Global Dynamics of a Delayed HIV-1 Infection Model with CTL Immune Response

A delayed HIV-1 infection model with CTL immune response is investigated. By using suitable Lyapunov functionals, it is proved that the infection-free equilibrium is globally asymptotically stable if the basic reproduction ratio for viral infection is less than or equal to unity; if the basic reproduction ratio for CTL immune response is less than or equal to unity and the basic reproduction ratio for viral infection is greater than unity, the CTL-inactivated infection equilibrium is globally asymptotically stable; if the basic reproduction ratio for CTL immune response is greater than unity, the CTL-activated infection equilibrium is globally asymptotically stable.


Introduction
Recently, many mathematical models have been developed to describe the infection with HIV-1 human immunodeficiency virus 1 .By investigating these models, researchers have gained much important knowledge about the HIV-1 pathogenesis and have enhanced progress in the understanding of HIV-1 infection see, e.g., 1-4 .It is pointed out by the work of 5 that immune response is universal and necessary to eliminate or control the disease during viral infections.In particular, as a part of innate response, cytotoxic T lymphocytes CTLs play a particularly important role in antiviral defense by attacking infected cells.Thus, many authors have studied the mathematical modelling of viral dynamics with CTL immune response see, e.g., 5-9 .In 7 , Nowak and Bangham considered an HIV-1 infection model with CTL immune response which is described by the following differential equations: Moreover, infection rate plays an important role in the modelling of epidemic dynamics.Holling type-II functional response seems more reasonable than the bilinear incidence rate see, 10 .In 11 , by stability analysis, Song and Avidan obtained that the system with the bilinear incidence rate was an extreme case of the model with Holling type-II functional response term.
In 3, 4, 7 , the researchers used ordinary differential equations to describe different aspects of the dynamics of the viral infections.However, in the real virus dynamics, infection processes are not instantaneous.Time delays are usually introduced for the purpose of accurate representations of this phenomena see, e.g., 6, 12-17 .As pointed out in 12 , there is a time delay between initial viral entry into a cell and subsequent viral production, and the effect of saturation infection of an HIV-1 model was studied.By using the Lyapunov-LaSalle type theorem, sufficient conditions were derived for the global stability of the infection-free equilibrium and the chronic-infection equilibrium.In addition, there is also a period between virions that have created within a cell, and the new virions are released from the cell see, e.g., 6, 13, 17 .In 13 , Zhu and Zou studied an HIV-1 model with discrete delays and found that large delays can help eliminate the virus.To the best of our knowledge, there are few works on the dynamics of HIV-1 system with CTL immune response, Holling type-II functional response, and two kinds of discrete delays.Therefore, we are concerned with the effect of the above factors on system 1.1 .Motivated by the works of Nowak and Bangham 7 , Song and Avidan 11 , in the present paper, we consider the following delay differential equations: where the parameters have the same meanings as in system 1.1 , τ 1 represents the time between viral entry into a target cell and the production of new virus particles and τ 2 stands for a virus production period for new virions to be produced within and released from the infected cells.The initial conditions for system 1.2 take the form where φ 1 θ , φ 2 θ , φ 3 θ , φ 4 θ ∈ C −τ, 0 , R 4 0 , the Banach space of continuous functions mapping the interval −τ, 0 into R 4 0 , where It is well known by the fundamental theory of functional differential equations 18 that system 1.2 has a unique solution x t , y t , v t , z t satisfying the initial conditions 1.3 .It is easy to show that all solutions of system 1.2 with initial conditions 1.3 are defined on 0, ∞ and remain positive for all t > 0.
This paper is organized as follows.In Section 2, by analyzing the basic reproduction ratio for viral infection and CTL immune response, the existence of three equilibria is established.Moreover, the ultimate boundedness of the solutions for system 1.2 is presented.In Section 3, by means of suitable Lyapunov functionals and LaSalle's invariant principle, we discuss the global stability of the infection-free equilibrium, the CTL-inactivated infection equilibrium, and the CTL-activated infection equilibrium, respectively.In Section 4, we carry out some numerical examples to illustrate the theoretical results.Finally, a discussion is given in Section 5 to end this work.

Preliminary Results
In this section, we discuss the existence of three equilibria and prove that all the solutions are positive and bounded.
Clearly, system 1.2 always has an infection-free equilibrium E 0 x 0 , 0, 0, 0 λ/d, 0, 0, 0 .Denote Here, R 0 and R 1 are called the basic reproduction ratios for viral infection and CTL immune response of system 1.2 , respectively.It is easy to see that R 0 > R 1 always holds.If R 0 > 1, system 1.2 has a CTL-inactivated infection equilibrium E 1 x 1 , y 1 , v 1 , z 1 besides the equilibrium E 0 , where Theorem 2.1.Supposing that x t , y t , v t , z t is a solution of system 1.2 with initial conditions 1.3 , then there exists M > 0, such that all the solutions satisfy x t < M, y t < M, v t < M, z t < M for sufficiently large time t.
Since all solutions of system 1.2 are positive, simple calculation leads to

2.5
Therefore, we get N t < λ/δ ε M for sufficiently large time t, where ε is an arbitrarily small positive constant.Finally, all the solutions of system 1.2 are ultimately bounded by some positive constant.This completes the proof.

Global Stability
In this section, we study the global stability of each equilibrium of system 1.2 by using suitable Lyapunov functionals which are inspired by Xu 12 and McCluskey 19 and LaSalle's invariant principle.Define the following function: Clearly, for x ∈ 0, ∞ , g x has the minimum at x 1 and g 1 0.
Proof.Let x t , y t , v t , z t be any positive solution of system 1.2 with initial conditions 1.3 .Define the following Lyapunov functional: Calculating the derivative of V 0 t along positive solutions of system 1.2 , it follows that

3.3
Noting that R 0 ≤ 1, we obtain au/k − βx 0 / 1 αv t ≥ au/k − βλ/d au/k 1− R 0 ≥ 0. Hence, from 3.3 , we have V 0 t ≤ 0. By Theorem 5.3.1 in 18 , solutions limit to M 0 , the largest invariant subset of {V 0 t 0}.Let x t , y t , v t , z t be the solution with initial function in M 0 .Then, from the invariance of M 0 , we obtain x t x 0 , v t 0, and z t 0 for any t.Further, from the third equation of system 1.2 , we obtain y t 0. Accordingly, it follows from LaSalle's invariance principal that the infection-free equilibrium E 0 is globally asymptotically stable for any positive time delays.This completes the proof.Theorem 3.2.If R 1 ≤ 1 < R 0 , the CTL-inactivated infection equilibrium E 1 of system 1.2 is globally asymptotically stable.
Proof.Let x t , y t , v t , z t be any positive solution of system 1.2 with initial conditions 1.3 .Define the following Lyapunov functional: where

3.5
For clarity, we will calculate the derivatives of g x t /x 1 , g y t /y 1 , g v t /v 1 , z t , V 11 t , and V 12 t along positive solutions of system 1.2 , respectively.

3.7
Since u ky 1 /v 1 holds, it follows that

3.8
Calculating the derivatives of V 11 t and V 12 t shows that

3.9
We therefore derive from 3.6 -3.9 that

3.10
Noting that R 1 ≤ 1, we derive that y 1 −b/c bkβ cdu bdkα /ck β dα R 1 −1 ≤ 0. Hence, from 3.10 , we have V 1 t ≤ 0. Similar to Theorem 3.1, solutions limit to M 1 , the largest invariant subset of {V 1 t 0}.Let x t , y t , v t , z t be the solution with initial function in M 1 .Then, we obtain that

3.11
It is readily to show that x t and  z t 0 for any t.Thus, it follows from LaSalle's invariance principal that the CTL-inactivated infection equilibrium E 1 is globally asymptotically stable for any positive time delays.This completes the proof.Theorem 3.3.If R 1 > 1, the CTL-activated infection equilibrium E 2 of system 1.2 is globally asymptotically stable.
Proof.Let x t , y t , v t , z t be any positive solution of system 1.2 with initial conditions 1.3 .We construct the following Lyapunov functional:

3.13
Next, we will calculate the derivatives of g x t /x 2 , g y t /y 2 , g v t /v 2 , g z t /z 2 , V 21 t , and V 22 t along positive solutions of system 1.2 , respectively.Similar to 3.6 , we derive that

3.15
Similar to 3.8 , we get that

3.16
Calculating the derivative of g z t /z 2 shows that 3.17 Similar to 3.9 , it follows that

3.18
We therefore derive from 3.14 -3.18 that

3.19
Hence, from 3.19 , we get V 2 t ≤ 0. Similar to Theorem 3.1, solutions limit to M 2 , the largest invariant subset of {V 2 t 0}.Let x t , y t , v t , z t be the solution with initial function in M 2 .Then, it holds that

3.20
It is easy to show that x t x t−τ 1 x 2 , y t y t−τ 2 y 2 , and v t v t−τ 1 v 2 for any time t.Moreover, from the second equation of system 1.2 , we obtain z t z 2 .Therefore, it follows from LaSalle's invariance principal that the CTL-activated infection equilibrium E 2 is globally asymptotically stable for any positive time delays.This completes the proof.

Numerical Simulations
In the following, we give three examples to illustrate the main theoretical results above.All the parameters were obtained from 9, 16, 20 . .It is easy to show that R 0 0.833 < 1, and that system 1.2 has an infection-free equilibrium E 0 10, 0, 0, 0 By Theorem 3.1, we get that the infection-free equilibrium E 0 of system 1.2 is globally asymptotically stable.Numerical illustrates this fact see Figure 1   The CTL-inactivated infection equilibrium E 1 of system 1.2 is globally asymptotically stable.Here τ 1 5, τ 2 9, and the initial value is x 0 10, y 0 1, v 0 50, z 0 2 .system 1.2 has a CTL-activated infection equilibrium E 2 7.4856, 1.5000, 72.0000, 2.9835 .By Theorem 3.3, we get that the CTL-activated infection equilibrium E 2 of system 1.2 is globally asymptotically stable.Numerical simulation illustrates this fact see Figure 3 .

Discussion
In this paper, we have studied the global dynamics of a delayed HIV-1 infection model with CTL immune response.By constructing suitable Lyapunov functionals, sufficient conditions have been derived for the global stability of three equilibria.It is easy to show that if the basic reproduction ratio for viral infection R 0 1, infection-free equilibrium E 0 is globally asymptotically stable, and the virus is cleared up; if the basic reproduction ratio for CTL immune response R 1 satisfies R 1 ≤ 1 < R 0 , the equilibrium E 1 is globally asymptotically stable, and the infection becomes chronic but without CTL immune response; if R 1 > 1, The CTL-activated infection equilibrium E 2 of system 1.2 is globally asymptotically stable.Here τ 1 4, τ 2 2, and the initial value is x 0 1, y 0 1, v 0 50, z 0 4 .system 1.2 has a CTL-activated infection equilibrium E 2 besides E 0 and E 1 , which is globally asymptotically stable, and the infection turns to chronic with CTL immune response.From Theorems 3.1, 3.2, and 3.3, we see that the delays τ 1 and τ 2 do not affect the global stability of the feasible equilibria and therefore do not induce periodic oscillations, and the possibility of Hopf bifurcations is therefore ruled out.On the other hand, if the basic reproduction ratio for CTL immune response R 1 > 1, we can get which shows that the number of infected cells and virions of the equilibrium E 1 is greater than the number of those of the equilibrium E 2 .Hence, the CTL immune response plays an important role in the reduction of the infected cells and the free virions.
, y t , v t , and z t represent the densities of uninfected target cells, infected cells, virions, and CTL cells at time t, respectively.Uninfected cells are produced at rate λ, die at rate d, and become infected cells at rate βxv.Infected cells are produced from uninfected cells at rate βxv and die at rate a.The parameter p accounts for the strength of the lytic component.Free virions are produced from uninfected cells at rate ky and are removed at rate u.The parameter b is the death rate for CTLs, and cyz describes the rate of CTL immune response activated by the infected cells. .