Global properties of a virus dynamics model with self-proliferation of CTLs

A viral infection model with self-proliferation of cytotoxic T lymphocytes (CTLs) is proposed and its global dynamics is obtained. When the per capita self-proliferation rate of CTLs is sufficient large, an infection-free but immunity-activated equilibrium always exists and is globally asymptotically stable if the basic reproduction number of virus is less than a threshold value, which means that the immune effect still exists though virus be eliminated. Qualitative numerical simulations further indicate that the increase of per capita self-proliferation rate may lead to more severe infection outcome, which may provide insight into the failure of immune therapy.


Introduction
Outbreaks of viral infection have become a major global health concern. Different kinds of virus, such as hepatitis B virus (HBV), hepatitis C virus (HCV), human immunodeficiency virus (HIV), Ebola Virus and Zika Virus, have been associated with severe outcomes. A great deal of effort has been put toward to understand the life cycle of these virus. With the development of biomedical research, mathematical models also play an increasingly important role to provide insights into virus infection and dynamics, as well as on how an infection can be reduced or even eradicated. For example, on HBV infection, Nowak et al. [13] first proposed a basic three-dimensional viral infection model within-host. Note that immune responses play a critical part in the process of viral infections. Nowak et al. [14] further proposed the following four-dimensional system with the cytotoxic T lymphocytes (CTLs) population based on the basic model: (1.1) Here x(t), y(t), v(t) and z(t) represent uninfected target cells, infected cells, free virus and CTLs, respectively. Uninfected cells are produced at a constant rate s, die at rate d 1 x, and become infected at rate βxv. Infected cells are produced at rate βxv and die at rate d 2 y. Free virus are produced from infected cells at rate ky and die at rate uv. CTLs are produced at rate cyz due to the stimulation of infected cells, and die at rate d 3 z. Infected cells are eliminated by CTLs at rate pyz. After that, based on the basic models, many studies were carried out to analysis of the dynamics of various virus infection within-host, such as [1,9,12,15,16,17,22,23,25] and the reference therein.
Recent studies on the production mechanism of immune cells show that its self-proliferation cannot be neglected besides the stimulation of infected cells in [7]. Thus, to understand the effect of selfproliferation, based on the system (1.1) and [7], we propose the following new virus infection model: Here the logistic proliferation term rz(1 − z/m) describes the self-proliferation of CTLs, in which parameter r denotes a per capita self-proliferation rate, and m means the capacity of CTLs population. When r = 0, i.e., without self-proliferation of CTLs, (1.2) has been completely analyzed in [10] if there is not explicit dynamics of free virus under a plausible quasi steady-state assumption. To explore the effects of the recruitment of immune responses on virus infection, the main contribution of the present paper is to obtain the complete global properties of (1.2) when r > 0.

Global dynamics analysis
Since we are interested in the dynamics of viral infection, and not the initial processes of infection, we assume that the initial condition of (1.2) has the form x(0) > 0, y(0) > 0, v(0) > 0 and z(0) > 0. Based on the initial conditions, it is easy to show that the solutions of system (1.2) are non-negative and ultimately bounded. The equilibria of (1.2) are the solutions of the following algebraic equations: (2.1) Clearly, system (1.2) always has infection-free equilibrium E 0 = ( s d1 , 0, 0, 0). According to the definition and algorithm of the basic reproduction number of virus in [4], we can obtain the basic reproduction number of virus R 0 = βsk d1d2µ . Using the fourth equation of (2.1), we have z = 0 or When z = 0, based on the first three equations of (2.1), it is easy to obtain that the immunity-inactivated infection equilibrium Clearly, f (+∞) < 0 and function f (y) is monotonically decreasing since f (y) < 0 always valid. As a result, f (y) = 0 has a unique positive root if i.e., R 0 > 1 + mp rd2 (r − d 3 ). When r < d 3 , according to (2.2), in order to keep the positive of z, we need is valid, i.e., R 0 > 1 + βk(d3−r) cd1µ . In summary, we have the following proposition.
Proposition 2.1. The following hold.
(i) If R 0 > 1, the immunity-inactivated infection equilibrium E 1 always exists. Especially, an infection-free but immunity-activated equilibrium E 2 will appear if r > d 3 .
and y 3 is the unique positive root of f (y) = 0 in this case.
and y 4 is the unique positive root of f (y) = 0 in this case.
In order to obtain the stability of above mentioned equilibria, we first give the Jacobian matrix J of system (1.2) at (x, y, v, z), (2.5) So we have the following results.
Theorem 2.2. The following hold.
Proof. According to (2.5), we have the characteristic equation of system (1.2) at E 0 . It is easy to show that λ 1 = −d 1 < 0 and λ 2 = r − d 3 are the roots of (2.6). Further, we can get all roots of H 0 (λ) are negative real part if R 0 < 1, and there is one positive real root if R 0 > 1.
(i) When 0 ≤ r < d 3 , we have λ 2 < 0. As a result, the infection-free equilibrium E 0 is locally asymptotically stable if R 0 < 1 and 0 ≤ r < d 3 , and is unstable if R 0 > 1. When r = d 3 , λ 2 = 0. Thus, the center manifold is a curve tangent to the z-axis. In this case, settling a transformation where we substitute x, y, v, z forx,ỹ,ṽ,z. To obtain the approximative expression of the center manifold, we set (2.9) Substituting (2.7) and (2.8) into (2.9), we have (2.10) Comparing the coefficients of z and z 2 in (2.10), we have As a result, substituting (2.8) into the last equation of (2.7), we have Thus, the zero point z = 0 of (2.11) is locally asymptotically stable, then E 0 is locally asymptotically stable if R 0 < 1 and r = d 3 . Let Taking the time derivative of L 0 along the solution of system (1.2), we have As a result, E 0 is globally asymptotically stable based on the LaSalle's invariance principle.
According to (2.5), we have the following characteristic equation of system (1.2) at E 1 where H 1 (λ) = λ 3 + a 1 λ 2 + a 2 λ + a 3 , and is an eigenvalue at E 1 of system (1.2), and the real parts of H 1 (λ) are negative according to Routh-Hurwitz criterion.
Theorem 2.5. The following hold.
In order to obtain the global stability of E * , let Taking the time derivative of L * along the solution of system(1.2), we get Using s = βx * v * + d 1 x * , βx * v * = d 2 y * + py * z * , ky * = µv * and y * = rz * + m(d 3 − r) mc , we have and L * = 0 only if x = x * , y y * = v v * and z = z * . In this case, it is easy to obtain that the maximal invariant subset in {(x, y, v, z) : L * | (1.2) = 0} is the singleton {E * }. As a result, E * is globally asymptotically stable if it exists based on the LaSalle's invariance principle.
For the convenience of reading, we summarize the complete global properties of system (1.2) as shown in Figure 1.

Numerical simulations
Although the complete global properties of system (1.2) have been obtained in Figure 1, it is noted that the immunity-activated infection equilibrium E 3 or E 4 is related to the parameters of selfproliferation of CTLs (r and m) from Proposition 2.1. When E 3 or E 4 is globally asymptotically stable, the infected cells (y 3 or y 4 ) and corresponding proportion of infected cells y3 x3+y3 or y4 x4+y4 are often used to describe the severity of the infection. In this section, we give numerical simulations to investigate the effect of self-proliferation of CTLs, and explore the potential significance in clinical practice. First, we fix s = 1.0 × 10 4 ml −1 · day −1 , β = 3.0 × 10 −4 ml −1 · day −1 , p = 0.5 ml −1 · day −1 , c = 9.6 × 10 −6 ml −1 · day −1 , d 1 = 0.01 day −1 , d 2 = 1.0 day −1 , d 3 = 0.035 day −1 , k = 8 virions/cell, µ = 2.4 day −1 , which are within the similar ranges as those ones employed in [5,12,17,22]. Then, let parameters r and m change to qualitatively explore their influence on the values of E 3 or E 4 , and the corresponding proportion of infected cells. When r ∈ [0.0, 0.035] and m ∈ [100, 1000], i.e., 0 ≤ r ≤ d 3 , Figure 2 shows the simulated surface of E 3 and the corresponding proportion of y 3 changing with parameters r and m. Along with the increase of parameter r, although the infected cells (y 3 ) and virus load (v 3 ) will decrease ( Figure 2B and Figure 2C), the proportion of infected cells ( y3 x3+y3 ) is gradually increase ( Figure 2E). In particular, the immune cells (z 3 ) also decrease with the increase of r ( Figure 2D). These qualitatively indicate that the severity of infection may increase with the increase of r in case of 0 ≤ r ≤ d 3 . Along with the increase of parameter m, Figure 2A and Figure 2D show that uninfected cells (x 3 ) and immune cells (z 3 ) will increase gradually, while infected cells (y 3 ), virus load (v 3 ) and the proportion of infected cells ( y3 x3+y3 ) are decrease gradually. These qualitatively indicate that increasing parameter m can reduce the severity of virus infection. When r ∈ [0.0351, 0.07] and m ∈ [100, 1000], i.e., r > d 3 , Figure 3 shows the simulated surface of E 4 and the corresponding proportion of y 4 changing with parameters r and m. Comparing Figure 3 with Figure 2, we can find that all components of the immunity-activated infection equilibrium E 4 will increase with the increase of parameter r, including the proportion of infected cells ( y4 x4+y4 ). Along with the increase of parameter m, the variation of all components is similar to that in Figure 2.

Discussion
In this paper, a viral infection model with self-proliferation of CTLs is proposed and its global dynamic behavior is obtained. From Figure 1, comparing with [10], we can find the dynamic behavior of (1.2) will not change if the per capita self-proliferation rate of CTLs is insufficient, i.e., 0 < r < d 3 . However, when r = d 3 , the immunity-inactivated infection equilibrium E 1 is always unstable if it appears. Especially, when r > d 3 , a new steady state (named as infection-free but immunity-activated equilibrium E 2 ) appears and is globally asymptotically stable if the basic reproduction number is less than a threshold, which means that the immune effect still exists though virus be eliminated. This is consistent with the clinical practice of virus infection because immune cells are usually not depleted after a patient recovers. In fact, memory T cells can be maintained in lymphoid and nonlymphoid organs through self-renewal [3], including central memory T cells and effector memory T cells [18]. In particular, recent study on HBV also find that HBV specific TNF-α CD4 T cells may be in the early stage of differentiation rather than depletion of T cells [19], which suggests that the stability of immunity-inactivated infection equilibrium E 1 may be impossible in clinical practice.
On the effect of self-proliferation of immune cells, qualitative numerical simulations (Figure 2 and Figure 3) indicate that although there are different shape mode under different intensity of self-proliferation, the increase of per capita self-proliferation rate (r) can worsen the infection, while the increase of the capacity of CTLs (m) can reduce the severity of infection. Thus, inappropriate intensity of per capita self-proliferation rate may lead to more severe infection outcome, which is similar to the effect of covalently closed circular DNA (cccDNA) self-amplification rate in HBV infection [8]. These may provide insight into the failure of immune therapy [2].
Recently, under a plausible quasi steady-state assumption, [6] ignored the direct effect of virus load, but introduced the delayed activation effect of immune cells, and the dynamics of the corresponding virus model was studied. Compared with the results of [6], we can find that quai steady-state assumption cannot affect the dynamic behavior, which is consistent with the conclusion of [20]. Note that the spatial migration of virus particles is an inherent characteristic of virus infection within-host [21,28]. We will further analyze the effect of delay and spatial migration on the process of virus infection, such as global stability, bifurcation and the invasion speed of virus particles based on the latest research results [11,24,26,27] in the future study.