figure 7 Time plots of population using the Euler-PC method with h 001
16 Journal of Mathematics
7 Conclusions
In this paper a normalized NSFDPC for the SEIR modelconcerning the transmission dynamics of measles is con-structed and analysed -is proposed numerical scheme isvery competitive It is qualitatively stable that is it doublerefines the solution and gives realistic results even for largestep sizes It is dynamically consistent with a continuoussystem and unconditionally convergent and satisfies the
positivity of the state variables involved in the systemSimulations are carried out and its usefulness is comparedwith a well-known numerical method of standard differenceschemes such as RK4 and Euler predictor-corrector method-e standard finite difference schemes are highly dependenton step sizes and initial value problems but NSFDPC isindependent of these two features which make it morepractical Also this method saves computation time andmemory
Euler-PC scheme (disease-free equilibrium)times1041
ndash3
ndash2
ndash1
0
1
2
3
Popu
latio
n fra
ctio
ns
01 02 03 04 05 06 07 08 09 10Time (years) h = 01
SusceptibleExposedInfected
(a)
Euler-PC scheme (endemic equilibrium)times1053
ndash6
ndash4
ndash2
0
2
4
6
Popu
latio
n fra
ctio
ns
01 02 03 04 05 06 07 08 09 10Time (years) h = 01
SusceptibleExposedInfected
(b)
Figure 8 Divergence graphs of the population using the Euler-PC method with step size h 01
NSFD-PCRK-4
times1092
Infe
cted
frac
tion
ndash5
ndash4
ndash3
ndash2
ndash1
0
1
02 04 06 08 1 12 14 16 18 20Time (years) h = 01
(a)
NSFD-PCEuler-PC
times10213
Infe
cted
frac
tion
ndash4
ndash35
ndash3
ndash25
ndash2
ndash15
ndash1
ndash05
0
05
02 04 06 08 1 12 14 16 18 20Time (years) h = 01
(b)
Figure 9 Comparative convergence graphs of the NSFDPC method
Journal of Mathematics 17
Data Availability
-e data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
-e authors declare no conflicts of interest
Acknowledgments
-e authors extend their appreciation to the Deanship ofScientific Research at Majmaah University for funding thiswork under Project no RGP-2019-6
References
[1] E Hunter B Namee and J Kelleher ldquoAn open-data-drivenagent-based model to simulate infectious disease outbreaksrdquoPLoS One vol 14 no 1 pp 1ndash35 2018
[2] E A Bakare Y A Adekunle and K O Kadiri ldquoModeling andsimulation of the dynamics of the transmission of measlesrdquoInternational Journal of Computer Trends and Trends andTechnology vol 3 no 1 pp 174ndash177 2012
[3] L Zhou Y Wang Y Li and Y Michael ldquoGlobal dynamics ofa discrete age-structured SIR epidemic model with applica-tions to measles vaccination strategiesrdquo Mathematical Bio-sciences vol 308 pp 27ndash37 2019
[4] P Widyaningsih D R S Saputro and A W NugroholdquoSusceptible exposed infected recovery (SEIR) model withimmigration equilibria points and its applicationrdquo AmericanInstitute of Physics vol 1 pp 1ndash7 2018
[5] M G Roberts and J A P HeesterbeekMathematical Modelsin Epidemiology Mathematical Models UNESCO-EOLSSAbu Dhabi UAE 2007
[6] J H He F Y Ji and H M Sedighi ldquoDifference equation vsdifferential equation on different scalesrdquo International Journalof Numerical Methods for Heat amp Fluid Flow vol 18pp 0961ndash5539 2020
[7] R E Mickens Nonstandard Finite Difference Models of Dif-ferential Equations World Scientific Singapore 1994
[8] R E Mickens Applications of Nonstandard Finite DifferenceSchemes World Scientific Singapore 2000
[9] R E Mickens ldquoNumerical integration of population modelssatisfying conservation laws NSFD methodsrdquo Journal ofBiological Dynamics vol 1 no 4 pp 427ndash436 2007
[10] J D Lambert Computational Methods in Ordinary Differ-ential Equations Wiley New York NY USA 1973
[11] S M Moghadas M E Alexander B D Corbett andA B Gumel ldquoA positivity-preserving mickens-type dis-cretization of an epidemic modelrdquo Journal of DifferenceEquations and Applications vol 9 no 11 pp 1037ndash10512003
[12] R E Mickens ldquoNonstandard finite difference schemes fordifferential equationsrdquo Journal of Difference Equations andApplications vol 8 no 9 pp 823ndash847 2002
[13] R E Mickens ldquoDynamic consistency a fundamental prin-ciple for constructing nonstandard finite difference schemesfor differential equationsrdquo Journal of Difference Equations andApplications vol 11 no 7 pp 645ndash653 2005
[14] L-I W Roeger and R W Barnard ldquoPreservation of localdynamics when applying central difference methods appli-cation to SIR modelrdquo Journal of Difference Equations andApplications vol 13 no 4 pp 333ndash340 2007
[15] K C Patidar ldquoOn the use of non-standard finite differencemethodsrdquo Journal of Difference Equations and Applicationsvol 11 no 7 pp 735ndash758 2005
[16] R Anguelov and J M-S Lubuma ldquoNonstandard finite dif-ference method by nonlocal approximationrdquo Mathematicsand Computers in Simulation vol 61 no 3-6 pp 465ndash4752003
[17] R Anguelov and J M-S Lubuma ldquoContributions to themathematics of the nonstandard finite difference method andapplicationsrdquo Numerical Methods for Partial DifferentialEquations vol 17 no 5 pp 518ndash543 2001
[18] B M Chen-Charpentier D T Dimitrov andH V Kojouharov ldquoNumerical simulation of multi-speciesbiofilms in porous media for different kineticsrdquo Mathematicsand Computers in Simulation vol 79 no 6 pp 1846ndash18612009
[19] D T Dimitrov and H V Kojouharov ldquoNonstandard finite-difference schemes for general two-dimensional autonomousdynamical systemsrdquo Applied Mathematics Letters vol 18no 7 pp 769ndash774 2005
[20] A J Arenas J A Morantildeo and J C Cortes ldquoNon-standardnumerical method for a mathematical model of RSV epide-miological transmissionrdquo Computers amp Mathematics withApplications vol 56 no 3 pp 670ndash678 2008
[21] L Jodar R J Villanueva A J Arenas and G C GonzalezldquoNonstandard numerical methods for a mathematical modelfor influenza diseaserdquo Mathematics and Computers in Sim-ulation vol 79 no 3 pp 622ndash633 2008
[22] W Piyawong E H Twizell and A B Gumel ldquoAn uncon-ditionally convergent finite-difference scheme for the SIRmodelrdquo Applied Mathematics and Computation vol 146no 2-3 pp 611ndash625 2003
[23] H Jansen and E H Twizell ldquoAn unconditionally convergentdiscretization of the SEIR modelrdquo Mathematics and Com-puters in Simulation vol 58 no 2 pp 147ndash158 2002
[24] A J Arenas G Gonzalez-Parra and B M Chen-CharpentierldquoA nonstandard numerical scheme of predictor-correctortype for epidemic modelsrdquo Computers amp Mathematics withApplications vol 59 no 12 pp 3740ndash3749 2010
[25] F Brauer and C Castillo-Chavez Mathematical Models inPopulation Biology and Epidemiology Springer Berlin Ger-many 2001
18 Journal of Mathematics
Euler-PC
Disease-free equilibriumtimes107
1
15
2
25
3
35
4
45
5Su
scep
tible
frac
tion
20 40 60 80 100 120 140 160 180 2000Time (years) h = 001
(a)
Euler-PC
Disease-free equilibrium
Expo
sed
fract
ion
times104
0
05
1
15
2
25
3
35
4
45
5
02 04 06 08 1 12 14 16 18 20Time (years) h = 001
(b)
Euler-PC
Disease-free equilibrium
Infe
cted
frac
tion
times104
0
05
1
15
2
25
3
35
02 04 06 08 1 12 14 16 18 20Time (years) h = 001
(c)
Euler-PC
X 5733Y 2435e + 007
Endemic equilibriumtimes107
1
15
2
25
3
35Su
scep
tible
frac
tion
100 200 300 400 500 6000Time (years) h = 001
(d)
Euler-PC
X 5662Y 1118e + 004
Endemic equilibrium
Expo
sed
fract
ion
times105
0
2
4
6
8
10
12
100 200 300 400 500 6000Time (years) h = 001
(e)
Euler-PC
X 5427Y 7025
Endemic equilibrium
Infe
cted
frac
tion
times105
0
1
2
3
4
5
6
7
100 200 300 400 500 6000Time (years) h = 001
(f )
Figure 7 Time plots of population using the Euler-PC method with h 001
16 Journal of Mathematics
7 Conclusions
In this paper a normalized NSFDPC for the SEIR modelconcerning the transmission dynamics of measles is con-structed and analysed -is proposed numerical scheme isvery competitive It is qualitatively stable that is it doublerefines the solution and gives realistic results even for largestep sizes It is dynamically consistent with a continuoussystem and unconditionally convergent and satisfies the
positivity of the state variables involved in the systemSimulations are carried out and its usefulness is comparedwith a well-known numerical method of standard differenceschemes such as RK4 and Euler predictor-corrector method-e standard finite difference schemes are highly dependenton step sizes and initial value problems but NSFDPC isindependent of these two features which make it morepractical Also this method saves computation time andmemory
Euler-PC scheme (disease-free equilibrium)times1041
ndash3
ndash2
ndash1
0
1
2
3
Popu
latio
n fra
ctio
ns
01 02 03 04 05 06 07 08 09 10Time (years) h = 01
SusceptibleExposedInfected
(a)
Euler-PC scheme (endemic equilibrium)times1053
ndash6
ndash4
ndash2
0
2
4
6
Popu
latio
n fra
ctio
ns
01 02 03 04 05 06 07 08 09 10Time (years) h = 01
SusceptibleExposedInfected
(b)
Figure 8 Divergence graphs of the population using the Euler-PC method with step size h 01
NSFD-PCRK-4
times1092
Infe
cted
frac
tion
ndash5
ndash4
ndash3
ndash2
ndash1
0
1
02 04 06 08 1 12 14 16 18 20Time (years) h = 01
(a)
NSFD-PCEuler-PC
times10213
Infe
cted
frac
tion
ndash4
ndash35
ndash3
ndash25
ndash2
ndash15
ndash1
ndash05
0
05
02 04 06 08 1 12 14 16 18 20Time (years) h = 01
(b)
Figure 9 Comparative convergence graphs of the NSFDPC method
Journal of Mathematics 17
Data Availability
-e data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
-e authors declare no conflicts of interest
Acknowledgments
-e authors extend their appreciation to the Deanship ofScientific Research at Majmaah University for funding thiswork under Project no RGP-2019-6
References
[1] E Hunter B Namee and J Kelleher ldquoAn open-data-drivenagent-based model to simulate infectious disease outbreaksrdquoPLoS One vol 14 no 1 pp 1ndash35 2018
[2] E A Bakare Y A Adekunle and K O Kadiri ldquoModeling andsimulation of the dynamics of the transmission of measlesrdquoInternational Journal of Computer Trends and Trends andTechnology vol 3 no 1 pp 174ndash177 2012
[3] L Zhou Y Wang Y Li and Y Michael ldquoGlobal dynamics ofa discrete age-structured SIR epidemic model with applica-tions to measles vaccination strategiesrdquo Mathematical Bio-sciences vol 308 pp 27ndash37 2019
[4] P Widyaningsih D R S Saputro and A W NugroholdquoSusceptible exposed infected recovery (SEIR) model withimmigration equilibria points and its applicationrdquo AmericanInstitute of Physics vol 1 pp 1ndash7 2018
[5] M G Roberts and J A P HeesterbeekMathematical Modelsin Epidemiology Mathematical Models UNESCO-EOLSSAbu Dhabi UAE 2007
[6] J H He F Y Ji and H M Sedighi ldquoDifference equation vsdifferential equation on different scalesrdquo International Journalof Numerical Methods for Heat amp Fluid Flow vol 18pp 0961ndash5539 2020
[7] R E Mickens Nonstandard Finite Difference Models of Dif-ferential Equations World Scientific Singapore 1994
[8] R E Mickens Applications of Nonstandard Finite DifferenceSchemes World Scientific Singapore 2000
[9] R E Mickens ldquoNumerical integration of population modelssatisfying conservation laws NSFD methodsrdquo Journal ofBiological Dynamics vol 1 no 4 pp 427ndash436 2007
[10] J D Lambert Computational Methods in Ordinary Differ-ential Equations Wiley New York NY USA 1973
[11] S M Moghadas M E Alexander B D Corbett andA B Gumel ldquoA positivity-preserving mickens-type dis-cretization of an epidemic modelrdquo Journal of DifferenceEquations and Applications vol 9 no 11 pp 1037ndash10512003
[12] R E Mickens ldquoNonstandard finite difference schemes fordifferential equationsrdquo Journal of Difference Equations andApplications vol 8 no 9 pp 823ndash847 2002
[13] R E Mickens ldquoDynamic consistency a fundamental prin-ciple for constructing nonstandard finite difference schemesfor differential equationsrdquo Journal of Difference Equations andApplications vol 11 no 7 pp 645ndash653 2005
[14] L-I W Roeger and R W Barnard ldquoPreservation of localdynamics when applying central difference methods appli-cation to SIR modelrdquo Journal of Difference Equations andApplications vol 13 no 4 pp 333ndash340 2007
[15] K C Patidar ldquoOn the use of non-standard finite differencemethodsrdquo Journal of Difference Equations and Applicationsvol 11 no 7 pp 735ndash758 2005
[16] R Anguelov and J M-S Lubuma ldquoNonstandard finite dif-ference method by nonlocal approximationrdquo Mathematicsand Computers in Simulation vol 61 no 3-6 pp 465ndash4752003
[17] R Anguelov and J M-S Lubuma ldquoContributions to themathematics of the nonstandard finite difference method andapplicationsrdquo Numerical Methods for Partial DifferentialEquations vol 17 no 5 pp 518ndash543 2001
[18] B M Chen-Charpentier D T Dimitrov andH V Kojouharov ldquoNumerical simulation of multi-speciesbiofilms in porous media for different kineticsrdquo Mathematicsand Computers in Simulation vol 79 no 6 pp 1846ndash18612009
[19] D T Dimitrov and H V Kojouharov ldquoNonstandard finite-difference schemes for general two-dimensional autonomousdynamical systemsrdquo Applied Mathematics Letters vol 18no 7 pp 769ndash774 2005
[20] A J Arenas J A Morantildeo and J C Cortes ldquoNon-standardnumerical method for a mathematical model of RSV epide-miological transmissionrdquo Computers amp Mathematics withApplications vol 56 no 3 pp 670ndash678 2008
[21] L Jodar R J Villanueva A J Arenas and G C GonzalezldquoNonstandard numerical methods for a mathematical modelfor influenza diseaserdquo Mathematics and Computers in Sim-ulation vol 79 no 3 pp 622ndash633 2008
[22] W Piyawong E H Twizell and A B Gumel ldquoAn uncon-ditionally convergent finite-difference scheme for the SIRmodelrdquo Applied Mathematics and Computation vol 146no 2-3 pp 611ndash625 2003
[23] H Jansen and E H Twizell ldquoAn unconditionally convergentdiscretization of the SEIR modelrdquo Mathematics and Com-puters in Simulation vol 58 no 2 pp 147ndash158 2002
[24] A J Arenas G Gonzalez-Parra and B M Chen-CharpentierldquoA nonstandard numerical scheme of predictor-correctortype for epidemic modelsrdquo Computers amp Mathematics withApplications vol 59 no 12 pp 3740ndash3749 2010
[25] F Brauer and C Castillo-Chavez Mathematical Models inPopulation Biology and Epidemiology Springer Berlin Ger-many 2001
18 Journal of Mathematics
7 Conclusions
In this paper a normalized NSFDPC for the SEIR modelconcerning the transmission dynamics of measles is con-structed and analysed -is proposed numerical scheme isvery competitive It is qualitatively stable that is it doublerefines the solution and gives realistic results even for largestep sizes It is dynamically consistent with a continuoussystem and unconditionally convergent and satisfies the
positivity of the state variables involved in the systemSimulations are carried out and its usefulness is comparedwith a well-known numerical method of standard differenceschemes such as RK4 and Euler predictor-corrector method-e standard finite difference schemes are highly dependenton step sizes and initial value problems but NSFDPC isindependent of these two features which make it morepractical Also this method saves computation time andmemory
Euler-PC scheme (disease-free equilibrium)times1041
ndash3
ndash2
ndash1
0
1
2
3
Popu
latio
n fra
ctio
ns
01 02 03 04 05 06 07 08 09 10Time (years) h = 01
SusceptibleExposedInfected
(a)
Euler-PC scheme (endemic equilibrium)times1053
ndash6
ndash4
ndash2
0
2
4
6
Popu
latio
n fra
ctio
ns
01 02 03 04 05 06 07 08 09 10Time (years) h = 01
SusceptibleExposedInfected
(b)
Figure 8 Divergence graphs of the population using the Euler-PC method with step size h 01
NSFD-PCRK-4
times1092
Infe
cted
frac
tion
ndash5
ndash4
ndash3
ndash2
ndash1
0
1
02 04 06 08 1 12 14 16 18 20Time (years) h = 01
(a)
NSFD-PCEuler-PC
times10213
Infe
cted
frac
tion
ndash4
ndash35
ndash3
ndash25
ndash2
ndash15
ndash1
ndash05
0
05
02 04 06 08 1 12 14 16 18 20Time (years) h = 01
(b)
Figure 9 Comparative convergence graphs of the NSFDPC method
Journal of Mathematics 17
Data Availability
-e data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
-e authors declare no conflicts of interest
Acknowledgments
-e authors extend their appreciation to the Deanship ofScientific Research at Majmaah University for funding thiswork under Project no RGP-2019-6
References
[1] E Hunter B Namee and J Kelleher ldquoAn open-data-drivenagent-based model to simulate infectious disease outbreaksrdquoPLoS One vol 14 no 1 pp 1ndash35 2018
[2] E A Bakare Y A Adekunle and K O Kadiri ldquoModeling andsimulation of the dynamics of the transmission of measlesrdquoInternational Journal of Computer Trends and Trends andTechnology vol 3 no 1 pp 174ndash177 2012
[3] L Zhou Y Wang Y Li and Y Michael ldquoGlobal dynamics ofa discrete age-structured SIR epidemic model with applica-tions to measles vaccination strategiesrdquo Mathematical Bio-sciences vol 308 pp 27ndash37 2019
[4] P Widyaningsih D R S Saputro and A W NugroholdquoSusceptible exposed infected recovery (SEIR) model withimmigration equilibria points and its applicationrdquo AmericanInstitute of Physics vol 1 pp 1ndash7 2018
[5] M G Roberts and J A P HeesterbeekMathematical Modelsin Epidemiology Mathematical Models UNESCO-EOLSSAbu Dhabi UAE 2007
[6] J H He F Y Ji and H M Sedighi ldquoDifference equation vsdifferential equation on different scalesrdquo International Journalof Numerical Methods for Heat amp Fluid Flow vol 18pp 0961ndash5539 2020
[7] R E Mickens Nonstandard Finite Difference Models of Dif-ferential Equations World Scientific Singapore 1994
[8] R E Mickens Applications of Nonstandard Finite DifferenceSchemes World Scientific Singapore 2000
[9] R E Mickens ldquoNumerical integration of population modelssatisfying conservation laws NSFD methodsrdquo Journal ofBiological Dynamics vol 1 no 4 pp 427ndash436 2007
[10] J D Lambert Computational Methods in Ordinary Differ-ential Equations Wiley New York NY USA 1973
[11] S M Moghadas M E Alexander B D Corbett andA B Gumel ldquoA positivity-preserving mickens-type dis-cretization of an epidemic modelrdquo Journal of DifferenceEquations and Applications vol 9 no 11 pp 1037ndash10512003
[12] R E Mickens ldquoNonstandard finite difference schemes fordifferential equationsrdquo Journal of Difference Equations andApplications vol 8 no 9 pp 823ndash847 2002
[13] R E Mickens ldquoDynamic consistency a fundamental prin-ciple for constructing nonstandard finite difference schemesfor differential equationsrdquo Journal of Difference Equations andApplications vol 11 no 7 pp 645ndash653 2005
[14] L-I W Roeger and R W Barnard ldquoPreservation of localdynamics when applying central difference methods appli-cation to SIR modelrdquo Journal of Difference Equations andApplications vol 13 no 4 pp 333ndash340 2007
[15] K C Patidar ldquoOn the use of non-standard finite differencemethodsrdquo Journal of Difference Equations and Applicationsvol 11 no 7 pp 735ndash758 2005
[16] R Anguelov and J M-S Lubuma ldquoNonstandard finite dif-ference method by nonlocal approximationrdquo Mathematicsand Computers in Simulation vol 61 no 3-6 pp 465ndash4752003
[17] R Anguelov and J M-S Lubuma ldquoContributions to themathematics of the nonstandard finite difference method andapplicationsrdquo Numerical Methods for Partial DifferentialEquations vol 17 no 5 pp 518ndash543 2001
[18] B M Chen-Charpentier D T Dimitrov andH V Kojouharov ldquoNumerical simulation of multi-speciesbiofilms in porous media for different kineticsrdquo Mathematicsand Computers in Simulation vol 79 no 6 pp 1846ndash18612009
[19] D T Dimitrov and H V Kojouharov ldquoNonstandard finite-difference schemes for general two-dimensional autonomousdynamical systemsrdquo Applied Mathematics Letters vol 18no 7 pp 769ndash774 2005
[20] A J Arenas J A Morantildeo and J C Cortes ldquoNon-standardnumerical method for a mathematical model of RSV epide-miological transmissionrdquo Computers amp Mathematics withApplications vol 56 no 3 pp 670ndash678 2008
[21] L Jodar R J Villanueva A J Arenas and G C GonzalezldquoNonstandard numerical methods for a mathematical modelfor influenza diseaserdquo Mathematics and Computers in Sim-ulation vol 79 no 3 pp 622ndash633 2008
[22] W Piyawong E H Twizell and A B Gumel ldquoAn uncon-ditionally convergent finite-difference scheme for the SIRmodelrdquo Applied Mathematics and Computation vol 146no 2-3 pp 611ndash625 2003
[23] H Jansen and E H Twizell ldquoAn unconditionally convergentdiscretization of the SEIR modelrdquo Mathematics and Com-puters in Simulation vol 58 no 2 pp 147ndash158 2002
[24] A J Arenas G Gonzalez-Parra and B M Chen-CharpentierldquoA nonstandard numerical scheme of predictor-correctortype for epidemic modelsrdquo Computers amp Mathematics withApplications vol 59 no 12 pp 3740ndash3749 2010
[25] F Brauer and C Castillo-Chavez Mathematical Models inPopulation Biology and Epidemiology Springer Berlin Ger-many 2001
18 Journal of Mathematics
Data Availability
-e data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
-e authors declare no conflicts of interest
Acknowledgments
-e authors extend their appreciation to the Deanship ofScientific Research at Majmaah University for funding thiswork under Project no RGP-2019-6
References
[1] E Hunter B Namee and J Kelleher ldquoAn open-data-drivenagent-based model to simulate infectious disease outbreaksrdquoPLoS One vol 14 no 1 pp 1ndash35 2018
[2] E A Bakare Y A Adekunle and K O Kadiri ldquoModeling andsimulation of the dynamics of the transmission of measlesrdquoInternational Journal of Computer Trends and Trends andTechnology vol 3 no 1 pp 174ndash177 2012
[3] L Zhou Y Wang Y Li and Y Michael ldquoGlobal dynamics ofa discrete age-structured SIR epidemic model with applica-tions to measles vaccination strategiesrdquo Mathematical Bio-sciences vol 308 pp 27ndash37 2019
[4] P Widyaningsih D R S Saputro and A W NugroholdquoSusceptible exposed infected recovery (SEIR) model withimmigration equilibria points and its applicationrdquo AmericanInstitute of Physics vol 1 pp 1ndash7 2018
[5] M G Roberts and J A P HeesterbeekMathematical Modelsin Epidemiology Mathematical Models UNESCO-EOLSSAbu Dhabi UAE 2007
[6] J H He F Y Ji and H M Sedighi ldquoDifference equation vsdifferential equation on different scalesrdquo International Journalof Numerical Methods for Heat amp Fluid Flow vol 18pp 0961ndash5539 2020
[7] R E Mickens Nonstandard Finite Difference Models of Dif-ferential Equations World Scientific Singapore 1994
[8] R E Mickens Applications of Nonstandard Finite DifferenceSchemes World Scientific Singapore 2000
[9] R E Mickens ldquoNumerical integration of population modelssatisfying conservation laws NSFD methodsrdquo Journal ofBiological Dynamics vol 1 no 4 pp 427ndash436 2007
[10] J D Lambert Computational Methods in Ordinary Differ-ential Equations Wiley New York NY USA 1973
[11] S M Moghadas M E Alexander B D Corbett andA B Gumel ldquoA positivity-preserving mickens-type dis-cretization of an epidemic modelrdquo Journal of DifferenceEquations and Applications vol 9 no 11 pp 1037ndash10512003
[12] R E Mickens ldquoNonstandard finite difference schemes fordifferential equationsrdquo Journal of Difference Equations andApplications vol 8 no 9 pp 823ndash847 2002
[13] R E Mickens ldquoDynamic consistency a fundamental prin-ciple for constructing nonstandard finite difference schemesfor differential equationsrdquo Journal of Difference Equations andApplications vol 11 no 7 pp 645ndash653 2005
[14] L-I W Roeger and R W Barnard ldquoPreservation of localdynamics when applying central difference methods appli-cation to SIR modelrdquo Journal of Difference Equations andApplications vol 13 no 4 pp 333ndash340 2007
[15] K C Patidar ldquoOn the use of non-standard finite differencemethodsrdquo Journal of Difference Equations and Applicationsvol 11 no 7 pp 735ndash758 2005
[16] R Anguelov and J M-S Lubuma ldquoNonstandard finite dif-ference method by nonlocal approximationrdquo Mathematicsand Computers in Simulation vol 61 no 3-6 pp 465ndash4752003
[17] R Anguelov and J M-S Lubuma ldquoContributions to themathematics of the nonstandard finite difference method andapplicationsrdquo Numerical Methods for Partial DifferentialEquations vol 17 no 5 pp 518ndash543 2001
[18] B M Chen-Charpentier D T Dimitrov andH V Kojouharov ldquoNumerical simulation of multi-speciesbiofilms in porous media for different kineticsrdquo Mathematicsand Computers in Simulation vol 79 no 6 pp 1846ndash18612009
[19] D T Dimitrov and H V Kojouharov ldquoNonstandard finite-difference schemes for general two-dimensional autonomousdynamical systemsrdquo Applied Mathematics Letters vol 18no 7 pp 769ndash774 2005
[20] A J Arenas J A Morantildeo and J C Cortes ldquoNon-standardnumerical method for a mathematical model of RSV epide-miological transmissionrdquo Computers amp Mathematics withApplications vol 56 no 3 pp 670ndash678 2008
[21] L Jodar R J Villanueva A J Arenas and G C GonzalezldquoNonstandard numerical methods for a mathematical modelfor influenza diseaserdquo Mathematics and Computers in Sim-ulation vol 79 no 3 pp 622ndash633 2008
[22] W Piyawong E H Twizell and A B Gumel ldquoAn uncon-ditionally convergent finite-difference scheme for the SIRmodelrdquo Applied Mathematics and Computation vol 146no 2-3 pp 611ndash625 2003
[23] H Jansen and E H Twizell ldquoAn unconditionally convergentdiscretization of the SEIR modelrdquo Mathematics and Com-puters in Simulation vol 58 no 2 pp 147ndash158 2002
[24] A J Arenas G Gonzalez-Parra and B M Chen-CharpentierldquoA nonstandard numerical scheme of predictor-correctortype for epidemic modelsrdquo Computers amp Mathematics withApplications vol 59 no 12 pp 3740ndash3749 2010
[25] F Brauer and C Castillo-Chavez Mathematical Models inPopulation Biology and Epidemiology Springer Berlin Ger-many 2001
18 Journal of Mathematics
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