Global analysis of virus dynamics model with logistic mitosis, cure rate and delay in virus...

Research Article

Received 17 September 2012 Published online in Wiley Online Library

(wileyonlinelibrary.com) DOI: 10.1002/mma.3096MOS subject classification: 34K05; 34K18; 34K20; 37B25

Global analysis of virus dynamics model withlogistic mitosis, cure rate and delay invirus production

Cruz Vargas-De-Leóna,b, Noé Chan Chíc and Eric Ávila Valesc*†

Communicated by E. Venturino

In this paper, we study a virus dynamics model with logistic mitosis, cure rate, and intracellular delay. By means ofconstruction of a suitable Lyapunov functionals, obtained by linear combinations of Volterra—type functions, compositequadratic functions and Volterra—type functionals, we provide the global stability for this model. If R0, the basic repro-ductive number, satisfies R0 � 1, then the infection-free equilibrium state is globally asymptotically stable. Our systemis persistent if R0 > 1. On the other hand, if R0 > 1, then infection-free equilibrium becomes unstable and a uniqueinfected equilibrium exists. The local stability analysis is carried out for the infected equilibrium, and it is shown that, if theparameters satisfy a condition, the infected equilibrium can be unstable and a Hopf bifurcation can occur. We also havethat if R0 > 1, then the infected equilibrium state is globally asymptotically stable if a sufficient condition is satisfied. Weillustrate our findings with some numerical simulations. Copyright © 2014 John Wiley & Sons, Ltd.

Keywords: global stability; permanence; lyapunov functionals; time delay; local stability; Hopf bifurcation

1. Introduction

The mathematical theory of viral infections has been developed in the last decades (See [1, 2]). Models in this class describe thenonlinear interaction between virus and uninfected target cell populations. One of the principles of the modeling of viral infections isthat the viral infection term is assumed to be non-linear as it is a function of the interaction between virus particles and target cells.Commonly, the infection rate is assumed to occur at a rate proportional to the product of the concentration of virus particles anduninfected target cells, this notion, the so-called ‘mass action principle’ is valid when the system is well mixed. Another principle of themodeling of viral infections is to describe the intrinsic growth rate of the uninfected target cells through a function, which includes pro-duction of new cells and natural mortality of cells. A reasonable model for population dynamics of target cells is proposed by Perelsonand Nelson [1] as

dT.t/

dtD g .T.t// , (1)

where

g.T.t// D s C aT.t/

�1 � T.t/

Tmax

�� dT.t/.

Here, T.t/ represents the number of uninfected target cells, s is the rate of creation of new T cells from source within the body; d is

the death rate per T cell; the mitosis of uninfected target cells is described by the logistic term aT.t/�

1 � T.t/Tmax

�, where a is the intrinsic

mitosis rate when T.t/ is small, and Tmax is the maximum population of uninfected cells maintained by homeostasis in the body.Recent studies have shown that several viral infections can be controlled by ‘curative’ mechanisms [3]. By hepatitis B virus (HBV) infec-

tion, according to the immunological basis found in [4], when a hepatocyte is infected with HBV, it is possible that infected cells may

a Unidad Académica de Matemáticas, Universidad Autónoma de Guerrero, Mexico, Mexicob Departamento de Matemáticas, Facultad de Ciencias, Universidad Nacional Autónoma de México, 04510 Mexico, Mexicoc Facultad de Matemáticas, Universidad Autónoma de Yucatán, Mexico, Mexico* Correspondence to: Eric Ávila Vales, Facultad de Matemáticas, Universidad Autónoma de Yucatán, Mexico, Mexico.† E-mail: [email protected]

Copyright © 2014 John Wiley & Sons, Ltd. Math. Meth. Appl. Sci. 2014

C. VARGAS-DE-LEÓN, N. CHAN CHÍ AND E. ÁVILA VALES

also revert to the uninfected state by loss of all cccDNA (covalently closed circular DNA) from their nucleus [4]. The immune mechanismto mediate cccDNA clearance is a noncytolytic antiviral mechanisms, that is to say, cytokine-induced ‘curing’ of infected cells. By humanimmunodeficiency virus (HIV-1) infection, according to the virological basis found in [5, 7], when a HIV RNA genome enters a quiescentCD4C T lymphocytes, the new viral RNA may not be completely reverse transcribed into DNA. If the lymphocyte is activated shortlyfollowing infection, reverse transcription can proceed to completion [7]. Actually, the unintegrated HIV-1 DNA harbored in quiescentcells may decay with time, and partial DNA transcripts are labile and degrade quickly [6, 7]. In other words, a proportion of infectedcells in the quiescent stage can revert to the uninfected state before the viral genome is integrated into the genome of the CD4C Tlymphocytes.

The ‘curative’ mechanisms of virus–infected cells is another innovation in the modeling of viral infections. Technically, a term pI.t/ isincorporated into the equation for uninfected cells, corresponding to the rate at which uninfected cells are created through ‘cure’. Thiskind models focus on ‘cure’ of infected cells for HBV infection [8–10] and HIV infection [7, 11–16]. The models with cure rate studiedin [7–9, 11, 15] can be described by a system of ordinary differential equations. Other class of virus dynamics models with cure rate[10, 12, 14, 16] have incorporated time delays of one type or another.

In particular, Yang, Wang, and Li [14] proposed a virus dynamics model with ‘cure’ of infected cells, which incorporates a logisticmitosis term for the uninfected cells and a discrete intracellular delay on the viral production term, namely,

dT.t/

dtD g .T.t// � ˇT.t/V.t/C pI.t/

dI.t/

dtD ˇT.t/V.t/ � .ı C p/I.t/ (2)

dV.t/

dtD qI.t � �/ � cV.t/.

Here, I.t/ represents the concentration of infected CD4C T cells at time t, and V.t/ represents the concentration of free HIV at time t.Free virus infects uninfected cells to produce infected cells at rate ˇT.t/V.t/. Infected cells die at rate ı. New virus is produced frominfected cells at rate q and dies at rate c. The average lifetime of infected cells is 1=ı. The average number of virus particles producedover the lifetime of a single infected cell (the burst size) is given by q=ı. p is cure rate from infected cells to healthy cells. The delay �represents the time necessary for the newly produced virions to become mature and then infectious particles.

All parameters are positive values.The initial conditions for system (2) take the form

T.�/ D '1.�/, I.�/ D '2.�/, V.�/ D '3.�/,

'i.�/ � 0 � 2 Œ�� , 0/, 'i.0/ > 0 .i D 1, 2, 3/,(3)

where .'1.�/,'2.�/,'3.�// 2 C�Œ�� , 0�,R3

C0

�, the Banach space of continuous functions mapping the interval Œ�� , 0� into R3

C0,where

R3C0 D ˚

.T , I, V/ 2 R3 : T � 0, I � 0, V � 0�

.

System (2) always has an infection-free equilibrium Eı D .Tı, 0, 0/, where

Tı D Tmax

2a

".a � d/C

s.a � d/2 C 4as

Tmax

#. (4)

The basic reproductive number of system (2) is

R0 D ˇqTı

c.ı C p/. (5)

Here, R0 denotes the average number of the free viruses released by the infected cells that are infected by the first virus. When R0 > 1,system (2) has a unique infected equilibrium E� D .T�, I�, V�/, where

T� D c.ı C p/

qˇD Tı

R0, I� D g .T�/

ı, V� D qg .T�/

cı. (6)

The question of global stability in population models is a very interesting mathematical problem. Many authors have studied theglobal stability of virus dynamics models without delay using the second Lyapunov method. Typically, the Lyapunov function candidatefor population biology models is the Volterra–type function. Korobeinikov used the Volterra–type Lyapunov function in [17, 18] toprove global stability of equilibria of viral infections models. Furthermore, Vargas-De-León used the linear combinations of Volterra–type functions and composite quadratic functions to prove the global stability of equilibria of virus dynamics models with ‘cure’ of



infected cells [9–11]. On the other hand, McCluskey study the global stability of the equilibrium states of epidemic models with delay(discrete and distributed) [19,20] and in [10,21] analyzed viral infection models with discrete delay, using a novel Lyapunov functionals,that we will call Volterra–type functionals. This functional is used also in a class of host-vector models with incubation period [22].

Yang, Wang, and Li [14] investigated the effect of this released period on the local stability of the equilibria, and the directionand stability of the Hopf bifurcation. Persistence and stability are important concepts in mathematical biology, relevant to study thepersistence and stability of model (2). We address the problem of global stability and persistence of delayed system (2). Assuming theexistence of an infected equilibrium, we relax the conditions for its local asymptotic stability given in [14]. We provide the conditionsfor global asymptotic stability of the infection–free equilibrium and infected equilibrium using Lyapunov’s direct method.

The paper is organized as follows: The global asymptotic stability of the infection-free equilibrium state is established in Section 2.In Section 3, we study the permanence. The local stability of the infected equilibrium is analyzed in Section 4. In Section 5, theHopf bifurcation at the infected equilibrium is determined. The global asymptotic stability of the infected equilibrium state is estab-lished in Section 6. Numerical simulations are presented in the Section 7 to illustrate our results. The paper ends with a discussionin Section 8.

2. Global stability of the infection-free equilibrium

We prove the global stability of the infection-free equilibrium Eı D .Tı, 0, 0/. The strategy of proof is to use suitable Lyapunovfunctionals and classical LaSalle’s invariance principle.

Theorem 2.1If R0 � 1, then the infection-free equilibrium state Eı of (2) is globally asymptotically stable for any time delay � � 0.

ProofLet .T.t/, I.t/, V.t// be any positive solution of system (2) with initial conditions (3). The global Lyapunov functional proposed here hasthe form

W.t/ D AıTı

�T

Tı� 1 � ln

T

Tı

�C Dı

�.T � Tı/C I

�2 C AıI

C BıV C Cı

Z �

0I.t � �/d� ,

where Aı, Bı, Cı and Dı are the positive constants

Aı D Dı Tı

p

�ı C 2

aTı

TmaxC d � a

�, Bı D Aı .ı C p/

qand Cı D Aı.ı C p/.

Using the relation

s

TıD aTı

TmaxC d � a. (7)

We have

Aı D Dı Tı

p

�ı C aTı

TmaxC s

Tı

�> 0.

Calculating the derivative of W along positive solutions of system (2), it follows that

dW

dtD Aı .T � Tı/

T

dT

dtC Dı

�.T � Tı/C I

� �dT

dtC dI

dt

�

C Aı dI

dtC Bı dV

dt� Cı

Z �

0

d

d�I.t � �/d� .

Using g.Tı/ D 0, we get

dW

dtD Aı .T � Tı/

T

g.T/ � g.Tı/ � ˇTV C pI

C Dı

�.T � Tı/C I

�.g.T/ � g.Tı/ � ıI/

C Aı .ˇTV � .ı C p/I/

C Aı .ı C p/

q.qI.t � �/ � cV/

� Aı.ı C p/ .I.t � �/ � I/ .



Noting that

g.T/ � g.Tı/ D ��

aTı

TmaxC d � a

�.T � Tı/ � aT

Tmax.T � Tı/,

p.T � Tı/

TI D �pI

.T � Tı/2

TTıC p

Tı.T � Tı/I,

a

TmaxT.T � Tı/I D a

Tmax.T � Tı/2I C a

TmaxTı.T � Tı/I.

We substitute the expressions

dW

dtD �Aı

�aTı

TmaxC d � a

�.T � Tı/

2

T� Aı a

Tmax

T � Tı

2

� AıpI.T � Tı/2

TTıC Aı p

Tı.T � Tı/I

� Dı

�aTı

TmaxC d � a

�.T � Tı/2 � Dı

�ı C 2

aTı

TmaxC d � a

�.T � Tı/I

� Dı aT

Tmax.T � Tı/2 � Dı a

Tmax.T � Tı/2I � DııI2

� Aı

�c.ı C p/

q� ˇTı

�V ,

D �Aı

�aTı

TmaxC d � a C p

I

Tı

�.T � Tı/

2

T

� Dı

�aTı

TmaxC d � a C a

TmaxI

�.T � Tı/2 � Dı aT

Tmax.T � Tı/2 � DııI2

� c.ı C p/

qAı

�1 � ˇqTı

c.ı C p/

�V .

Using (7) and rewriting dW=dt in terms of basic reproductive number (5), we have

dW

dtD �Aı .s C pI/

.T � Tı/2

TıT� Dı

�s

TıC a

TmaxI

�.T � Tı/2

� Dı aT

Tmax.T � Tı/2 � DııI2 � c.ı C p/

qAı .1 � R0/ V .

Clearly, dW=dt � 0. It follows from the classical Lyapunov-LaSalle invariance principle [23] that solutions converge to the largestinvariant set fdW=dt D 0g. We note that dW=dt D 0 holds if and only if T.t/ D Tı and I.t/ D V.t/ D 0 for all t, and so the largestinvariant set consists of the single point .Tı, 0, 0/. Thus, .Tı, 0, 0/ is globally asymptotically stable. This proves Theorem 2.1.

3. Permanence

In this section, we follow closely the ideas of the proof of permanence in [24] and [25]. Yang, Wang, and Li [14] has established thefollowing results for system (2)

Theorem 3.1The solution of system (2) is positive and bounded.

In this section, we discuss the permanence of system (2). From the work of Yang, Wang, and Li [14] we known that the solution ofsystem (2) is bounded, say by Mc,

� D f.T , I, V/j0 < T , I, V � Mcgtherefore, the solutions satisfy

lim supt!1

T.t/ � Mc, lim supt!1

I.t/ � Mc, lim supt!1

V.t/ � Mc

Let us introduce the definition of uniform persistence.

Definition 1System (2) is said to be uniformly persistent in the bounded set V� if there exist a constant � > 0 independent of initial value in V�,such that

lim inft!1

T.t/ � � , lim inft!1

I.t/ � � , lim inft!1

V.t/ � � ,

hold for any positive solution .T.t/, I.t/, V.t// of (2).



Note that our system is dissipative (or equivalently uniformly ultimately bounded), and for a dissipative system uniform persistenceis equivalent to the permanence.

To prove permanence of system (2), we present the permanence theory for infinite dimensional system from Hale and Waltman [26].Assume that the complete metric space X is the closure of an open set X0; that is, X D X0

S@X0, where @X0 (assumed to be non-empty)

is the boundary of X0. Suppose that �.t/ is a C0-semigroup on X satisfying

�.t/ : X0 ! X0, �.t/ : @X0 ! @X0. (8)

Set �@.t/ D �.t/j@X0 and let H@ be the global attractor for �@.t/. The semigroup � is said to be point dissipative in X if there is a boundednon-empty set B1 in X such that, for any x 2 X , there is a t0 D t0.x, B1/ such that �.t/x 2 B1 for t � t0.

A set M � @X0 is said to be chained to another (not necessarily distinct) set N � @X0, written by M ! N, if there is some y 2 @X0,y 62 M [ N, and a full orbit through in @X0 whose ı-limit set is contained in M and whose !-limit set is contained in N. A finite coveringM D Sk

iD1 Mi is called cyclic if, after possible renumbering, M1 ! M1 or M1 ! M2 ! � � � ! Mi ! M1 for some i 2 .2, : : : , k/, M iscalled an acyclic covering otherwise. Consider the particular invariants sets

QH@ D[

x2H@

!.x/.

QH@ is isolated if there exist a covering M D SkiD1 Mi of QH@ by pairwise disjoint, compact, isolated invariant sets M1, M2, : : : , Mk for �@

such that each Mi , is also an isolated invariant set for �.

Lemma 3.2Suppose that �.t/ satisfies (8) and following conditions.

1. There is a t0 � 0 such that �.t/ is compact for t > t0,2. �.t/ is point dissipative in X ,3. QH@ D [x2H@

!.x/ is isolated and has an acyclic covering M D SkiD1 Mi .

Then, �.t/ is uniformly persistent if and only if for eachMi � M, Ws.Mi/ \ X0 D ;, i D 1, 2, : : : , k.

Theorem 3.3If R0 > 1, then the system is permanent

ProofWe define

X2 D ˚. ,�1,�2/ 2 C

Œ�� , 0�, R3

C

: .�/ 6D 0,�1.�/ D �2.�/ D 0, .� 2 Œ�� , 0�/

�.

Let X1 D intC�Œ�� , 0�, R3

C

�, which satisfies the conditions stated in lemma 3.2. We know that the solutions are uniformly and ultimately

bounded, thus the semigroup �.t/ is point dissipative. We need to determine the following set

�1 D[

y2Y2

!.y/,

where Y2 D fy0 2 X2jy.t, y0/ 2 X2, 8t > 0g and !.y/ is the !-limit set of the solution y.t/ of system (2) starting at y0. It follows fromsystem (2) that all solutions starting in X2 but not in T-axis will leave X2 and that positive half T-axis is an invariant set, that is to say,Y2 D f.T , I, V/j.T , I, V/ 2 X2, I D V D 0g. In the set Y2, system (2) becomes

dT

dtD s � dT.t/C aT.t/

�1 � T.t/

Tmax

�.

Clearly, E0 is one fixed point of �.t/ in Y2. Hence, any solution .T.t/, I.t/, V.t// of system (2) initiating from Y2 ultimately converges to E0,that is�1 D fE0g. E0 is a covering of�1, which is isolated (because E0 is a hyperbolic equilibrium state when R0 > 1) and nonperiodic(because there is no nontrivial solution in X2 that links E0 to itself ).

Now, we show that Ws.E0/ \ X1 D ;, where Ws.E0/ denotes the stable manifold of E0. Therefore, it follows from lemma 3.2 thatsystem (2) is uniformly persistent. Assuming the contrary, then there exists a positive solution

QT.t/, QI.t/, QV.t/ of system (2) such that QT.t/, QI.t/, QV.t/ ! .Tı, 0, 0/ as t ! 1. Note that T.t/ � Tı, let us choose � > 0 small enough and t0 > 0 be sufficiently large enoughsuch that

Tı � � < QT.t/ < Tı C �

for t > t0 � � . Then, we have for t > t0 ( PQI.t/ � ˇ.Tı � �/ QV.t/ � .ı C p/QI.t/,PQV.t/ D qI.t � �/ � cV.t/,

let us consider the matrix



A� D� �.ı C p/ ˇ.Tı � �/

q �c

�.

Because A� admits positive off-diagonal elements, Perron-Frobenius theorem implies that there is a positive eigenvector V for themaximum eigenvalue 1 of A� . Moreover, we see that 1 is positive, because we have c.ı C p/ < ˇq.Tı � �/.

Let us consider PI.t/ D ˇ.Tı � �/V.t/ � .ı C p/I.t/

PV.t/ D qI.t � �/ � cV.t/(9)

Let v D .v1, v2/ and l > 0 be small enough such that

lv1 < QI.t0 C �/, lv2 < QV.t0 C �/

for � 2 Œ�� , 0�, if .I.t/, V.t// is a solution of (9) satisfying I.t/ D lv1, V.t/ D lv2 for t0 � � � t � t0.Because semiflow of system (9) is monotone and A�v > 0, it follows that I.t/ and V.t/ are strictly increasing and I.t/ ! 1, V.t/ ! 1

as t ! 1. Note that QI.t/ � I.t/, QV.t/ � V.t/ for t > t0. We have QI.t/ ! 1, QV.t/ ! 1 as t ! 1. At this time, we are able to concludefrom the bounded of solutions that X2 repels the positive solutions of system (2) uniformly. Incorporating this into lemma 3.2 and thebounded solutions, we know that system (2) is permanent.

4. Local stability of the infected equilibrium state

We now study the local stability behavior of the infected equilibrium state E� when R0 > 1 for system (2) with delay. Thus, lineariz-ing the system (2) at infected equilibrium state E�.T�, I�, V�/, we obtain that the associated transcendental characteristic equation isgiven by

det

0B@ C a

TmaxT� C ˇV� Cƒ �p ˇT�

�ˇV� ı C p Cƒ �ˇT�

0 �qe�ƒ� c Cƒ

1CA D 0,

that can be rewritten as

det

0BB@ C a

TmaxT� C .ı C p/ I�

T� Cƒ �p .ı C p/ I�

V�

�.ı C p/ I�

T� ı C p Cƒ �.ı C p/ I�

V�

0 �qe�ƒ� q I�

V� Cƒ

1CCA D 0,

when we take into account the identities

ˇ D .ı C p/I�

T�V�and c D q

I�

V�. (10)

Let

:D d C a

TmaxT� � a. (11)

By computation, we have

P.ƒ, �/ D ƒ3 C a1ƒ2 C a2ƒC a3 C .b1ƒC b2/e

�ƒ� D 0, (12)

where

a1 D ı C p C C a

TmaxT� C .ı C p/

I�

T�C q

I�

V�,

a2 D�ı C q

I�

V�

�� C a

TmaxT� C .ı C p/

I�

T�

�C p

� C a

TmaxT�

�C q.ı C p/

I�

V�,

a3 D q.ı C p/

� C a

TmaxT� C ı

I�

T�

�I�

V�,

b1 D �q.ı C p/I�

V�,

b2 D �� C a

TmaxT�

�q.ı C p/

I�

V�.

When � D 0, we have from (12) that

P.ƒ, 0/ D ƒ3 C a1ƒ2 C .a2 C b1/ƒC a3 C b2 D 0, (13)

where

a2 C b1 D�ı C q

I�

V�

�� C a

TmaxT� C .ı C p/

I�

T�

�C p

� C a

TmaxT�

�, (14)



a3 C b2 D ıq.ı C p/.I�/2

T�V�. (15)

If � 0 (or equivalently d C aT�=Tmax � a) then a1 > 0, a3 C b2 > 0 and

a1.a2 C b1/ � .a3 C b2/ D�

p C C a

TmaxT� C .ı C p/

I�

T�C q

I�

V�

�.a2 C b1/

C ı

�ı

� C a

TmaxT� C .ı C p/

I�

T�

�C� C a

TmaxT�

��p C q

I�

V�

��> 0,

from the well-known Routh–Hurwitz criterion, it follows that any root of (13) has negative real part for � D 0. Hence, we get thefollowing conclusion.

Theorem 4.1If R0 > 1 and

d C aT�

Tmax� a, (16)

then the infected equilibrium state E� of (2) is locally asymptotically stable when � D 0.

In the following, for � > 0, we investigate the existence of purely imaginary rootsƒ D i$ ($ > 0) to equation (13).Substitutingƒ D i$ into equation (13) and separating the real and imaginary parts, we obtain

$3 � a2$ D b1$ cos$� � b2 sin$� ,

a1$2 � a3 D b1$ sin$� C b2 cos$� .

(17)

Taking moduli in the above equation and grouping in terms of the powers of$ gives

Q.$ , �/ D $6 C c1$4 C c2$

2 C c3 D 0, (18)

where c1 D a21 � 2a2, c2 D a2

2 � 2a1a3 � b21 and c3 D a2

3 � b22.

Then,ƒ D i$ ($ > 0) is a root of the (13) if and only if Q.$ , �/ D 0. Let z D $2, then (18) becomes

G.z, �/ D z3 C c1z2 C c2z C c3 D 0. (19)

Straightforward calculations shows that

c1 D ı2 C 2ıp C� C a

TmaxT�

�2

C 2

� C a

TmaxT�

�.ı C p/

I�

T�

C�

p C .ı C p/I�

T�

�2

C�

qI�

V�

�2

,

D .ı C p/2 C� C a

TmaxT� C p C .ı C p/

I�

T�

�2

C 2p.ı C p/I�

T�C�

qI�

V�

�2

> 0,

(20)

c2 D ı2 C

�q

I�

V�

�2!�

C a

TmaxT� C .ı C p/

I�

T�

�2

C p2

� C a

TmaxT�

�2

C 2ıp

� C a

TmaxT�

�� C a

TmaxT� C .ı C p/

I�

T�

�C 2p.ı C p/

I�

T�

�q

I�

V�

�2

,

D�

qI�

V�

�2 � C a

TmaxT� C .ı C p/

I�

T�

�2

C 2p.ı C p/I�

T�

�q

I�

V�

�2

C�

p

� C a

TmaxT�

�C ı

� C a

TmaxT� C .ı C p/

I�

T�

��2

> 0,

(21)

a3 C b2 D ıq.ı C p/.I�/2

T�V�> 0, (22)

a3 � b2 D q.ı C p/

�2

� C a

TmaxT�

�C ı

I�

T�

�I�

V�. (23)

Clearly, if d C aT�=Tmax � a (or equivalently � 0) then c3 D .a3 C b2/.a3 � b2/ D a23 � b2

2 > 0.



In order to show that the infected equilibrium state E� is locally asymptotically stable, we have to show that the equation (19) has nopositive roots, and thus the characteristic equation (13) has no purely imaginary roots. In addition, P.0, �/ D a3 C b2 > 0 for all � � 0implies that zero is not the root of (13).

Finally, we show that the condition (16) is equivalent to the condition (24). Using R0 D Tı

T� , the inequality (16) may be written

aTı

R0TmaxC d � a, so that

aTı

R0Tmax� .a � d/, and R0 � aTı

.a � d/Tmax.

Hence, the sufficient conditions, in terms of R0, can be written

1 < R0 � aTı

.a � d/Tmax. (24)

Summarizing what was mentioned earlier, we have obtained the following theorem.

Theorem 4.2If R0 > 1 and (16) holds, then the infected equilibrium state E� of (2) is locally asymptotically stable. In particular, R0 > 1 and (16) areequivalent to (24)

5. Hopf bifurcation

In this section, we present the Hopf bifurcation analysis for model (2). Note that Theorem 4.2 indicates that if their conditionsare satisfied, then the infected equilibrium is asymptotically stable independently of the delay. We point out that if conditions inTheorem 4.2 are not satisfied and if c3 < 0 holds, where c3 is a coefficient in (19), then the stability of the infected equilibrium dependson the delay value and as the delay varies the infected equilibrium con lose stability, which can lead to oscillations.

Now, we turn to the bifurcation analysis. We use the delay � as bifurcation parameter. We view the solutions of (19) as functions of thebifurcation parameter � . Let .�/ D �.�/C i!.�/ be the eigenvalue of equation (19) such that for some initial value of the bifurcationparameter �0, we have �.�0/ D 0, and !.�0/ D !0 (without loss of generality we may assume !0 > 0). From equation (17), we canobtain

�0 D 1

!0arccos

�b1!

40 C .a1b2 � a2b1/!

20 � a3b2

b22 C b2

1!20

�

To establish the Hopf bifurcation at � D �0, we need to show thatRe.�/

d�

ˇ̌̌�D�0

> 0. Differentiating (12) with respect to � , we get

32 C 2a1C a2

d

d�� .b1C b2/e

�� .b1C b2/e�� d

d�C b1e�� d

d�D 0

clearingd

d�, we have

d

d�D e�� .b1C b2/

32 C 2a1C a2 C b1e�� e�� .b1C b2/, this gives

�d

d�

��1

D 32 C 2a1C a2 C b1e�� e�� .b1C b2/

e�� .b1C b2/D 32 C 2a1C a2 C b1e��

e�� .b1C b2/� �

D 33 C 2a12 C a2C b1e��

2e�� .b1C b2/� �

using equation (12), we can rewrite the last equality as

�d

d�

��1

D 23 C a12 � a3 � b2e��

2e�� .b1C b2/� �

D 23 C a1

2 � a3

�2.3 C a12 C a2C a3/� b2

2.b1C b2/� �

.

Thus,

sign

d.Re/

d�

��Di!0

D sign

(Re

�d

d�

��1)

i!0

D sign

(Re

�23 C a1

2 � a3

�2.3 C a12 C a2C a3/

�i!0

� Re

�b2

2.b1C b2/

�i!0

� Reh �

ii!0

)

D sign

(Re

"�2!3

0 i � a1!20 � a3

!20

�!30 i � a1!

20 C a2!0i C a3

#

C Re

�b2

!20 .b1!i C b2/

�� Re

��

!0i

�)

D sign

8<: 2!6

0 C .a21 � 2a2/!

20 � a2

3

!20

ha2!0 � !3

0

2 C .a3 � a1!0/2i C b2

2

!20

b2

2 C b21!

20

9=; ,



using system (17) in the last equality we obtain

sign

d.Re/

d�

��Di!0

D sign

8<: 2!6

0 C a2

1 � 2a2

!4

0 � a23 C b2

2

!20

ha2!0 � !3

0

2 C .a3 � a1!0/2i9=;

using now (12) we obtain

sign

d.Re/

d�

��Di!0

D sign

8<:3!6

0 C 2

a21 � 2a2

!4

0 C a2 � 2a1a3 � b2

1

!2

0

!20

ha2!0 � !3

0

2 C .a3 � a1!0/2i

9=;

D sign

(3!4

0 C 2

a21 � 2a2

!2

0 C a2 � 2a1a3 � b2

1

a2!0 � !30

2 C .a3 � a1!0/2

)

Because

G.z/ D z3 C c1z2 C c2z C c3

Thus,dG.z/

dzD 3z2 C 2˛z C ˇ D 3z2 C 2

a2

1 � 2a2

z C a2

2 � 2a1a3 � b21

As !0 is the largest positive simple root of (18), from lemma 3.3.2 in [27], we have

dG.z/

dz

ˇ̌̌zD!2

0> 0,

hence,

d Re

d�

ˇ̌̌ˇ̌̌ˇ!D!0, �D�0 D

dG

!20

dz

a2!0 � !30

2 C .a3 � a1!0/2> 0

The aforementioned analysis can be summarized into the following theorem:

Theorem 5.1Suppose that R0 > 1. If 2

� C a

TmaxT��

C ı I�

T� < 0 is satisfied, and !0 is the largest positive simple root of (18), then the infected

equilibrium E� of the delay model (2) is asymptotically stable when � < �0 and unstable when � > �0, where

�0 D 1

!0arccos

�b1!

40 C .a1b2 � a2b1/!

20 � a3b2

b22 C b2

1!20

�

when � D �0, a Hopf bifurcation ocurrs; that is a family of periodic solutions bifurcates from E� as � passes through the critical value �0

Using time delay as a bifurcation parameter, the previous theorem indicates that model could exhibit Hopf bifurcation at a certain

value �0 of the delay if the parameters satisfy the condition 2� C a

TmaxT��

C ı I�

T� < 0.

6. Global stability of the infected equilibrium state

In this section, by the Lyapunov direct method and using the technology of constructing Lyapunov functionals, we obtain a sufficientcondition for global asymptotic stability of the unique infected equilibrium E� when R0 > 1.

Theorem 6.1Let R0 > 1. If the condition

d C a

TmaxT� � a C max

p

T�,

a

Tmax

�I�, (25)

holds then the unique infected equilibrium, E�, of (2) is globally asymptotically stable for any time delay � � 0.

To prove our Theorem 6.1, we need the following lemma.

Lemma 6.2If the following condition

d C a

TmaxT� � a C max

p

T�,

a

Tmax

�I�, then ı C d C 2

a

TmaxT� > a.



ProofFirst, we have that

d C a

TmaxT� D a C max

p

T�,

a

Tmax

�I�, then max

p

T�,

a

Tmax

�I� C ı C a

TmaxT� > 0.

Second, we have that

d C a

TmaxT� > a C max

p

T�,

a

Tmax

�I�, then

max

p

T�,

a

Tmax

�I� C ı C d C 2

a

TmaxT� > d C a

TmaxT� > a C max

p

T�,

a

Tmax

�I�.

Now, we present the proof of Theorem 6.1

ProofLet .T.t/, I.t/, V.t// be any positive solution of system (2) with initial conditions (3). Define the global Lyapunov functional for E�,

QL.t/ D L.t/C ˇT�V�A�LC.t/

where

L.t/ D A�T�

�T

T�� 1 � ln

T

T�

�C A�I�

�I

I�� 1 � ln

I

I�

�

C B�V�

�V

V�� 1 � ln

V

V�

�

C C�

2

�.T � T�/C .I � I�/

�2,

(26)

LC.t/ DZ �

0

�I.t � �/

I�� 1 � ln

I.t � �/I�

�d� . (27)

At the infected equilibrium state, we have

0 D �g.T�/C ˇT�V� � pI�, (28)

.ı C p/ D ˇT�V�

I�, (29)

c D qI�

V�, (30)

0 D �g.T�/C ıI�. (31)

The derivative of (26) along the solution curves of (2) is given by the expression

dL

dtD A�

�1 � T�

T

�dT

dtC A�

�1 � I�

I

�dI

dt

C B�

�1 � V�

V

�dV

dtC C�

�.T � T�/C .I � I�/

� �dT

dtC dI

dt

�

D A�

�1 � T�

T

�.g.T/ � ˇTV C pI/C A�

�1 � I�

I

�.ˇTV � .ı C p/I/

C B�

�1 � V�

V

�.qI.t � �/ � cV/

C C��.T � T�/C .I � I�/

�.g.T/ � ıI/.



Using (28)–(31), we obtain:

dL

dtD A�

�1 � T�

T

��g.T/ � g.T�/C ˇT�V�

�1 � TV

T�V�

�C p.I � I�/

�

C A�ˇT�V�

�1 � I�

I

��TV

T�V�� I

I�

�

C B�qI��

1 � V�

V

��I.t � �/

I�� V

V�

�C C�

�.T � T�/C .I � I�/

�.g.T/ � g.T�/ � ı.I � I�//

D A�

�1 � T�

T

� g.T/ � g.T�/

C A�p

�1 � T�

T

�.I � I�/

C A�ˇT�V�

�1 � TV

T�V�� T�

TC V

V�

�

C A�ˇT�V�

�TV

T�V�� I

I�� TI�V

T�IV�C 1

�

C B�qI��

I.t � �/I�

� V

V�� I.t � �/V�

I�VC 1

�C C�

�.T � T�/C .I � I�/

�.g.T/ � g.T�/ � ı.I � I�//.

Choosing the constants

A� D C� T�

p

�ı C 2

aT�

TmaxC d � a

�,

B� D A� ˇT�V�

qI�,

and C� > 0, by the Lemma 6.2, we have A� > 0. We obtain

dL

dtD A�

�1 � T�

T

� g.T/ � g.T�/

C A�p

�1 � T�

T

�.I � I�/

C C��.T � T�/C .I � I�/

�.g.T/ � g.T�/ � ı.I � I�//

� A�ˇT�V�

�I

I�� I.t � �/

I�C T�

TC I.t � �/V�

I�VC TI�V

T�IV�� 3

�.

Note that

g.T/ � g.T�/ D ��

aT�

TmaxC d � a

�.T � T�/ � aT

Tmax.T � T�/,

p

�1 � T�

T

�.I � I�/ D �p.I � I�/

.T � T�/2

TT�C p

T�.T � T�/.I � I�/.

We substitute the expressions

dL

dtD �A�

�1 � T�

T

��aT�

TmaxC d � a

�.T � T�/C aT

Tmax.T � T�/

�

� A�p.I � I�/.T � T�/2

TT�C A� p

T�.T � T�/.I � I�/

C C��.T � T�/C .I � I�/

� ��

aT�

TmaxC d � a

�.T � T�/ � aT

Tmax.T � T�/ � ı.I � I�/

�

� A�ˇT�V�

�I

I�� I.t � �/

I�C T�

TC I.t � �/V�

I�VC TI�V

T�IV�� 3

�

D �A�

�aT�

TmaxC d � a

�.T � T�/

2

T� A� a

Tmax

T � T�

2



� A�p.I � I�/.T � T�/2

TT�C A� p

T�.T � T�/.I � I�/

� C�

�aT�

TmaxC d � a

�.T � T�/2 � C�

�ı C aT�

TmaxC d � a

�.T � T�/.I � I�/

� C� aT

Tmax.T � T�/2 � C� aT

Tmax.T � T�/.I � I�/ � C�ı.I � I�/2

� A�ˇT�V�

�I

I�� I.t � �/

I�C T�

TC I.t � �/V�

I�VC TI�V

T�IV�� 3

�.

Note thata

TmaxT.T � T�/.I � I�/ D a

Tmax.T � T�/2.I � I�/C a

TmaxT�.T � T�/.I � I�/.

Figure 1. Dynamics of system with the cure rate p D 1.5 with this values R0 D 0.975 and the infection–free equilibrium Eı.863.32, 0, 0/.

Figure 2. Dynamics of system with the cure rate p D 1.5 with these values R0 D 1.9 and the infected equilibrium E�.525, 37.5, 3745/.



Thus,dL

dtD �A�

�aT�

TmaxC d � a

�.T � T�/

2

T� A� a

Tmax

T � T�

2

� A�p.I � I�/.T � T�/2

TT�C A� p

T�.T � T�/.I � I�/

� C�

�aT�

TmaxC d � a

�.T � T�/2 � C�

�ı C 2

aT�

TmaxC d � a

�.T � T�/.I � I�/

� C� aT

Tmax.T � T�/2 � C� a

Tmax.T � T�/2.I � I�/ � C�ı.I � I�/2

� A�ˇT�V�

�I

I�� I.t � �/

I�C T�

TC I.t � �/V�

I�VC TI�V

T�IV�� 3

�

D �A�

�aT�

TmaxC d � a � p

I�

T�C p

I

T�

�.T � T�/

2

T

� C�

�aT�

TmaxC d � a � a

TmaxI� C a

TmaxI

�.T � T�/2 � C� aT

Tmax.T � T�/2 � C�ı.I � I�/2

� A�ˇT�V�

�I

I�� I.t � �/

I�C T�

TC I.t � �/V�

I�VC TI�V

T�IV�� 3

�.

(32)

We now calculate the derivative of (27),

dLC

dtD d

dt

Z �

0

�I.t � �/

I�� 1 � ln

I.t � �/I�

�d�

DZ �

0

d

dt

�I.t � �/

I�� 1 � ln

I.t � �/I�

�d�

D �Z �

0

d

d�

�I.t � �/

I�� 1 � ln

I.t � �/I�

�d�

D ��

I.t � �/I�

� 1 � lnI.t � �/

I�

��

�D0

D � I.t � �/I�

C I

I�C ln

I.t � �/I�

� lnI

I�

D � I.t � �/I�

C I

I�C ln

I.t � �/V�

I�VC ln

TI�V

T�IV�C ln

T�

T.

(33)

We derive from (32) and (33) that

dQLdt

D dL

dtC ˇT�V�A� dLC

dt

D �A�

�aT�

TmaxC d � a � p

I�

T�C p

I

T�

�.T � T�/

2

T

� C�

�aT�

TmaxC d � a � a

TmaxI� C a

TmaxI

�.T � T�/2 � C� aT

Tmax.T � T�/2 � C�ı.I � I�/2

� ˇT�V�A�

�T�

T� 1 � ln

T�

T

�� ˇT�V�A�

�I.t � �/V�

I�V� 1 � ln

I.t � �/V�

I�V

�

� ˇT�V�A�

�TI�V

T�IV�� 1 � ln

TI�V

T�IV�

�.

(34)

The terms between the brackets in the expression of (34) are Volterra-type form. These forms are positive definite. If the followingconditions are satisfied

d C a

TmaxT� � a C p

I�

T�and d C a

TmaxT� � a C a

TmaxI� (35)

then dQL=dt � 0 for all T , I, V > 0. The conditions (35) are equivalent to (25). By Corollary 5.2 of [23], solutions limit to M, thelargest invariant subset of

˚dQL=dt D 0

�. Furthermore, dQL=dt D 0 if and only if T.t/ D T�, I.t/ D I� and I.t��/V�

I�V.t/ D T.t/I�V.t/T� I.t/V� D 1.

Therefore, the largest compact invariant set in M is the singleton fE�g, where E� is the infected equilibrium state. This shows thatlimt!1.T.t/, I.t/, V.t// D .T�, I�, V�/. By the classical Lyapunov–LaSalle invariance principle (Theorem 5.3 of [23]), if (25) is satisfied,then E� is globally asymptotically stable. This completes the proof.

Now, we analyzed the global stability of delay virus dynamics model (2) without cure rate, p D 0.



Theorem 6.3Let p D 0 and R0 > 1. If the condition (16) holds, then the unique infected equilibrium, E�, of (2) is globally asymptotically stable forany time delay � � 0. In particular, R0 > 1 and (16) are equivalent to (24).

Figure 3. Dynamics of virus and phase space for system 2 with the cure rate p D 0, R0 D 2.9 and the infected equilibrium E�.216.6, 33.9, 1697.3/.

Figure 4. Dynamics of virus and phase space for system 2 with the cure rate p D 1, R0 D 1.02 and the infected equilibrium E�.750, 2.4, 168.2/.



Figure 5. Dynamics of virus and phase space for system 2 with the cure rate p D 0, R0 D 49.7 and the infected equilibrium E�.28.8, 60.5, 22723.8/, the periodicsolutions appear for values of the delay higher than �0 D 0.72.

ProofThe proof is similar to the proof of Theorem 6.1, but with the Lyapunov functional

QU.t/ D L.t/C ˇT�V�ALC.t/. (36)

The functionals L.t/ and LC.t/ are previously defined in this Section by (26) and (27), respectively. Choosing the constants A > 0 and

B D AˇT�V�

qI�.

Calculating the derivative of (36) along positive solutions of system (2), it follows that

d QUdt

D �A

�aT�

TmaxC d � a

�.T � T�/

2

T� A

a

Tmax

T � T�

2

� ˇT�V�A

�T�

T� 1 � ln

T�

T

�� ˇT�V�A

�I.t � �/V�

I�V� 1 � ln

I.t � �/V�

I�V

�

� ˇT�V�A

�TI�V

T�IV�� 1 � ln

TI�V

T�IV�

�.

If (16), then d QU=dt � 0 for all T , I, V > 0. By the classical Lyapunov-LaSalle invariance principle (Theorem 5.3 of [23]), if (16) is satisfied,then E� is globally asymptotically stable.

7. Numerical simulations

In this section, we consider the parameters as suggested by Culshaw and Ruan [28] and Hattaf and Yousfi [12] to illustrate the resultsobtained in the previous sections. In order to explore the behavior of the system and illustrate the stability of the equilibria solutions,we have used dde23 [29], based on Runge-Kutta methodsFor Figure 1, we take s D 10, d D 0.02, a D 0.03, Tmax D 1200, ˇ D 0.000024, p D 1.2, ı D 0.5, q D 240, c D 3, with this parameterschoice R0 D 0.975 and the infection-free equilibrium Eı.Tı, 0, 0/, with Tı D 863.32, is globally stable according to theorem 2.1. Weillustrate this result for some values of � .

In Figure 2, we consider the parameters of [28] s D 10, d D 0.02, a D 0.03, Tmax D 1500, ˇ D 2.4 � 10�5, p D 1, ı D 0.26, q D 240and c D 2.4. With these values, the infected equilibrium is E�.525, 37.5, 3745/ and R0 D 1.9 and satisfies condition (16) of theorem 4.2,



Figure 6. Dynamics of system with the cure rate p D 0.1 with this values R0 D 13.9 and the infected equilibrium E�.83.3, 84.2, 25264.8/, the periodic solutionsappear for values of the delay higher than �0 � 1.1.

Figure 7. Dynamics of system with the cure rate p D 1 with these values R0 D 5.57 and the infected equilibrium E�.208.3, 173.8, 52149/, and the critical valueof delay is �0 D 4.5.

Figure 8. Dynamics of system with the cure rate p D 1.5 with these values R0 D 4.18 and the infected equilibrium E�.277.8, 212.4, 63710/, and the critical valueof delay is �0 D 7.98.



therefore we have local stability of infected equilibrium for all � , in simulation, we illustrate the dynamics for some values of � . Alsonote that these values for the parameters do not satisfies condition (iii) of theorem 2.3 of Yang, Wang, and Li [14] for local stability butthe result is true.

For simulations in Figure 3, we consider a treatment rate p D 0 and the following values for the rest of parameters s D 10,d D 0.03, a D 0.03, Tmax D 1200, ˇ D 0.000024, ı D 0.26, q D 120, c D 2.4 with these values, R0 D 2.9 and the equilibrium isE�.216.6, 33.9, 1697.3/. These values satisfy conditions of theorem 6.3, therefore the equilibrium will be globally stable.

In Figure 4, the parameters are s D 10, d D 0.02, a D 0.03, Tmax D 1000, ˇ D 0.000024, p D 1, ı D 0.26, q D 168, c D 2.4 and weconsider several values for � . With these values, R0 D 1.02 and conditions (16) of theorem 4.2 and (25) of theorem 6.1 are satisfied thenthe infected equilibrium E�.750, 2.4, 168.2/will be globally stable in this case for all � .

In Figure 5, we consider again a treatment rate p D 0 and the values for parameters as s D 5, d D 0.02, a D 0.4, Tmax D 1500,ˇ D 0.000024, ı D 0.26, q D 900, c D 2.4 with these values R0 D 49.7 and the equilibrium is E�.28.8, 60.5, 22723.8/. These values do

not satisfy conditions of theorem 6.3 and satisfy the condition 2� C aT�

Tmax

�C ıI�

T� D �0.18 < 0, therefore we expect a periodic solution

for some value of �0 as established in theorem 5.1, with the values of these parameters, the coefficient c3 of (18) is c3 D �0.039 andthe largest positive simple root is !0 D 1.31 therefore �0 � 0.72. To verify this statement, we consider the values of delay 0.6, �0 and 1in our simulations, note that for higher values of � , the periodic solutions are evident, the oscillations seem to be small but the scale isorder 104.

We consider for the Figure 6 the parameters of Hattaf and Yousfi [12], we take s D 5, d D 0.02, a D 0.5, Tmax D 1200, ˇ D 2.4 � 10�5,p D 0.1, ı D 0.5, q D 900, c D 3 and several values for the delay � for the simulation in Figure 6 with these values R0 D 13.9 andconditions (16) of theorem 4.2 and (25) of theorem 6.1 are not satisfied then the infected equilibrium E�.83.3, 84.2, 25264.8/will not be

stable for some values of � in this case, with the values of this parameters, the condition 2� C aT�

Tmax

�C ıI�

T� < 0 of Theorem 5.1 is equal

to �0.31 and the coefficient c3 of (18) is c3 D �0.51 and the largest positive simple root is !0 D 0.81 therefore �0 � 1.1.In Figure 6 (a)–(b), the infected equilibrium E� of system is stable for a value of 0.5 for � , but this can not be guaranteed for highervalues of �0 � 1.1. In (c)–(d), we consider a value of �0 for � , with this value for the delay, we can see a periodic solution for system (2).In (e)–(f ), for a delay of � D 1.5, we also can see a periodic solution with the amplitude of periodic solution affected by the delay.

In the Figures 7–8 we consider as values for the parameters as in [12], s D 5, d D 0.02, a D 0.5, Tmax D 1200, ˇ D 2.4 � 10�5, ı D 0.5,q D 900, c D 3, several values for the delay � and for the cure rate p. We can see the effect of different values of the cure rate p, we seethe periodic solution appear for a higher value of � when p increases.

8. Concluding remarks

In this paper, we presented a mathematical analysis of the global dynamics for a virus dynamics model with cure rate and an intracellulardelay and we gave conditions to ensure permanence of system. The intracellular delay corresponds to the time necessary between viraltranscription and viral release and maturation.

By the Lyapunov direct method and using the technology of constructing Lyapunov functionals, we obtain conditions for globalstability of the equilibria of delay model. Lyapunov functionals are obtained by linear combinations of Volterra–type functions,composite quadratic functions and Volterra–type functionals.

The qualitative analysis of the delay model (2) with cure of infected cells reveals the existence of three scenarios possible of aviral infection.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

1

2

3

4

5

6

7

8

9

Cure rate of infected cells (p)

Bas

ic r

epro

du

ctio

n n

um

ber

Figure 9. Plot of the basic reproduction number R0 in (5) as a function of the cure rate p. We consider the values for parameters as s D 5, d D 0.02, a D 0.5,Tmax D 1200, ˇ D 2.4 � 10�5, ı D 0.5, q D 450, c D 3.



A first scenario is that the viral population is eventually totally eliminated. Mathematically, if the basic reproduction number R0 is lessthan unity, the infection–free equilibrium is globally asymptotically stable.

Biologically, the uniform persistence characterizes chronic infection of cells. It follows from persistence theory for infinite dimensionalsystems, we showed that uniform persistence occurs if the basic reproduction number is greater than unity.

A second scenario is that the infection becomes established, and that the virus population grows with damped oscillations, orunimodal growth, is that the infected equilibrium (positive interior equilibrium) is asymptotically stable. Formally, we prove that if thebasic reproduction number R0 is greater than unity and the condition (16) is satisfied, then the infected equilibrium is locally asymp-totically stable. We improve the local stability result obtained in [14]. Also, we gave a sufficient condition on the parameters for theglobal stability of the infected equilibrium. In particular, by Theorems 4.2 and 6.3, we saw that the conditions to the local stability andglobal stability of infected equilibrium of viral infection model (2) without cure rate are equivalent. We conjecture the global stabilityof positive equilibrium if we assume condition (16).

A third scenario is that the virus population grows with undamped oscillation or sustained. Mathematically, it is shown that theundamped oscillation onset occurs through a Hopf bifurcation. Theorem 5.1 indicates that the model with intracellular delay could

exhibit Hopf bifurcation at certain value of the time delay if the parameters values satisfy the condition 2� C aT�

Tmax

�C ıI�

T� < 0. From

the previous condition, we observe that the mitosis (logistic) term can lead to Hopf bifurcation. Numerically, we can also observe fromour simulations how the delay affects the onset of periodic solutions and when in combination with different values of the cure rate,affects the amplitude of such kind of solutions.

Studying the reversion rate of infected cells into the uninfected state as a control parameter of a viral infection is very important. Weplot R0 versus p, and vary the values of p between .0, 5/. In Figure 9, we note that if p D 0 then R0 D 8.3686 and if it increases the valueof cure rate of the infected cells, then the basic reproductive number decrease below one. A result relevant of model (2) is that the viralinfection gets eradicated if p is larger.

According to the immunological basis found in [30], the curative effect of the cytotoxic T–lymphocyte response is much moreefficient than its destructive effect on infected cells. Possible future work could investigate the ‘curative’ mechanisms of infected cellsreported by Guidotti and co-workers [3, 4, 30].

Acknowledgements

This work has been partially supported by SNI under grant 15284.

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