Mathematical analysis of an HTLV-I infection model with the mitosis of CD4+ T cells and delayed CTL immune response

Abstract. In this paper, we consider an improved Human T-lymphotropic virus type I (HTLV-I) infection model with the mitosis of CD4 T cells and delayed cytotoxic T-lymphocyte (CTL) immune response by analyzing the distributions of roots of the corresponding characteristic equations, the local stability of the infection-free equilibrium, the immunity-inactivated equilibrium, and the immunity-activated equilibrium when the CTL immune delay is zero is established. And we discuss the existence of Hopf bifurcation at the immunity-activated equilibrium. We define the immune-inactivated reproduction ratio R0 and the immune-activated reproduction ratio R1. By using Lyapunov functionals and LaSalle’s invariance principle, it is shown that if R0 < 1, the infection-free equilibrium is globally asymptotically stable; if R1 < 1 < R0, the immunityinactivated equilibrium is globally asymptotically stable; if R1 > 1, the immunity-activated equilibrium is globally asymptotically stable when the CTL immune delay is zero. Besides, uniform persistence is obtained when R1 > 1. Numerical simulations are carried out to illustrate the theoretical results.


Introduction
Human T-lymphotropic virus type I (HTLV-I) is a pathogenic retrovirus. About 10 million to 20 million people worldwide are infected [1,13,19]. It is closely linked to two main types of viral diseases: adult T cell leukaemia/lymphoma (ATL), an aggressive blood cancer; and HTLV-I associated myelopathy/tropical spastic paraparesis (HAM/TSP), a progressive neurological and inflammatory disease [8,9,20,21]. However, there is no definite mechanism for the development of HTLV-I related diseases and no effective treatment. Besides, most of the infected persons are asymptomatic carriers, and only 0.1-4 % of the infected persons develop from long-term asymptomatic carriers to ATL or HAM/TSP [21].
Similar to Human Immunodeficiency Virus (HIV), the target cell of HTLV-I is CD4 + T cells. For the infected CD4 + T cells, a hypothesis was put forward by Asquith and Bangbam that only a small part of infected CD4 + cells express Tax [5]. Based on this, the infected CD4 + T cells were divided into actively infected CD4 + T cells and latently infected CD4 + T cells according to whether Tax is expressed or not [5]. And the hypothesis of dynamic interaction between CD4 + T cells without Tax expression and CD4 + T cells with Tax expression proposed by Asquith and Bangbam explains that HTLV-I infected individuals have a sustained activation of the specific immune response to HTLV-I, while the viral load increases [3]. Therefore, it is very important to distinguish latently infected CD4 + T cells from actively infected CD4 + T cells. For mitosis, it should be pointed out that although mitosis is a natural process that occurs in all CD4 + T cells, normal homeostatic mitosis occurs at a much slower rate than that of actively infected CD4 + T cells proliferation. To avoid unnecessarily complicating the mathematical analysis, the mitosis of the healthy and latently CD4 + T cells is ignored [15].
Based on the above discussion, in order to explore the dynamic interaction between latently infected CD4 + T cells and actively infected CD4 + T cells in HTLV-I infection, in [15], Lim and Li proposed the following mathematical model: u (t) = δβxy + ry 1 − x + u k − (µ + σ)u, where x(t) denotes the concentration of healthy CD4 + T cells, which are produced at rate λ and die at rate d, u(t) denotes the concentration of latently infected CD4 + T cells, and y(t) denotes the concentration of actively infected CD4 + T cells; β is the transmission coefficient; a and µ represent the death rates of actively infected CD4 + T cells and latently infected CD4 + T cells, respectively; σ is the rate at which latently infected CD4 + T cells translate into actively infected CD4 + T cells. δβxy and ry(1 − (x + u)/k) are used to describe the newly infected CD4 + T cells entering the latently infected CD4 + T cells compartment through infection and mitosis or vertical transmission, respectively. In model (1), logistic growth term ry(1 − (x + u)/k) is used to describe the mitosis of actively infected CD4 + T cells. However, the numerical results show that there is no qualitative difference between exponential growth term and logical growth term in the behaviour of trajectories [17]. Therefore, in order to avoid the complication of the model equation, it is reasonable to assume that x(t)+u(t) k, the proliferation of actively infected CD4 + T cells follows an exponential growth term ry instead of logical growth term ry(1−(x+u)/k). The exponential growth term ry has been used by Lim and Maini [17].
In the process of HTLV-I infection, a strong cytotoxic T lymphocyte (CTL) immune response was established to fight infection [5,12]. In the most virus infections, CTL http://www.journals.vu.lt/nonlinear-analysis immune response can lower the proviral load and consequently lower the risk of disease [1]. However, experiments have shown that the cytotoxicity of CTL ultimately leads to demyelination of HAM/TSP central nervous system and the development of HAM/TSP disease [16]. It can be seen that the effect of CTL immune response on HTLV-I infection is much more complicated. Therefore, the consideration of CTL immune response in the HTLV-I infection model is of great significance to the study of the development and treatment of ATL or HAM/TSP.
Since antigenic stimulation to generate HTLV-I specific CTLs involves a series of events that require a time delay [7]. Therefore, it is necessary to consider the effect of time delay in the model, and the form of CTL immune function f (y, z) = cy(t − τ ) has been used in [14]. Motivated by the works of Li and Shu [16], Lim and Maini [17], in this paper, we consider the following HTLV-I infection model with actively infected CD4 + T cells mitosis and delayed CTL immune response: where z(t) denotes the concentration of the specific CD8 + CTLs, b is the death rate of specific CD8 + CTLs; pyz describes actively infected CD4 + T cells being lysed by specific CD8 + CTLs, cy(t − τ )z(t − τ ) represents that the CTLs produced at time t depends on the concentration of CTLs and actively infected CD4 + T cells at time t − τ [16,23]. Experiments have shown that the mitosis of actively CD4 + T cells is usually lower than the removal rate caused by natural death [17], hence, in the following, we assume that a > r.
The initial condition for system (2) takes the form where This paper is organized as follows. In Section 2, we show the positivity and boundedness of solutions to system (2). In Section 3, the existence of feasible equilibria, the local stability of the infection-free equilibrium, the immunity-inactivated equilibrium, and the immunity-activated equilibrium (when τ = 0) is established. And we discuss the existence of Hopf bifurcation at the immunity-activated equilibrium. In Section 4, by constructing suitable Lyapunov functionals and using LaSalle's invariance principle, the global stability of the infection-free equilibrium, the immune-inactivated equilibrium, and the immune-activated equilibrium (when τ = 0) is established. In Section 5, we analyze the uniform persistence of system (2) when the immune-activated reproduction ratio is greater than one. In Section 6, we give some numerical simulations to illustrate the theoretical results. Finally, a brief remark is given in Section 7 to conclude this work. Proof. Let (x(t), u(t), y(t), z(t)) be any solution of system (2) with the initial condition (3). First, by the first equation of system (2) we haveẋ| x=0 = λ > 0. This implies that x(t) > 0 for all t > 0 as long as x(0) = φ 1 (0) > 0.
Theorem 2. There is a positive constant M such that for any positive solution (x(t), u(t), y(t), z(t)) of system (2) with the initial condition (3), Proof. Let (x(t), u(t), y(t), z(t)) be any positive solution of system (2) with the initial condition (3). Define Calculating the derivative of N (t) along positive solution of system (2), it follows thaṫ yielding lim sup t→+∞ N (t) λ/m, where m = min{d, µ, a − r, b}.

Equilibria, local stability, and Hopf bifurcation
In this section, we study the existence of feasible equilibria, the local stability of the infection-free equilibrium, the immunity-inactivated equilibrium, and the immunity-activated equilibrium (when τ = 0) of system (2) by analyzing the distributions of roots of the corresponding characteristic equations, and we show the existence of Hopf bifurcation at the immunity-activated equilibrium. System (2) always has an infection-free equilibrium E 0 (λ/d, 0, 0, 0). We can obtain the immune-inactivated reproduction ratio by using the method of the next generation matrix [24]: is the spectrum radius of F V −1 . R 0 represents the expected number of newly infected cells generated by a single infected cell in its life span. If R 0 > 1, in addition to the infection-free equilibrium, system (2) has an immunity-inactivated equilibrium E 1 (x 1 , u 1 , y 1 , 0), where Further, by calculation we obtain the immune-activated reproduction ratio If R 1 > 1, in addition to E 0 and E 1 , system (2) has an immunity-activated equilibrium Proof. The characteristic equation of system (2) at the equilibrium E 0 is Clearly, (4) has negative real roots s 1 = −b, s 2 = −d, and other roots of (4) are determined by the following equation: If R 0 < 1, it is easy to show that all roots of (5) have only negative real parts. Therefore, the equilibrium E 0 is locally asymptotically stable.
Hence, (5) has at least one positive real root. Accordingly, E 0 is unstable.
Proof. The characteristic equation of system (2) at the equilibrium E 1 is where .
We first claim that all roots of the following transcendental equation have negative real parts. Otherwise, there exists a root s 1 = a 1 + ib 1 with a 1 0. If R 1 < 1 < R 0 , we have It follows that which contradicts (7). Hence, all roots of (7) have negative real parts, and other roots of (6) are determined by the following equation: Now, we claim that all roots of (8) have negative real parts. Otherwise, there exists a root s 2 = a 2 + ib 2 with a 2 0. In this case, we have It follows that which contradicts (8). Therefore, if R 1 < 1 < R 0 , all roots of (8) have negative real parts. Further, all roots of (6) have negative real parts. Accordingly, E 1 is locally asymptotically stable.
If R 1 > 1, it only needs to consider (7), it is clear that . Therefore, (6) has at least one positive real root. Accordingly, E 1 is unstable.
The characteristic equation of system (2) at the immunity-activated equilibrium E * is where p 0 = β 2 x * y * bσ, When τ = 0, (9) reduces to Now, we claim that all roots of (10) have negative real parts. Otherwise, (10) has at least one root s 1 = x 1 + iy 1 with x 1 0. Noting that which contradicts (10). Thus, all roots of (10) have negative real parts. Hence, when τ = 0, the equilibrium E * is locally asymptotically stable.

Global stability
In this section, we study the global stability of the infection-free equilibrium, the immuneinactivated equilibrium, and the immune-activated equilibrium (when τ = 0) of system (2) by using Lyapunov functionals and LaSalle's invariance principle.

Uniform persistence
In this section, we verify the uniform persistence of system (2) when R 1 > 1.
Let T ∂ (t) = T (t)| X0 , A ∂ be the global attractor for T ∂ (t). The following results was developed in [10].
Lemma 2. (See [10].) Suppose that T (t) satisfies (17) and that the following conditions are valid: http://www.journals.vu.lt/nonlinear-analysis (i) There is a t 0 0 such that T (t) is compact for t > t 0 .
is isolated and has an acyclic coveringM , whereM = Then T (t) is uniformly persistent in the sense that there is an ε > 0 such that, for any where d is the distance of T (t)x from X 0 .
where X 0 is a positive invariant set for system (2), X 0 is a positive invariant set for system (2) when any initial component is zero. Therefore, X satisfies (17). Denote where (x(t), u(t), y(t), z(t)) is a positive solution of system (2) with the initial condition (3). Then {T (t)} t 0 is a C 0 semigroup generated by (2). It is easy to prove that conditions (i)-(iv) of Lemma 2 is satisfied when R 1 > 1. Therefore, all solutions of system (2) in X 0 are uniform repellers with respect to X 0 [25]. In other words, there is an ε 0 > 0 such that for any solution Φ(t) := (x(t), u(t), y(t), z(t)) of system (2) with initial condition in X 0 , we have where d is the distance of Φ(t) from X 0 . Thus, there exists an ε 1 > 0 such that for any solution of system (2) with the initial condition in X 0 . We obtain from the first equation of system (2) that By comparison we have x(t) > λ/2(d + βM ) for sufficiently large t, for any solution of system (2) with initial condition in X 0 . Let y ∞ = lim inf t→+∞ y(t), u ∞ = lim inf t→+∞ u(t). By the fluctuations lemma in [11] there exists a sequence τ n → ∞, y(τ n ) → y ∞ and y (τ n ) = 0. By the third equation of system (2) we have and hence a + pM σ y ∞ + y ∞ u ∞ + y ∞ ε 1 .
LHS allows an un-biased estimate for each parameter of R 0 and R 1 , a probability density function is defined and divided into N equal probability intervals. N represents the sample size. The choice for N should be at least k + 1, where k is the parameters varied, but usually much larger to ensure accuracy. A single value is then selected randomly from every interval [18]. In this way, an input value from each sampling interval is used only once in the analysis, but the entire parameter space is equitably sampled in an efficient manner.
Through analysis of the sample derived from LHS, we can obtain large efficient data in respect to different parameters of R 0 and R 1 . Figure 7 shows that r, σ are both positive correlative variables with R 0 and R 1 . It is clear that σ contributes more to R 0 and R 1 than r, hence, σ is a more important factor.

Conclusions
In this paper, we considered an improved HTLV-I infection model with CD4 + T cells mitosis and delayed cytotoxic T-lymphocyte (CTL) immune response. Through a rigorous mathematical analysis, the threshold dynamical of the model was established, and it can be determined by the immune-inactivated reproduction ratio R 0 and the immune-activated reproduction ratio R 1 . If R 0 < 1, the infection-free equilibrium is globally asymptotically stable; if R 1 < 1 < R 0 , the immunity-inactivated equilibrium is globally asymptotically stable; if R 1 > 1, the immunity-activated equilibrium is globally asymptotically stable when τ = 0. Besides, we established the existence of Hopf bifurcation at the immunityactivated equilibrium. By Theorem 5 we found that when the delay varies, the immunityactivated steady state loses its stability and Hopf bifurcation occurs. That is to say, the time delay can destabilize the immunity-activated equilibrium and lead to periodic oscillation through Hopf bifurcation. Numerical simulations showed the occurrence of bifurcating periodic oscillation when the delay passes the critical value. Sensitivity analysis showed that σ has a great influence on the threshold parameter R 0 and R 1 , which can provide some suggestions for clinical treatment of HTLV-I related diseases.