Dynamics of a Hiv-1 Infection Model with Cell-mediated Immune Response and Intracellular Delay

In this paper, we consider a mathematical model for HIV-1 infection with intracellular delay and cell-mediated immune response. A novel feature is that both cytotoxic T lymphocytes (CTLs) and the intracellular delay are incorporated into the model. We obtain a necessary and sufficient condition for the global stability of the infection-free equilibrium and give sufficient conditions for the local stability of the two infection equilibria: one without CTLs being activated and the other with. We also perform some numerical simulations which support the obtained theoretical results. These results show that larger intracellular delay may help eradicate the virus, while the activation of CTLs can only help reduce the virus load and increase the healthy CD + 4 cells population in the long term sense. 1. Introduction. In the past decade, there has been much interest in mathematical modeling of HIV dynamics (see, for example, [14, 18, 19]). This is because HIV mathematical models can provide insights into the dynamics of viral load in vivo and may play a significant role in the development of a better understanding of HIV/AIDs and drug therapies. A simple, standard yet classic model (probably the first) for HIV dynamics was proposed by Perelson et al. in [18, 19] as follows:


1.
Introduction. In the past decade, there has been much interest in mathematical modeling of HIV dynamics (see, for example, [14,18,19]). This is because HIV mathematical models can provide insights into the dynamics of viral load in vivo and may play a significant role in the development of a better understanding of HIV/AIDs and drug therapies.
A simple, standard yet classic model (probably the first) for HIV dynamics was proposed by Perelson et al. in [18,19] as follows: The variables ( ), ( ), and ( ) denote the concentrations of uninfected cells, infected cells and virus, respectively. The parameter is the rate at which new target cells are generated, is the death rate of the susceptible cells and is the constant 512 HUIYAN ZHU AND XINGFU ZOU characterizing the infection rate. The infected cells are assumed to die at a rate (say, via lysis) due to the action of virus, each releasing new virus particles as the lysis of infected cells occurs. Thus, on average, virus is instantaneously produced at rate ( ). Lastly, virus particles are cleared from the system at rate per virion.
Note that the immune response after viral infection is universal and necessary to eliminate or control the disease. Antibodies, cytokines, natural killer cells, B cells and T cells are all essential components of a normal immune response to viral infection. However, in HIV-1 infection, cytotoxic T lymphocytes (CTLs) play a critical role in antiviral defense by attacking virus-infected cells. Indeed, it is believed that CTLs are the main host immune factor that determines virus load (e.g., [1]). Therefore, the dynamics of HIV-1 infection with CTLs response has recently drawn much attention of researchers in the related areas (see, e.g., [1,15,21,7] and the references therein), and is also the main concern of this research. Letting ( ) be the concentration of CTLs, model (1) can be modified to where accounts for the strength of the lytic component and is the death rate for CTLs. The function ( , , ) describes the rate of immune response activated by the infected cells. Arnaout et al. [1] and Wang et al. [21] assumed that the production of CTLs depends only on the concentration of infected cells and chose the linear dependent former ( , , ) = ( ). In [15], Nowak and Bangham assumed that the production of CTLs is also dependent upon the concentration of CTL cells, and used ( , , ) = ( ) ( ), leading to the following concrete model: Using symbolic calculation software on computers, Liu [13] gave stability analysis of model (3). By simple algebraic manipulations, Kajiwara and Sasaki [11] presented pure theoretic results on the stability of model (3). Culshaw et al. [7] further assumed that the production of CTLs is also related the healthy cells' help and accordingly chose ( , , ) = ( ) ( ) ( ). With the extra assumption that the viral load is proportional to the level of infected cells, A HIV-1 INFECTION MODEL WITH DELAY   513 they proposed and studied the following HIV-1 infection model:

DYNAMICS OF
Here the ratio ′ : is the proportion of infected cells that survive the incubation period ( average time between the new infection of a + 4 T cell and the time it becomes infectious). Model (4) may have up to three equilibria and the local stability of the three equilibria were analyzed in [7].
As pointed out in [22], time delays can not be ignored in models for immune response, since antigenic stimulation generating CTLs may need a period of time, that is, the activation rate of CTL response at time may depend on the population of antigen at a previous time. Based on such a reality, in [5,6,22], a time delay was incorporated into the the immune activation term ( , , ) for ( , , ) = and ( , , ) = respectively, and the effects of the delay on the dynamics of the corresponding models was investigated. It has been found (in [22]) that the delay in activating immune response could lead to very complicated dynamics including stable periodic solutions and chaos, and such complicated dynamical behaviors may well explain irregular real time series data for the immune state of a patient.
The aforementioned models can capture some essential features of the immune system and are able to produce a variety of immune responses. On the other hand, it has been realized recently that there are also delays in the process of cell infection and virus production, and thus, delays should be incorporated into the infection equation and/or the virus production equation of a model. In the absence of immune responses, Nelson et al. [16] added a discrete delay in the equation of model (1), and presented detailed analysis of the resulting delay differential equation model. By comparing their results to those from the corresponding model without delay (i.e., (1)), they showed that the predicted rate of decline in plasma virus concentration depends on the length of the delay. Nelson and Perelson [17] further generalized the model by considering a general delay distribution, and suggested incorporating delays in both cell infection equation and virus replication. As a follow-up of [17], Zhu and Zou [23] investigated a model with a discrete delay in the infection equation and another discrete delay in the virus production equation, by analyzing the two delay model, they found that large delays can help eliminate the virus.
In this paper, following the line of [16], we incorporate a delay into the cell infection equation in model (3). That is, we propose the following model: where denotes the lag between the time the virus contacts a target cell and the time the cell becomes actively infected (including the steps of successful attachment of virus to the cell, and penetration of virus into the cell). The novelty of the model (5) is that it includes both the main immune response factor CTLs in HIV infection and the intracellular delay in virus production. For other viral infections, as long as there is an intracellular delay in virus replication and the CTLs plays the main role in the immune response to the virus, we believe the model is also appropriate. In the rest of this paper, we investigate the impact of the delay and CTLs on the the dynamics of model (5). In Section 2, the positivity and boundedness of solutions of the system (5) are presented. The stability analysis for the three equilibria are given in Section 3, and some numerical simulations are given in Section 4. Finally, in Section 5, some conclusions are drawn from the obtained results in previous sections, revealing both qualitatively and quantitatively the positive role of the CTLs and the intracellular delay. (5) is a system of delay differential equations. For such a system, initial functions need to be specified and well-posedness needs to be addressed. Let = ([− , 0]; 4 ) be the Banach space of continuous mapping from [− , 0] to equipped with the sup-norm. By the fundamental theory of FDEs (see Hale and Verduyn Lunel [10]), we know that there is a unique solution ( ( ), ( ), ( ), ( )) to system (5) with initial conditions

Positivity and boundedness of solutions. Model
For biological reasons, the initial functions ( ), ( ), ( ) and ( ) are assumed to be non-negative: The following theorem establish the positivity and boundedness of solutions of (5) with initial functions satisfying (6) and (7). Proof. Note that from (5), we have . Positivity immediately follows from the above integral forms and (6) and (7).
Remark 1. (i) From the proof, one can see that in addition to (6) and (7) However, if ℛ 0 > 1, in addition to 0 , there is another biologically meaningful equilibrium which corresponds to positive levels of healthy + 4 T cells, infected + 4 T cells and virus, but no CTL response. If or equivalently system (5) also has an interior equilibrium accounting for the presence of all four components: uninfected The characteristic equation for (8) is Obviously, = − and = − are eigenvalues for (8), and hence, the stability of 0 is determined by the distribution of the roots of equation If ℛ 0 < 1, then = 0 is not a root of the equation (10) In this case, ℛ 0 < 1 reduces to < 1. Clearly, if < 1, then − / > 0 under which all roots of (11) have negative real parts. Note that all roots of (10) depend continuously on (see [3]). Notice also that the assumption (ii) of [4] holds and this ensures ( ) < +∞ for any root of (10). Therefore, as the delay increases, the roots of (10) can only enter the right-half in complex plane by crossing the imaginary axis. Let = with > 0 be a purely imaginary root of (10), then, Taking moduli in both sides of the above equation gives If ℛ 0 < 1, then (12) has no non-negative real root. Therefore, there is no root = with ≥ 0 for (10), implying that the roots of (10) can not cross the purely imaginary axis. Hence all roots of (10) have negative real parts provided ℛ 0 < 1. On the other hand, it is easy to see that (10) has a real positive root if ℛ 0 > 1.
Summarizing the above, we have established the following Theorem 3.1. If ℛ 0 < 1, then the infection-free equilibrium 0 is locally asymptotical stable; if ℛ 0 > 1, then the infection-free equilibrium 0 becomes unstable and there occurs the equilibrium 1 .
Theorem 3.1 only establishes local stability of 0 under ℛ 0 < 1. By constructing a Lyapunov functional, we can actually obtain globally asymptotic stability of the infection-free equilibrium 0 under the condition ℛ 0 < 1.
Calculating the time derivative of along the solution of (5), we obtain Here we have used the fact that ( ), ( ), ( ) and ( ) are non-negative and ℛ 0 < 1. The globally asymptotic stability of the infection-free equilibrium 0 follows from the above inequality and LaSalle Invariance principle (see e.g., [12]). 1 . In this subsection, we assume ℛ 0 > 1. Thus, 1 exists and the linearization of (5) at 1 is

HUIYAN ZHU AND XINGFU ZOU
The characteristic equation for (13) is The first factor on the left hand side of (14) gives a real root which is negative if ℛ 1 < 1 and positive if ℛ 1 > 1. The remaining roots of (14) are obtained by considering Rewrite equation (15) as Clearly, under ℛ 0 > 1, = 0 is not a root of (16) since to which, the Routh-Hurwitz Theorem [9] for cubic polynomials is applicable. Note that under ℛ 0 > 1, Hence all roots of (16) have negative real parts when = 0. Note that a root of (16) depends continuously on (see [3]). Notice also that the assumption (ii) of [4] holds and this ensures that ( ) < +∞ for any root of (16). Therefore, as delay increases, a root of (16) can only enter the right-half of the complex plane by crossing the imaginary axis. Let = with ≥ 0 be a purely imaginary root of (16). Then, Straightforward calculations shows that Thus the function ( ) is monotonically increasing in ∈ [0, ∞) with (0) = ( ) > 0 and hence (18) has no real non-negative root. This implies that no root can cross the imaginary axis as increases, ensuring that under ℛ 0 > 1 all roots of (15) have negative real parts for all ≥ 0. Summarizing the above, we have obtained the following , then 1 becomes unstable and there occurs the interior equilibrium 2 .

3.3.
Stability of the CTL-activated infection equilibrium 2 . In this subsection, we consider the case ℛ 1 > 1 (equivalently ℛ 0 > 1 + / ) and discuss the stability of the CTL-activated infection equilibrium 2 = (¯ ,¯ ,¯ ,¯ ). To this end, we linearize (5) The characteristic equation of (19) is given by At 2 , we have ¯ − = 0, and ¯ = − ( + ¯ ). Let ′ = ¯ , = + ′ > ′ and = + ¯ . Then the characteristic equation of system (5) at 2 can be calculated as When = 0, 2 has been proved in [11,13] to be asymptotically stable, implying that all roots of (20) with = 0 have negative real parts. Next, we show that 2 is also asymptotically stable for small > 0. Rewrite equation (20) as Since the characteristic equation (21) is a transcendental equation of degree four, the discussion of the distributions of its roots becomes harder. At least, excluding crossing of its roots over the the pure imaginary axis in the complex plane (as was done to (10) and (15)  In what follows, we choose to apply the theory developed in [20] to obtain conditions for the stability of 2 . To this end, we first introduce some notations corresponding to Corollary 2.38 in [20]. For the characteristic function ( ) defined by (21), let ( ) and ( ) be, respectively, the real and pure imaginary parts of ( ), i.e., Then, it is easy to see that Note that when we have Hence, + ( ) has two real positive zeros + 1 and + 2 , where Thus, under (22), both − ( ) and + ( ) have exactly the same number (two) of real zeros. It is also easy to verify the following: . Hence, the intervals 2 and 1 are disjoint.
Furthermore, when <¯ , we have Hence, + ( ) only has one positive zero + = √ 1+ 1 3− 2 , and − ( ) also only has one positive zero − = Combining the above, we see that Corollary 2.38 in [20] is applicable, provided that one can verify the following two conditions (23) and the formula for − , we know that Therefore, by Corollary 2.38 in [20], we have established the following

Conclusions.
We have studied a HIV-1 infection model with cell-mediated immune response and intracellular delay, that is, model (5). By combining the analysis of the characteristic equation and the Lyapunov-LaSalle method, we have proved that the infection-free equilibrium 0 , corresponding to the absence of virus, is globally asymptotically stable when the basic reproduction number ℛ 0 < 1. In this case the virus is unable to maintain the infection and will go extinct ( the uninfected cell population will converge to the value ). When ℛ 0 > 1, 0 becomes unstable and there occurs the CTL-inactivated infection equilibrium 1 . The stability of 1 depends on how much ℛ 0 is larger than 1: when 0 ∈ (1, 1 + / ), 1 is asymptotically stable; when ℛ 0 > 1 + / , 1 becomes unstable and there occurs the third biologically meaningful equilibrium, that is, the CTL-activated infection equilibrium 2 . We have proved and numerically confirmed the stability of 2 under additional conditions (22) and Δ( ) > 0. For the case Δ( ) < 0 under (22), we are unable to make a conclusion, but numeric simulations have shown the possibility that 2 may still be stable.
From the theoretical and numeric results summarized above, we see that the basic reproduction number ℛ 0 determines the dynamics of the model. Considering ℛ 0 = ℛ 0 ( ) = − as a function of , we see that it is decreasing in with ℛ 0 (∞) = 0. An implication of this observation is that the intracellular delay plays a positive role in preventing the virus, because with all other parameters fixed, larger can bring ℛ 0 to a level lower than 1 (regardless of either ℛ 0 (0) < 1 or ℛ 0 (0) > 1), making the infection free equilibrium globally asymptotically stable.
We point out that ℛ 0 is independent of the CTLs related parameters , and , meaning that CTLs does not help eliminate the virus. However, the activation of CTLs does help reduce the virus load and increase the healthy cell population, the latter being crucial because when the population of healthy CD + 4 cells drops below certain level (e.g., [18] ), an HIV carrying person becomes an AIDS patient. This can be seen by comparing the virus load components and the healthy cell population components in the immune-inactivated infection equilibrium 1 = ( 1 , 1 , 1 , 0) and the immune-activated infection equilibrium 2 = (¯ ,¯ ,¯ ,¯ ): under the condition ℛ 0 > 1 + / , simple calculations show that 1 = − >¯ = and 1 = <¯ = + .
Notice that 1 and 1 are independent of and , while¯ is increasing in and decreasing in , and¯ is decreasing in and increasing in . These dependence and independence explicitly explain, both qualitatively and quantitatively, the positive role of CTLs in maintaining the level of the healthy cells as well as in controlling the load of HIV virus.