Global dynamics of an age-structured HIV infection model incorporating latency and cell-to-cell transmission

In this paper, we are concerned with an age-structured HIV infection model incorporating latency and cell-to-cell transmission. The model is a hybrid system consisting of coupled ordinary differential equations and partial differential equations. First, we address the relative compactness and persistence of the solution semi-flow, and the existence of a global attractor. Then, applying the approach of Lyapunov functionals, we establish the global stability of the infection-free equilibrium and the infection equilibrium, which is completely determined by the basic reproduction number.


1.
Introduction. Acquired immunodeficiency syndrome (AIDS) is a condition in human in which progressive failure of the immune system allows lifethreatening opportunistic infections and cancers to thrive. AIDS is considered as a pandemic-a disease outbreak which is present over a large area and is actively spreading. By 2010, more than 30 million people worldwide had died of AIDS (UNAIDS2010). In 2013, it resulted in about 1.34 million deaths.
In 1984, the human immunodeficiency virus (HIV) infection was identified as the cause of AIDS. HIVs are intracellular parasites that depend on the host cells, CD4 + T-cells, to survive and replicate. CD4 + T-cells can be damaged either directly by the virus or by immune responses to the virus [22]. It is believed that CTLs, also known as CD8 + T-cells or killer T-cells, are the main host immune factor that limits the virus replication in vivo, blocks up virus into target cells, and determines the viral load [23]. These cells cannot become infected with the virus, but do destroy infected cells.
The last few decades have witnessed a substantial increase in the application of mathematical models to understand HIV infection at both the population level and within-host level. Models of HIV infection at the population level can be used to predict the future prevalence and suggest possible control strategies while viral dynamic models characterizing the interaction between CD4 + T cells and viruses and studying the effects of drug therapy are aimed to guide treatment strategies. In this paper, we focus on within-host HIV models. We refer to Perelson and Nelson [25] for a review of such models built and analyzed before 1995.
Most within-host HIV models are described by ordinary differential equations (ODEs) for uninfected CD4 + T cells, actively infected CD4 + T cells, free infectious viruses, (CTLs response). For example, Perelson et al. [24] examined a model for the interaction of HIV with CD4 + T cells that consists of four populations: uninfected T cells (T (t)), latently infected T cells (T * (t)), actively infected T cells (T * * (t)), and free viruses (V (t)). The model is as follows, (1.1) . Denote by N the total number of new virus particles produced by each infected cell during its life time. It is shown that if N < N crit , then the uninfected steady state is the only steady state in the nonnegative orthant and this steady state is stable; while if N > N crit , then the uninfected steady state is unstable and the endemically infected steady state can be either stable, or unstable and surrounded by a stable limit cycle. Recently, Wang et al. [36] studied the following HIV-1 infection model with the mass-action infection rate and latent cells, (1.2) (Note that we have changed the symbols for state variables in order to be consistent with (1.1).) If the basic reproduction number R 0 = kλβ(eq+δ) adu(e+δ) ≤ 1 then the infectionfree equilibrium of (1.2) is globally asymptotically stable while if R 0 > 1 then its chronic-infection equilibrium is globally asymptotically stable.
As in the above just mentioned references, most of the existing work on withinhost HIV infection only considers the cell-free spread. However, it is reported recently that cell-to-cell transmission of viruses also occurs when a healthy cell comes into contact with an infected cell [27]. Productive cell-to-cell infection requires interaction between the viral envelop glycoproteins on the surface of the infected cell and HIV receptors on the surfaces of target cells, leading to the formation of virological synapses. For details on the infection mechanism via cell-to-cell contact, we refer the readers to [16,19] and the references therein. Though direct cell-tocell transmission of HIV is a more potent and efficient means of virus propagation than infection by cell-free particles, this mode of viral propagation is less studied by using mathematical models [13,14,18,28,42]. The models in [13,14,28] consist of equations for three populations (uninfected, infected and virus) while those in [18,42] also include another population, the latently infected cells. Roughly, there is a threshold dynamics with/without additional conditions besides those on the existence of endemic equilibria [13,18,28,42]. Sometimes, Hopf bifurcation can occur [14].
It is well-known that the HIV virus may not reveal any symptoms for many years. According to health professionals, this could be around 10 years. However, the virus will still be active, infecting new cells and making copies. Due to the virus persisting in reservoirs, there will be a long-term low viral load persistence in patients on antiretroviral therapy. It has been reported in [2] that latently infected CD4+ T cells are the best known reservoir. The latently infected CD4+ T cells are neither interfered with antiretroviral therapy nor affected by immune responses. However, they can produce virus when activated by relevant antigens. Over time this will cause a lot of damage to the immune system. Moreover, in the study of HIV/AIDS epidemic, early infectivity experiments and the measurements of antigen and antibody titers suggest the possibility of an early infectivity peak (a few weeks after exposure) and a late infectivity plateau (one year or so before the onset of "full-blown" AIDS) for HIV-infected individuals. One realistic way to describe such phenomena is to introduce age structures. The idea of using age-structured models to study the spread of infectious diseases already has a long history. At the heart of an age-structured epidemic model is a coupled system of hyperbolic partial differential equations (PDEs) with/without ODEs, which is a generalization of those of standard differential equations and of delay differential equations. The introduction of PDEs enhances the interconnectivity and accuracy of the model. The resulting system is substantially complicated to analyze. For some background of age-structured epidemic models, see the books by Cushing [5] and Webb [41]. By now, lots have been done for age-structured HIV infection models only with cell-free spread (to name a few, see [3,6,8,10,12,21,29,34,38,39]). In spite of this, to the best of our knowledge, only Wang et al. [37] studied the following age-structured HIV infection model with both cell-free infection and cell-to-cell transmission, where T (t) denotes the concentration of uninfected target T cells at time t, i(a, t) denotes the density of infected T cells of infection age a at time t, and V (t) denotes the concentration of infectious virus at t. Here L 1 + (0, ∞) is the set of all nonengative integrable functins on R + := [0, ∞). A threshold dynamics is established by applying the technique of Lyapunov functionals.
Considering that antigen specificity plays an important role in the activation of latently infected cells. Mathematical models have been formulated to study the decay dynamics of the latent reservoir. See, for example, [11,20,4]. It has been reported in [31,32] that "cells specific to frequently encountered antigens can be activated quickly while cells specific to rarely encountered antigens need more time to be activated. Thus, the more time elapsed since the establishment of latent infection, the more likely the latently infected cell is specific to a rarer antigen, and the less likely it is to be activated". In a recent work [1], the authors assumed that the activation rate of latently infected cells should depend on the age of latent infection (the time elapsed since latent infection). They incorporated this structure into infection model with latency proposed by Perelson et al. [26] to study the second phase viral load decline during combination therapy. Some stability results of the infection-free and infected steady states are obtained in terms of the basic reproductive number.
Motivated by the above discussions and based on (1.2) and (1.3), in this paper, we propose and study an age-structured HIV infection model with latency and both cell-free and cell-to-cell infection modes. Let e(a, t) be the density of infected T cells of latency age a at time t. The model to be analyzed is as follows, and initial conditions Infection rate of CD4 + T cells by latently infected T cells β 2 Infection rate of CD4 + T cells by infectious T cells q 1 (a) Infectivity of a latently infected T cell with latency age a q 2 (b) Infectivity of an infectious T cell with infection age b θ 1 (a) Sum of death rate and activation rate ξ(a) of latently infected T cells with latency age a θ 2 (b) Death rate of infectious T cells with infection age b p(b) Viral production rate of an infectious T cell with infection age b c Clearance rate of virions ξ(a) Activation rate of latently infected T cells with latency age a transmission diagram is depicted in Fig. 1.
In what follows, we always make the following assumptions on parameters in (1.4) and (1.5).
The remaining of this paper is organized as follows. First, in Section 2, we present some preliminary results including properties of solutions and the existence of equilibria. Then we show the relative compactness of orbits in Section 3 and the uniform persistence of (1.4) in Section 4. The coming three sections are devoted to the global dynamics of (1.4). First, we obtain the local stability of the infectionfree equilibrium and the infection equilibrium in Section 5. Then we establish their global attractivity in Section 6 and Section 7, respectively. The main results are demonstrated with numerical examples in Section 8. The paper concludes with a brief summary and discussion.

Preliminaries.
2.1. The solution semi-flow. From the biological background of the model, define the state space of (1.4) as then (1.4) is well-posed under Assumption 1.1 due to Iannelli [9] and Magal [15]. In fact, for such solutions, it is not difficult to show that (T (t), e(·, t), i(·, t), V (t)) ∈ Y for each t ∈ R + . In what follows, we always assume that the initial values satisfy the coupling equations. Thus we can get a continuous solution semi-flow Φ : In what follows, we denote For ease of notations, associated with each solution of (1.4), we introduce the following functions on R + , Integrating the second and third equations of (1.4) respectively along the characteristic lines t − a = const and t − b = const gives us Then Ξ is a positively invariant and attractive subset of Φ.
Proof. Clearly, for a solution (T (t), e(, ·, t), i(·, t), V (t)) of (1.4) with initial condition X 0 ∈ Ξ, we have T (t) > 0 for t > 0. It follows from (2.2) and changes of variables that With the help of (2.1) and changing of variables, one can easily get Therefore, with the help of (iv) of Assumption 1.1, we have for t ∈ R + . It follows that, for t ∈ R + , This, combined with the fourth equation of (1.4), tells us that, for t ∈ R + , and consequently Then the required results follow easily from (2.4) and the above inequalities.
According to Proposition 2.1, we only need to consider (1.4) with initial conditions in Ξ for the limiting behavior. The next result is a direct consequence of Proposition 2.1.
be given. If X 0 ∈ Ξ, then the following statements hold for all t ∈ R + .
By Proposition 2.2, Assumption 1.1 and [40, Proposition 4.1], we can obtain the following basic properties of the functions P (t), Q(t), M (t), and N (t).
be given. Then, for any solution of (1.4) with initial condition in Ξ, the associated functions P (t), Q(t), M (t), and N (t) are Lipschitz continuous on R + with Lipschitz constants respectively.
(2.5) It follows that if one of e * , i * , and V * is not zero then the others are not either, that is, an equilibrium must be an infection equilibrium if it is not the infection-free equilibrium. Denote Note that K > 0 and J > 0 by Assumption 1.1 (iii). After a simple calculation, we see that (1.4) has infection equilibria if and only if ℜ 0 > 1 and in this case it only has a unique infection equilibrium P * = (T * , e * , i * , V * ) with where ℜ 0 is called the basic reproduction number of (1.4). Biologically, βT 0 KJ c accounts for the total number of newly infected cells resulted from the cell-free infection mode, which is the basic reproduction number for the corresponding model with cell-free infection only. Similarly, β 1 T 0 H and β 2 T 0 KL denote the total numbers of newly infected cells that arise from any one latently infected cell and from the cell-to-cell transmission, respectively. β 2 T 0 KL is also the basic reproduction number for the corresponding model with the cell-to-cell transmission only. As we will see later, ℜ 0 serves as a sharp threshold parameter for (1.4), which completely determines the global behavior of (1.4).
In summary, we have shown the following result.
3. Asymptotic smoothness of Φ(t, X 0 ). To give the proof of the global attractivity of equilibria, we will use the Lyapunov functional technique combined with the invariance principle. To this end, according to [35, Theorem 4.2 of Chapter IV], we have to ensure the relative compactness of the orbit {Φ(t, X 0 ) | t ∈ R + } in Y as our problem is an infinite dimensional dynamical system. To achieve this, we decompose Φ : R + × Y → Y into the following two operators Θ, Ψ : Proof. Obviously, lim t→∞ ∆(t, r) = 0. By (2.2) and (2.3), Then, for X 0 ∈ Y satisfying X 0 ≤ r and for t ∈ R + , we have This completes the proof. Thus it suffices to show thatẽ(·, t) andĩ(·, t) remain in precompact subsets of L 1 + (0, ∞), which is independent of X 0 ∈ Ξ. To achieve it, we only need to verify the following conditions forẽ (a, t) and similar ones forĩ (b, t) whose proofs are omitted (see, for example, [30,Theorem B.2]).
(i) The supremum of ẽ (·, t) L 1 with respect to X 0 ∈ Ξ is finite; (ii) lim h→∞ ∞ hẽ (a, t) da = 0 uniformly with respect to X 0 ∈ Ξ; (iii) lim h→0+ ∞ 0 |ẽ (a + h, t) −ẽ (a, t)| da = 0 uniformly with respect to X 0 ∈ Ξ; (iv) lim h→0+ h 0ẽ (a, t) da = 0 uniformly with respect to X 0 ∈ Ξ. It follows from (2.2) and (2. This, together with Proposition 2.2 and (2.1), gives directly from which conditions (i), (ii) and (iv) follow. Now, we are in a position to verify condition (iii). For sufficiently small h ∈ (0, t), we have Due to the fact that 0 ≤ Ω(a) ≤ 1 and Ω is a non-increasing function, we have Hence, from Proposition 2.2, we can conclude that Next we estimate ∆ 2 . Firstly, we have It follows from Proposition 2.2 and the first and fourth equations of (1.4) that T (t) and V (t) are both Lipschitz continuous on R + with Lipschitz constants M T = h + dA + βA 2 + β 1q1 A 2 + β 2q2 A 2 and M V = (p + c)A, respectively. Further, from Proposition 2.3, both functions P (t) and Q(t) are Lipschitz continuous with Lipschitz constants M P and M Q , respectively. According to [17,Proposition 6], we conclude that T (t)V (t), T (t)P (t), and T (t)Q(t) are Lipschitz continuous with Lipschitz constants M T V = AM V + AM T , M T P = AM P +q 1 AM T , and M T Q = AM Q +q 2 AM T , respectively. Denote that M = βM T V + β 1 M T P + β 2 M T Q . It follows from the above estimate that and condition (iii) directly follows. This completes the proof.

The uniform persistence. The aim of this section is to show that (1.4) is uniformly persistent under the condition that
Proof. By way of contradiction, for any ε > 0, there exists X ε 0 ∈ Y 0 such that lim sup t→∞ ρ(Φ(t, X ε 0 )) ≤ ε. Since ℜ 0 > 1, there exists a sufficiently small ǫ 0 > 0 such that  Then there exists X ε 0 2 0 ∈ Y 0 (for simplicity, denoted by X 0 in the remaining of the proof) such that lim sup t→∞ ρ(Φ(t, X 0 )) ≤ ε 0 2 .
Without loss of generality (with possible replacing of the initial condition), we can assume that ρ(Φ(t, X 0 )) ≤ ε 0 , t ∈ R + . We shall obtain a contradiction as follows.
It follows from the first equation of (1.4) that which implies that lim inf t→∞ T (t) ≥ h−ε0 d . As before, we can assume that By the fourth equation of (1.4), we have Then e(0, t) for all t ∈ R + . Taking the Laplace transforms of the above inequality gives us Here L[e(0, ·)] denotes the Laplace transform of e((0, t), which is strictly positive because of X 0 ∈ Y 0 . Dividing both sides of the above inequality by L[e(0, ·)] and letting λ → 0 give us  A total trajectory of Φ is a function X : R → Y such that Φ(t, X(r)) = X(t + r) for t ∈ R + and r ∈ R. We know that the global attractor A can only contain points with total trajectories through them since it must be invariant. For a total trajectory X(t) = (T (t), e(·, t), i(·, t), V (t)), e(a, r) = e (0, r − a) Ω(a) for r ∈ R and a ∈ R + , i(b, r) = i (0, r − b) Γ(b) for r ∈ R and b ∈ R + .
The alpha limit of a total trajectory X passing through X(0) = X 0 is Proof. On the one hand, by the first equation of (1.4) and Proposition 2.1, This provides a lower bound ε 1 for the T -coordinate for any point in A ∩ Y 0 . On the other hand,, by Theorem 4.3, there exists ε 2 > 0 such that e(0, t) ≥ ε 2 for all t ∈ R. Then, for t ∈ R, which implies that lim inf t→∞ V (t) ≥ ε2KJ c ε 3 . This gives a lower bound ε 3 for the V -coordinate for any point in A∩Y 0 . Letting ε = min{ε 1 , ε 2 , ε 2 K, ε 3 } completes the proof. 5. The local stability of equilibria. We first investigate the local stability of the infection-free equilibrium P 0 .
Theorem 5.1. The infection-free equilibrium P 0 is locally asymptotically stable if ℜ 0 < 1 and it is unstable if ℜ 0 > 1.
Proof. This time, the characteristic equation at P * is It is sufficient to show that (5.2) has no roots with non-negative real parts. By way of contradiction, suppose that it has a root λ = µ + νi with µ ≥ 0. Then we have Separating the real parts of the above equality gives Noticing that C 1 (0) = T * ℜ0 T0 = 1 and C 1 is a decreasing function, we have which contradicts with (5.3). This completes the proof. 6. Global stability of the infection-free equilibrium. In this section, we study the global stability of the infection-free equilibrium P 0 . In the discussion, we need the important function g on (0, ∞) defined by g(x) = x − 1 − ln x for x ∈ (0, ∞). This function is continuous and concave up. Moreover, it only attains its global minimum at x = 1 with g(1) = 0.

Proof. Consider a candidate Lyapunov functional defined by
Here the nonnegative kernel functions φ and ψ will be determined later. By Proposition 2.1, without loss of generality, we can assume that T 0 > 0 and hence L IF E is well defined. The derivative of L 1 along the solutions of (1.4) is calculated as follows, Using integration by parts, we have Similarly, It is easy to see that Therefore, we have

Now we choose
Then ψ(0) = βT 0 J c + β 2 T 0 L, φ(0) = ℜ 0 , and ψ and φ satisfy respectively. It follows that the derivative of L IF E along solutions of (1.4) is Notice that dLIF E (t) dt = 0 implies that T = T 0 . It can be verified that the largest invariant set where dLIF E (t) dt = 0 is the singleton {P 0 }. Therefore, by the invariance principle, P 0 is globally attractive when ℜ 0 ≤ 1.
The following result immediately follows from Theorem 5.1 and Theorem 6.1.

Global stability of the infection equilibrium.
To establish the global stability of the infection equilibrium, we need the following properties of solutions to (1.4).
Proof. By Theorem 5.2, it suffices to show that A ∩ Y 0 = {P * }. We show this by applying the Lyapunov functional technique again. Define It is easy to see that G is non-negative on (0, ∞) × (0, ∞) with the minimum value 0 only when x = y. Furthermore, it is easy to verify that xG Let X(t) = (T (t), e(·, t), i(·, t), V (t)) be a total trajectory in A ∩ Y 0 . Consider a candidate Lyapunov functional defined as follows, where e(a, t), e * (a) da, (This reason of this choice is similar to that in the Proof of Theorem 6.1). One can easily see that φ 1 (0) = 1, ψ 1 (0) = βT * J c + β 2 T * L, and Note that T (t), e(0, t), i(0, t), and V (t) are bounded. Moreover, they are also bounded away from 0 by Corollary 4.1. Hence L EE (t) is well-defined and is bounded on X(t). Now, we show that dLEE (t) dt is non-positive. Firstly, Secondly, using (2.2), we have Differentiating and using e * (a) = e * (0)e − a 0 θ1(ω)dω and (7.4) produce Finally, From the second equation of (1.5) and the sixth equation of (2.5), we have ∞ 0

Thus (7.5) reduces to
With the help of the first equation of (1.5) and the fifth equation of (2.5), we see Therefore, L EE is a non-increasing function. Since L EE is bounded on X(t), α(X 0 ) must be contained in the largest invariant subset M in { dLEE dt = 0}. Clearly, Then it is not difficult to check that M = {P * }.
From the above discussion, we see that α(X 0 ) = {P * } and hence L EE (X(t)) ≤ L EE (P * ) for all t ∈ R. It follows that X(t) ≡ P * and so A ∩ Y 0 = {P * }. This completes the proof.
with boundary conditions and initial conditions The basic reproduction number of (8.1) is given by where J and L are defined in (2.6). According to Theorem 1.1 in [37], ℜ = 0.724973. Hence, from Theorem 6.2, we see that the infection-free equilibrium is globally asymptotically stable. In fact, in Figure 2, the numbers of infected cells of (8.1) and (1.4), and the number of latent cells Hence, from Theorem 6.2, we see that the infection-free equilibrium of (1.4) is globally asymptotically stable, which is supported by Figure 3(a). However, the infected cells of (8.1) will converge to positive distribution according to Theorem 1.1 in [37]. Again, Figure 3  , ℜ 0 does not altered the global dynamics qualitatively. On the other hand, from the examples of parameter values, if latency-age structured cells are ignored, then the basic reproduction number will be over-estimated. In practice, this would lead to wrong diagnosis. 9. Summary. In this paper, an age-structured HIV infection model incorporating latency and cell-to-cell transmission is proposed. We focus on the mathematical analysis of the model, which is formulated as a hybrid system consisting of coupled ordinary differential equations and partial differential equations. After addressing the relative compactness and persistence of the solution semi-flow and the existence of a global attractor, we establish a threshold dynamics completely determined by the basic reproduction number ℜ 0 . Namely, if ℜ 0 < 1 then the infection-free equilibrium is globally asymptotically stable, which means that the virus can be cleared; if ℜ 0 > 1 then the infection equilibrium is globally attractive to solutions with initial effective infection and hence the virus is persistent.  The threshold dynamics is supported with numerical simulations, which also indicate how the inclusion of latent cells affects the distribution of infected cells. On the one hand, including latent cells will decrease the basic reproduction number corresponding with the case without it and hence it can make the infection less persistent. On the other hand, when the infection is persistent, the inclusion of latent cells can significantly decrease the level of infected cells at the infection equilibrium. In other words, without considering latent cells will overestimate the infection. From the perspective of therapy, this may lead to overdose.
However due to the virus persistence in cellular compartments or reservoirs, how to design and solve the situation of long-term low viral load persistence in patients with antiretroviral therapy remains an interesting question to address in the future.