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1-s2.0-S1007570414002123-main

The stability analysis of a general viral infection model

with distributed delays and multi-staged infected

progression

Jinliang Wang a ,1,Shengqiang Liu b ,?,2

a School of Mathematical Science,Heilongjiang University,Harbin 150080,China

b

The Academy of Fundamental and Interdisciplinary Science,Harbin Institute of Technology,3041#,2Yi-Kuang Street,Harbin 150080,China

a r t i c l e i n f o Article history:

Received 3May 2013

Received in revised form 27April 2014Accepted 30April 2014

Available online 15May 2014Keywords:

Distributed delays General incidence Virus dynamics Global stability

Lyapunov functional

a b s t r a c t

We investigate an in-host model with general incidence and removal rate,as well as distributed delays in virus infections and in productions.By employing Lyapunov functionals and LaSalle’s invariance principle,we de?ne and prove the basic reproductive number R 0as a threshold quantity for stability of equilibria.It is shown that if R 0>1,then the infected equilibrium is globally asymptotically stable,while if R 061,then the infection free equilibrium is globally asymptotically stable under some reasonable assumptions.Moreover,n t1distributed delays describe (i)the time between viral entry and the transcription of viral RNA,(ii)the n à1-stage time needed for activated infected cells between viral RNA transcription and viral release,and (iii)the time necessary for the newly produced viruses to be infectious (maturation),respectively.The model can describe the viral infection dynamics of many viruses such as HIV-1,HCV and HBV.

ó2014Elsevier B.V.All rights reserved.

1.Introduction

From the initial models of Anderson and May [1,21],several modeling frameworks toward to systematic method to describe the vivo infection process such as human immunode?ciency virus type 1(HIV-1),hepatitis C virus (HCV)and hep-atitis B virus (HBV)have been developed in the literature (see,e.g.,[23–30]).Classical HIV models have played a signi?cant role in the development of a better understanding of the disease and various drug therapy strategies used against it.It is well known that viruses are intracellular parasites that depend on the host cells to survive and duplicate.The parasite needs some target cells to proliferate,for example,Lymphocyte T cells for HIV-1(see,e.g.,[23]and references cited therein).These in-host models are useful for exploring possible mechanisms and outcomes of the viral infection process.It is often of interest from both a mathematical and a biological point to explore possible mechanisms of viral infection process and to estimate main parameter values such as virion clearance rate,life span of infected cells,average viral generation time and shoulder time (see,e.g.,[25,26,30]).

The question of whether Hopf bifurcations in in-host models are the result of intracellular delays and incidence form for the infection,or a combination of both is often of interest to be investigated.In most in host models,a bilinear incidence rate b x et Tv et Twith a constant b >0is frequently used (see,e.g.,[4,13,17,24,26,28]).In reality,bilinear incidence rate associated https://www.doczj.com/doc/1c17042590.html,/10.1016/https://www.doczj.com/doc/1c17042590.html,sns.2014.04.027

1007-5704/ó2014Elsevier B.V.All rights reserved.

?Corresponding author.Tel.:+8645186402559.

E-mail addresses:jinliangwang@https://www.doczj.com/doc/1c17042590.html, (J.Wang),sqliu@https://www.doczj.com/doc/1c17042590.html, (S.Liu).

1

The research was partially supported by National Natural Science Foundation of China (No.11201128),and the Science and Technology Innovation Team in Higher Education Institutions of Heilongjiang Province (No.2014TD005).2

The research was partially supported by the Fundamental Research Funds for the Central Universitys (No.HIT.IBRSEM.A.201401)and the NNSF of China (No.11301453).

with the mass action principle is insuf?cient to describe the infection process in detail,and some nonlinear transmissions

were proposed(for details one can refer[2,5,32]).We summarize previous studies in the literature related to incidence functions.

(i)Korobeinikov[14,15]assumed that the incidence rate is given by a general nonlinear function of target cells and free

virus concentration Fex;vT,which satis?es some conditions.

(ii)Li and Shu[18]studied the dynamics of an in-host model with general form of target-cell dynamics,nonlinear inci-dence hex;vTand distributed intracellular delays.

(iii)Huang et al.[11]have studied a class of virus infection model with a nonlinear incidence rate Fex;vT,a removal rate of infected cells,and two discrete delays.

(iv)Georgescu and Hsieh[5]considered the nonlinear incidence rate cexTfevT,where cexTdenotes the contact rate function at concentration of the susceptible cells x and fevTdenotes the force of infection by virus at concentration v.

The removal rate of infected cells may be different under different conditions and different virulence.The typical assump-tion is that infected cells die at a linear rate d y and viruses are produced at a rate N d y,where N represents the average num-ber of virus particles produced by an infected T cell.Considering infected cells with saturated loss rate,Song and Cha[33] introduced a term py=e1tayTdescribing the removal rate of infected cells.Without loss of generality,Georgescu and Hsieh [5]denoted the removal of infected cells by c3peyT,which includes the mortality of infected cells,and the function p is non-linear function.It is assumed in[11]that the death rate of the infected cells depends on their concentration,that is,on the function aGeyT,where a is a positive constant.

On the other hand,in virus dynamics,time delays are usually incorporated for the purpose of accurate representations of the biological phenomena.Time delays are intrinsic to the viral infection and replication processes.It has been assumed that new virus particles are produced after the initial infection within a time interval and this leads mathematical models to be governed by delay differential equations.Two kinds of delays are encountered in the literature,namely discrete constant delays and continuous distributed delays.It is advocated in[22]that?xed delays are not biologically realistic,and in the context of compartmental systems continuous probability functions of delays are far more important than discrete delays

[12].Many works have investigated whether intracellular delays can cause periodic oscillations through Hopf bifurcations,

i.e.,the effect of the time delay on the stability of the endemically infected equilibrium.For details,we refer readers to ref-erences[3,9,18,35]and the references therein.The mathematical analysis of these models is necessary to obtain an compre-hensive view for in-host model.In particular,the global stability of a steady state for these models will give us a detailed information and enhances our understanding about the virus dynamics[27].

It was pointed in[6]that the progression of the productively-infected lymphocyte from infection to death should be lik-ened to an aging process,in which death probability sharply increases when a certain age is reached.Viral cytopathicity, anti-HIV cytotoxic T lymphocytes(CTLs),or activation-associated apoptosis cause cell death at that stage.

Moreover,as reported by Gross et al.[7]and Zhou et al.[36]initially,the newly released virions are immature;subse-quently,they undergo a proteolytic maturation step to become infectious(mature).An immature virus is not infectious and may fail to enter into target cells,and only a mature virus may contact the cells.To model the delay between viral RNA transcription and viral release and further maturation,we assume that target cells remains in the activated state for

a certain period,passing through n stages y

1

!y2!...!y n before producing virus particles.In this paper,we assume that the infected cells appear after the initial infection within a time interval x.Let x be the random variable that describes the time between viral entry and the transcription of viral RNA and x is distributed according to a probability distribution fexT:?0;h !Rt,where h is the limit superior of the infection delay.In addition,we assume that a time is needed for the virus production after the virions enter a cell(see also[18]).Thus,we also assume the production delay s i be the random variable that is the period among these n-stages events with a probability distribution f ies iT:?0;h i !Rt,where h i is the limit superior of the stages delay.

Let eàk i s i;i?1;2;...;n be the survival probability of an infected cells in the i th stage.Thus the average time spent by an infected cell in the i th stage is given by1=k i.It is assumed that the death rate of infected cell depends on their concentration, k i G iey iT,where k i is a positive constant.For this case,on average~k i=k i represents the transfer rate of productively infected cell to next stage during its i-stage lifetime.

Due to ongoing viral replication in HIV infection proceeding,the time from productive infected of CD4cells to their death, is modeled by dividing the process into several short stages.Motivated by recent studies by[5,11,18,27],incorporating?nite intracellular delays and nonlinear incidence and removal rate into Grossman’s ongoing viral replication model[6],we arrive at the new model of delay differential equations,

x0etT?sàdxetTàFexetT;vetTT;

y0 1etT?

R h

fexTFexetàxT;vetàxTTd xàk1G1ey1etTT;

y0 i etT?~k ià1

R h ià1

f ià1es ià1TG ià1ey ià1etàs ià1TTd s ià1àk i G iey ietTT;

v0etT?~k n R h n

0f nes nTG ney netàs nTTd s nàl vetT;i?2;3;...;n;

8 >>> >>>

>< >>> >>> >:e1:1T

264J.Wang,S.Liu/Commun Nonlinear Sci Numer Simulat20(2015)263–272

where x et T;y i et Tand v et Talso represent the population of target cells,n -stage infected cells and free virus,respectively.All recruitment is into target cells,and occurs at a constant rate s ,while d is the speci?c death rate.On adequate contact with virus,target cells becomes activated but not produce viral particles.The concentration of newly infected target cells per unit time is given by F ex et T;v et TT.

The terms on the right-hand of (1.1)can be explained as follows.As in [18],we assume that cells infected at time t will be activated and produce viral materials at time t tx .Hence the number of actively infected cells at time t is given by

Z

h

f ex TF ex et àx T;v et àx TTd x :

To model the delay between viral RNA transcription and viral release and maturation,we assume that before producing virus particles,actively infected cells need to pass n -stages,and each stage has a production delay s i with distribution f i es i Ton the interval ?0;h i ;i ?1;2;...;n .Thus,the concentration of the i th stage infected cells and the mature viral particles pro-duced at time t are given by

~k i à1

Z

h i à1

f i à1es i à1TG i à1ey i à1et às i à1TTd s i à1;

i ?2;3;...;n

and

~k

n Z

h n

f n es n TG n ey n et às n TTd s n ;

respectively.Free viruses are cleared from the system at a rate l .

Generally,it is not easy to investigate delay differential equations with multiple time delays and nonlinearity.The Lyapu-nov functional and the LaSalle Invariance Principle theorem in [8,16]provide a direct and effective method to establish glo-bal dynamical properties for the nonlinear system of delay differential equations.Recently,McCluskey [19,20],Li and Shu [17,18]and Huang et al.[10,11]elegantly developed a class of Lyapunov functionals for delay epidemic models and viral infection models.

The main purpose of this paper is to establish the complete global dynamics of (1.1).The model (1.1)incorporates with a nonlinear incidence rate,a nonlinear removal rate of infected cells,and two distributed time delays.Stability analysis for (1.1)with discrete intracellular delay was carried out by Huang et al.[11],but the stability analysis of (1.1)still remains open.Based on works by Nakata [27],Li and Shu [18]and Georgescu and Hsieh [5],we obtain that,the basic reproductive number is de?ned and proved to be threshold parameter for the global stability of infection free equilibrium and infected equilibrium by constructing suitable Lyapunov functionals and LaSalle’s invariance principle.The global stability conditions,which depend only on the properties of nonlinear functions are established.Moreover,it is shown that if R 061,then the infection free equilibrium E 0is globally asymptotically stable,and the virus will be cleared,while if R 0>1,all positive solu-tions converge to the unique infected equilibrium E ?.The global stability of E ?rules out any possibility for Hopf bifurcations and existence of sustained oscillations.

The paper is organized as follows.In Section 2,some preliminaries and hypothesis are given.Following the technique of constructing Lyapunov functionals,we show that the global asymptotic stability of the model depends only on the basic reproductive number under some assumptions in Section 3.In last section,we give some discussions and conclusions about our model.

2.Preliminaries and assumptions 2.1.Well-posedness

To investigate the global dynamics of (1.1),we set a suitable phase space.Let

r ?max f h ;h i g ;i ?1;2;...;n .Denote by

C ?C e?àr ;0 ;R n t2

Tthe Banach space of continuous functions mapping the interval ?àr ;0 into R n t2Twith the sup-norm.

The nonnegative cone of C is de?ned by C t?C e?àr ;0 ;R n t2

tT.

The initial condition of (1.1)are given as

x eh T?u 1eh T;y i eh T?u 2i eh T;v eh T?u 3eh T;for h 2?àr ;0 ;

e2:2T

where u ?eu 1;u 21;...;u 2n ;u 3TT 2C t.The sup-norm de?ned in C takes the form k u k ?sup àr 6h 60j u eh Tj .Standard theory of functional differential equations [8]can be used to show that solutions of (1.1)exist and are differentiable for all t >0.Throughout this paper,we assume the functions F ex ;v Tand G i ey Tin (1.1)are always positive,differentiable,monotoni-cally increasing for all x >0;y >0;v >0and F ex ;v Tis concave with respect to v ,that is,satisfy (H 1)F ex ;v T;F 0x ex ;v T;F 0v ex ;v Tand àF 00vv ex ;v T

are positive

for

any

x >0

and

v >0.

Furthermore,

F e0;v T?F ex ;0T?0;F 0

v ex ;0T>0for x >0and v >0.eH 2TG i e0T?0;G 0i ey i T>0for y i >0;i ?1;2;...;n .

J.Wang,S.Liu /Commun Nonlinear Sci Numer Simulat 20(2015)263–272265

2.2.Basic reproduction number

In system (1.1),without infection ey i ?0;v ?0T;i ?1;2;...;n ,uninfected target cells stabilize at the equilibrium x 0?s =d .The basic reproductive number R 0for in-host models (see,e.g.,Perelson et al.[31]and van den Driessche and Watmough [34])measures the average number virus-producing target cells produced by an single virus-producing target cell during its entire infectious period in an entirely uninfected target-cell population.The basic reproductive number for (1.1)is given by

R 0?Y n i ?1~k i

k i Z h 0f ex Td x á@F ex 0;0T@v 1l áY n i ?1Z h i

f i es i Td s i :e2:3T

Each virus-producing target cell produces Q n i ?1~

k i

i virus particles over its entire infectious period Q n i ?1k i ,and each virus infects R h 0f ex Td x á@F ex 0;0T@v 1

l target cells over the life span 1=l .Among newly infected target cells,a fraction of Q n i ?1R h i

0f i es i Td s i sur-vives the n -stages delay period s i to become virus producing.

2.3.Existence of the equilibria

System (1.1)always has an infection free equilibrium E 0?ex 0;0;...;0;0T;x 0?s =d .In addition to infection free equilib-rium,when R 0>1,(1.1)has an infected equilibrium E ??ex ?;y ?1;...;y ?n ;v ?T,which satis?es

0?s àdx ?

àF ex ?;v ?T;0?R h 0

f ex TF ex ?;v ?Td x àk 1G 1ey ?

1T;0?~k i R h i 0f i es i TG i ey ?i Td s i àk i t1G i t1ey ?i t1T;i ?1;2;...;n à1;

0?~k n R h n 0

f n es n TG n ey ?n Td s n àl v ?:8>>>>><>>>>>:e2:4T

In what follows,we give a lemma which establishes the existence condition of an infected equilibrium.Proceeding,we

give the assumptions on the function G i ey i T:

eH 3Tlim y i !t1G i ey i T?W 6t1;W P Q n à1i ?1~k i áR h 0

f ex Td x áQ n à1i ?1R h i

0f i es i Td s i ás Q n à1i ?1

k i :Lemma 2.1.Suppose that the functions F ex ;v Tand G i ey i Tsatisfy the conditions eH 1T;eH 2Tand eH 3T.If R 0>1,then system (1.1)

has an infected equilibrium

E ?ex ?;y ?1;y ?2;...;y ?n ;v ?

T:

Proof.Let the right-hand sides of three equations in system (1.1)equal zero.It follows that

s àdx ?F ex ;v T?k 1G 1ey 1TR h 0f ex Td x ?k 1k 2G 2ey 2TR h 0f ex Td x á~k 1R h 10

f 1es 1Td s 1

...;?Q n

i ?1k i l v Q n i ?1~k i áR h 0f ex Td x áQ n i ?1R h i 0f i es i Td s i :e2:5T

For the sake of clarity,we denote by

Y n i ?1k i

~k

i

?k ;Z

h

f ex Td x á

Y n i ?1

Z

h i

f i es i Td s i ?A ;

Y n i ?1

Z

h i

f i es i Td s i ?B :

After substituting the expression of x by v ,we obtain the following equation for v :

H ev T?F sA àk l v dA ;v àk l v

A ?0:

It is obvious that H e0T?0,and when

v ?v 0?sA =k l ,

H ev 0T?F e0;v 0Tàs ?às <0:

Since H ev Tis continuous for

v P 0,we have that

H 0e0T?lim v !0

t

H ev TàH e0Tv

?

@F ex 0;0Tv

à

k l à

k l @F ex 0;0T

?

k l

eR 0à1T:Thus,R 0>1ensures that H 0e0T>0,and there exists some v ?2e0;v 0Tsuch that H ev ?T?0.Knowing the value of v ?,from the monotonicity of the function G i ey i Tand Eq.(2.5),the values of x ?and y ?can be computed.It is easy to check that

266J.Wang,S.Liu /Commun Nonlinear Sci Numer Simulat 20(2015)263–272

g ex T?s àdx àF ex ;v ?Thas a positive solution x ?since g e0T>0and g e1T?à1.Further,note that

h n ey n T?

G n ey n Tàl v

?

~k n

R h n 0

f n es n Td s n ?0has a positive solution since h n e0T<0and

lim y n !t1h n ey n T>lim y n !t1G n ey n Tàl v 0~k n R h n 0f n

es n Td s n ?W àsa

k ~k n R h n 0f n es n Td s n ?W àQ n à1i ?1~k i áR h 0f ex Td x áQ n à1i ?1R h i

0f i es i Td s i ás Q n à1

i ?1k i

P 0:Therefore,the existence of the infected equilibrium E ?ex ?;y ?1;y ?2;...;y ?n ;v ?

Tfor system (1.1)under the condition R 0>1is guaranteed.This completes the proof.h

Further,for i ?1;2;...;n ,we assume the following assumptions hold true.

eH 4Tx àx 0à

Z x

x 0lim v !0

t

F ex 0;v T

r v d r !t1as x !0t;

eH 5Tx àx ?

à

Z x

x ?F ex ?;v ?T

F er ;v ?T

d r !t1as x !0t

or x !t1;

eH 6Ty i ày ?i

à

Z

y i

y ?

i

G ey ?i TG i er T

d r !t1as y i !0tor y i !t1:

Similar arguments to the proof of Theorem 1of Huang et al.[11],we can show that solutions of system (1.1)are non-negative.

Theorem 2.1.Let ex et T;y i et T;v et TTT be the solution of system (1.1)with the initial conditions (2.2).Then ex et T;y i et T;v et TTT exists and be nonnegative on ?0;t1Twhen eH 1TàeH 6Tare satis?ed.Hence we discuss system (1.1)in the closed set

X ?eu 1;u 2i ;u 3TT

j k u 1k 6x 0;u 2i ;u 3P 0;i ?1;2;...;n n o

:

It is easy to show that X is positively invariant with respect to system (1.1).We also assume the following:(H 7)F 0v ex ;0Tis increasing with respect to x >0.

From the above assumption eH 7T,it is easy to know the following inequalities hold,

F 0v ex 0;0TF 0v ex ;0T>1for x 2e0;x 0Tand F 0v ex 0;0T

F 0v ex ;0T

<1for x >x 0:

3.Global stability results

In this section,we shall state our main results concerning the global dynamics of (1.1)by constructing suitable Lyapunov

functionals and LaSalle’s invariance principle.

Before giving the proof of main result,we introduce some notations.For the construction of Lyapunov functionals,it is very important to highlight the fact that for u 2e0;t1T,

h eu T?u à1àln u ;

e3:7T

has the global minimum at u ?1and h e1T?0.

Theorem 3.1.Suppose that conditions eH 1TàeH 7Tare satis?ed.

(i)If R 061,the infection free equilibrium E 0?ex 0;0;...;0;0Tis globally asymptotically stable;

(ii)If R 0>1,the infected equilibrium E ?ex ?;y ?1;y ?2;...;y ?n ;v ?

Tis globally asymptotically stable.

Proof.

(i)We construct the following Lyapunov functional for E 0?ex 0;0;...;0;0T,

L 1?L 1tL ttL à;e3:8T

J.Wang,S.Liu /Commun Nonlinear Sci Numer Simulat 20(2015)263–272267

where

L1?xetTàx0à

Z xetT

x0lim

v!0t

Fex0;vT

r v d r

á

Z h

0fexTd xá

Y n

i?1

Z h i

f ies iTd s i

!

t

X n

i?1

Y ià1

j?1

k j

~k

j

!

y

i

etT

áY n

j?i

Z h j

f jes jTd s j

!

t

Y n

j?1

k j

~k

j

!

vetTe3:9T

and

Lt?

Z h

0/exTFexetàxT;vetàxTTd xá

Y n

i?1

Z h i

f ies iTd s i

!

;e3:10T

Là?

X n

i?1

Y i

j?1

k j

Y

j?1

~k

j

B@

1

C A

Z h i

u

i

es iTG iey ietàs iTTd s iá

Y n

j?it1

Z h j

f jes jTd s j

!

;e3:11T

with the convention Q0

j?1

k j

~k

j

?1;

Q0

j?1

~k

j

?1and

Q n

j?nt1

R h j

f jes jTd s j?1.We denote

/exT?Z h

x

fexTd x;u ies iT?

Z h

s i

fes iTd s i;i?1;2;...;n:

ByeH1T–eH4T,it is obvious that L1is de?ned and continuous for all xetT;y ietT;vetT>0,and L1?0atex0;0;...;0;0T.Calcu-lating the time derivative of1along solution of system(1.1),we obtain

dL1 dt ?1àlim

v!0t

Fex0;vT

Fex;vT

ádx0àdxàFex;vT

eTáAty0

1

áBt

k1

~k

1

áy0

2

á

Y n

j?2

Z h j

f jes jTd s j

!

t

k1k2

~k

1

~k

2

áy0

Y n

j?3

Z h j

f jes jTd s j

!

t...t

Y nà1

j?1

k j

~k

j

!

áy0

n

á

Z h n

f nes nTd s nt

Y n

j?1

k j

~k

j

!

áv0etT:

Using the facts that

d

Z h

0/exTFexetàxT;vetàxTTd x?à

Z h

/exTd FexetàxT;vetàxTT

?

?à/exTFexetàxT;vetàxTTj h x?0t

Z h

FexetàxT;vetàxTTd/exT

?

Z h

fexTFexetT;vetTTàFexetàxT;vetàxTT

? d xe3:12T

and

d

Z h i

0u

i

es iTG iey ietàs iTTd s i?à

Z h i

u

i

es iTd?G iey ietàs iTT

?àu ies iTG iey ietàs iTTj h i s

i

?0

t

Z h i

G iey ietàs iTTd u ies iT

?

Z h i

f ies iTG iey iTàG iey ietàs iTT

? d s i;i?1;2;...;n;e3:13T

we can easily obtain

dLtdt ?

Z h

fexTFexetT;vetTTàFexetàxT;vetàxTT

? d xá

Y n

i?1

Z h i

f ies iTd s i

!

e3:14T

and

dLàdt ?

X n

i?1

Q i

j?1

k j

Q ià1

j?1

~k

j

!

á

Z h

f ies iTG iey iTàG iey ietàs iTT

? d s iá

Y n

j?it1

Z h j

f jes jTd s j

!

:e3:15T

Consequently,adding(3.9),(3.14)and(3.15)yields

dL1 dt ?

dL1

dt

t

dLt

dt

t

dLà

dt

?1àlim

v!0t

Fex0;vT

Fex;vT

dx0àdx

? à

Y n

i?1

k i

~k

i

l vtFex;vTálim

v!0t

Fex0;vT

Fex;vTáA

?1àlim

v!0t

Fex0;vT

Fex;vT

dx0àdx

? t

Y n

i?1

k i

~k

i

l vá

Y n

i?1

~k

i

k i

á

1

Fex;vT

válim

v!0t

Fex0;vT

Fex;vTáAà1

"#

:

268J.Wang,S.Liu/Commun Nonlinear Sci Numer Simulat20(2015)263–272

ByeH7T,the following inequality holds:

x0 x à1

1àlim

v!0tFex0;vTFex;vT

?

x0

x

à1

F0vex0;0T

F0vex;0T

60:e3:16T

Note that F0vex0;0T=F0vex;0T–1for x–x0;x>0and v>0,and the equality in(3.16)holds only if x?x0because of F0vex;0T>0andeH7T.It follows from the concavity of Fex;vTwith respect to v that

Fex;vT

válim

v!0tFex0;vT

Fex;vT?

Fex;vT

@Fex0;0T

@v

@Fex;0T

v

6

@Fex0;0T

@v:

Therefore,dL160under the condition R061.Hence,every solution of(1.1)tends to J

1,where J

1

is the largest invariant sub-

set in dL1

dt ?0

n o

with respect to(1.1).We show that J

1consists of only the equilibrium E0.LetexetT;y

i

etT;vetTTbe the solution

with initial function in J

1.Then,from the invariance of J

1

,we have xetT?x0and it then follows that vetT?0for any t from the

?rst equation of(1.1).From the second and third equation of(3.21),we obtain y

i

etT?0for any t;i?1;2;...;n.Therefore,by the LaSalle’s invariance principle for delay differential systems(see[8,16]),E0is globally attractive.Furthermore,it can be veri?ed that E?is locally stable using the same proof as that for Corollary5.3.1in[8].Therefore,E0is globally asymptotically stable under the conditions R061.

(ii)From Lemma2.1,there exists at least one infected equilibrium E?when R0>1.De?ne a Lyapunov functional L2?L2tFex?;v?TLttLà;e3:17Twhere

L2?xetTàx?à

Z xetT

x?Fex?;v?Tr v?d r

áAt

X n

i?1Y ià1

j?1

k j

~k

j

!

y

i

etTày?

i

à

Z y ietT

y?

i

G iey?

i

T

i r

d r

!

á

Y n

j?i

Z h j

f jes jTd s j

!

tY n

j?1

k j

~k

j

!

vetTàv?àv?ln v v

?

e3:18T

and

Lt?

Z h

0/exTh

FexetàxT;vetàxTT

Fex?;v?T

d xá

Y n

i?1

Z h i

f ies iTd s i

!

;

Là?

X n

i?1Q i

j?1

k j

Q ià1

j?1

~k

j

!

áG iey?

i

Z h i

ues iTh G iey ietàhTT

i i

d s iá

Y n

j?it1

Z h j

f jes jTd s j

!

;

where,heáTis de?ned in(3.7).Calculating the time derivative of L2along solution of system(1.1),we obtain

dL2 dt ?1à

Fex?;v?T

Fex;v?T

ádx?tFex?;v?TàdxàFex;vT

eTáAt

X n

i?1

Y ià1

j?1

k j

~k

j

!

á1à

G iey?

i

T

G iey iT

áy0

i

á

Y n

j?i

Z h j

f jes jTd s j

!t

Y n

j?1

k j

~k

j

!

v?

v

v0etT:

Using the facts

d

Z h

0/exTh

FexetàxT;vetàxTT

?v?

d x?à

Z h

/exT

d

x h

FexetàxT;vetàxTT

?v?

d x

?à/exTh

FexetàxT;vetàxTT

Fex?;v?T

h

x?0

t

Z h

h

FexetàxT;vetàxTT

Fex?;v?T

d/exT

?

Z h

fexTà

FexetàxT;vetàxTT

Fex?;v?Tt

FexetT;vetTT

Fex?;v?Ttln

FexetàxT;vetàxTT

FexetT;vetTT

d x

and

d

Z h i

0ues iTh G iey ietàhTT

i i

d s i?

Z h i

f ies iTG iey ietTT

i i

à

G iey ietàs iTT

i i

tln

G iey ietàs iTT

i i

d s i;

one can obtain

dLtdt ?

Z h

fexTà

FexetàxT;vetàxTT

Fex?;v?Tt

FexetT;vetTT

Fex?;v?Ttln

FexetàxT;vetàxTT

FexetT;vetTT

d xá

Y n

i?1

Z h i

f ies iTd s i

!

e3:19TJ.Wang,S.Liu/Commun Nonlinear Sci Numer Simulat20(2015)263–272269

and

dL àdt ?X n i ?1

Q i j ?1k j Q i à1j ?1

~k j "#áG i ey ?

i TáZ h i 0f i es i TG i ey i et TTG i ey ?i TàG i ey i et às i TTG i ey ?i Ttln G i ey i et às i TTG i ey i et TT d s i áY n j ?i t1Z h j 0f j es j Td s j !:e3:20T

Combining (3.19)and (3.20),and using

F ex ?

;v ?

T?Q n

i ?1k i G n ey ?

n T

Q n à1i ?1~

k i R 0f ex Td x

Q n à1i ?1R i

0f i es i Td s

i

?Y n i ?1k i ~k i l v ?

R 0f ex Td x Q n i ?1R i

f i es i Td s i ;we obtain

dL 2?dx ?1àx 1àF ex ?;v ?Tv áA tF ex ?;v ?TZ h 0f ex TY n i ?1Z h i 0f i es i Tln F ex et àx T;v et àx TTv t

X n i ?1

ln G i ey i

et às i TTi i "#(ten t2TàF ex ?;v ?TF ex ;v ?TàG 1ey ?1TG 1ey 1TF ex et àx T;v et àx TTF ex ?;v ?TàX n i ?2

G i ey ?i TG i ey i TG i à1ey i à1et às i à1TTG i à1ey ?i à1Tà

G n ey n et às n TTG n ey ?n Tv ?v "#

tàv v ?tF ex ;v TF ex ;v ?T

d x d s i :

By the properties of the logarithm function,we have that

ln F ex et àx T;v et àx TTv tX n i ?1ln G i ey i et às i TTi i ?ln F ex ?;v ?Tv ?tln G 1ey ?1T11F ex et àx T;v et àx TT?v ?tX n i ?2

ln G i ey ?i Ti i ?

G i à1ey i à1et às i à1TTG i à1ey ?

i à1Ttln G n ey n et às n TT

G n ey ?n Tv ?v tln v v ?F ex ;v ?T

F ex ;v T

:Therefore,

2dt ?dx ?

1àx x ? 1àF ex ?;v ?TF ex ;v ?T áZ h 0

f ex Td x áY n i ?1Z h i

0f i es i Td s i

!àF ex ?;v ?

TZ h

f ex TY n i ?1Z h i

0f i es i Th

F ex ?;v ?TF ex ;v ?T d x d s i àF ex ?;v ?TZ h

0f ex TY n i ?1Z h i

0f i es i Th

G 1ey ?1

T11F ex et àx T;v et àx TT?v ? d x d s i àF ex ?;v ?TZ h 0f ex TY n i ?1Z h i 0f i es i TX n i ?2

h G i ey ?i TG i ey i TG i à1ey i à1et às i à1TTG i à1ey i à1T d x d s i àF ex ?;v ?

TZ h

0f ex TY n i ?1Z h i

0f i es i Th

G n ey n et às n TTG n ey ?n Tv ?v d x d s i àF ex ?;v ?

TZ h

0f ex TY n i ?1Z h i

f i es i Th v v ?

F ex ;v ?TF ex ;v T d x d s i tF ex ?;v ?

TZ h

0f ex TY n i ?1Z h i

f i es i Tv v ?àF ex ;v Tv ? F ex ;v ?Tv à1 d x d s i :

e3:21T

It follows from the monotonicity of the function F ex ;v Ton x that

1àx x ? 1à

F ex ?;v ?T

F ex ;v ?T 60:From the concavity and monotonicity of the function F ex ;v Ton v ,we have the following inequalities holds,

1P F ex ;v T=F ex ;v ?TP v =v ?;for 0

for

v P v ?;

e3:22T

which implies that

v v ?à

F ex ;v TF ex ;v ?T

F ex ;v ?T

F ex ;v T

à1 60:

Therefore,dL 2

dt 60holds from (3.19)–(3.21).Hence,every solution of (1.1)tends to J 2,where J 2is the largest invariant subset in dL 2

?0n o

with respect to (1.1).We show that J 2consists of only the equilibrium E ?.Let ex et T;y i et T;v et TTbe the solution with initial function in J 2,then it holds that x et T?x ?;G 1

ey ?1TG 1ey 1

T

F ex et àx T;v et àx TT

F ex ?;v ?T?1for almost x 2?0;h ;

G i ey ?i TG i ey i T

G i à1ey i à1et às i à1TT

G i à1ey ?i à1

Tfor almost

270J.Wang,S.Liu /Commun Nonlinear Sci Numer Simulat 20(2015)263–272

J.Wang,S.Liu/Commun Nonlinear Sci Numer Simulat20(2015)263–272271 x2?0;h i and vetT?v?.From the invariance of J2,we have x0etT?0and it then follows that vetT?v?for any t from the?rst

etT?y1etàs1T?y?1for any t;y ietT?y ietàs iT?y?i;i?2;3;...;n,for any t and equation of(1.1).From(3.21),we obtain y

1

then,vetT?vetàxT?v?follows from the third equation of(1.1).Therefore,by the LaSalle’s invariance principle for delay systems(see[8,16]),E?is globally attractive.Furthermore,it can be veri?ed that E?is locally stable using the same proof as that for Corollary5.3.1in[8].Therefore,E?is globally asymptotically stable under the conditions R0>1andeH1T–eH7T.This completes the proof.h

4.Discussions and conclusions

We have studied an nt2dimensional in-host model with?nite distributed delays,general incidence rate and general removal rate.By constructing Lyapunov functionals and LaSalle’s invariance principle,we have identi?ed the basic reproduc-tion number R0as threshold quantity for stability of equilibria.More precisely,it is shown that,if R061,then the infection free equilibrium is globally asymptotically stable,while if R0>1,then there exists a unique infected equilibrium which is globally asymptotically stable(see Theorem3.1).nt1distributed delays describe(i)the time between viral entry and the transcription of viral RNA,(ii)the nà1-stage time needed for activated infected cells between viral RNA transcription and viral release,and(iii)the time necessary for the newly produced viruses to be infectious(maturation),respectively.

The construction and computation of global Lyapunov functionals are motivated by the recent works by[11,18–20].Such form function of heuT?uàu?àu?lneu=u?Twas usually used to construct Lyapunov functions for the Lotka–Volterra system at?rst,then it was successfully applied to ODEs epidemics models by Korobenikov[14].On the other hand,incorporating the ?nite intracellular delays would not arise periodic oscillations.

The LaSalle’s invariance principle in[8,16]provides a direct and effective method to establish global dynamical properties for the system of delay differential equations.It should be pointed out that,from Theorem3.1,the global stability of equi-libria only depends on the basic reproductive number when the conditionseH1T–eH7Tare satis?ed.Actually,the nonlinear incidence rate Fex;vTand the virus producing rate G iey iTdo not destabilize two https://www.doczj.com/doc/1c17042590.html,paring the model with discrete delays studied in[11],our results show that distributed delay in initial infection and the distributed delays in infected host have no impact for the global stability of equilibria.

By using the method of Lyapunov functionals,LaSalle’s invariance principle,a general incidence rate and a general removal rate,we can resolve global dynamics for a class of in host viral infection model.If the system(1.1)is modi?ed to incorporate in?nite distributed delays,then the computations may still be valid,provided that the kernel functions f and f i are bounded from above by decaying exponential function and that a suitable fading memory space is chosen.These aspects,however,are left for future consideration.

Acknowledgments

The authors thank the editor and the anonymous referees for the valuable suggestions and numerous comments which helped to improve our manuscript considerably.

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