Threshold dynamics of a HCV model with virus to cell transmission in both liver with CTL immune response and the extrahepatic tissue

In this paper, a deterministic model characterizing the within-host infection of Hepatitis C virus (HCV) in intrahepatic and extrahepatic tissues is presented. In addition, the model also includes the effect of the cytotoxic T lymphocyte (CTL) immunity described by a linear activation rate by infected cells. Firstly, the non-negativity and boundedness of solutions of the model are established. Secondly, the basic reproduction number and immune reproduction number are calculated, respectively. Three equilibria, namely, infection-free, CTL immune response-free and infected equilibrium with CTL immune response are discussed in terms of these two thresholds. Thirdly, the stability of these three equilibria is investigated theoretically as well as numerically. The results show that when , the virus will be cleared out eventually and the CTL immune response will also disappear; when , the virus persists within the host, but the CTL immune response disappears eventually; when , both of the virus and the CTL immune response persist within the host. Finally, a brief discussion will be given.


Introduction
Viral hepatitis affects approximately 500 million people around the world -more than 10 times the number affected by HIV/AIDS [1]. Different viruses can cause various forms of viral hepatitis. It is estimated that about 71 million people or 1% of the global population are chronically infected with hepatitis C according to the World Health Organization(WHO) Global Hepatitis Report in 2017 [21]. Therefore, it is important to understand the dynamics of HCV infection in order to manage control programmes efficiently.
Hepatitis C virus (HCV) infection can lead to two different outcomes [8]: in a small fraction of patients (15% of cases), the infection can be controlled and cleared from the blood; the rest of the patients become chronic. Chronic HCV is the main cause of chronic liver diseases and cirrhosis leading to liver transplantation (LT) or death [2]. Unfortunately, the early results of transplantation for patients with chronic HCV were discouraging. Mortality rate of liver transplant is very high, and reinfection of the liver graft often occurs [20]. This inevitable post-transplant infection may be related to the existence of an auxiliary compartment [11]. The presence of HCV replicative intermediates has been reported in serum [7], oral mucosa [3] and gastric mucosa [5]. Dahari et al. [4] studied viral loads of 30 patients undergoing liver transplantation and observed the existence of a second replication compartment.
Virus clearance after acute HCV infection is associated with strong and polyclonal CD4 T cell responses, as well as sustained CTL responses. Recently, molecular techniques have provided fundamental insights into the molecular mechanisms of the immune system for HCV infection [16,18]. In 1996, Nowak and Bangham [15] proposed a simple mathematical model to explore the relation between antiviral immune response and virus load. In 2003, Wodarz [22] extended the model in [15] to investigate the role of CTL and antibody response in HCV infection dynamics and pathology. Zhou et al. [24,25] considered the CTL immune response against HCV infection. However, the mechanism of CTL action in HCV infection is still not fully understood [6,9].
Mathematical models have become important tools in analysing the spread and control of HCV epidemic. Dahari et al. [4] constructed a few within-host HCV infection models to describe HCV viral dynamics from the beginning of the anhepatic phase until the first viral increase data point. These models included two compartments of infection, but did not describe the asymptotical viral dynamics after transplantation of the liver. Qesmi et al. [17] proposed a mathematical model of ordinary differential equations to describe the dynamics of the HBV/HCV and its interaction with both liver and blood cells based on [4,13], and found that the system undergoes either a transcritical or a backward bifurcation. Wodarz and Jansen [23] proposed a model containing infected cells, non-acitived antigen presenting cells (APCs), acitived APCs and CTL, and analysed its complex dynamics.
However, there are very few HCV infection models with two compartments. Based on the existence of a second replication compartment for HCV and the role of CTL immune response against HCV infection, in this paper, we propose a new mathematical model containing another compartment of HCV infection. Then by the analysis of golbal dynamics, these theoretical results will reveal the interaction between HCV and CLT immune response more completely. This paper is organized as follows. In Section 2, we formulate a new HCV infection model with CTL immune response and give a positively invariant set. Section 3 deals with the existence of equilibria for the model and two important parameters thresholds will be defined. In Section 4, the global stability of equilibria is investigated by using the Routh-Hurwitz criterion and Lyapunov functions. Some numerical examples are shown in Section 5. Finally, the epidemiological meanings of the obtained results are discussed, and the basic reproduction numbers of HCV infection and CTL immune response are given in Section 6.

Model formulation
In this section, we formulate a dynamical model with two proliferative compartments of HCV, one of which is the liver, the other is the extrahepatic compartment including serum, peripheral blood mononuclear cells (PBMC), and perihepatic lymph nodes (PLN). No experiments have shown that CTL immune response has effect or no effect on the extrahepatic compartment, here, it is assumed that the CTL immune response takes part in clearing infected hepatocytes and plays no role for the second proliferative compartment. The flowchart of HCV infection is shown in Figure 1. Here, we denote the liver and the second proliferative compartment (extrahepatic compartment) as compartments C 1 and C 2 , respectively. In compartment C 1 , there are uninfected hepatocytes (x 1 (t)), infected hepatocytes (y 1 (t)) and the CTL immune response (z(t)). In compartment C 2 , there are uninfected extrahepatic cells (x 2 (t)), infected extrahepatic cells (y 2 (t)) and free virus (v(t)). Following the transmission diagram in Figure 1, our model takes the form in (1) Here, λ i is the recruitment rate of healthy cells and 1 d i is the average lifespan of uninfected cells in compartment C i (i = 1, 2). The healthy cells become infected by free virus at a rate β i x i v; infected cells in compartment C i (i = 1, 2) die at a rate a i y i , and infected cells in compartment C 1 are cleared by the CTL immune response at a rate py 1 z; the CTL immune response is triggered at a rate qy 1 z, which in turn decays a rate rz. We assume that the parameters are positive and a i ≥ d i (i = 1, 2) [14] according to the biological meaning. Note that a saturated nonlinear function was used in Wodarz and Jansen [23] to describe the activation of the CTL immune response by the virus. Since we are interested in the global dynamics of the model, for the sake of simplicity we use a linear function here.
We can see that solutions of model (1) with the nonnegative initial conditions remain nonnegative. From the first equation of (1), we have then lim sup t→∞ x 1 (t) ≤ λ 1 /d 1 . From the first two equations of (1), we obtain since a 1 ≥ d 1 , then lim sup t→∞ x 1 (t) + y 1 (t) ≤ λ 1 /d 1 . Similarly, from the middle two equations of (1), we have lim sup t→∞ x 2 (t) ≤ λ 2 /d 2 , and lim sup t→∞ When And z = 0 always satisfies the last equation in (1). Therefore, the region is positively invariant with respect to system (1). Therefore, it is sufficient to study the dynamics of model (1) with initial conditions in .

Existence of equilibria
In this section, we discuss the existence of equilibria of model (1) satisfying the following equations on the set .

Model (1) always has an infection-free equilibrium
From the first and third equations of (2), we obtain Substituting them into the second and fourth equations of (2), they yields respectively When v = 0 and z = 0, substituting y 1 and y 2 of (4) into the fifth equation of (2) yields We can see that function h(v) is decreasing with respect to v. Note that Furthermore, the corresponding x (1) i and y (1) i (i = 1, 2) can be obtained from (3) and (4). Thus, (1) has a boundary equilibrium , 0 when h(0) > 1. When z = 0, from the last equation of (2) we have y 1 = r q := y (2) 1 . Substituting it and (3) into the second equation of (2) gives Then a necessary condition on the existence of the positive equilibrium is λ 1 > a 1 r q , and, for the positive equilibrium E 2 x On the other hand, substituting y 1 = r q and Under the case that λ 1 > a 1 r q (i.e. r q < λ 1 a 1 ) and for a i ≥ d i (i = 1, 2), we have Applying again the inequality that 1 1+m + 1 1+n < 1 holds for m, n > 0 if and only if mn > 1, we know that g 1 Then, according to the monotonicity of function On the other hand, direct calculation shows that g(v) = h(v). Hence, according the existence of the equilibrium E 1 , model (1) has a unique positive equilibrium E 2 as 1

Theorem 3.1: Denote
The existence of equilibria in system (1) can be summarized below.

Stability of equilibria
In this section, we discuss the global stability of equilibria of (1). We first present two propositions for the infection-free equilibrium E 0 .

Proposition 4.2: When R
Moreover, direct calculation shows that R 01 < 1 is equivalent to the following inequality So we can choose a positive number m 1 satisfying the inequality that is, 1 γ Further, for the given m 1 , we choose a positive number m 2 satisfying the inequality 1 γ When m 1 and m 2 are given, we define a function Then x 1 ≤ λ 1 d 1 and x 2 ≤ λ 2 d 2 imply that the derivative of V 1 along solutions of model (1) is given by It follows from (9) and (10) that For the global stability of equilibria of (1), we have the following results.

Conclusion and discussion
The novelty of our study is that we introduced a new compartment (extrahepatic compartment)) into the classical within-host hepatitis C virus infection models (see Wozard and Jansen [23]) and provided results on the gobal dynamics of the model. According to Theorems 3.1 and 4.1, R 01 and R 02 are two thresholds determining the dynamical behaviours of system (1) (see Figure 5). When R 01 < 1, system (1) has a unique equilibrium E 0 , which is globally stable in the set ; when R 02 < 1 < R 01 , besides the boundary equilibrium E 0 , system (1) also has another boundary equilibrium E 1 , which is globally stable in the set ; when R 02 > 1, in addition to the boundary equilibria E 0 and E 1 , system (1) has a unique infection equilibrium E 2 which is globally stable in the set .
Notice that is the basic reproduction number of Hepatitis C virus infection within the host. In fact,  compartment C 1 , k 1 a 1 is the number of new released virion by an infected cell, 1 γ is the average infectious period of virus, then k 1 β 1 λ 1 a 1 d 1 γ is the basic reproduction number of Hepatitis C virus within compartment C 1 . Similarly, k 2 β 2 λ 2 a 2 d 2 γ is the one within compartment C 2 . Notice that although R 02 is the threshold determining the existence and stability of the positive equilibrium E 2 , it is not the basic reproduction number. From Theorem 3.1, qy (1) 1 r , denoted by R 02 , can also be thought as a threshold when R 01 > 1, which plays the same role as the threshold R 02 in determining the dynamical behaviours of system (1). Since y (1) 1 is the number of infected cells in compartment C 1 at the steady state when R 01 > 1, qy (1) 1 is the CTL immune response existed in a unit time by per CTL response, and 1 r is the average decaying period of CTL response, then qy (1) 1 r represents the basic reproduction number of the CTL immune response.
For system (1), we have discussed the existence and stability of three equilibria E 0 , E 1 and E 2 . In the sense of viral dynamics, the boundary equilibrium E 0 represents the infectionfree steady state within the host; the other boundary equilibrium E 1 represents the steady state at which the host is infected, and the CTL immune response plays no role in protecting the host from infection; the positive equilibrium E 2 represents the steady state at which the host is infected and the CTL immune response plays a certain role in protecting the host from infection, but it cannot clear completely the infected cells in compartment C 1 . Therefore, by Theorems 3.1 and 4.1, when R 01 < 1, the virus will be cleared eventually and the CTL immune response also disappears; when R 02 < 1 < R 01 , the virus persists within the host, but the CTL immune response disappears eventually, this implies that the immune system in the body of patient has no effect on neutralizing the infection of the virus; when R 02 > 1, both the virus and the CTL immune response persist within the host, but the CTL immune response does not suffice to clear the virus completely.