Research ArticleThreshold Dynamics in a Periodic Three-Patch Rift Valley FeverVirus Transmission Model

Buyu Wen, Zhidong Teng , and Wenlin Liu

College of Mathematics and Systems Science, Xinjiang University, Urumqi 830046, China

Correspondence should be addressed to Zhidong Teng; zhidong_teng@sina.com

Received 29 April 2018; Revised 17 August 2018; Accepted 29 August 2018; Published 9 January 2019

Academic Editor: Peter Giesl

Copyright © 2019 Buyu Wen et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

This paper investigates a three-patch Rift Valley fever virus transmission model with periodic coefficients. The basic reproductionnumber Ri0 i = 1,2,3 is calculated for the model by using the next infection operator method. The threshold conditions on theextinction and permanence in the mean of the disease with probability one. The Rift Valley fever disease will be permanent inthe j-th j = 1,2,3 patch if j ≥ i, Ri0 > 1 and dies out in the j-th patch if j < i. The numerical simulations are given to confirm thetheoretical results.

1. Introduction

Rift Valley fever (RVF) is an acute viral zoonosis caused bythe Rift Valley fever virus (RVF virus). This disease is pri-marily transmitted among domestic animals, such as sheep,cattle, buffalo, goat, and camels, with the main route ofinfection being mosquito bites (see [1–4]). RVF virus infec-tion in livestock animals can produce high rate of abortionin pregnant animals and significant morbidity and mortalityrate in young ruminants. During the disease epidemicamong animals, humans may be infected with Rift Valleyfever virus through bites from infected mosquitoes or con-tacts with the blood, body fluids, and organs of infected ani-mals (see [5, 6]).

In the 1930s, RVF virus was discovered for the first timenear Naivasha Lake in the Rift Valley of Kenya. In 1977, ahuman RVF virus epidemic is reported in Egypt, which isthe largest epidemic. Subsequently, RVF virus outbreaks havebeen reported in Saharan and North Africa. After that, thisdisease occurred in Saudi Arabia and Yemen outside theAfrica in the 2000s. These disease outbreaks result in highmortality and abortion in young ruminants and case

significant economic losses in Africa and Middle East. Withthe enlargement of scope of RVF virus infection, there is agrowing concern that this disease will outbreak further inother parts of Asia and Europe. In recent years, numerousmathematical models have been developed to investigateRVF virus outbreaks (see [1, 3, 6–15]).

Research results show that, in Northeastern Africa,human activities, including those associated with the Eidal-Adha feast, along with a combination of climatic factorssuch as rainfall level and hydrological variations, contributeto the transmission and dispersal of the disease pathogen.Moreover, sporadic outbreaks may occur when the twoevents occur together: (1) abundant livestock is recruited intoareas at risk from RVF due to the demand for the religiousfestival and (2) abundant numbers of mosquitoes emerge.In [8, 14, 15], the authors consider the religious festival anddiscuss three-patch model for RVF virus transmission.

On the one side, a religious festival, the Eid al-Adha feast,at which time large numbers of livestock are driven towardsthe site of the feast. In Africa and the Middle East, the inva-sion path follows the same route that people use to travel tothe Nile Delta, where the important Islamic festival Greater

HindawiComplexityVolume 2019, Article ID 7896946, 18 pageshttps://doi.org/10.1155/2019/7896946

Bairam is held. Geographically, the disease seems to be alonga path from southwest to northeast of Africa. This is consis-tent with the way to Mecca, Saudi Arabia’s capital (see [15]).

On the other side, the number of mosquitoes is affectedby temperature and humidity, the greater the humidity, themore the number of mosquitoes, the humidity at about thirtydegrees is the most suitable for mosquitoes to grow and prop-agate, and lower than ten degrees will enter hibernation. Dueto rainfall and temperature, in fact, the mosquito populationdensities follow seasonal fluctuations, achieving maximumnumbers during the rainy season and reaching lowest levelduring the winter season (see [4, 7, 8]). The abundant num-bers of mosquitoes corresponds to an increased number ofinfected ruminants. Consequently, RVF virus infections fluc-tuate over time and highly link to seasonal variations.

The autonomous models have been discussed in[3, 11–15]. At the same time, we can see that Rift Valley feverwill change periodically with the religious festivals and mos-quitoes’ periodicity. Recently, in [8], Xiao et al. proposed athree-patch periodic model for RVF virus transmission withthe periodic recruitment rate of livestock in first patch, theperiodic carrying capacities for mosquitoes in three patches,the periodic natural death rate for livestock in the last patch,and the periodic migration rate. In order to seek the seasonaland festival-driven impacts for RVF outbreaks, the authors in[8] investigated the dynamical behaviors of the model just byusing the numerical simulation method.

The main purpose of this paper is to discuss the thresholddynamics of a periodic three-patch Rift Valley fever virustransmission model, and thereout, we will make up the defi-ciency for the research given in [8]. By applying the methodsin given [16–21], we calculate basic reproduction numberRi0 i = 1,2,3 in i-patch of model 1 i By using the theory ofpersistence for dynamical systems, we will establish that Ri0serves as a threshold value for Rift Valley fever virus dynam-ics in periodic case.

The organization of this paper is as follows. In the nextsection, we develop a periodic three-patch model todescribe the transmission of Rift Valley fever virus andpresent some useful lemmas. In Section 3, we will proposethe main results, namely, we will obtain the extinction orpermanence of all positive solutions of model 1 i. In Section4, we will give the numerical simulations to verify our the-oretical results.

2. Model

We investigated the festival-driven and seasonal impacts onthe patterns of RVF outbreaks among livestock in Africaand Middle East.

We assume that a disease invades and subdivides thetarget livestock into four classes: the susceptible class Si , theexposed class Ei , the infectious class Ii , and the recoveredclass Ri ; the female mosquitoes are divided into three com-partments: uninfected compartment Ui , exposed compart-ment Li , and infected compartment Vi in each patch.

We assume that r1 t is the number of livestock importeddaily at time t in 1-patch, αi t is the transmission rate of dis-ease from mosquitoes to livestock at time t in i-patch, βi t is

the transmission rate of disease from livestock to mosquitoesat time t in i-patch, μi t is the time-dependent natural deathrate for livestock in i-patch (average death rate for differentlivestock, i.e., cattle and sheep), νi t is the time-dependentnatural death rate for female mosquitoes in i-patch, ξi t isthe rate at which immunity is lost after recovery for livestockat time t in i-patch, εi t is the rate of becoming infectious forlivestock at time t in i-patch, ηi t is the rate of becominginfectious for mosquitoes at time t in i-patch, γi t is therecovery rate of leaving the infectious class for livestock attime t in i-patch, bi t is the time-dependent birth rate formosquitoes in i-patch, Mi t is the time-dependent carryingcapacity of mosquitoes in i-patch, δi t is the disease-induceddeath rate for livestock at time t in i-patch, Ci t is momentspeed of animals from i-patch to i + 1-patch, and di is thelength of journey for animals within patch i. Motivated bythe above assumption, we propose the following periodicthree-patch RVF virus transmission model.

dSi tdt

= ri t +Ci−1 tdi−1

Si−1 t − αi t Si t Vi t

− μi t Si t + ξi t Ri t −Ci tdi

Si t ,

dEi tdt

=Ci−1 tdi−1

Ei−1 t + αi t Si t Vi t

− μi t + εi t Ei t −Ci tdi

Ei t ,

dIi tdt

=Ci−1 tdi−1

Ii−1 t + εi t Ei t

− μi t + γi t + δi t Ii t −Ci tdi

Ii t ,

dRi tdt

=Ci−1 tdi−1

Ri−1 t + γi t Ii t

− μi t + ξi t Ri t −Ci tdi

Ri t ,

dUi tdt

= bi t − νi t Ui t 1 −Ui t + Li t +Vi t

Mi t

− βi t Ii t Ui t ,

dLi tdt

= βi t Ii t Ui t − νi t + ηi t Li t ,

dVi tdt

= ηi t Li t − νi t Vi t , i = 1,2,3,

1 i

where r2 t = r3 t = C0 t = C3 t = 0.For any continuous and bounded function f t de-

fined for R+ = 0,∞ , we denote f i = inf t∈R+f t and f u =

supt∈R+f t .

Based on the biological background of model 1 i, theinitial condition for model 1 i (for any i = 1,2,3) has inthe following form:

2 Complexity

Si 0 ≥ 0, Ei 0 ≥ 0, Ii 0 ≥ 0, Ri 0≥ 0,Ui 0 > 0, Li 0 ≥ 0, Vi 0 ≥ 0

2

For model 1 i, we give assumptions as follows:H1 functions r1 t , αi t , βi t , μi t , C1 t , C2 t , νi

t , ξi t , εi t , ηi t , γi t , bi t ,Mi t and δi t i = 1,2,3are continuous and ω-periodic defined on R+ = 0,∞ .Furthermore, functions αi t , βi t μi t , εi t , νi t , bi t ,γi t , ηi t , and Mi t are positive, and functions C1 t ,C2 t , ξi t , and δi t are nonnegative.

H2ω0 r1 t dt > 0, ω

0 bi t − νi t dt > 0 and bi t ≥ νit i = 1,2,3 for all t ≥ 0.

Consider a ω-periodic linear equation as follows:

dxdt

= a t − c t x, 3

where a t and c t are ω-periodic continuous functionsdefined for t ≥ 0. We have the following lemma.

Lemma 1 (see [17]). Assume that ω0 a t dt > 0, ω

0 c t dt > 0and c t ≥ 0 for all t ≥ 0. Then, equation (3) has a unique pos-itive ω-periodic solution x∗ t which is globally attractive.

Furthermore, we consider a ω-periodic logistic equationas follows:

dxdt

= x a t − c t x , 4

where a t and c t are ω-periodic continuous functionsdefined for t ≥ 0. We have the following lemma.

Lemma 2 (see [18]). Assume that ω0 a t dt > 0, ω

0 c t dt > 0and c t ≥ 0 for all t ≥ 0. Then, equation (4) has a unique pos-itive ω-periodic solution x∗ t which is globally attractive.

When E1 t = 0, I1 t = 0, L1 t = 0,V1 t = 0 and R1t = 0, we obtain the disease-free subsystem of model 1 1:

dS1 tdt

= r1 t − μ1 t + C1 td1

S1 t ,

dU1 tdt

= b1 t − ν1 t U1 t 1 −U1 tM1 t

5

As consequence of Lemma 1 and Lemma 2, we have theresult on the existence of disease-free periodic solution ofmodel 1 1.

Lemma 3. System (5) has a unique positive ω-periodic solu-tion S∗1 t ,U∗

1 t which is globally attractive.

From Lemma 3, model 1 1 has a unique disease-freeperiodic solution E10 = S∗1 t , 0,0,0,U∗

1 t ,0,0 .Let A t be a continuous, cooperative, irreducible, and ω-

periodic n × n matrix function; we consider a linear system:

dx tdt

= A t x t 6

Let ΦA t be the fundamental solution matrix of system(6) satisfying initial condition ΦA 0 = I, where I is an unitmatrix. Let ρ ΦA ω be the spectral radius of matrixΦA ω . Since A t is continuous, cooperative, and irreduc-ible, we obtain that ΦA t is nonnegative for all t ≥ 0. ByPerron-Frobenius theorem, ρ ΦA ω is the principal eigen-value ofΦA ω in the sense that it is simple and admits a pos-itive eigenvector υ∗. We present the following lemma.

Lemma 4 (see [19]). Let p = 1/ω ln ρ ΦA ω . Then, thereexists a positive ω-periodic function υ t such that x t =eptυ t is a solution of system (6).

Now, we calculate the basic reproduction number ofmodel 1 1 by applying the next-generation matrix approachwhich is given in [20]. Firstly, we easily validate that model1 1 satisfies the conditions A1 − A7 given in [20]. Let

F = F1 t =

0 0 0 α1 t S∗1 t

0 0 β1 t U∗1 t 0

0 0 0 0

0 0 0 0

,

V = B1 t =

μ1 t + ε1 t +C1 td1

0 0 0

0 ν1 t + η1 t 0 0

−ε1 t 0 μ1 t + γ1 t + δ1 t +C1 td1

0

0 −η1 t 0 ν1 t

7

3Complexity

Let Y t, s with t ≥ s be the evolution operator of linearω-periodic system:

dy tdt

= −B1 t y t 8

Then for any s ∈ℝ and t ≥ s, dY t, s /dt = −B1 tY t, s and Y s, s = I, where I is the unit matrix. Let ℂω bethe ordered Banach space of all ω periodic continuous func-tions ϕ R→ R4. Assume that ϕ s ∈ℂω is the initial distri-bution of infectious individuals. Then, F1 s ϕ s is the rateat which new infections are produced by infected individualswho were introduced into the population at time s. Whent ≥ s, then Y t, s F1 s ϕ s represents the distribution ofthose infected individuals who were newly infected individ-uals at time s and remain in the infected compartments attime t. Hence, the cumulative distribution of new infectionsat time t produced by all those infected individuals ϕ sintroduced before t is given by a linear operator as follows:

ℒ1ϕ t =t

−∞Y t, s F1 s ϕ s ds

=∞

0Y t, t − a F1 t − a ϕ t − a da

9

Applying the results obtained in [20], the basic reproduc-tion number R10 for model 1 1 is defined as spectral radiusρ ℒ1 of operator ℒ1, that is, R10 = ρ ℒ1 .

Furthermore, using Theorem 2.2 given in [20], wealso can obtain the following results on basic reproductionnumber R10.

Lemma 5 (see [20]). The following conclusions hold.

(a) R10 = 1 if and only if ρ ΦF1−B1ω = 1

(b) R10 > 1 if and only if ρ ΦF1−B1ω > 1

(c) R10 < 1 if and only if ρ ΦF1−B1ω < 1

(d) Disease-free periodic solution E10 for model 1 1 islocally asymptotically stable if R10 < 1 and unstable ifR10 > 1

Remark 1. In particular, when periodic systems (1) degener-ate into autonomous systems, by the next-generation matrixmethod (see [22]), we can obtain

F =

0 0 0 α1S∗1

0 0 β1U∗1 0

0 0 0 0

0 0 0 0

,

V =

μ1 + ε1 +C1d1

0 0 0

0 ν1 + η1 0 0

−ε1 0 μ1 + γ1 + δ1 +C1d1

0

0 −η1 0 ν1

,

10

where S∗1 = r1/ μ1 + C1/d1 and U∗1 =M1. Then,

3. Main Results

For model 1 i, we investigate the persistence and extinctionof positive solutions on the three patches.

3.1. The First Patch. In this section, we discuss model 1 1.Firstly, on the nonnegativity, positivity, and boundednessof solutions for model 1 1, we have the following result.

Theorem 1. Let S1 t , E1 t , I1 t , R1 t ,U1 t , L1 t , V1 tbe the solution of model 1 1 with initial condition (2).Then, S1 t , E1 t , I1 t , R1 t ,U1 t , L1 t , V1 t is non-negative for all t ≥ 0 and ultimately bounded, and whenS1 0 > 0, E1 0 > 0, I1 0 > 0, R1 0 > 0,U1 0 > 0, L1 0 > 0

and V1 0 > 0, then S1 t , E1 t , I1 t , R1 t ,U1 t , L1 t ,V1 t is also positive for all t > 0.

Proof 1. In fact, by the continuous dependence of solutionswith respect to initial values, we only need to prove thatwhen S1 0 > 0, E1 0 > 0, I1 0 > 0, R1 0 > 0,U1 0 > 0, L10 > 0 and V1 0 > 0, then S1 t , E1 t , I1 t , R1 t ,U1 t ,L1 t , V1 t is positive for all t > 0. From the fifth equationof model 1 1, we have for any t ≥ 0.

dU1 tdt

= b1 t − ν1 t 1 −U1 t + L1 t + V1 t

M1 t

− β1 t I1 t U1 t

12

R10 = ρ FV−1 =α1r1ε1η1β1M1

ν1 μ1 + C1/d1 μ1 + γ1 + δ1 + C1/d1 μ1 + ε1 + C1/d1 ν1 + η111

4 Complexity

Since U1 0 > 0, we get

U1 t =U1 0 expt

0b1 s − ν1 s

1 −U1 s + L1 s + V1 s

M1 s− β1 s I1 s ds > 0

13

Define m1 t =mint≥0 S1 t , E1 t , I1 t , R1 t , L1 t , V1t . Obviously, m1 0 = min S1 0 , E1 0 , I1 0 , R1 0 , L10 , V1 0 > 0. We only need to prove m1 t > 0 for allt ≥ 0. Suppose that there exists a t0 > 0 such that m1 t0 = 0andm1 t > 0 for all t ∈ 0, t0 . Then, we only need to discussthe following six cases: (1) m1 t0 = S1 t0 , (2) m1 t0 =E1 t0 , (3) m1 t0 = I1 t0 , (4) m1 t0 = R1 t0 , (5) m1 t0 =L1 t0 , and (6) m1 t0 =V1 t0 .

Now, we only give the proof for case (2). The remainingcases can be proved in a similar manner. Let m1 t0 =E1 t0 . Since m1 t > 0 for all t ∈ 0, t0 , then V1 t > 0,S1 t > 0, and hence,

dE1 tdt

≥ − μ1 t + ε1 t + C1 td1

E1 t , for all t ∈ 0, t0

14

Integrating from 0 to t0, we get

0 = E1 t0 ≥ E1 0 exp −t0

0μ s + ε1 s +

C1 sd1

ds > 0,

15

which leads to a contradiction.Next, we show the boundedness of nonnegative solutions

of model 1 1. Let N1 t = S1 t + E1 t + I1 t + R1 t , thenwe get

dN1 tdt

= r1 t − μ1 t +C1 td1

N1 t − δ1 t I1 t

≤ r1 t − μ1 t +C1 td1

N1 t16

Hence, we have lim supt→∞N1 t ≤ ru1/ μi1 + C1/d1i ,

which implies that S1 t , E1 t , I1 t and R1 t are ultimatelybounded. Since for any t ≥ 0,

dU1 tdt

≤ b1 t − ν1 t U1 t 1 −U1 tM1 t

, 17

using Lemma 3; we further have lim supt→∞U1 t ≤lim supt→∞U∗

1 t =max0≤t≤ω U∗1 t Hence, for any ϵ0 >

0, there is a T0 > 0 such that I1 t ≤ ru1/ μi1 + C1/d1i +

ϵ0 and U1 t ≤ P1 + ϵ0 for all t ≥ T0, where P1 = max0≤t≤ωU∗

1 t . From the sixth equation of model 1 1, we have

dL1 tdt

≤ β1 t P2 − ν1 t + η1 t L1 t , 18

for all t ≥ T0, where P2 = ru1/ μi1 + C1 t /d1i + ϵ0

P1 + ϵ0 . Hence, we can obtain lim supt→∞L1 t ≤ βu1P2/

ν1 + η1i From this, for any ϵ1 > 0, there is a T1 > 0 such

that L1 t ≤ βu1P2/ ν1 + η1

i + ϵ1 for all t ≥ T1. Hence,from the last equation of model 1 1, we further have

dV1 tdt

≤ −ν1 t V1 t + P3η1 t , 19

for all t ≥ T1, where P3 = βu1P2/ ν1 + η1

i + ϵ1 . This showsthat lim supt→∞V1 t ≤ P3η

u1/νi1 . From above discussions,

we finally obtain that all solutions of model 1 1 with initialcondition (2) are ultimately bounded. This completes theproof.

Now, we show that R10 serves as a threshold parameter.We prove that when R10 < 1, then disease-free periodic solu-tion E10 is globally asymptotically stable, that is to say, theRVF virus is extinct in the first patch, and when R10 > 1, thenmodel 1 1 is permanent, that is to say, the RVF virus is exis-tent in the population in the first patch.

Theorem 2. If R10 < 1, then disease-free periodic solution E10of model 1 1 is globally asymptotically stable, and if R10 > 1,then E10 is unstable.

Proof 2. From conclusion d of Lemma 5, it follows that ifR10 > 1, then E10 is unstable, and if R10 < 1, then E10 is locallyasymptotically stable.

We now prove that if R10 < 1, then E10 is globallyattractive. Let R10 < 1, from conclusion c of Lemma 5, thenρ ΦF1−B1

ω < 1. We can choose constant ς > 0 smallenough such that ρ ΦF1−B1+ςH ω < 1, where

H t =

0 0 0 α1 t

0 0 β1 t 0

0 0 0 0

0 0 0 0

20

From model 1 1, we have

dN1 tdt

= r1 t − μ1 t + C1 td1

N1 t − δ1 t I1 t ≤ r1 t − μ1 t + C1 td1

N1 t ,

dU1 tdt

≤ b1 t − ν1 t U1 t 1 −U1 tM1 t

21

5Complexity

From Lemma 3, for above given ς > 0, there exists aT1 > 0 such that N1 t = S1 t + I1 t + E1 t + R1 t ≤ S∗1 t+ ς and U1 t ≤U∗

1 t + ς for any t ≥ T1. Hence, from model1 1, we obtain that for all t ≥ T1

dE1 tdt

≤ α1 t S∗1 t + ς V1 t − μ1 t + ε1 t +C1 td1

E1 t ,

dL1 tdt

≤ β1 t U∗1 t + ς t I1 t − ν1 t + η1 t L1 t ,

dI1 tdt

= ε1 t E1 t − μ1 t + γ1 t + δ1 t +C1 td1

I1 t ,

dV1 tdt

= −ν1 t V1 t + η1 t L1 t

22

Consider the following auxiliary system:

dE1 tdt

= α1 t S∗1 t + ς V1 t − μ1 t + ε1 t +C1 td1

E1 t ,

dL1 tdt

= β1 t U∗1 t + ς t I1 t − ν1 t + η1 t L1 t ,

dI1 tdt

= ε1 t E1 t − μ1 t + γ1 t + δ1 t +C1 td1

I1 t ,

dV1 tdt

= −ν1 t V1 t + η1 t L1 t

23

It follows from Lemma 4 that there is a positive ω-peri-odic function υ t = υ1 t , υ2 t , υ3a t , υ4 t T such that

E1 t , L1 t , I1 t , V1 tT = eptυ t is a solution of system

(23), where p = 1/ω ln ρ ΦF1−B1+ςH ω .

Denote J t = E1 t , L1 t , I1 t , V1 t T We canchoose constant τ > 0 such that J T1 ≤ τepT1υ T1According to the comparison theorem of the vector form(see [23]), we obtain J t ≤ τeptυ t for any t ≥ T1. By ρΦF1−B1+ςH ω < 1, we know p = 1/ω ln ρ ΦF1−B1+ςH ω

< 0 Then, we conclude that limt→∞J t = 0, that is to say,limt→∞ E1 t , L1 t , I1 t , V1 t = 0,0,0,0 . By the equa-tions of R1, S1, and U1 of model 1 1, we also obtain limt→∞S1 t , R1 t ,U1 t = S∗1 t , 0,U∗

1 t . Hence, disease-freeperiodic solution E10 is globally attractive. This shows thatdisease-free periodic solution E10 is globally asymptoticallystable. This completes the proof.

Theorem 3. If R10 > 1, then the disease in model 1 1 ispermanent.

Proof 3. From conclusion b of Lemma 5, we know thatR10 > 1 if and only if ρ ΦF1−B1

ω > 1. We can choose asmall enough constant σ > 0 such that ρ ΦF1−B1+ςH ω > 1,where N t is given in (20). From H1 and H2 , we canchoose ϵ1 > 0 small enough such that b1 t − ν1 t − β1 t ϵ1> 0 for all t ≥ 0. Consider the following auxiliary equations:

dS1 tdt

= r1 t − α1 t S1 t ϵ1 − μ1 t +C1 td1

S1 t 24

and

dU1 tdt

= b1 t − ν1 t U1 t 1 −U1 tM1 t

− β1 t U1 t ϵ1,

25

from Lemma 1 and Lemma 2 we can get equations (24) and(25) admit globally uniformly attractive positive ω-periodicsolution S1ϵ1 t and U1ϵ1 t , respectively. By the continuityof solutions with respect to parameter ϵ1, for above σ > 0,there exists a 0 < ϵ0 < ϵ1 such that for all t ∈ 0, ω

S1ϵ0 t > S∗1 t −σ

2,U1ϵ0 t >U∗

1 t −σ

226

Define the sets as follows:

X1 = S1, E1, I1, R1,U1, L1, V1 : S1 > 0, E1

≥ 0, I1 ≥ 0, R1 ≥ 0,U1 > 0, L1 ≥ 0, V1 ≥ 0 ,

X10 = S1, E1, I1, R1,U1, L1, V1

∈ X1 E1 > 0, I1 > 0, L1 > 0, V1 > 0 ,

∂X10 =X1X10

= S1, E1, I1, R1,U1, L1, V1

∈ X1 E1I1L1V1 = 0

27

Define further the Poincaré map P X → X1 as follows:

P x0 = u ω, x0 , x0 = S01, E01, I

01, R

01,U

01, L

01, V

01 ∈ X1, 28

where u t, x0 = S1 t , E1 t , I1 t , R1 t ,U1 t , L1 t , V1 tis the solution of model 1 1 satisfying initial conditionu 0, x0 = x0. It follows from Theorem 1 that all solutionsof model 1 1 are ultimately bounded. Therefore, map P ispoint dissipative and compact on X1.

We define

M∂ = x0 = S01, E01, I

01, R

01,U

01, L

01, V

01

∈ ∂X10 Pm x0 ∈ ∂X10, forallm ≥ 0 ,29

where P0 x0 = x0, P1 x0 = P x0 and Pm x0 = P Pm−1 x0

for m = 1, 2,… . Now, we claim M∂ = S1, 0, 0, R1,U1,0,0 :S1 > 0, R1 ≥ 0,U1 > 0 In fact, from model 1 1, we directlyobtain that the solution of model 1 1 with initial valueS01, 0, 0, R0

1,U01,0,0 ∈ ∂X10 has the form S1 t , 0, 0, R1 t ,

U1 t ,0,0 . Hence,

Pm S01, 0, 0, R01,U

01,0,0 = S1 mω , 0, 0, R1 mω ,

U1 mω ,0,0 ∈ ∂X0,30

for all m ≥ 0. This shows S01, 0, 0, R01,U0

1,0,0 ∈M∂. There-fore, S1, 0, 0, R1,U1,0,0 : S1 > 0, R1 ≥ 0,U1 > 0 ⊆M∂.

6 Complexity

Next, we prove M∂ ⊆ S1, 0, 0, R1,U1,0,0 : S1 > 0, R1 ≥0,U1 > 0 . Suppose that there is a S01, E0

1, I01, R01,U0

1, L01,V0

1 ∈M∂ such that S01, E01, I01, R0

1,U01, L01, V0

1 ∉ S1, 0, 0,R1,U1,0,0 : S1 > 0, R1 ≥ 0,U1 > 0 .

We firstly have max E01, I01, L01, V0

1 > 0. Let S1 t , E1t , I1 t , R1 t ,U1 t , L1 t , V1 t be the solution of model1 1 with initial value S01, E0

1, I01, R01,U0

1, L01, V01 at t = 0. From

Theorem 1, we firstly have that S1 t , E1 t , I1 t , R1 t ,U1t , L1 t , V1 t is nonnegative for all t ≥ 0. Furthermore,by S01 > 0 and U0

1 > 0, from the first and fifth equationsof model 1 1, we also have that S1 t > 0 and U1 t > 0 forall t ≥ 0, respectively.

If E01 > 0, then by the second equation of model 1 1,

we obtain

dE1 tdt

≥ − μ1 t + ε1 t +C1 td1

E1 t , for all t ≥ 0 31

Integrating this inequality from 0 to t, we have

E1 t ≥ E01 exp −

t

0μ1 s + ε1 s +

C1 sd1

ds > 0,

 for all t ≥ 032

Further, from the third equation of model 1 1 wecan obtain

I1 t > I01 exp −t

0μ1 s + γ1 s + δ1 s +

C1 sd1

ds ≥ 0,

 for all t > 033

From the sixth equation of model 1 1, we get

dL1 tdt

> − ν1 t + η1 t L1 t , for all t ≥ 0 34

Hence,

L1 t > L01 exp −t

0ν1 s + η1 s ds ≥ 0, for all t > 0

35

Lastly, from the equation of V1 for model 1 1, we alsohave V1 t > 0 for all t > 0. Therefore, we finally obtainE1 t I1 t L1 t V1 t > 0 for all t > 0.

Similarly, if I01 > 0 or L01 > 0 or V01 > 0, we also can obtain

E1 t I1 t L1 t V1 t > 0 for all t > 0. This shows that S01,E01, I01, R0

1,U01, L01, V0

1 ∉M∂, which leads to a contradiction.This implies that M∂ ⊆ S1, 0, 0, R1,U1,0,0 : S1 > 0, R1 ≥ 0,U1 > 0 . Hence, M∂ = S1, 0, 0, R1,U1,0,0 : S1 > 0, R1 ≥ 0,U1 > 0 .

In M∂, model 1 1 degenerates to

dS1 tdt

= r1 t − μ1 t +C1 td1

S1 t + ξ1 t R1 t ,

dR1 tdt

= − μ1 t + ξ1 t +C1 td1

R1 t ,

dU1 tdt

= b1 t − ν1 t U1 t 1 −U1 tM1 t

36

We easily prove that system (36) has a globally asymp-totically stable ω-periodic solution S∗1 t , 0,U∗

1 t , whereS∗1 t ,U∗

1 t is given in Lemma 3. It is clear that mapP has a unique globally attractive fixed point restrictedin M∂, which is E1 = S∗1 0 , 0,0,0,U∗

1 0 ,0,0 .Now, we define Ws E1 = x0 Pm x0 → E1,m→∞ ,

which is said to be stable set of E1. We show that

Ws E1 ∩ X0 = 0 37

According to the continuity of solutions with respect tothe initial values, for above given constant ϵ0, there existsδ > 0 such that for any x0 = S01, E0

1, I01, R01,U0

1, L01, V01 ∈ X0

with x0 − E1 ≤ δ, it follows that u t, x0 − u t, E1 ≤ ϵ0for all t ∈ 0, ω , where u t, x0 = S1 t , E1 t , I1 t , R1 t ,U1 t , L1 t , V1 t is the solution of model 1 1 with ini-tial condition S1 0 , E1 0 , I1 0 , R1 0 ,U1 0 , L1 0 , V1 0= x0 and u t, E1 = S∗1 t , 0,0,0,U∗

1 t ,0,0 . Since u t, E1is ω-periodic, we have u t1, E1 = u t2, E1 for any t1, t2

≥ 0 with t2 = kω + t1, where k is an integer. We claim that

lim supm→∞

Pm x0 − E1 ≥ δ,

 for all x0 = S01, E01, I

01, R

01,U

01, L

01, V

01 ∈ X0

38

Suppose that (38) does not hold. Then, we have

lim supm→∞

Pm x0 − E1 < δ,

 for some x0 = S01, E01, I

01, R

01,U

01, L

01, V

01 ∈ X0

39

Without loss of generality, we assume that Pm x0 −E1 < δ for all m ≥ 0. Thus,

u t, Pm x0 − u t, E1 ≤ ϵ0, for all t ∈ 0, ω ,m ≥ 040

For any t ≥ 0, let t = t′ + nω, where t′ ∈ 0, ω andn = t/ω are the greatest integers less than or equalto t/ω. Then, we have u t, x0 − u t, E1 = u t′, Pm x0 −u t′, E1 < ϵ0, for any t ≥ 0

It follows that 0 < E1 t < ϵ0, 0 < L1 t < ϵ0, 0 < I1 t < ϵ0,and 0 <V1 t < ϵ0 for all t ≥ 0. Thus, from model 1 1, wehave the following:

7Complexity

dS1 tdt

≥ r1 t − α1 t S1 t ϵ0 − μ1 t +C1 td1

S1 t ,

dU1 tdt

≥ b1 t − ν1 t U1 t 1 −U1 tM1 t

− β1 t +2 b1 t − ν1 t

M1 tU1 t ϵ0,

41

for any t ≥ 0. Using the comparison principle, we obtain

S t ≥ S1ϵ0 t ,U t ≥U1ϵ0 t , 42

for any t ≥ 0, where S1ϵ0 t and U1ϵ0 t are the solutionsof equations (20) and (23) with respect to parameter ϵ0satisfying initial conditions S1ϵ0 0 = S1 0 and U1ϵ0 0 =U1 0 , respectively.

From Lemma 1 and Lemma 2, equations (24) and (25)with respect to the parameter ϵ0 have globally uniformlyattractive solutions S1ϵ0 t and U1ϵ0 t , respectively. Hence,there exists t2 > 0 such that

S1ϵ0 t ≥ S1ϵ0 t −σ

2,U1ϵ0 t ≥U1ϵ0 t −

σ

2, 43

for all t ≥ t2. By (26), (42), and (43), we obtain S1 t ≥ S1ϵ0t ≥ S1ϵ0 t − σ/2 > S∗1 t − σ and U1 t ≥U1ϵ0 t ≥U1ϵ0t − σ/2 >U∗

1 t − σ for all t ≥ t2. Thus, for all t ≥ t2, itholds that

dE1 tdt

≥ α1 t S∗1 t − σ V1 t − μ1 t + ε1 t +C1 td1

E1 t ,

dL1 tdt

≥ β1 t U∗1 t − σ I1 t − ν1 t + η1 t L1 t ,

dI1 tdt

= ε1 t E1 t − μ1 t + γ1 t + δ1 t +C1 td1

I1 t ,

dV1 tdt

= −ν1 t V1 t + η1 t L1 t

44

Consider the following auxiliary system:

dE1 tdt

= α1 t S∗1 t − σ V1 t − μ1 t + ε1 t +C1 td1

E1 t ,

dL1 tdt

= β1 t U∗1 t − σ I1 t − ν1 t + η1 t L1 t ,

dI1 tdt

= ε1 t E1 t − μ1 t + γ1 t + δ1 t +C1 td1

I1 t ,

dV1 tdt

= −ν1 t V1 t + η1 t L1 t

45

System (45) can be rewritten in terms of the matricesF1 t and B1 t as follows:

ddt

E1

L1

I1

V1

= F1 t − B1 t + ςH t

E1

L1

I1

V1

46

It follows from Lemma 4 that there is a positive ω-peri-odic function w t = w1 t ,w2 t ,w3a t ,w4 t T such that

E1 t , L1 t , I1 t , V1 tT = eθtw t is a solution of system

(46), where θ = 1/ω ln ρ ΦF1−B1+ςH ω .

Now, we denote J t = E1 t , L1 t , I1 t ,V1 t T. Wecan choose small enough constant τ > 0 such that J t2 ≥ τeθt2w t2 . Thus, by comparison theorem of the vector form,it follows that J t ≥ τeθtw t for all t ≥ t2. By ρ ΦF1−B1+ςHω > 1, we obtain θ = 1/ω ln ρ ΦF1−B1+ςH ω > 0. Hence,limt→∞J t =∞, which implies that

limt→∞

E1 t =∞, limt→∞

L1 t =∞,

limt→∞

I1 t =∞, limt→∞

V1 t =∞,47

which leads to a contradiction. Hence, claim (25) holds. Thisshows Ws E1 ∩ X0 = 0.

It is clear that fixed point E1 of P is globally attractiverestricting on M∂. Hence, E1 is acyclic in M∂. From the the-orems established in [19] on the uniform persistence ofdynamical systems, we obtain that P is uniformly persistentwith respect to X0, ∂X0 , which implies that model 1 1 ispermanent. This completes the proof.

3.2. The Second Patch. In this section, we discuss model 1 2.First of all, we have the following result on the positivity andultimate boundedness of solutions in model 1 2.

Theorem 4. Let S2 t , E2 t , I2 t , R2 t , R2 t , L2 t , V2 tbe the solution of model 1 2 with initial condition (2). Then,S2 t , E2 t , I2 t , R2 t , R2 t , L2 t , V2 t is nonnegativefor all t ≥ 0 and ultimately bounded, and when S2 0 > 0, E20 > 0, I2 0 > 0, R2 0 > 0, R2 0 > 0, L2 0 > 0 and V2 0> 0, then S2 t , E2 t , I2 t , R2 t , R2 t , L2 t , V2 t is alsopositive for all t > 0.

Proof 4. Using the same method given in Theorem 1, wecan prove the positivity of solutions for model 1 2. Next,we show the boundedness of solutions for model 1 2. LetN2 t = S2 t + E2 t + I2 t + R2 t , then we get

dN2 tdt

=C1 td1

N1 t − μ2 t +C2 td2

N2 t − δ2 t I2 t

≤C1 td1

N1 t − μ2 t +C2 td2

N2 t

48

8 Complexity

By the comparison theorem and Lemma 1, we get thatthere exists constant M1 > 0 such that lim supt→∞N2 t ≤M1, which implies that S2 t , E2 t , I2 t and R2 t areultimately bounded. Since for any t ≥ 0,

dU2 tdt

≤ b2 t − ν2 t U2 t 1 −U2 tM2 t

, 49

then there exists constant M2 > 0 such that lim supt→∞U2 t ≤M2. Hence, for any ϵ > 0, there is a T0 > 0 suchthat I2 t ≤M1 + ϵ0 ≔Q1 and U2 t ≤M2 + ϵ0 ≔Q2 for allt ≥ T0. From the sixth equation of model 1 2, we have

dL2 tdt

≤ β2 t Q1Q2 − ν2 t + η2 t L2 t , 50

for all t ≥ T0. Hence, we can obtain that there exists M3 > 0such that lim supt→∞L2 t ≤M3. From this, for any ϵ1 > 0,there is a T1 ≥ T0 such that L2 t ≤M3 + ϵ1 ≔Q3 for all

t ≥ T1. Hence, from the last equation of model 1 2, wefurther have

dV2 tdt

≤ −ν2 t V2 t +Q3η2 t , 51

for all t ≥ T1. This shows that there exists M4 > 0 such thatlim supt→∞V2 t ≤M4. Hence, for any ϵ2 > 0, there is aT2 ≥ T1 such that V2 t ≤M4 + ϵ2 ≔Q4 for all t ≥ T2.

From above discussions, we finally obtain that all solu-tions of model 1 2 with initial condition (2) are ultimatelybounded. This completes the proof.

Assume firstly R10 > 1. From above result, without loss ofgenerality, we can assume that there are positive constantsm1 and K2 such that min S1 t , E1 t , I1 t , R1 t ,U1 t , L1t , V1 t ≥m1 for any solution S1 t , E1 t , I1 t , R1 t ,U1 t , L1 t , V1 t of model 1 1 and max S1 t , E1 t ,I1 t , R1 t ,U1 t , L1 t , V1 t ≤ K2 for any solution S2t , E2 t , I2 t , R2 t ,U2 t , L2 t , V2 t of model 1 2.Thus, from model 1 2, we further have

Consider auxiliary equations as follows:

dS2 tdt

≥C1 td1

m1 − α2 t S2 t K2 − μ2 t S2 t −C2 td2

S2 t ,

dE2 tdt

≥C1 td1

m1 − μ2 t + ε2 t E2 t −C2 td2

E2 t ,

dI2 tdt

≥C1 td1

m1 − μ2 t + γ2 t + δ2 t I2 t −C2 td2

I2 t ,

dR2 tdt

≥C1 td1

m1 − μ2 t + ξ2 t R2 t −C2 td2

R2 t ,

dU2 tdt

≥ b2 t − ν2 t U2 t 1 −U2 t + 2K2

M2 t− β2 t K2U2 t ,

dL2 tdt

= β2 t I2 t U2 t − ν2 t + η2 t L2 t ,

dV2 tdt

= η2 t L2 t − ν2 t V2 t

52

dS2 tdt

=C1 td1

m1 − α2 t K2 + μ2 t +C2 td2

S2 t ,

dE2 tdt

=C1 td1

m1 − μ2 t + ε2 t +C2 td2

E2 t ,

dI2 tdt

=C1 td1

m1 − μ2 t + γ2 t + δ2 t +C2 td2

I2 t ,

dR2 tdt

=C1 td1

m1 − μ2 t + ξ2 t +C2 td2

R2 t ,

dU2 tdt

= b2 t − ν2 t 1 −2K2M2 t

− β2 t K2 U2 t −b2 t − ν2 t

M2 tU2

2 t ,

dL2 tdt

= β2 t I2 t U2 t − ν2 t + η2 t L2 t ,

dV2 tdt

= η2 t L2 t − ν2 t V2 t

53

9Complexity

Firstly, we consider the livestock equations of model (53)by Lemma 1, and we have

lim inft→∞

S2 t =C1/d1 m1

i

α2K2 + μ2 + C2/d2u ,

lim inft→∞

E2 t =C1/d1 m1

i

μ2 + ε2 + C2/d2u ,

lim inft→∞

I2 t =C1/d1 m1

i

μ2 + γ2 + δ2 + C2/d2u ,

lim inft→∞

R2 t =C1/d1 m1

i

μ2 + ξ2 + C2/d2u

54

Next, we consider the mosquito system. From the fifthequation of model (53), we have

lim inft→∞

U2 t =b2 − ν2 1 − 2K2/M2 − β2K2

i

b2 − ν2/M2u 55

Therefore, for any ϵ3 > 0, there is a T3 > 0 such that for allt ≥ T3,

I2 t ≥C1/d1 m1

i

μ2 + γ2 + δ2 + C/d2u − ϵ3 ≔Q5,

U2 t ≥b2 − ν2 1 − 2K2/M2 − β2K2

i

b2 − ν2/M2u − ϵ3 ≔Q6

56

Considering the sixth equation of model (53), we have

dL2 tdt

≥ β2 t Q5Q6 − ν2 t + η2 t L2 t , 57

then we can obtain lim inf t→∞L2 t = β2Q5Q6i/ ν2 + η2

u.Thus, for any ϵ4 > 0, there is a T4 > 0 such that L2 t ≥β2Q5Q6

i/ ν2 + η2u − ϵ4 ≔Q7. Considering the last

equation of model (53), we have

dV2 tdt

≥ η2 t Q7 − ν2 t V2 t , 58

and then, we can obtain lim inf t→∞V2 t = η2Q7i/νu2 .

Using comparison theorem, we can obtain the theoremas follows.

Theorem 5. If R10 > 1, the disease in model 1 2 is permanent.

Next, we assume if R10 < 1, then each nonnegative solu-tion of model 1 1 converges to disease-free periodic solutionE10 = S∗1 t , 0,0,0,U∗

1 t ,0,0 , and we further obtain the lim-iting system of model 1 2 as follows:

From a similar argument as in the above to model 1 1, wecan obtain that model (59) has a unique disease-free periodic

solution E20 = S∗2 t , 0,0,0,U∗2 t ,0,0 , where S∗2 t ,U∗

2 tis the unique positive ω-periodic solution.

dS2 tdt

=C1 td1

S∗1 t − α2 t S2 t V2 t − μ2 t S2 t + ξ2 t R2 t −C2 td2

S2 t ,

dE2 tdt

= α2 t S2 t V2 t − μ2 t + ε2 t E2 t −C2 td2

E2 t ,

dI2 tdt

= ε2 t E2 t − μ2 t + γ2 t + δ2 t I2 t −C2 td2

I2 t ,

dR2 tdt

= γ2 t I2 t − μ2 t + ξ2 t R2 t −C2 td2

R2 t ,

dU2 tdt

= b2 t − ν2 t U2 t 1 −U2 t + L2 t + V2 t

M2 t− β2 t I2 t U2 t ,

dL2 tdt

= β2 t I2 t U2 t − ν2 t + η2 t L2 t ,

dV2 tdt

= η2 t L2 t − ν2 t V2 t

59

10 Complexity

dS2 tdt

=C1 td1

S∗1 t − μ2 t +C2 td2

S2 t ,

dU2 tdt

= b2 t − ν2 t U2 t 1 −U2 tM2 t

60

Furthermore, by the same way as in the above formodel 1 1, we can obtain the basic reproduction numberR20 = ρ ℒ2 for model (59), where the operator ℒ2 can bedefined similarly to ℒ1.

It is clear that model (59) is very similar with model1 1. Therefore, using the theory of limiting systems andthe nearly same arguments as in the above, we can provethe following theorems.

Theorem 6. Let R10 < 1 if R20 < 1, then disease-free periodicsolution E20 of model (59) is globally asymptotically stable formodel 1 2, and if R20 > 1, then E20 is unstable for model 1 2.

Theorem 7. If R10 < 1 and R20 > 1, then the disease in model1 2 is permanent.

3.3. The Third Patch. In this section, we discuss model1 3. For model 1 3, the discussions are nearly similarto model 1 2. We here directly state the followingmain results.

Theorem 8. Let S3 t , E3 t , I3 t , R3 t ,U3 t , L3 t , V3 tbe the solution of model 1 3 with initial condition (2). Then,S3 t , E3 t , I3 t , R3 t ,U3 t , L3 t , V3 t is nonnegativefor all t ≥ 0 and ultimately bounded, and when S3 0 > 0,E3 0 > 0, I3 0 > 0, R3 0 > 0,U3 0 > 0, L3 0 > 0 and V30 > 0, then S3 t , E3 t , I3 t , R3 t ,U3 t , L3 t , V3 t isalso positive for all t > 0.

Theorem 9. If R10 > 1 or R10 < 1, R20 > 1, the disease in model1 3 is permanent.

If R10 < 1 and R20 < 1, then any nonnegative solutionS2 t , E2 t , I2 t , R2 t ,U2 t , L2 t , V2 t of model 1 2converges to disease-free periodic solution S∗2 t , 0,0,0,U∗

2 t ,0,0 of model (59). Thus, we obtain the limiting sys-tem of model 1 3 as follows:

From a similar argument as in the above to model 1 2,we can obtain that model (61) has a unique disease-free peri-odic solution E30 = S∗3 t , 0,0,0,U∗

3 t ,0,0 , where S∗3 t ,U∗

3 t is the unique positive ω-periodic solution.

dS3 tdt

=C2 td2

S∗2 t − μ3 t S3 t ,

dU3 tdt

= b3 t − ν3 t U3 t 1 −U3 tM3 t

62

By the same way as in the above for model 1 1, we canobtain the basic reproduction number R30 = ρ ℒ3 for model(61), where the operator ℒ3 can be defined similarly to ℒ1.Thus, we further have the following results.

Theorem 10. Let R10 < 1 and R20 < 1, if R30 < 1, then disease-free periodic solution E30 of model (61) is globally asymptoti-cally stable for model 1 3, and if R30 > 1, then E30 is unstablefor model 1 3.

Theorem 11. If R10 < 1, R20 < 1 and R30 > 1, then the diseasein model 1 3 is permanent.

4. Numerical Simulation

In this section, we will present some examples and thenumerical simulations to confirm our theoretical results.The numerical values of most parameters in the followingexamples are adopted from Xiao et al. [8], Gao et al. [13],and Gaff et al. [14]. And the numerical simulation method

dS3 tdt

=C2 td2

S∗2 − α3 t S3 t V3 t − μ3 t S3 t + ξ3 t R3 t ,

dE3 tdt

= α3 t S3 t V3 t − μ3 t + ε3 t E3 t ,

dI3 tdt

= ε3 t E3 t − μ3 t + γ3 t + δ3 t I3 t ,

dR3 tdt

= γ3 t I3 t − μ3 t + ξ3 t R3 t ,

dU3 tdt

= b3 t − ν3 t U3 t 1 −U3 t + L3 t + V3 t

M3 t− β3 t I3 t U3 t ,

dL3 tdt

= β3 t I3 t U3 t − ν3 t + η3 t L3 t ,

dV3 tdt

= η3 t L3 t − ν3 t V3 t

61

11Complexity

of basic reproduction number is presented of Posny andWang [24]. We also give our numerical simulation withreference to [25].

In the following numerical examples, we choose the initialvalue S1 0 = 1800, E1 0 = 0, I1 0 = 50, R1 0 = 100, U1 0= 1000, L1 0 = 0, V1 0 = 100, S2 0 = 0, E2 0 = 0, I2 0 =0, R2 0 = 0, U2 0 = 8000, L2 0 = 0, V2 0 = 0, S3 0 = 0,

E3 0 = 0, I3 0 = 0, R3 0 = 0, U3 0 = 1500, L3 0 = 0and V3 0 = 0 in model 1 1– 1 3 which is given in [8].

Example 1. We assume that in model 1 1– 1 3, only thecapacity of mosquitoes is periodic. Thus, we take r1 t = 300,C1 t = C2 t = 200, αi t = 0 003, β1 t = 0 008, β2 t = β3

0 1 2 3 4 5 6 7 8 9 10 11 12 13(Year)

S1(t)S2(t)S3(t)

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V2(t)

V3(t)

500100015002000250030003500

Infe

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r)

(g)

Figure 1: The graphs of solution Si t , Ei t , Ii t , Ri t ,Ui t , Li t , Vi t , i = 1,2,3 of model 1 1– 1 3. From (c) and (g), we obtain that thedisease is permanent in model 1 1– 1 3 in three patches.

12 Complexity

t = 0 0008, εi t = 0 6, μ1 t = μ2 t = 0 0012, μ3 t = 0 4,ξi t = 0 005, d1 = 100, d2 = 800, δi t = 0 1, ηi t = 0 5, γit = 0 4, bi t = 0 45, νi t = 0 06, where i = 1,2,3,M1 t =

1000 1 + a1 sin 2πt/T2 + ϕ , M2 t = 8000 1 + a1 sin2πt/T2 + ϕ and M3 t = 1500 1 + a1 sin 2πt/T2 + ϕ ,where a1 = 0 3, T2 = 365, and ϕ = π.

00

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S2(t)

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E2(t)

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(g)

Figure 2: The graphs of solution Si t , Ei t , Ii t , Ri t ,Ui t , Li t , Vi t , i = 1,2,3 of model 1 1– 1 3. From (c) and (g), we obtain that thedisease is permanent in model 1 1– 1 3 in three patches.

13Complexity

It is clear that H1 and H2 are satisfied. By thenumerical calculations, we get the basic reproduction num-ber R10 = ρ ℒ1 = 2 1374 > 1 Therefore, from Theorems 3,

5, and 9, we obtain that the disease in model 1 1– 1 3 inthree patches is permanent. The numerical simulations aregiven in Figure 1.

0 1 2 3 4 5 6 7 8 9 10 11 12 13

(Year)

S1(t)

S2(t)

S3(t)

100200300400500600700800900

10001100

Susc

eptib

le p

opul

atio

n (h

ost)

(a)

0 1 2 3 4 5 6 7 8 9 10 11 12 13

50100150200250300350400450

(Year)

Expo

sed

popu

latio

n (h

ost)

E1(t)

E2(t)

E3(t)

(b)

0 1 2 3 4 5 6 7 8 9 10 11 12 13

50

100

150

200

250

300

(Year)

Infe

ctio

us p

opul

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n (h

ost)

I1(t)

I2(t)

I3(t)

(c)

0 1 2 3 4 5 6 7 8 9 10 11 12 13

50100150200250300350400450500

(Year)

Reco

vere

d po

pula

tion

(hos

t)

R1(t)

R2(t)

R3(t)

(d)

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50100150200250300350400450500

(Year)

Uni

nfec

ted

popu

latio

n (v

ecto

r)

U1(t)

U2(t)

U3(t)

(e)

0 1 2 3 4 5 6 7 8 9 10 11 12 13

200400600800

1000120014001600

(Year)

Expo

sed

popu

latio

n (v

ecto

r)

L1(t)

L2(t)

L3(t)

(f)

0 1 2 3 4 5 6 7 8 9 10 11 12 13

500100015002000250030003500

(Year)

Infe

ctio

us p

opul

atio

n (v

ecto

r)

V1(t)

V2(t)

V3(t)

(g)

Figure 3: The graphs of solution Si t , Ei t , Ii t , Ri t ,Ui t , Li t , Vi t , i = 1,2,3 of model 1 1– 1 3. From (c) and (g), we obtain that thedisease is permanent in model 1 1– 1 3 in three patches.

14 Complexity

Example 2. We only consider the capacity of mosquitoes,and the parameters incorporating both seasonal and festi-val impacts are periodic. In model 1 1– 1 3, we take r1 tin the following form:

r1 j =

r1 j − 1 + dr1, k − n < j ≤ k,

r1 j − 1 − dr1, k < j ≤ k +m,

r10, otherwise,

63

with k = 315, n =m = 10, dr1 = dr1 = 20, r10 = 300 and j ∈1,354 . We also take C1 t , C2 t , and μ3 t using thesame definition as in (63) with dCi = dCi = 98 i = 1, 2 ,dμ3 = dμ3 = 0 02, C10 = C20 = 200 and μ30 = 0 4. The otherparameters in model 1 1– 1 3 are chosen as in Example 1.

It is clear that H1 and H2 are satisfied. By thenumerical calculations, we get the basic reproduction num-ber R10 = ρ ℒ1 = 2 0891 > 1 Therefore, from Theorems 3,5, and 9, we obtain that the disease in model 1 1– 1 3 inthree patches is permanent. The numerical simulations aregiven in Figure 2.

Example 3. In model 1 1– 1 3, we take β1 t = 0 008 1 +a1 sin 2πt/T2 + ϕ , β2 t = β3 t = 0 0008 1 + a1 sin 2πt/T2 + ϕ , bi t = 0 45 1 + a1 sin 2πt/T2 + ϕ , ηi t= 0 5 1 + a1 sin 2πt/T2 + ϕ and νi t = 0 06 1 + a1 sin2πt/T2 + ϕ , where a1, T , and ϕ are given as in Example

1, where i = 1,2,3. We also take αi t , εi t , ξi t , δi t , γi t ,μ1 t and μ2 t using the same definition as in (63) with dαi = dαi = 0 0001, dεi = dεi = 0 01, dξi = dξi = 0 0001, dδi = dδi = 0 01, dγi = dγi = 0 04, dμ1 = dμ1 = dμ2 = dμ2 = 0 00005,αi0 = 0 003, εi0 = 0 6, ξi0 = 0 005, δi0 = 0 1, γi0 = 0 4 andμ10 = μ20 = 0 0012, where i = 1,2,3. We further choose r1 t ,

C1 t , C2 t and μ3 t as in Example 2 and Mi t i = 1,2,3 ,d1, and d2 as in Example 1.

It is clear that H1 and H2 are satisfied. By thenumerical calculations, we get the basic reproduction num-ber R10 = ρ ℒ1 = 2 1334 > 1 Therefore, from Theorems 3,5, and 9, we obtain that the disease in model 1 1– 1 3 inthree patches is permanent. The numerical simulationsare given in Figure 3.

Example 4. In model 1 1– 1 3, take βi t = 0 00008 1 +a1 sin 2πt/T2 + ϕ and bi t = 0 25 1 + a1 sin 2πt/T2+ ϕ for i = 1,2,3. We also take C1 t , C2 t , and αi t usingthe same definition as in (63) with dC1 = dC1 = dC2 = dC2= 98, dαi = dαi = 0 00001, C10 = C20 = 40 and αi0 = 0 0003,where i = 1,2,3. The other parameters in model 1 1– 1 3are chosen as in Example 3.

It is clear that H1 and H2 are satisfied. By thenumerical calculations, we get the basic reproduction num-bers R10 = ρ ℒ1 = 0 4158 < 1 and R20 = ρ ℒ2 = 5 2205 >1. Therefore, from Theorem 9, we obtain that the disease inmodel 1 1– 1 3 dies out in the first patch and is permanentin the second and third patches. The numerical simulationsare given in Figure 4.

Example 5. In model 1 1– 1 3 take β3 t = 0 0008 1 + a1sin 2πt/T2 + ϕ , η2 t = 0 1 1 + a1 sin 2πt/T2 + ϕ .We also take αi t and εi t i = 1,2,3 using the same defi-nition as in (63) with dα1 = dα1 = dα2 = dα2 = 0 00001, dα3 = dα3 = 0 0001, dε2 = dε2 = 0 001, dε3 = dε3 = 0 01, α10 =α20 = 0 00003, α30 = 0 003 and ε20 = 0 02ε30 = 0 9. The otherparameters in model 1 1– 1 3 are chosen as in Example 4.

It is clear that H1 and H2 are satisfied. By the numer-ical calculations, basic reproduction numbers are R10 =

0 1 2 3 4 5 6 7 8 9 10 11 12 13

100

200

300

400

500

600

(Year)

Infe

ctio

us p

opul

atio

n (h

ost)

I1(t)

I2(t)

I3(t)

(a)

0 1 2 3 4 5 6 7 8 9 10 11 12 13

500

1000

1500

2000

2500

(Year)

Infe

ctio

us p

opul

atio

n (v

ecto

r)

V1(t)

V2(t)

V3(t)

(b)

Figure 4: The graphs of components Ii t and Vi t i = 1,2,3 of the solution for model 1 1– 1 3. We see that the disease dies out in the firstpatch and is permanent in the second and third patches.

15Complexity

ρ ℒ1 = 0 1315 < 1, R20 = ρ ℒ2 = 0 7547 < 1, and R30 =ρ ℒ3 = 5 5289 > 1 Therefore, from Theorem 11, we obtainthat the disease in model 1 1– 1 3 dies out in first and sec-ond patches and is permanent in third patch. The numericalsimulations are given in Figure 5.

Example 6. In model 1 1– 1 3, take η2 t = 0 1 1 + a1 sin2πt/T2 + ϕ for i = 1,2,3. We also take αi t and ε2 t

using the same definition as in (63) with dαi = dαi =

0 00001 i = 1,2,3 , dε2 = dε2 = 0 001, αi0 = 0 00003 i = 1,2,3and ε20 = 0 02. The other parameters in model 1 1– 1 3are chosen as in Example 4.

It is clear that H1 and H2 are satisfied. By the nu-merical calculations, basic reproduction numbers are R10 =ρ ℒ1 = 0 1315 < 1, R20 = ρ ℒ2 = 0 7546 < 1, and R30 = ρℒ3 = 0 1664 < 1 Therefore, from Theorem 10, we obtainthat the disease in model 1 1– 1 3 in three patches diesout. The numerical simulations are given in Figure 6.

0 1 2 3 4 5 6 7 8 9 10 11 12 13

100

200

300

400

500

600

(Year)

Infe

ctio

us p

opul

atio

n (h

ost)

I1(t)

I2(t)

I3(t)

(a)

0 1 2 3 4 5 6 7 8 9 10 11 12 13

100

200

300

400

500

600

(Year)

Infe

ctio

us p

opul

atio

n (v

ecto

r)

V1(t)

V2(t)

V3(t)

(b)

Figure 5: The graphs of components Ii t and Vi t i = 1,2,3 of the solution for model 1 1– 1 3. We see that the disease dies out in first andsecond patches and is permanent in third patch.

0 1 2 3 4 5 6 7 8 9 10 11 12 13

5

10

15

20

25

30

35

40

(Year)

Infe

ctio

us p

opul

atio

n (h

ost)

I1(t)

I2(t)

I3(t)

(a)

0 1 2 3 4 5 6 7 8 9 10 11 12 13

10

20

30

40

50

60

70

80

90

(Year)

Infe

ctio

us p

opul

atio

n (v

ecto

r)

V1(t)

V2(t)

V3(t)

(b)

Figure 6: The graphs of components Ii t andVi t i = 1,2,3 of the solution ofmodel 1 1– 1 3. We see that the disease dies out in three patches.

16 Complexity

5. Discussion

From the analysis and simulations of the model, we foundthat the basic reproduction number Ri0 i = 1,2,3 which wascalculating by using the next infection operator approachdetermined whether the disease was persistent or extinct.

For the model 1 i, if Ri0 > 1, the disease is persistent inthe patch; if the time-dependent birth rate or the transmis-sion rate of disease of the disease or the rate of becominginfectious for livestock and female mosquitoes decreases,the basic reproductive number Ri0 < 1, then the disease diesout. This means that importation of livestock (especiallythe major festivals such as Eid al-AdhA) results in morelivestock and increases the chance of disease outbreaks; thetransportation of livestock from one patch to the other patchincreases the chance of disease outbreaks in the other pathbut reduces the chance of disease outbreaks in the one path.Also, as summer temperatures rise, rain is abundant, mos-quitoes are increasing, and disease carriers are increasingwhich increases the chance of disease outbreaks. That isabundances of hosts and vectors increase the probability oflarge disease outbreaks.

According to our work, diseases will occur during reli-gious festivals which hold in summer. But this is not entirelyconsistent with the actual situation. In the model, humanactivity is reflected by periodic parameters, and human com-partments are not directly considered, but real-time changesin human activity have different effects on the pattern of dis-ease. Since we assumed well-mixed populations of livestockin each patch and identical movement rates regardless theclinical stage of livestock during transportation, the modelmay overestimate the real situation. Implementations ofvarious kinds of vaccination strategies and unpredicted reli-gious festival preparations or other stochastic events will alsochange the pattern of disease outbreaks. These factors willalso be our future research work.

We see that some basic theoretical results are establishedfor the Rift Valley fever virus transmission model. The defi-ciency presented in [8] is made up. There is a shortage thatin this paper we do not establish a theoretical result on thelocal and global stability of the model when R0 > 1. It willbe studied in our future works.

Data Availability

The data used to support the findings of this study areincluded within the article.

Conflicts of Interest

The authors have declared that no competing interests exist.

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (Grant no. 11771373) and the Open-ing Project of Key Laboratory of Xinjiang (Grant no.2016D03022).

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