Chaos, Solitons and Fractals 137 (2020) 109870

Contents lists available at ScienceDirect

Chaos, Solitons and Fractals

Nonlinear Science, and Nonequilibrium and Complex Phenomena

journal homepage: www.elsevier.com/locate/chaos

Stability analysis of the hiv model through incommensurate

fractional-order nonlinear system

Bahatdin DA S BA S I

Kayseri University, Faculty of Applied Sciences, TR-38039, Kayseri, Turkey

a r t i c l e i n f o

Article history:

Received 24 February 2020

Revised 3 May 2020

Accepted 4 May 2020

Available online 11 May 2020

Keywords:

HIV mathematical model

Incommensurate fractional-order

differential equation

Stability analysis

Equilibrium points

34A08

34D20

34K60

92C50

92D30

a b s t r a c t

In this study, it is employed a new model of HIV infection in the form of incommensurate fractional

differential equations systems involving the Caputo fractional derivative. Existence of the model’s equi-

librium points has been investigated. According to some special cases of the derivative-orders in the

proposed model, the asymptotic stability of the infection-free equilibrium and endemic equilibrium has

been proved under certain conditions. These stability conditions related to the derivative-orders depend

on not only the basic reproduction rate frequently emphasized in the literature but also the newly ob-

tained conditions in this study. Qualitative analysis results were complemented by numerical simulations

in Matlab, illustrating the obtained stability result.

© 2020 Elsevier Ltd. All rights reserved.

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. Introduction

Even though fractional-order calculus (FOC) and differential

quations (FODEs) have nearly the same history as those of ordi-

ary differential equations (ODEs), they did not attract much at-

ention till recent decades [1] . FOC, expressed as a generalization

f ordinary differentiation and integration to arbitrary non-integer

rder and extensively used in different fields of science recently, is

branch of mathematical analysis [ 2 , 3 ]. Most important feature of

OC is memory concept. If the output of a system at each time t

epends only on the input at time t , then such systems are said to

e memoryless systems. Otherwise, if the system has to remem-

er previous values of the input to specify the current value of the

utput, then such systems are called memory systems [4] . In mod-

ling of various memory phenomena, it is mentioned that a mem-

ry process usually consists of two stages. One is short with per-

anent retention, and the other is governed by a simple model of

ractional derivative [5] . Numerous literature has been developed

n the applications of FODEs and their systems (FOSs), a new and

owerful tool that has recently been employed to model complex

tructures with nonlinear behavior and long-term memory [6–8] .

E-mail addresses: dasbasi_bahatdin@hotmail.com , bdasbasi@kayseri.edu.tr

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ttps://doi.org/10.1016/j.chaos.2020.109870

960-0779/© 2020 Elsevier Ltd. All rights reserved.

specially, biological systems are also rich source for mathematical

odeling through FOSs [9] .

Considering the recent mathematical modeling process of dis-

ases, many scientists have used FOC and FODEs to describe differ-

nt variety of diseases such as Ebola [10] , tuberculosis [11] , hep-

titis [12] , dengue fever [13] , MERS-Cov [14] , chickenpox [15] , Zika

irus [16] , measles [17] , rubella [18] , etc. Moreover, the modelling

f epidemic diseases assits understanding the main mechanisms

ffecting the spread of the disease, so that the control strategies

re proposed through the modeling process [19] . These models are

ombined under two main headings, as the first is modeling the

pread of infected individuals in a population [20] and the second

s modeling the density of the infectious pathogen such as virus,

acteria, etc. in an individual [ 21 , 22 ] as in this paper.

Viruses are the main cause of common human diseases such as

nfluenza, cold, chicken pox and cold sores. Currently, there exist

1 families of viruses expressed to cause diseases in humans. Some

f these diseases are very seriously infectious diseases such as AIDS

acquired immuno deficiency syndrome), Hepatitis, Herpes Sim-

lex, Measles, avian influenza, SARS and SARS- or MERS-like coron-

virus [23] . They have common features, such as they are all highly

athogenic to humans or livestock [24] . In particular AIDS, most se-

ere stage of HIV (human immunodeficiency virus) infection, is re-

arkable as a fatal disease [25] . Considering the World Health Or-

anization’s report on the global situation and HI V/AI DS trends in

2 B. DA S BA S I / Chaos, Solitons and Fractals 137 (2020) 109870

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2018, there was globally about 37.9 million people living with HIV ,

23.3 million people accessed antiretroviral therapy, 1.7 million peo-

ple newly became infected with HIV and 770 thousand people died

from AIDS-related illnesses [26] . HIV spreads only through certain

body fluids such as blood, semen, pre-seminal fluids, rectal fluids,

vaginal fluids, and breast milk, from an HIV -infected person. The

immune response plays an important role to control the dynamics

of viral infections such as HIV [21] . Mathematical models aimed

to understand the host-virus interaction in case of HIV can supply

non-intuitive information about the dynamics of the host response

to the viruses and they can also offer new ways for the theraphy.

In recent years, the HIV models with cure rate has received a great

deal of attention.

A general mathematical model considered the basic dynamics

of virus-host cell interaction was developed by Nowak et al. in [27] .

In their study, they formulated the HIV model by using the follow-

ing ODEs:

dx dt

= λ − dx − βx v dy dt

= βx v − ay − ρyz dv dt

= ky − u v dz d t

= cyz − bz

(1)

with positive initial conditions. In here, x (t) , y (t) , v (t) , and z(t) are

the concentrations of uninfected (susceptible) host cells, infected

host cells, free viruses, and CT L cells at time t , respectively. The

production of uninfected cells is at a constant rate, λ. When un-

infected cells encounter with free virus particles, they become in-

fected at a rate βx v . d and b are rates of the natural death of unin-

fected cells and CT L cells, respectively. The infected cells die at an

additional rate ay , which is the viral caused cell death (cytopathic-

ity or cytotoxicity). Infected cells produce new virus particles with

a rate ky , and the free virus particles that have been released from

the cells decay with a rate u . The proliferation rate of CT L cells in

the presence of infected cells is c. Finally, CT L cells cleans infected

cells with the ratio ρ from the host. They explained the stabilities

of the infection-free equilibrium and the positive equilibrium ac-

cording to the basic reproduction number of the virus. Thus, they

stimulated a model to work, aimed at interpreting experimental

data, and led to the development of a new field of study called as

viral dynamics.

Considering Eqs. (1) , several nonlinear models, given in [ 28 , 29 ]

through ODEs and [ 12 , 21 , 30–33 ] through FODEs, were studied

by researchers. In this sense, they analyzed qualitatively and/or

numerically their models by developing Eqs. (1) under vari-

ous assumptions. Also, these models include 3-dimensional time-

dependent variables, where T (t) , I(t) and V (t) represents the con-

centration of healthy CD 4 + T -cells at time t , the concentration of

infected CD 4 + T -cells at time t and the concentration of free HIV

at time t , respectively.

Mascio et al. [34] also considered the effect of antiviral drugs

for Eqs. (1) , and so, the efficacy of these drugs was estimated by

mathematical modeling for retroviruses such as HIV − 1 . While a

protease inhibitor causes infected cells to produce immature non-

infectious virus particles, a reverse transcriptase inhibitor effec-

tively blocks the successful infection of a cell. In this sense, they

assumed that when an antiretroviral drug such as a protease in-

hibitor or a reverse transcriptase inhibitor are applied to a patient

in steady state as the viral load falls. To model this fall, the effec-

tiveness of the drug is introduced into the model. Therefore, the

virus dynamics is reformed to the following ODEs:

dx dt

= λ − dx − ( 1 − εRT ) βx v I dy dt

= ( 1 − εRT ) βx v I − ay d v I dt

= ( 1 − εPI ) ky − u v I d v NI = εPI ky − u v NI

(2)

dt

ith positive initial conditions. Also, εRT and εPI are the effica-

ies of the reverse transcriptase inhibitor and protease inhibitor,

espectively, and v I and v NI , denote infectious and non-infectious

irions, respectively. This division of virions is because of the use of

rotease inhibitors. They performed stability analysis of the equi-

ibrium points of their model. Based on Eqs. (2) , several nonlinear

odels given in [35–37] through ODEs and [ 38 , 39 ] through FODEs

ave been developed to describe the dynamics of the HIV − 1

irus, which take into account the dynamics of the HIV infection

hrough antiretroviral therapies with different cell populations.

According to the derivative-orders in the system, FOSs can be

onsidered in two parts commensurate and incommensurate and

ommensurate FOSs is a special case of incommensurate FOSs.

herefore, studies on incommensurate FOSs as in [40–52] are in-

reasingly included in the literature.

The proposed model in this study has the following innova-

ions:

• It is assumed that both the infected cells and free virus parti-

cles have cleared by CT L cells and some neutralizing antibodies.

Also, immune system cells have logistic growth rules. • Model has created by using a incommensurate FOS in Caputo

sense. • In model, infected cells die at an additional rate called as the

natural death rate.

In the qualitative analysis, specific conditions on the develop-

ent of host cells (infected / uninfected) and viral particles (in-

ectious / non-infectious) are obtained, which are under the pres-

ure of the CT L response of the host and inhibitors. Additionally,

he numerical simulations of the model are given as a detailed de-

cription of the dynamical behaviors of the proposed system. To do

forementioned, the rest of the paper is organized as follows.

• In Section II, some preliminary definitions related to fractional

derivative operators are described. The asymptotic stability con-

ditions of the equilibrium point not only for incommensurate

but also for commensurate FOSs are given. • The Section III presents the mathematical formulation of the

proposed HIV infection model. • The Section IV discusses biological existence of the equilibrium

points for the proposed model as well as its stability analysis. • Section V suggests numerical simulations to support the quali-

tative analysis results of the proposed FOS. • In Section VI, the paper finishes with some concluding remarks.

. Preliminaries and definitions

efinition 2.1. Based on Riemann-Liouville definition, the α-th

( α > 0 ) order fractional derivative of function f (t) with respect to

is given by

d α f ( t )

d t α=

1

�( m − α)

d m

d t m

t

∫ 0 ( t − τ )

m −α−1 f ( τ ) dτ, (3)

where m is the first integer larger than α such that m − 1 ≤ α <

[53] .

efinition 2.2. Considering the Caputo sense definition, the α-th

( α > 0 ) order fractional derivative of function f (t) with respect to

is described as the following:

d α f ( t )

d t α=

{

1 �( m −α)

t

∫ 0

f ( m ) ( τ )

( t−τ ) α−m +1 dτ f or m − 1 < α < m

d m f ( t ) d t m

f or α = m

(4)

where m is the first integer larger than α [54] .

In the rest of this paper, the notation

d α

d t αrepresents the Caputo

ractional derivative of order α.

B. DA S BA S I / Chaos, Solitons and Fractals 137 (2020) 109870 3

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Fig. 1. The stable and unstable regions for incommensurate FOS in Eqs. (7) .

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emark 2.1. In this paper, we have consider the following nonlin-

ar FOS:

d αX ( t )

d t α= F ( t , X ( t ) ) , (5)

with suitable initial conditions X(0) = X 0 , where X(t) = x 1 (t) , x 2 (t) , . . . , x n (t) ] T ∈ R

n is the state vectors of Eqs (5) ,

= [ f 1 , f 2 , . . . , f n ] T ∈ R

n , f i : [ 0 , + ∞ ) x R

n → R , ( i = 1 , 2 , . . . , n ) ,

¯ = [ α1 , α2 , . . . , αn ] T is the multi-order of Eqs (5) , d αX(t)

d t α=

d α1 x 1 (t)

d t α1 ,

d α2 x 2 (t)

d t α2 , . . . ,

d αn x n (t) d t αn

] T [47] .

Throughout rest of the paper, it has been accepted that αi is a

ational number in the interval ( 0 , 1 | . efinition 2.3. In particular, if α1 = α2 = . . . = αn = α, then Eqs

5) can be written as

d αX ( t )

d t α= F ( t , X ( t ) ) . (6)

We call Eqs (6) as the commensurate FOS, otherwise, call Eqs.

5) as incommensurate FOS [45] .

efinition 2.4. The autonomous form of incommensurate FOS in

qs. (5) is shown as

d αX ( t )

d t α= F ( X ( t ) ) , (7)

with initial conditions X(0) = X 0 . Also, the equilibrium point of

qs. (7) is the point X = ( x 1 , x 2 , . . . , x n ) obtained from equations

( X ) = 0 .

emma 2.1. Eigenvalues λi for i = 1 , 2 , . . . , m ( α1 + α2 + . . . + αn )

f Eqs (7) are obtained from the charasteristic equation given as

et (d iag ( λm α1 , λm α2 , . . . , λm αn ) − J

(X

))= 0 (8)

where m is the smallest of the common multiples of the de-

ominators of rational numbers α1 , α2 , . . . , αn and J( X ) =

∂F ∂X

| X= X .f all eigenvalues λi obtained from Eq. (8) satisfy

arg ( λi ) | >

π

2 m

, (9)

hen X is asymptotically stable for incommensurate FOS in Eqs.

7) [55] .

The stable and unstable regions for incommensurate and com-

ensurate forms of Eqs. (7) are shown in Figs. 1 and 2 .

According to some special cases of fractional derivative orders,

he stability analysis has summarized below:

i Let α1 = α2 = . . . = αn = α < 1 in Eqs (7) . If all eigenvalues λi

for i = 1 , 2 , . . . , n obtained from

Det (

λI nxn − J (X

))= 0 (10)

atisfies either the Routh–Hurwitz stability conditions or the fol-

owing conditions:

arg ( λi ) | >

απ

2

f or i = 1 , 2 , . . . n, (11)

hen X is asymptotically stable point [56] . Here, the matrix I nxn is

n identity matrix.

Additionally, the charasteristic equation obtained from

q. (10) can be showed by

( λ) = λn + a 1 λn −1 + . . . + a n −1 λ + a n , (12)

here coefficients a i for i = 1 , ..., n are real constants. The Routh-

urwitz stability conditions for polynomial of degree n = 2 and 3

an be summarized as

1 , a 2 > 0 f or n = 2

1 , a 3 > 0 and a 1 a 2 > a 3 f or n = 3 . (13)

Above mentioned criteria has supplied necessary and sufficient

onditions for all roots of P (λ) to lie in the left half of the complex

lane [57] .

i Let α1 = α2 = . . . = αn = 1 in Eqs (7) . It is presumed that the

characteristic equation is as showing in Eq. (12) . If all eigenval-

ues λi for i = 1 , 2 , . . . , n obtained from Eq. (12) satisfy Routh-

Hurwitz stability conditions, then X is asymptotically stable

point [58] .

. The HIV model through incommensurate FOS

In this study, the new HIV infection model in an individual

ased on Eqs. (1) and (2) have been analyzed by incommensurate

OS. Let us denote by x (t) population size of uninfected (or suscep-

ible) cells of host at time t , by y (t) population size of the emerged

nfected cells when x (t) meet free viruses at time t , by v I (t) popu-

ation size of the infectious viral particles concentration at time t ,

y v NI (t) population size of the noninfectious viral particles con-

entration at time t and by z(t) population size of CT L response of

ost at time t . The recruitment of CT L responses have been classi-

ally associated with the control of HIV replication and CT L is very

mportant for the clearance of HIV . The newly produced virus par-

icles are separated into two parts as v I (t) and v NI (t) , to analyze

he effect of protease inhibitor. Therefore, we have incommensu-

ate FOS given by

d α1 x ( t ) d t α1

= γ − ρx − ( 1 − εRT ) βx v I d α2 y ( t )

d t α2 = ( 1 − εRT ) βx v I − ( ρ + ω ) y − δyz

d α3 v I ( t ) d t α3

= ( 1 − εPI ) ky − u v I − σv I z d α4 v NI ( t )

d t α4 = εPI ky − u v NI − σv NI z

d α5 z ( t ) d t α5

= rz (1 − z

C

)(14)

here t ≥ 0 , αi ∈ ( 0 , 1 ] for i = 1 , 2 , . . . , 5 and the parameters have

he properties given as

, ρ, β, ω, δ, k, u, σ, r, C ∈ R

+

< εRT < 1 and 0 < εPI < 1

(15)

We also have positive initial conditions x ( t 0 ) = x 0 , y ( t 0 ) = y 0 ,

I ( t 0 ) = v I 0 , v NI ( t 0 ) = v NI 0 and z( t 0 ) = z 0 . The meanings of biologi-

al parameters in Eqs. (14) are given in Table 1 .

The abovementioned scenario for Eqs. (14) has been graphically

emonstrated in Fig. 3 .

4 B. DA S BA S I / Chaos, Solitons and Fractals 137 (2020) 109870

Fig. 2. The stable and unstable regions for commensurate FOS form of Eqs. (7) .

Table 1

Meanings of parameters used in Eqs. (14) .

γ : Concentration of the uninfected target cells (x ) produced at a constant rate

ρ : Rate of natural death of uninfected cells and infected cells ( x and y )

β : Encounter rate of uninfected cells (x ) with free virus particles ( v I ) εRT : The efficacy of the therapy with reverse transcriptase inhibitors

εPI : The efficacy of the therapy with reverse protease inhibitors

δ : Removed rate of infected cells by CT L cells

ω : Rate of death of the infected cells due to cytopathicity or cytotoxicity of free virus particles

k : Rate of the produce of new virus particles by infected cells

u : Rate of natural death of viral particles

σ : Removed rate of virus particles by CT L cells

r : The proliferate rate of CT L cells

C : The carrying capacity of CT L cells

Fig. 3. Schematic demonstration of interaction among variables in Eqs. (14) .

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4. Qualitative analysis of the proposed HIV model

In this section, the threshold parameters given as R 0 and R 1 are

first introduced to ease the qualitative analysis. Then it is discussed

the existence and stability of equilibrias of the model in Eqs. (14) .

Definition 4.1. Let

R 0 =

γ βk ( 1 − εRT ) ( 1 − εPI )

ρ( u + σC ) ( ρ + ω + δC ) and R 1 =

u ( ρ + ω )

( u + σC ) ( ρ + ω + δC )

(16)

for reduce the complexity of operations. Considering In Eqs.

15) , it is clear that

< R 0 and 0 < R 1 < 1 . (17)

In here, the R 0 threshold parameter, sometimes called basic re-

roduction rate or basic reproductive ratio, is used to measure the

ransmission potential of a disease. Biologically, this parameter is

he average number of newly infected cells produced by a single

nfected cell when almost all cells are still uninfected. Also, the

arameter R 1 has been given only to reduce the processing com-

lexity in the analysis.

B. DA S BA S I / Chaos, Solitons and Fractals 137 (2020) 109870 5

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roposition 4.1. According to the biological existence conditions

f the equilibrium points of Eqs. (14) , it is obtained the following

esults:

i E 0 ( γρ , 0 , 0 , 0 , 0 ) always exists.

ii E 1 ( γ R 1 R 0 ρ

, γ ( R 0 −R 1 ) R 0 ( ρ+ ω ) ,

ρ( R 0 −R 1 ) β( 1 −εRT ) R 1

, γ k εPI ( R 0 −R 1 )

R 0 u ( ρ+ ω ) , 0 ) exists, when R 0 >

R 1 .

iii E 2 ( γρ , 0 , 0 , 0 , C ) is the infection-free equilibrium point and al-

ways exists.

iv E 3 ( γρ

1 R 0

, γ

[ ( ρ+ ω )+ δC ] ( 1 − 1

R 0 ) ,

ρ( R 0 −1 ) β( 1 −εRT )

, ρεPI ( R 0 −1 )

β( 1 −εRT )( 1 −εPI ) , C ) is the

endemic equilibrium point and exists when R 0 > 1 .

In here, R 0 and R 1 are in Definition 4.1 .

roof. The steady state solution of Eqs. (14) is a point

( x , y , v I , v NI , z ) satisfying the following equations: d α1 x d t α1

= 0 ,

d α2 y

d t α2 = 0 ,

d α3 v I d t α3

= 0 , d α4 v NI

d t α4 = 0 and

d α5 z d t α5

= 0 . Therefore, it is ob-

ained the system given by

− ρx − ( 1 − εRT ) β x v I = 0

( 1 − εRT ) β x v I − ( ρ + ω ) y − δy z = 0

( 1 − εPI ) k y − u v I − σv I z = 0

PI k y − u v NI − σv NI z = 0

z (1 − z

C

)= 0 .

(18)

By fifth equation of Eqs. (18) , it is z = 0 or z = C. Therefore we

ave the followings:

a) Firstly, let z = 0 . Then,

v I =

( 1 − εPI ) k

u

y and v NI =

εPI k

u

y (19)

re found from the third and fourth equations in Eqs. (18) . If

qs. (19) are written their place in the first and second equations

f Eqs. (18) , then it is acquired equations given as

− ρx − x y βk ( 1 −εRT ) ( 1 −εPI ) u

= 0

¯

(x βk ( 1 −εRT ) ( 1 −εPI )

u − ( ρ + ω )

)= 0

(20)

From the second equation of Eqs. (20) , y = 0 or x =u ( ρ+ ω )

βk ( 1 −εRT )( 1 −εPI ) are obtained.

1 Let y = 0 . We have x =

γρ by first equation of Eqs. (20) and v I =

v NI = 0 by Eqs. (19) . Hence, E 0 ( γρ , 0 , 0 , 0 , 0 ) is found. This point

always exists, because it is γρ > 0 in accordance with Ineqs (15) .

1 Let x =

u ( ρ+ ω ) βk ( 1 −εRT )( 1 −εPI )

. It is obtained y =( γβk ( 1 −εRT )( 1 −εPI ) −ρu ( ρ+ ω )

βk ( 1 −εRT )( 1 −εPI )( ρ+ ω ) ) from the first equation of

Eqs. (20) . Therefore, it is v I =

γ βk ( 1 −εRT )( 1 −εPI ) −ρu ( ρ+ ω ) β( 1 −εRT ) u ( ρ+ ω )

and v NI = εPI ( γβk ( 1 −εRT )( 1 −εPI ) −ρu ( ρ+ ω )

β( 1 −εRT )( 1 −εPI ) u ( ρ+ ω ) ) from Eqs. (19) .

Let us consider Eqs. (16) . In this case, we have

E 1 ( γ R 1 R 0 ρ

, γ ( R 0 −R 1 ) R 0 ( ρ+ ω ) ,

ρ( R 0 −R 1 ) β( 1 −εRT ) R 1

, γ k εPI ( R 0 −R 1 )

R 0 u ( ρ+ ω ) , 0 ) . If R 0 > R 1 , then

E 1 exists in regard to Ineqs. (15) and (17) .

a) On the other hand, let z = C. Eqs. (18) translates system given

by

γ − ρx − ( 1 − εRT ) β x v I = 0

( 1 − εRT ) β x v I − ( ρ + ω ) y − Cδy = 0

( 1 − εPI ) k y − u v I − Cσv I = 0

εPI k y − u v NI − Cσv NI = 0 .

(21)

The following equations:

I =

( 1 − εPI ) k

( u + Cσ ) y and v NI =

εPI k

( u + Cσ ) y (22)

s obtained from third and fourth equations in (21) . When the

qualities in Eqs. (22) have rewritten in first and second equations

f (21) ,it is obtained the system given as

− ρx − x y βk ( 1 −εRT ) ( 1 −εPI ) ( u + Cσ )

= 0

¯

(x βk ( 1 −εRT ) ( 1 −εPI )

( u + Cσ ) − ( ( ρ + ω ) + Cδ)

)= 0

(23)

By the second equation of Eqs. (23) , it is either y = 0 or x =( ( ρ+ ω )+ Cδ)( u + Cσ ) βk ( 1 −εRT )( 1 −εPI )

.

i Let us consider as y = 0 . In this case, it is obtained the fol-

lowing equations: v I = v NI = 0 by Eqs. (22) and x =

γρ by the

first equation of Eqs. (23) . Therefore, we obtain the equilibrium

point E 2 ( γρ , 0 , 0 , 0 , C ) . This equilibrium point is the infection-

free equilibrium point and it exists always according to Ineqs

(15) .

ii Lastly, let x =

( ( ρ+ ω )+ Cδ)( u + Cσ ) βk ( 1 −εRT )( 1 −εPI )

. Similarly to a)-

ii, v I =

kγ ( 1 −εPI ) ( u + Cσ )( ( ρ+ ω )+ Cδ)

( 1 − ρ( u + Cσ )( ( ρ+ ω )+ Cδ) βkγ ( 1 −εRT )( 1 −εPI )

) , v NI =kγ εPI

( u + Cσ )( ( ρ+ ω )+ Cδ) ( 1 − ρ( u + Cσ )( ( ρ+ ω )+ Cδ)

βkγ ( 1 −εRT )( 1 −εPI ) ) by Eqs. (22) and

y =

γ( ( ρ+ ω )+ Cδ)

( 1 − ρ( u + Cσ )( ( ρ+ ω )+ Cδ) βkγ ( 1 −εRT )( 1 −εPI )

) by the first equa-

tion of Eqs (23) are found. If the threshold parame-

ter R 0 in Eqs. (16) is taken into consideration, then

E 3 ( γρ

1 R 0

, γ ( 1 − 1

R 0 )

[ ( ρ+ ω )+ δC ] ,

ρ( R 0 −1 ) β( 1 −εRT )

, ρεPI ( R 0 −1 )

β( 1 −εRT )( 1 −εPI ) , C ) , called as the

endemic equilibrium point, is obtained. Considering Ineqs.

(15) and (17) , if R 0 > 1 , then this point exists.

Thus, the Proposition is proved.

roposition 4.2. Let us consider Eqs. (14) . For all αi ’s for i = , 2 , . . . , 5 are rational numbers between 0 and 1 . Assume m be

he lowest common multiple of the denominators m i ’s of αi ’s,

here αi =

k i m i

, ( k i , m i ) = 1 , k i , m i εZ

+ . Under aforementioned as-

umptions, it is provided the followings:

i E 0 is always unstable point.

ii When R 0 > R 1 , E 1 exists. However, it is an unstable point under

this condition.

iii Let us consider infection-free equilibrium point E 2 , which al-

ways exists. It is obtained the following cases:

• Let α2 � = α3 < 1 . If R 0 < 1 and eigenvalues obtained from

m ( α2 + α3 ) + ( u + σC ) λm α2 + ( ( ρ + ω ) + δC ) λm α3 + ( u + σC ) ( ( ρ +meet conditions given as | arg( λn ) | >

π2 m

for n = , 2 , . . . , m ( α2 + α3 ) , then it is asymptotically stable point

or Eqs. (14) .

• Let α2 = α3 = α ≤ 1 . If R 0 < 1 , it is asymptotically stable

point for Eqs. (14) .

i Let us consider endemic equilibrium point E 3 , which exists for

R 0 > 1 . The following cases are obtained.

• Let α1 � = α2 � = α3 < 1 . If eigenvalues obtained from

m ( α1 + α2 + α3 ) + ( u + σC ) λm ( α1 + α2 ) + ( ( ρ + ω ) + δC ) λm ( α1 + α3 ) + ρR

eet conditions given by | arg( λn ) | >

π2 m

for n = , 2 , . . . , m ( α1 + α2 + α3 ) , then it is asymptotically stable point

or Eqs. (14) .

• Let α1 = α2 = α3 = α ≤ 1 . It is asymptotically stable point for

Eqs. (14) .

roof. To perform stability analysis, the functions in Eqs. (14) are

etermined by

d α1 x d t α1

= f 1 ( x, y, v I , v NI , z ) = γ − ρx − ( 1 − ∫ RT ) βx v I d α2 y d t α2

= f 2 ( x, y, v I , v NI , z ) = ( 1 − ε RT ) βx v I − ( ρ + ω ) y − δyz d α3 v I d t α3

= f 3 ( x, y, v I , v NI , z ) = ( 1 − ε PI ) ky − u v I − σv I z d α4 v NI

d t α4 = f 4 ( x, y, v I , v NI , z ) = ε PI ky − u v NI − σv NI z

d α5 z d t α5

= f 5 ( x, y, v I , v NI , z ) = rz (1 − z

C

).

(24)

6 B. DA S BA S I / Chaos, Solitons and Fractals 137 (2020) 109870

−( u

ρ)

0 ] =

, 0 )

ω ) +

a

t

s

a

e

R

t

i

t

E

t

g

a

a

E

λ − R 0 ) = 0

aa

t

n

p

s

i

e

δC ) )

( λ

w

a

e

λ

s

l

λρ+

o

(

+

That jacobian matrix obtained from Eqs. (24) is

J =

⎛

⎜ ⎜ ⎜ ⎝

−( βv I ( 1 − εRT ) + ρ) 0 −βx ( 1 − εRT ) βv I ( 1 − εRT ) −( ( ρ + ω ) + δz ) βx ( 1 − εRT )

0 k ( 1 − εPI ) −( u + σ z ) 0 k εPI 0

0 0 0

For the jacobian matrix evaluated at equilibrium point

E j ( x , y , v I , v NI , z ) for j = 0 , 1 , 2 , 3 , the characteristic equation

have found from

( λm α4 + ( u + σ z ) )

(λm α5 − r

(1 − 2

z

C

))∣∣∣∣∣λm α1 + ( βv I ( 1 − εRT ) +

−βv I ( 1 − εRT ) 0

(25)

with respect to d et( d iag( λm α1 , λm α2 , λm α3 , λm α4 , λm α5 ) − J( E j ) ) .

i By Eq. (25) calculated at E 0 ( γρ , 0 , 0 , 0 , 0 ) , some of the eigenval-

ues are achieved from equations given as λm α1 = −ρ, λm α4 =−u and λm α5 = r and the remained eigenvalues are obtained

from

λm ( α2 + α3 ) + u λm α2 + ( ρ + ω ) λm α3 + ( u + σC ) ( ρ + ω + δC ) [ R 1 − R

where R 0 and R 1 are in Eqs. (16) . In here, λm α5 is positive real

number according to Ineqs (15) . Considering De-Moivre formulas,

the roots of λm α5 are obtained from

λn =

m α5 √

r cis

(2 ( n + 1 ) π

m α5

)εR

+ for n = 0 , 1 , 2 , . . . , ( m α5 − 1 ) ,

(26)

such that cisπ = cos π + i sin π, i =

√ −1 . Angles, | arg( λn ) | , at-

tained from Eq. (26) are found out as 0 , 2 πm α5

, 4 πm α5

, . . . , 2( m α5 −1 ) π

m α5 .

Clearly, these angles are not greater than

π2 m

, due to the definition

of derivative-orders in Eqs (14) . Considering Ineqs. (9) , the stability

condition is not supplied. Therefore, E 0 is unstable point.

i Let R 0 > R 1 . In this case, E 1 ( γ R 1 R 0 ρ

, γ ( R 0 −R 1 ) R 0 ( ρ+ ω ) ,

ρ( R 0 −R 1 ) β( 1 −εRT ) R 1

, γ k εPI ( R 0 −R 1 )

R 0 u ( ρ+ ω ) exists. When E 1 is calculated in Eq. (25) , the eigenvalues are

obtained from the equations given as λm α4 = −u and λm α5 = r

and the following determinant: ∣∣∣∣∣∣∣λm α1 +

(ρ(

R 0 R 1

− 1

)+ ρ

)0

γ ( 1 −εRT ) β

ρR 0 R 1

−ρ(

R 0 R 1

− 1

)λm α2 + ( ρ + ω ) − γ ( 1 −εRT ) β

ρR 0 R 1

0 −( 1 − εPI ) k λm α3 + u

∣∣∣∣∣∣∣ = 0 ,

where R 0 and R 1 are in Eqs. (16) . There is the similar state to the

unstability of E 0 , because λm α5 is positive real number. In this case,

E 1 is unstable point.

i By Eq. (25) evaluated at E 2 ( γρ , 0 , 0 , 0 , C ) , the eigenvalues obtain

from the following equations: λm α1 = −ρ, λm α4 = −( u + σC ) ,

λm α5 = −r and

λm ( α2 + α3 ) + ( u + σC ) λm α2 + ( ( ρ + ω ) + δC ) λm α3 + ( u + σC ) ( ( ρ +(27)

where R 0 is in Eqs. (16) . It is clearly that λm α1 , λm α4 , λm α5 εR

− in

accordance with Ineqs (15) . By De-Moivre formulas, we have

λn 1 =

m α1 √

ρcis ( 2 n 1 +1 ) πm α1

for n 1 = 0 , 1 , . . . , ( m α1 − 1 )

λn 2 =

m α4

√

( u + σC ) cis ( 2 n 2 +1 ) πm α4

for n 2 = 0 , 1 , . . . , ( m α4 − 1 )

λn 3 =

m α5 √

r cis ( 2 n 3 +1 ) πm α5

for n 3 = 0 , 1 , . . . , ( m α5 − 1 )

(28)

such that cisπ = cos π + i sin π, i =

√ −1 . Angles given as

| arg( λn 1 ) | =

πm α1

, 3 πm α1

, . . . so on, | arg( λn 2 ) | =

πm α4

, 3 πm α4

, . . . so on

0 0

0 −δy 0 −σv I + σ z ) −σv NI

0 r (1 − 2

z C

)

⎞

⎟ ⎟ ⎟ ⎠

.

0 β x ( 1 − εRT ) λm α2 + ( ( ρ + ω ) + δz ) −β x ( 1 − εRT )

−k ( 1 − εPI ) λm α3 + ( u + σ z )

∣∣∣∣∣ = 0 .

0

δC ) ( 1 − R 0 ) = 0 ,

nd | arg( λn 3 ) | =

πm α5

, 3 πm α5

, . . . so on, are greater than

π2 m

, due to

he definition of derivative-orders in Eqs (14) . In this respect, the

tability conditions of E 2 for these eigenvalues do not deteriorate

ccording to Ineqs. (9) . Accordingly, the roots of Eq. (27) must be

xamined. Let’s remember Descartes’ rule of sign [59] . If

0 < 1 , (29)

hen all coefficients of Eq. (27) are positive real number accord-

ng to Ineqs. (15) and (17) . Eq. (27) has no positive root, since

he sign change number of its coefficients is zero. In this sense,

q. (27) have not positive real root, and so, the roots of this equa-

ion are composed of negative real numbers and/or complex conju-

ate numbers. To show the stability of E 2 , these roots are examined

ccording to Ineqs. (9) .

As a consequence, we have the following results:

• Let α2 � = α3 < 1 . If eigenvalues obtained from Eq. (27) have met

conditions given as

| arg ( λn ) | >

π

2 m

f or n = 1 , 2 , . . . , m ( α2 + α3 ) , (30)

nd Ineq. (29) is satisfied, then infection-free equilibrium point

2 ( γρ , 0 , 0 , 0 , C ) is asymptotically stable.

• Let α2 = α3 = α ≤ 1 . When Eq. (27) is regulated to Eq. (12) , the

characteristic equation is obtained as

2 + ( ( ( ρ + ω ) + δC ) + ( u + σC ) ) λ + ( u + σC ) ( ( ρ + ω ) + δC ) ( 1

(31)

According to n = 2 in Ineqs (13) , it is

1 = ( ( ρ + ω ) + δC ) + ( u + σC ) 2 = ( u + σC ) ( ( ρ + ω ) + δC ) ( 1 − R 0 ) .

(32)

Considering Ineqs (15) , if Ineq. (29) is satisfied, then it is clear

hat a 1 > 0 and a 2 > 0 . The eigenvalues of Eq. (31) either are the

egative real number or the complex number with negative real

arts (Routh-Hurwitz Criteria). Consequently, E 2 is asymptotically

table in terms of Lemma 2.1 -i.

i Lastly, let

R 0 > 1 . (33)

In this case, E 3 ( γρ

1 R 0

, γ ( 1 − 1

R 0 )

[ ( ρ+ ω )+ δC ] ,

ρ( R 0 −1 ) β( 1 −εRT )

, ρεPI ( R 0 −1 )

β( 1 −εRT )( 1 −εPI ) , C ) ex-

sts. When Eq. (25) is evaluated at this equilibrium point, the

igenvalues are obtained from the following equation:

( λm α4 + ( u + σC ) ) ( λm α5 + r )

∣∣∣∣∣∣( λm α1 + ρR 0 ) 0

−ρ( R 0 − 1 ) ( λm α2 + ( ( ρ + ω ) +0 −( 1 − εPI ) k

(34)

here R 0 is in Eqs. (16) . Therefore, some of the eigenvalues is

cquired from λm α4 = −( u + σC ) and λm α5 = −r. Considering In-

qs (15) , we have λm α4 , λm α5 εR

−. That eigenvalues λm α4 andm α5 does not influence the stability conditions of E 3 , is previously

tated through De-Moivre formulas. Accordingly, we have the fol-

owing characteristic equation:

m ( α1 + α2 + α3 ) + ( u + σC ) λm ( α1 + α2 ) + ( ( ρ + ω ) + δC ) λm ( α1 + α3 ) +

R 0 λm ( α2 + α3 ) + ρR 0 ( u + σC ) λm α2 + ( ( ρ + ω ) + δC ) ρR 0 λ

m α3 +

ρ( R 0 − 1 ) ( u + σC ) ( ρ + ω + δC ) = 0

(35)

btained from determinant in Eq. (34) . Considering Ineqs. (15) ,

17) and (33) , the signs of the coefficients of Eq. (35) are + + + + + + , respectively. According to Descartes’ rule of sign, these

B. DA S BA S I / Chaos, Solitons and Fractals 137 (2020) 109870 7

Table 2

The existence and asymptotically stable conditions for infection-free and endemic equilibrium points of Eqs. (14) .

Equilibrium Point Namely The Existence Condition The Asymptotically Stable Condition

E 2 Infection-

free

equi-

lib-

rium

point

Always In case of α2 = α3 In other cases

If R 0 < 1 , If R 0 < 1 and Eq. (27) meet conditions Ineqs (30) ,

E 3 Endemic

equi-

lib-

rium

point

R 0 >

1

In case of α1 = α2 = α3 In other cases

Stable point. If Eq. (35) meet conditions Ineqs (36) ,

e

c

o

s

v

t

w

a

ρ

d

a

ρ ( u

a

u

b

s

a

4

l

d

l

t

w

E

p

t

b

t

t

e

o

s

t

p

i

E

e

p

c

o

a

c

α

o

5

p

i

i

R

a

a

r

i

w

λ 1 . 9072

λ

λ

−

λ

f

b

a

a

(

r

w

r

E

i

s

E

igenvalues are not compose of positive real numbers, since the

hange number of these signs is zero. Thus, eigenvalues consist

f negative real numbers and/or complex conjugate numbers. To

how the stability of E 3 , it must be demonstrated that the eigen-

alues achieved through Eq. (35) provide Ineqs (9) .

Consequently, we obtain the following results:

• Let α1 � = α2 � = α3 < 1 . If the eigenvalues obtained from

Eq. (35) meet the conditions given as

| arg ( λn ) | >

π

2 m

for n = 1 , 2 , . . . , m ( α1 + α2 + α3 ) , (36)

hen E 3 is asymptotically stable.

• Let α1 = α2 = α3 = α ≤ 1 . If Eq. (35) regulated with respect to

Eq. (12) , it is found the following characteristic equation:

λ3 + a 1 λ2 + a 2 λ + a 3 = 0 (37)

here a 1 = ( ( ρ + ω + δC ) + ( u + σC ) + ρR 0 ) ,

2 = ρR 0 ( ( ρ + ω + δC ) + ( u + σC ) ) and a 3 =

( R 0 − 1 )( u + σC )( ρ + ω + δC ) . In Eq. (37) , it is a 1 , a 2 , a 3 > 0 ,

ue to Ineqs. (15) , (17) and (33) . On the other hand, it is

1 a 2 − a 3 = [(( ρ + ω + δC )

2 + ( u + σC ) 2 +

( ρ + ω + δC ) ( u + σC )

)R 0 + ρR 0

2

(( ρ + ω + δC ) +

( u + σC )

)+

(38)

nd a 1 a 2 − a 3 > 0 . In accord with n = 3 in Ineqs (13) , all eigenval-

es of Eq. (37) are either negative real numbers or complex num-

ers having negative real parts. As a result, E 3 is asymptotically

table.

The proof is accomplished. The obtained results about stability

nalysis sum up briefly in Table 2 .

.1. Qualitative analysis results and discussion

For the proposed model in this study, the possible stable equi-

ibrium point is either infection-free equilibrium point E 2 or en-

emic equilibrium point E 3 . Also, it is clear that these two equi-

ibrium points are not stable under the same conditions according

o Table 2 . While the equilibrium point E 2 represents the state in

hich an individual is free of viral particles, the equilibrium point

3 shows the state in which an individual continues to fight viral

articles. In this sense, the infected individual heals or the infec-

ion continues.

Considering the derivative-orders of Eqs. (14) , the rational num-

ers α1 , α2 , α3 , α4 and α5 are derivative-orders in the system of

ime-dependent variables x (t) , y (t) , v I (t) , v NI (t) and z(t) , respec-

ively.

Provided that R 0 is less than one, the stability of infection-free

quilibrium point varies only depending on whether the derivative

rders α2 and α3 are equal or not. In this sense, the infection-free

tatus depends on the derivative-orders of equations expressing

he population size of infected cells of host and the infectious viral

article concentration in the proposed model. In case of α2 = α3 ,

nfection-free equilibrium point is stable and in case of α2 � = α3 , if

q. (27) meet Ineqs (30) , it is stable.

+ σC ) ( ρ + ω + δC )

]

Let’s assume that R 0 is greater than one. In this case, endemic

quilibrium point E 3 exists. The stability of this point varies de-

ending on the states α1 = α2 = α3 and α1 � = α2 � = α3 . In this

ontext, the endemic infection status depends on the derivative-

rders of equations expressing the population sizes of uninfected

nd infected cells of host and the infectious viral particle con-

entration in the proposed model. This point is stable in case of

1 = α2 = α3 , and it is stable if Eq. (35) meet Ineqs (36) in case

f α1 � = α2 � = α3 .

. Numerical simulation of the proposed HIV model

To support the results of the qualitative analysis of the pro-

osed HIV infection model in Eqs. (14) , we have given numerical

llustrations here. The parameter values used in model for numer-

cal study are given in Table 3 .

Numerical Study 1: From Table 3 , the basic reproduction rate

0 is calculated as 52 . 762 . Also, infection-free equilibrium point

nd endemic equilibrium point are found as E 2 ( 10 6 , 0 , 0 , 0 , 3 )

nd E 3 ( 1 . 8953e + 04 , 6 . 3912e + 03 , 2 . 3964e + 04 , 2 . 6627e + 03 , 3 ) ,

espectively. It is clear that R 0 > 1 . According to Table 2 , E 3 ex-

sts and E 2 is unstable point. Therefore, it can only be examined

hether E 3 is stable or not.

a Let [ αi ] = [ 4 5 4 5

4 5

19 20

9 10 ] for i = 1 , 2 , . . . , 5 . Because α1 = α2 =

α3 , E 3 is asymptotically stable in terms of Table 2 . Fig. 4 shows

this situation.

b Let [ αi ] = [ 1 2 3 4

5 8

19 20

9 10 ] for i = 1 , 2 , . . . , 5 . In here, it is α1 � =

α2 � = α3 . Also, it is m = 8 , which is the smallest of the common

multiples of the denominators of rational numbers α1 , α2 and

α3 . Therefore, Eq. (35) translates to

15 + 0 . 5276 λ11 + 2 . 4003 λ10 + 1 . 535 λ9 + 1 . 2664 λ6 + 0 . 8099 λ5 +(39)

From here, the solutions for eigenvalues are given as

1 = −1 . 0938 , λ2 = −0 . 9390 + 0 . 3856 i , λ3 = −0 . 9390 − 0 . 3856 i ,

4 = −0 . 6817 + 0 . 7612 i , λ5 = −0 . 6817 − 0 . 7612 i , λ6 =0 . 2719 + 1 . 1668 i , λ7 = − 0 . 2719 − 1 . 166 8 i , λ8 = − 0 . 04 80 +

0 . 9273 i , λ9 = −0 . 0480 − 0 . 9273 i , λ10 = 1 . 0688 + 0 . 7239 i ,

11 = 1 . 0688 − 0 . 7239 i , λ12 = 0 . 5729 + 0 . 7876 i , λ13 = 0 . 5729 −0 . 7876 i , λ14 = 0 . 8458 + 0 . 3362 i and λ15 = 0 . 8458 − 0 . 3362 i

or i =

√ −1 . It is satisfied Ineqs (36) due to Re { λ j } < 0 for

j = 1 , 2 , . . . , 9 . Thus, eigenvalues λ j do not impair the sta-

ility conditions of E 3 . In addition that, arg{ λ10 } = 33 . 94 0 ,

rg{ λ11 } = 326 . 06 0 , arg{ λ12 } = 54 . 19 0 , arg{ λ13 } = 305 . 81 0 ,

rg{ λ14 } = 21 . 80 0 and arg{ λ15 } = 338 . 20 0 . Considering Ineqs

36) , it is | arg( λk ) | >

π2 m

=

π16 = 11 . 25 0 for k = 10 , 11 , . . . , 15 . As a

esult, the endemic equilibrium point E 3 is asymptotically stable

ith respect to Table 2 . This situation is observed in Fig. 5 .

Numerical Study 2: Lastly, the values of the basic reproductive

atio and the equilibrium point are calculated as R 0 = 0 . 794 and

2 ( 10 6 , 0 , 0 , 0 , 10 ) , respectively. Considering Table 2 , E 3 is not ex-

sts due to R 0 < 1 . Consequently, only the E 2 point can or not be

table according to the different states of the derivative-orders in

qs. (14) .

8 B. DA S BA S I / Chaos, Solitons and Fractals 137 (2020) 109870

Table 3

Parameter values used in the numerical simulations of the optimal control for Eqs. (14) .

Notation Value Reference

γ 10 4 m l −1 da y −1 [60]

ρ 0 . 01 da y −1 [60]

β 0 . 0 0 0 024 m l −1 da y −1 [29]

εRT 0 . 1 ∗ and 0 . 8 ∗∗ Assumed

εPI 0 . 1 ∗ and 0 . 8 ∗∗ Assumed

ω 0 . 025 da y −1 [ 34 , 60 ]

δ 0 . 5 ml . da y −1 [60]

k 10 da y −1 [29]

u 2 . 4 da y −1 [29]

σ 0 . 0 0 01 ml . da y −1 Assumed

r 0 . 6 da y −1 [61]

C 3 ml ∗ and 10 ml ∗∗ Assumed

[ αi ] = [ α1 α2 α3 α4 α5 ] [ 4 5

4 5

4 5

19 20

9 10

] ∗, [ 1 2

3 4

5 8

19 20

9 10

] ∗ , [ 8 9

5 8

5 8

19 20

9 10

] ∗∗ [ 1 2

5 8

5 8

19 20

9 10

] ∗∗ Assumed

∗: Only the used value for first numerical study, ∗∗: Only the used value for second numerical study,

Other values are commonly used in numerical studies.

Fig. 4. According to [ αi ] = [ 4 5

4 5

4 5

19 20

9 10

] for i = 1 , 2 , . . . , 5 , the temporary trajectory of population sizes of the variables in Eqs. (14) with initial conditions

( 10 0 0 , 10 , 10 0 0 , 10 0 , 2 ) for values ∗ in Table 3 .

b

a

d

n

I

ε

p

E

l

endemic case would occur after at least 600 days.

a Let [ αi ] = [ 8 9 5 8

5 8

19 20

9 10 ] for i = 1 , 2 , . . . , 5 . Because R 0 < 1 and

α2 = α3 , E 2 is asymptotically stable in terms of Table 2 . This

situation shows in Figs. 6–7 .

b Lastly, let us consider as [ αi ] = [ 1 2 3 4

5 8

19 20

9 10 ] for i = 1 , 2 , . . . , 5 .

In here, it is α2 � = α3 . Because α2 =

3 4 and α3 =

5 8 , it is m = 8 .

Eq. (27) translates to

λ11 + 2 . 401 λ6 + 5 . 035 λ5 + 2 . 489 = 0 . (40)

Therefore, we obtain that λ1 = −0 . 1138 + 1 . 3105 i , λ2 =−0 . 1138 − 1 . 3105 i , λ3 = −1 . 0763 + 0 . 5670 i , λ4 = −1 . 0763 −0 . 5670 i , λ5 = −0 . 9362 , λ6 = −0 . 2140 + 0 . 8523 i , λ7 = −0 . 2140 −0 . 8523 i , λ8 = 1 . 1893 + 0 . 7449 i , λ9 = 1 . 1893 − 0 . 7449 i , λ10 =0 . 6829 + 0 . 4651 i and λ11 = 0 . 6829 − 0 . 4651 i . Since Re { λ j } < 0

for j = 1 , 2 , . . . , 7 and arg{ λ8 } = 31 . 88 0 , arg{ λ9 } = 328 . 12 0 ,

arg{ λ10 } = 34 . 65 0 and arg{ λ11 } = 325 . 35 0 , we have | arg( λk ) | >π

2 m

=

π16 = 11 . 25 0 for k = 1 , 2 , . . . , 11 . According to Table 2 , E 2 is

asymptotically stable as seen Figs. 8–9 .

5.1. Numerical simulation results and discussion

In this part, we have given some numerical simulations for the

presented model in Eqs. (14) . For this model, we used the values of

iological parameters and derivative-orders from Table 3 (Values ∗

nd

∗∗), and so the dynamics of Eqs. (14) with different initial con-

itions ( x 0 , y 0 , v I 0 , v NI 0 , z 0 ) are plotted in Figs. 4–9 . Two different

umerical studies have been done by using the values of Table 3 .

n this sense, different scenarios have been tried to be obtained.

In the first study, the values indicated by ∗, where εRT =PI = 0 . 1 and C = 3 ml , are used. While infection-free equilibrium

oint E 2 ( 10 6 , 0 , 0 , 0 , 3 ) always exists, endemic equilibrium point

3 ( 1 . 8953e + 04 , 6 . 3912e + 03 , 2 . 3964e + 04 , 2 . 6627e + 03 , 3 ) bio-

ogically exits due to R 0 = 52 . 762 > 1 .

• For derivative-orders [ 4 5 4 5

4 5

19 20

9 10 ] ( α1 = α2 = α3 ) , E 3 has

been shown to meet asymptotic stability conditions for

Eqs. (14) according to Table 2 . In this context, the Fig. 4 has

drawn. Approximately at least 400 days later, the infection pro-

cess will approach a positive equilibria and the disease will

continue endemically. • Eqs. (14) has been considered for derivative-orders

[ 1 2 3 4

5 8

19 20

9 10 ] ( α1 � = α2 � = α3 ) . As a result of providing the

related conditions in Table 2 , the stability of E 3 was shown in

Fig. 5 . In this sense, it was graphically represented that this

B. DA S BA S I / Chaos, Solitons and Fractals 137 (2020) 109870 9

Fig. 5. According to [ αi ] = [ 1 2

3 4

5 8

19 20

9 10

] for i = 1 , 2 , . . . , 5 , the temporary trajectory of population sizes of the variables in Eqs. (14) with initial conditions

( 10 0 0 , 10 , 10 0 0 , 10 0 , 2 ) for values ∗ in Table 3 .

Fig. 6. According to [ αi ] = [ 8 9

5 8

5 8

19 20

9 10

] for i = 1 , 2 , . . . , 5 , the temporary trajectory of population sizes of uninfected cells in Eqs. (14) with initial conditions

( 10 0 0 , 10 , 100 , 100 , 2 ) for values ∗∗ in Table 3 .

ε

e

r

o

p

a

6

fi

f

In the second study, it is used the values indicated by ∗∗, where

RT = εPI = 0 . 8 and C = 10 ml . Here, there is a situation where the

fficacy of the therapy with reverse transcriptase inhibitors and

everse protease inhibitors is increased and the carrying capacity

f CT L response of host is greater too. Infection-free equilibrium

oint and basic reproduction rate are found as E 2 ( 10 6 , 0 , 0 , 0 , 10 )

nd R 0 = 0 . 794(< 1) , respectively.

• Let us considered the derivative-orders as

[ 8 9 5 8

5 8

19 20

9 10 ] ( α2 = α3 ) . Taking into consideration Table 2 , it

is shown that E 2 meet asymptotic stability conditions, and thus

Figs. 6–7 is drawn. In about 200 days, while infected cells and

viral particles disappear, CT L response of host approaches its

carrying capacity. On the other hand, it takes a long time for

the uninfected cells to approach its equilibrium value. • Derivative-orders were considered as [ 1 2

3 4

5 8

19 20

9 10 ] ( α2 � = α3 ) .

In here, it was shown that the conditions related to the stability

of E 2 in the Table 2 were satisfied, and it was supported by Figs.

8–9 . As can be seen from these figures, clearing the infection

takes at least 200 days.

. Conclusions

In this study, we proposed the new HIV model including the

ve time-dependent variables: the host cells as susceptible and in-

ected, the viral particles as infectious and noninfectious and the

10 B. DA S BA S I / Chaos, Solitons and Fractals 137 (2020) 109870

Fig. 7. According to [ αi ] = [ 8 9

5 8

5 8

19 20

9 10

] for i = 1 , 2 , . . . , 5 , the temporary trajectory of population sizes of the variables exceptly uninfected cells in Eqs. (14) with initial

conditions ( 10 0 0 , 10 , 100 , 100 , 2 ) for values ∗∗ in Table 3 .

Fig. 8. According to [ αi ] = [ 1 2

3 4

5 8

19 20

9 10

] for i = 1 , 2 , . . . , 5 , the temporary trajectory of population sizes of the variables in Eqs. (14) with initial conditions

( 10 0 0 , 10 , 10 0 0 , 10 0 , 2 ) for values ∗∗ in Table 3 .

e

O

o

{3 .

{= α3 .

If ( 4 . 20 ) meet conditions ( 4 . 21 ) in other cases .

host’s immune system response as CT L cells. This model proposed

in Eqs. (14) is the form of incommensurate fractional-order nonlin-

ear system (FOS) with the Caputo fractional derivative. In addition,

the derivative-orders of these dependent variables in the system

are as follows α1 , α2 , α3 , α4 and α5 in interval ( 0 , 1 ] , respectively.

Considering the HIV models in the literature, the main innovations

in our model are follows:

• We built the model by using incommensurate FOS consisting of

five equations. • We have assumed that CT L cells of the host have the effect of

destroying both infected cells and viral particles, and CT L cells

have followed the logistic growth model.

Our model exhibits two equilibria, namely, disease-free equilib-

rium and the endemic equilibrium points. In general, the HIV mod-

ls in literature trying to explain the infection process with the

NLY parameter basic reproduction rate R 0 . By qualitative analysis

f our model, what we found are as follows

• Disease-free equilibrium point always exists and is asymptoti-

cally stable,

If R 0 < 1 in case of α2 = αIf R 0 < 1 and ( 4 . 12 ) meet conditions ( 4 . 15 ) in other cases .

• Endemic equilibrium point exists when R 0 > 1 . This point is

asymptotically stable,

If R 0 > 1 ( also the existence condition ) in case of α1 = α2

B. DA S BA S I / Chaos, Solitons and Fractals 137 (2020) 109870 11

Fig. 9. According to [ αi ] = [ 1 2

3 4

5 8

19 20

9 10

] for i = 1 , 2 , . . . , 5 , the temporary trajectory of population sizes of the variables in Eqs. (14) with initial conditions

( 10 0 0 , 10 , 10 0 0 , 10 0 , 2 ) for values ∗∗ in Table 3 .

c

n

t

y

i

s

D

c

i

C

V

v

R

[

[

[

We have achieved the abovementioned stability conditions that

an be seen in Table 2 . To support the stability analysis results, the

umerical simulations of our model have been made in the light of

he parameter values taken from the literature. The obtained anal-

sis results of model demonstrate the simplicity and the productiv-

ty of this model, when the progress of the infection is considered.

In future studies, the progress of the infection may be de-

cripted better by considering such as the following factors:

• acquiring disease through gene transfer between infected and

uninfected cells, • effect of regional conditions (For example, the progression

times of HIV between people living in Europe and African con-

tinent can vary considerably.) and

• different inhibitor treatment strategies.

eclaration of Competing Interest

The authors declare that they have no known competing finan-

ial interests or personal relationships that could have appeared to

nfluence the work reported in this paper.

RediT authorship contribution statement

Bahatdin DA S BA S I: Conceptualization, Methodology, Software,

alidation, Formal analysis, Writing - original draft, Writing - re-

iew & editing, Visualization, Supervision.

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