Global Analysis of a Model of Viral Infection with Latent Stage and Two Types of Target Cells

Abstract

By introducing the probability function describing latency of infected cells, we unify some models of viral infection with latent stage. For the case that the probability function is a step function, which implies that the latency period of the infected cells is constant, the corresponding model is a delay differential system. The model with delay of latency and two types of target cells is investigated, and the obtained results show that when the basic reproduction number is less than or equal to unity, the infection-free equilibrium is globally stable, that is, the in-host free virus will be cleared out finally; when the basic reproduction number is greater than unity, the infection equilibrium is globally stable, that is, the viral infection will be chronic and persist in-host. And by comparing the basic reproduction numbers of ordinary differential system and the associated delayed differential system, we think that it is necessary to elect an appropriate type of probability function for predicting the final outcome of viral infection in-host.

1. Introduction

The dynamical models of virus infection have played an important role in understanding the action of in-host free virus on target cells. Nowak et al. [1, 2] proposed one of the earliest of these models: 𝑥^′𝑦=𝜆−𝑑𝑥−𝛽𝑥𝑣,_′𝑣=𝛽𝑥𝑣−𝑎𝑦,_′=𝑘𝑎𝑦−𝛾𝑣, (1) where 𝑥=𝑥(𝑡), 𝑦=𝑦(𝑡), and 𝑣=𝑣(𝑡) are the concentrations of uninfected cells, infected cells, and viral particles (virions) at time 𝑡, respectively. In model (1), uninfected target cells are assumed to be produced at a constant rate 𝜆 and die at a rate 𝑑𝑥. Infection of target cells by in-host free virus is assumed to occur at a bilinear rate 𝛽𝑥𝑣. Infected cells are lost at a rate 𝑎𝑦. Free virus are produced by infected cells at a rate 𝑘𝑎𝑦 in which 𝑘 is the average number of viral particles produced by a single infected cell over its lifetime and die at a rate 𝛾𝑣. Model (1) is a basic model of viral infection, which has been used widely to investigate infection of some viruses (such as, HIV, HBV, HCV, and HLTV). However, following infection of virus, within a cell the provirus may remain latent, giving no sign of its presence for months or years [3]. According to this fact, in order to investigate HIV-1 dynamics in vivo, Perelson et al. incorporated the latently infected cells into the basic model (1) [4, 5]. It implies that the development of infected cell should include latent and active two stages. That is, once infected, a cell first becomes a latently infected cell, but does not produce virus; after a period of time a latently infected cell turns active and begins to produce virus. The global stability of some ODE models of viral infection with latent stage is considered in [6, 7].

In 1997, Perelson et al. [8] observed that HIV attacks two types of target cells, CD4⁺ T cells and macrophages. On the other hand, it was also detected that, except for liver tissue, HCV may be produced in some extrahepatic tissues, such as bone marrow [9], peripheral blood mononuclear cells (PBMC) [10], brain [11], and lymph nodes [12]. Then, according to these virological findings, based on model (1), some viral dynamical models with two types of target cells were proposed [5, 13, 14], which are all expressed by ordinary differential equations.

Since the dynamics of viral infection in-host is not well understood, in order to investigate the mechanism of viral infection, some reasonable assumptions are often incorporated into mathematical models describing the interaction between target cells and viral particles. In this paper, we first introduce the probability functions describing the latency of infected cells to unify some models of viral infection with latent stage and then analyze dynamics of viral infection model with constant latency period and two types of target cells. The discussed model is a system of delay differential equations.

The organization of this paper is as follows. In Section 2, we unify some models of viral infection with latent stage by introducing the probability function describing latency of infected cells and propose a model of virus infection with latent delay and two types of target cells. The global stability is analyzed in Section 3. At last, the conclusion on the model is summarized, and the basic reproduction numbers of ordinary differential system and the associated delay differential system are compared.

2. Models

Let 𝑄(𝑡) for 𝑡≥0 denote the probability that an infected cell is in the latent stage at least 𝑡 time units before becoming the actively infected cell, and then, when the infection rate of virus is assumed to be 𝛽𝑥(𝑡)𝑣(𝑡), the concentration of the latently infected cells at time 𝑡 can be expressed by the following equation: 𝑤(𝑡)=𝑤₀(𝑡)+^𝑡₀𝛽𝑥(𝜃)𝑣(𝜃)𝑄(𝑡−𝜃)𝑒^{−𝑑(𝑡−𝜃)}𝑑𝜃, (2) where 𝑤₀(𝑡) is the concentration of the latently infected cells which are in the latent stage at time 0 and still at the same state at time 𝑡. Function 𝑤₀(𝑡) is a nonnegative, non-increasing, and piecewise continuous function. Thus, when incorporating the latent stage of infected cells into the basic model (1), we have the model of viral infection with latent stage: 𝑥^′𝑤(𝑡)=𝜆−𝑑𝑥−𝛽𝑥𝑣,=𝑤₀(𝑡)+^𝑡₀𝛽𝑥(𝜃)𝑣(𝜃)𝑄(𝑡−𝜃)𝑒^{−𝑑(𝑡−𝜃)}𝑦𝑑𝜃,_′𝑣=𝐹𝑥,𝑦,𝑤−𝑎𝑦,_′=𝑘𝑎𝑦−𝛾𝑣, (3) where the term 𝐹(𝑥,𝑦,𝑤) in the third equation of (3) represents the recruitment rate of actively infected cells; its expression should depend on the form of function 𝑄(𝑡).

Usually, function 𝑄(𝑡) is elected as one of the following two types, the exponential function (i.e., 𝑄(𝑡)=exp(−𝜀𝑡)) and the step function (i.e., 𝑄(𝑡) equals to 1 for 𝑡∈[0,𝜏) and 0 for 𝑡∈[𝜏,+∞)) [15, 16]. Here, the exponential function means that the transfer of the infected cells from the latent state to the active one follows the exponential distribution, and the step function means that the time length of staying at the latent state for the infected cells is constant 𝜏 and that they become active after the time period 𝜏. It is easy to know that the average latency period of the infected cells is 1/𝜀 for the exponential function and 𝜏 for the step function.

When 𝑄(𝑡)=exp(−𝜀𝑡), (2) becomes 𝑤(𝑡)=𝑤₀𝑒^{−(𝑑+𝜀)𝑡}+^𝑡₀𝛽𝑥(𝜃)𝑣(𝜃)𝑒^{−(𝑑+𝜀)(𝑡−𝜃)}𝑑𝜃, (4) where 𝑤₀ is the concentration of the latently infected cells at 𝑡=0, then we have 𝑤^′(𝑡)=𝛽𝑥(𝑡)𝑣(𝑡)−(𝑑+𝜀)𝑤(𝑡). (5) From the equation of 𝑤, we know that the removed rate of the latently infected cells is (𝑑+𝜀)𝑤, where 𝑑 is the death rate coefficient of uninfected cells; then 𝜀𝑤 is the infection-induced transfer rate to actively infected cells, that is, 𝐹(𝑥,𝑦,𝑤)=𝜀𝑤. So, system (3) becomes 𝑥^′𝑤=𝜆−𝑑𝑥−𝛽𝑥𝑣,_′()𝑦=𝛽𝑥𝑣−𝑑+𝜀𝑤,_′𝑣=𝜀𝑤−𝑎𝑦,_′=𝑘𝑎𝑦−𝛾𝑣. (6)

When 𝑄(𝑡) is a step function, that is, 𝑄(𝑡)=[),[),1for𝑡∈0,𝜏0for𝑡∈𝜏,+∞ (7)𝑤₀(𝑡)=0 for 𝑡≥𝜏. Thus, for 𝑡≥𝜏 the integral equation (2) becomes 𝑤(𝑡)=^𝑡_𝑡−𝜏𝛽𝑥(𝜃)𝑣(𝜃)𝑒^{−𝑑(𝑡−𝜃)}𝑑𝜃. (8) It is equivalent to the following delay differential equation: 𝑤^′(𝑡)=𝛽𝑥(𝑡)𝑣(𝑡)−𝛽𝑒^−𝑑𝜏𝑥(𝑡−𝜏)𝑣(𝑡−𝜏)−𝑑𝑤(𝑡), (9) with 𝑤(0)=∫⁰_−𝜏𝛽𝑥(𝜃)𝑣(𝜃)𝑒^𝑑𝜃𝑑𝜃, where the term 𝛽𝑒^−𝑑𝜏𝑥(𝑡−𝜏)𝑣(𝑡−𝜏) represents the recruitment rate of the actively infected cells for 𝑡>𝜏. Thus, when investigating the long-term behavior of model (3), the corresponding model with constant latent period is given by 𝑥^′𝑤=𝜆−𝑑𝑥−𝛽𝑥𝑣,_′=𝛽𝑥𝑣−𝛽𝑒^−𝑑𝜏𝑥()𝑣()𝑦𝑡−𝜏𝑡−𝜏−𝑑𝑤,_′=𝛽𝑒^−𝑑𝜏𝑥()𝑣()𝑣𝑡−𝜏𝑡−𝜏−𝑎𝑦,_′=𝑘𝑎𝑦−𝛾𝑣. (10) From the inference above, we may see that models (6) and (10) can be unified into model (3) and are two special cases of (3). This is due to the introduction of the probability function 𝑄(𝑡). Global properties of models (6) and (10) were investigated in [6, 17], respectively.

When considering the case that virus attacks two types of target cells, we denote the corresponding quantities by the same letters as model (3) with the subscript 1 or 2. The subscript represents the type of target cells. Thus, we have the following model with latent stage and two types of target cells: 𝑥^′₁=𝜆₁−𝑑₁𝑥₁−𝛽₁𝑥₁𝑤𝑣,₁(𝑡)=𝑤₁₀(𝑡)+^𝑡₀𝛽₁𝑥₁(𝜃)𝑣(𝜃)𝑄₁(𝑡−𝜃)𝑒^{−𝑑1(𝑡−𝜃)}𝑦𝑑𝜃,_′1=𝐹₁𝑥₁,𝑦₁,𝑤₁−𝑎₁𝑦₁,𝑥^′₂=𝜆₂−𝑑₂𝑥₂−𝛽₂𝑥₂𝑤𝑣,₂(𝑡)=𝑤₂₀(𝑡)+^𝑡₀𝛽₂𝑥₂(𝜃)𝑣(𝜃)𝑄₂(𝑡−𝜃)𝑒^{−𝑑2(𝑡−𝜃)}𝑦𝑑𝜃,_′2=𝐹₂𝑥₂,𝑦₂,𝑤₂−𝑎₂𝑦₂,𝑣^′=𝑘₁𝑎₁𝑦₁+𝑘₂𝑎₂𝑦₂−𝛾𝑣. (11)

Similarly, when the probability functions in (11) are exponential function, model (11) can become 𝑥^′₁=𝜆₁−𝑑₁𝑥₁−𝛽₁𝑥₁𝑤𝑣,_′1=𝛽₁𝑥₁𝑣−𝑑₁+𝜀₁𝑤₁,𝑦^′₁=𝜀₁𝑤₁−𝑎₁𝑦₁,𝑥^′₂=𝜆₂−𝑑₂𝑥₂−𝛽₂𝑥₂𝑤𝑣,_′2=𝛽₂𝑥₂𝑣−𝑑₂+𝜀₂𝑤₂,𝑦^′₂=𝜀₂𝑤₂−𝑎₂𝑦₂,𝑣^′=𝑘₁𝑎₁𝑦₁+𝑘₂𝑎₂𝑦₂−𝛾𝑣. (12) Its dynamical behavior was analyzed in [7].

When the probability functions in (11) are step function, for 𝑡>𝜏∶=max{𝜏₁,𝜏₂}, model (11) can become 𝑥^′₁(𝑡)=𝜆₁−𝑑₁𝑥₁(𝑡)−𝛽₁𝑥₁(𝑡)𝑣(𝑡),𝑤^′₁(𝑡)=𝛽₁𝑥₁(𝑡)𝑣(𝑡)−𝛽₁𝑒^{−𝑑1𝜏1}𝑥₁𝑡−𝜏₁×𝑣𝑡−𝜏₁−𝑑₁𝑤₁(𝑡),𝑦^′₁(𝑡)=𝛽₁𝑒^{−𝑑1𝜏1}𝑥₁𝑡−𝜏₁𝑣𝑡−𝜏₁−𝑎₁𝑦₁(𝑡),𝑥^′₂(𝑡)=𝜆₂−𝑑₂𝑥₂(𝑡)−𝛽₂𝑥₂(𝑡)𝑣(𝑡),𝑤^′₂(𝑡)=𝛽₂𝑥₂(𝑡)𝑣(𝑡)−𝛽₂𝑒^{−𝑑2𝜏2}𝑥₂(𝑡−𝜏)×𝑣𝑡−𝜏₂−𝑑₂𝑤₂(𝑡),𝑦^′₂=𝛽₂𝑒^{−𝑑2𝜏2}𝑥₂𝑡−𝜏₂𝑣𝑡−𝜏₂−𝑎₂𝑦₂(𝑡),𝑣^′=𝑘₁𝑎₁𝑦₁(𝑡)+𝑘₂𝑎₂𝑦₂(𝑡)−𝛾𝑣(𝑡). (13) For system (13), variables 𝑤₁ and 𝑤₂ do not appear in the equations of 𝑥_𝑖, 𝑦_𝑖 (𝑖=1,2), and 𝑣, then denoting 𝑏₁=𝑒^{−𝑑1𝜏1}, and 𝑏₂=𝑒^{−𝑑2𝜏2}, gives a subsystem of (13) as follows: 𝑥^′₁(𝑡)=𝜆₁−𝑑₁𝑥₁(𝑡)−𝛽₁𝑥₁(𝑡)𝑣(𝑡),𝑦^′₁(𝑡)=𝛽₁𝑏₁𝑥₁𝑡−𝜏₁𝑣𝑡−𝜏₁−𝑎₁𝑦₁(𝑡),𝑥^′₂(𝑡)=𝜆₂−𝑑₂𝑥₂(𝑡)−𝛽₂𝑥₂(𝑡)𝑣(𝑡),𝑦^′₂=𝛽₂𝑏₂𝑥₂𝑡−𝜏₂𝑣𝑡−𝜏₂−𝑎₂𝑦₂(𝑡),𝑣^′=𝑘₁𝑎₁𝑦₁(𝑡)+𝑘₂𝑎₂𝑦₂(𝑡)−𝛾𝑣(𝑡). (14) In this paper, we will investigate the global behaviors of system (14).

For system (14), we set a suitable phase space. Denote the Banach space of continuous functions mapping the interval [−𝜏,0] into ℝ⁵ with the sup-norm for 𝜙=(𝜙₁,𝜙₂,𝜙₃,𝜙₄,𝜙₅)^𝑇∈𝐶 by 𝐶=𝐶([−𝜏,0],ℝ⁵), where 𝜏=max{𝜏₁,𝜏₂}. The nonnegative cone of 𝐶 is defined as 𝐶₊=𝐶([−𝜏,0];ℝ⁵₊). From the biological meaning, the initial conditions for system (14) are given as follows: 𝑥₁(𝜃)=𝜙₁(𝜃),𝑦₁(𝜃)=𝜙₂(𝜃),𝑥₂(𝜃)=𝜙₃(𝜃),𝑦₂(𝜃)=𝜙₄(𝜃),𝑣(𝜃)=𝜙₅(𝜃),𝜃∈[−𝜏,0], (15) where (𝜙₁(𝜃),𝜙₂(𝜃),𝜙₃(𝜃),𝜙₄(𝜃),𝜙₅(𝜃))^𝑇∈𝐶₊ and 𝜙_𝑖(0)>0, 𝑖=1,2,3,4,5.

Under the initial conditions (15), it is easy to see that all solutions of system (14) are positive on [0,+∞). Furthermore, we have the following statement with respect to the boundedness of solutions of system (14).

Theorem 1. All solutions of system (14) under the initial conditions (15) are ultimately bounded.

Proof. Define a Lyapunov functional 𝐿₀₁=𝑏₁𝑥₁(𝑡)+𝑦₁(𝑡+𝜏₁) then from the first two equations of (14), we have 𝐿^′₀₁=𝑏₁𝜆₁−𝑏₁𝑑₁𝑥₁(𝑡)+𝑎₁𝑦₁𝑡+𝜏₁≤𝑏₁𝜆₁−𝜌₁𝐿₀₁, (16) where 𝜌₁=min{𝑑₁,𝑎₁}. It follows that limsup_𝑡→+∞[𝑏₁𝑥₁(𝑡)+𝑦₁(𝑡+𝜏₁)]≤𝑏₁𝜆₁/𝜌₁, that is, for any positive number 𝜀 there is 𝑇₁>0 such that 𝑏₁𝑥₁(𝑡)+𝑦₁(𝑡+𝜏₁)<𝑏₁𝜆₁/𝜌₁+𝜀 for 𝑡>𝑇₁.
Similarly, from the third and fourth equations of (14) we know that for any positive number 𝜀 there is 𝑇₂>0 such that 𝑏₂𝑥₂(𝑡)+𝑦₂(𝑡+𝜏₂)<𝑏₂𝜆₂/𝜌₂+𝜀 for 𝑡>𝑇₂, where 𝜌₂=min{𝑑₂,𝑎₂}.
Therefore, for 𝑡>𝑇=max{𝑇₁+𝜏,𝑇₂+𝜏}, we have 𝑦₁(𝑡)<𝑏₁𝜆₁/𝜌₁+𝜀 and 𝑦₂(𝑡)<𝑏₂𝜆₂/𝜌₂+𝜀. Thus, from the last equation of (14), it follows that 𝑣^′(𝑡)<𝑘₁𝑎₁𝑏₁𝜆₁𝜌₁+𝜀+𝑘₂𝑎₂𝑏₂𝜆₂𝜌₂+𝜀−𝛾𝑣(𝑡), (17) for 𝑡>𝑇. It implies that limsup_𝑡→+∞𝑣(𝑡)≤1𝛾𝑘₁𝑎₁𝑏₁𝜆₁𝜌₁+𝜀+𝑘₂𝑎₂𝑏₂𝜆₂𝜌₂+𝜀. (18)
Summarizing the above inference, Theorem 1 holds.

Since the positive number 𝜀 in the proof of Theorem 1 is arbitrary, we can know that the set Ω=𝑥^𝑇(𝑡)∈𝐶₊∶𝑏_𝑖𝑥_𝑖(𝑡)+𝑦_𝑖𝑡+𝜏_𝑖≤𝑏_𝑖𝜆_𝑖𝜌_𝑖,𝑖=1,2,𝑣(𝑡)≤𝜌 (19) is positively invariant to system (14), where 𝑥(𝑡)=(𝑥₁(𝑡),𝑦₁(𝑡),𝑥₂(𝑡),𝑦₂(𝑡),𝑣(𝑡)),  𝜌=(𝑘₁𝑎₁𝑏₁𝜆₁/𝜌₁+𝑘₂𝑎₂𝑏₂𝜆₂/𝜌₂)/𝛾. Therefore, we will consider system (14) on the set Ω.

3. Global Stability

In this section, we will investigate the existence and stability of equilibria of system (14).

Obviously, (14) always has the infection-free equilibrium 𝐸₀(𝑥₁₀,0,𝑥₂₀,0,0), where 𝑥₁₀=𝜆₁/𝑑₁ and 𝑥₂₀=𝜆₂/𝑑₂. The infection equilibrium 𝐸^∗(𝑥^∗₁,𝑦^∗₁,𝑥^∗₂,𝑦^∗₂,𝑣^∗)(𝑣^∗>0) is determined by the following equations: 𝜆₁−𝑑₁𝑥₁−𝛽₁𝑥₁𝛽𝑣=0,₁𝑏₁𝑥₁𝑣−𝑎₁𝑦₁𝜆=0,₂−𝑑₂𝑥₂−𝛽₂𝑥₂𝛽𝑣=0,₂𝑏₂𝑥₂𝑣−𝑎₂𝑦₂𝑘=0,₁𝑎₁𝑦₁+𝑘₂𝑎₂𝑦₂−𝛾𝑣=0. (20)

From the first and third equations of (20), we have 𝑥₁=𝜆₁𝑑₁+𝛽₁𝑣,𝑥₂=𝜆₂𝑑₂+𝛽₂𝑣, (21) respectively. Substituting them into the second and fourth equations of (20) yields 𝑦₁=𝛽₁𝑏₁𝜆₁𝑣𝑎₁𝑑₁+𝛽₁𝑣,𝑦₂=𝛽₂𝑏₂𝜆₂𝑣𝑎₂𝑑₂+𝛽₂𝑣, (22) respectively. When 𝑣≠0, substituting the above 𝑦₁ and 𝑦₂ into the last equation of (20) gives 𝑘₁𝛽₁𝑏₁𝜆₁𝑑₁+𝛽₁𝑣+𝑘₂𝛽₂𝑏₂𝜆₂𝑑₂+𝛽₂𝑣=𝛾. (23)

Since the function of 𝑣 at the left hand side of (23) is strictly decreasing, it is easy to see that (23) has a positive root if and only if 𝑘₁𝛽₁𝑏₁𝜆₁/𝑑₁+𝑘₂𝛽₂𝑏₂𝜆₂/𝑑₂>𝛾 and that the positive root is unique, denoted by 𝑣^∗. Therefore, with respect to the existence of equilibria of (14), we have the following result.

Theorem 2. Denote 𝑅₀=𝑘₁𝛽₁𝑏₁𝜆₁𝑑₁+𝑘₂𝛽₂𝑏₂𝜆₂𝑑₂1𝛾; (24) then, when 𝑅₀≤1, system (14) only has the infection-free equilibrium 𝐸₀(𝑥₁₀,0,𝑥₂₀,0,0), where 𝑥₁₀=𝜆₁/𝑑₁ and 𝑥₂₀=𝜆₂/𝑑₂; when 𝑅₀>1, besides 𝐸₀, system (14) also has a unique infection equilibrium 𝐸^∗(𝑥^∗₁,𝑦^∗₁,𝑥^∗₂,𝑦^∗₂,𝑣^∗), where 𝑥^∗₁=𝜆₁𝑑₁+𝛽₁𝑣^∗,𝑥^∗₂=𝜆₂𝑑₂+𝛽₂𝑣^∗,𝑦^∗₁=𝛽₁𝑏₁𝜆₁𝑣^∗𝑎₁𝑑₁+𝛽₁𝑣^∗,𝑦^∗₂=𝛽₂𝑏₂𝜆₂𝑣^∗𝑎₂𝑑₂+𝛽₂𝑣^∗, (25) and 𝑣^∗ is determined by (23).

Note that 𝑅₀ is the basic reproduction number describing the viral infection within host.

In the following, we consider the global stability of equilibria of (14).

In order to simplify the proof of the global stability of the infection equilibrium 𝐸^∗, we first introduce an inequality as lemma, which was proved in [18].

Lemma 3. For 𝑛 positive numbers 𝑐_𝑖 (𝑖=1,2,…,𝑛), the inequality 𝑛−𝑐₁−𝑐₂−⋯−𝑐_𝑛+ln𝑐₁𝑐₂⋯𝑐_𝑛≤0 (26) is true, and the equality holds if and only if 𝑐₁=𝑐₂=⋯=𝑐_𝑛=1.

Theorem 4. When 𝑅₀≤1, the infection-free equilibrium 𝐸₀ of system (14) is globally stable on Ω; when 𝑅₀>1, the infection equilibrium 𝐸^∗ of (14) is globally stable in the region Ω.

Proof. In order to prove the global stability of the infection-free equilibrium 𝐸₀ of (14), we define a Lyapunov function: 𝐿₁₁=𝑘₁𝑏₁𝑥₁−𝑥₁₀−𝑥₁₀𝑥ln¹𝑥₁₀+𝑘₁𝑦₁+𝑘₂𝑏₂𝑥₂−𝑥₂₀−𝑥₂₀𝑥ln²𝑥₂₀+𝑘₂𝑦₂+𝑣, (27) then 𝑑𝐿₁₁𝑑𝑡=𝑘₁𝑏₁𝑥₁−𝑥₁₀𝜆₁𝑥₁−𝑑₁−𝛽₁𝑣+𝑘₁𝛽₁𝑏₁𝑥𝑡−𝜏₁𝑣𝑡−𝜏₁−𝑎₁𝑦₁+𝑘₂𝑏₂𝑥₂−𝑥₂₀𝜆₂𝑥₂−𝑑₂−𝛽₂𝑣+𝑘₂𝛽₂𝑏₂𝑥𝑡−𝜏₂𝑣𝑡−𝜏₂−𝑎₂𝑦₂+𝑘₁𝑎₁𝑦₁+𝑘₂𝑎₂𝑦₂−𝛾𝑣=𝑘₁𝑏₁𝑥₁−𝑥₁₀𝜆₁1𝑥₁−1𝑥₁₀−𝛽₁𝑣+𝑘₁𝛽₁𝑏₁𝑥𝑡−𝜏₁𝑣𝑡−𝜏₁−𝑎₁𝑦₁+𝑘₂𝑏₂𝑥₂−𝑥₂₀𝜆₂1𝑥₂−1𝑥₂₀−𝛽₂𝑣+𝑘₂𝛽₂𝑏₂𝑥𝑡−𝜏₂𝑣𝑡−𝜏₂−𝑎₂𝑦₂+𝑘₁𝑎₁𝑦₁+𝑘₂𝑎₂𝑦₂−𝛾𝑣=𝑘₁𝑏₁𝜆₁𝑥2−¹𝑥₁₀−𝑥₁₀𝑥₁+𝑘₂𝑏₂𝜆₂𝑥2−²𝑥₂₀−𝑥₂₀𝑥₂+𝑘₁𝑏₁𝛽₁𝑥₁𝑡−𝜏₁𝑣𝑡−𝜏₁−𝑥₁(𝑡)𝑣(𝑡)+𝑘₂𝑏₂𝛽₂𝑥₂𝑡−𝜏₂𝑣𝑡−𝜏₂−𝑥₂(𝑡)𝑣(𝑡+𝑘₁)𝑏₁𝛽₁𝑥₁₀+𝑘₂𝑏₂𝛽₂𝑥₂₀−𝛾𝑣. (28) Let 𝐿₁=𝐿₁₁+𝑘₁𝑏₁𝛽₁∫^𝑡_𝑡−𝜏1𝑥₁(𝜃)𝑣(𝜃)𝑑𝜃+𝑘₂𝑏₂𝛽₂∫^𝑡_𝑡−𝜏2𝑥₂(𝜃)𝑣(𝜃)𝑑𝜃; then 𝑑𝐿₁𝑑𝑡=𝑘₁𝑏₁𝜆₁𝑥2−¹𝑥₁₀−𝑥₁₀𝑥₁+𝑘₂𝑏₂𝜆₂𝑥2−²𝑥₂₀−𝑥₂₀𝑥₂+𝛾𝑅₀−1𝑣, (29) where 𝑅₀=(𝑘₁𝑏₁𝛽₁𝑥₁₀+𝑘₂𝑏₂𝛽₂𝑥₂₀)/𝛾 is used.
It is easy to see that 2≤𝑥_𝑖/𝑥_𝑖0+𝑥_𝑖0/𝑥_𝑖 (𝑖=1,2), then 𝑑𝐿₁/𝑑𝑡≤0 as 𝑅₀≤1. Note that, for 𝑅₀<1, 𝑑𝐿₁/𝑑𝑡=0 if and only if 𝑥_𝑖=𝑥_𝑖0 (𝑖=1,2) and 𝑣=0; for 𝑅₀=1, 𝑑𝐿₁/𝑑𝑡=0 if and only if 𝑥_𝑖=𝑥_𝑖0 (𝑖=1,2). No matter the case 𝑅₀<1 or 𝑅₀=1, the largest invariant set of (14) on the set Ω is the singleton {𝐸₀}. Since any solution of (14) is bounded, it follows from the Lyapunov-LaSalle Invariance Principle for functional differential equations that the infection-free equilibrium 𝐸₀ is globally stable on the set Ω when 𝑅₀≤1 [19].
In order to prove the global stability of the infection equilibrium 𝐸^∗(𝑥^∗₁,𝑦^∗₁,𝑥^∗₂,𝑦^∗₂,𝑣^∗), define the following Lyapunov functions and functionals: 𝐿₂₁=𝑘₁𝑏₁𝑥₁−𝑥^∗₁−𝑥^∗₁𝑥ln¹𝑥^∗₁+𝑘₁𝑦₁−𝑦^∗₁−𝑦^∗₁𝑦ln¹𝑦^∗₁,𝐿₂₂=𝑘₂𝑏₂𝑥₂−𝑥^∗₂−𝑥^∗₂𝑥ln²𝑥^∗₂+𝑘₂𝑦₂−𝑦^∗₂−𝑦^∗₂𝑦ln²𝑦^∗₂,𝐿₂₃=𝑣−𝑣^∗−𝑣^∗𝑣ln𝑣^∗,𝐿₂₄=𝑘₁𝑏₁𝛽₁𝑥^∗₁𝑣^∗×^𝑡_𝑡−𝜏1𝑥₁(𝜃)𝑣(𝜃)𝑥^∗₁𝑣^∗𝑥−1−ln¹(𝜃)𝑣(𝜃)𝑥^∗₁𝑣^∗𝑑𝜃+𝑘₂𝑏₂𝛽₂𝑥^∗₂𝑣^∗×^𝑡_𝑡−𝜏2𝑥₂(𝜃)𝑣(𝜃)𝑥^∗₂𝑣^∗𝑥−1−ln²(𝜃)𝑣(𝜃)𝑥^∗₂𝑣^∗𝑑𝜃. (30) According to (20), we have 𝑑₁=𝜆₁𝑥^∗₁−𝛽₁𝑣^∗,𝑎₁=𝛽₁𝑏₁𝑥^∗₁𝑣^∗𝑦^∗₁,𝑑₂=𝜆₂𝑥^∗₂−𝛽₂𝑣^∗,𝑎₂=𝛽₂𝑏₂𝑥^∗₂𝑣^∗𝑦^∗₂,𝑘𝛾=¹𝑎₁𝑦^∗₁𝑣^∗+𝑘₂𝑎₂𝑦^∗₂𝑣^∗. (31) Then, system (14) can be rewritten as 𝑥^′₁(𝑡)=𝑥₁(𝑡)𝜆₁1𝑥₁(𝑡)−1𝑥^∗₁−𝛽₁𝑣(𝑡)−𝑣^∗,𝑦^′₁(𝑡)=𝛽₁𝑏₁𝑦₁(𝑡𝑥)¹𝑡−𝜏₁𝑣𝑡−𝜏₁𝑦₁(𝑡)−𝑥^∗₁𝑣^∗𝑦^∗₁,𝑥^′₂(𝑡)=𝑥₂(𝑡)𝜆₂1𝑥₂(𝑡)−1𝑥^∗₂−𝛽₂𝑣(𝑡)−𝑣^∗,𝑦^′₂(𝑡)=𝛽₂𝑏₂𝑦₂(𝑡𝑥)²𝑡−𝜏₂𝑣𝑡−𝜏₂𝑦₂(𝑡)−𝑥^∗₂𝑣^∗𝑦^∗₂,𝑣^′(𝑡)=𝑣(𝑡)𝑘₁𝑎₁𝑦₁(𝑡)𝑣(𝑡)−𝑦^∗₁𝑣^∗+𝑘₂𝑎₂𝑦₂(𝑡)𝑣(𝑡)−𝑦^∗₂𝑣^∗. (32)
By using 𝜆₁=𝑑₁𝑥^∗₁+𝛽₁𝑥^∗₁𝑣^∗ and 𝜆₂=𝑑₂𝑥^∗₂+𝛽₂𝑥^∗₂𝑣^∗, direct computations show that 𝐿^′₂₁=𝑘₁𝑏₁𝑑₁𝑥^∗₁𝑥2−^∗1𝑥₁−𝑥₁𝑥^∗₁+𝑘₁𝑏₁𝛽₁𝑥^∗₁𝑣^∗𝑥2−^∗1𝑥₁−𝑥₁𝑣𝑥^∗₁𝑣^∗+𝑣𝑣^∗+𝑘₁𝑏₁𝛽₁𝑥^∗₁𝑣^∗𝑥₁𝑡−𝜏₁𝑣𝑡−𝜏₁𝑥^∗₁𝑣^∗−𝑦₁𝑦^∗₁−𝑥₁𝑡−𝜏₁𝑦^∗₁𝑣𝑡−𝜏₁𝑥^∗₁𝑦₁𝑣^∗,𝐿^′₂₂=𝑘₂𝑏₂𝑑₂𝑥^∗₂𝑥2−^∗2𝑥₂−𝑥₂𝑥^∗₂+𝑘₂𝑏₂𝛽₂𝑥^∗₂𝑣^∗𝑥2−^∗2𝑥₂−𝑥₂𝑣𝑥^∗₂𝑣^∗+𝑣𝑣^∗+𝑘₂𝑏₂𝛽₂𝑥^∗₂𝑣^∗𝑥₂𝑡−𝜏₂𝑣𝑡−𝜏₂𝑥^∗₂𝑣^∗−𝑦₂𝑦^∗₂−𝑥₂𝑡−𝜏₂𝑦^∗₂𝑣𝑡−𝜏₂𝑥^∗₂𝑦₂𝑣^∗,𝐿^′₂₃=𝑘₁𝑎₁𝑦^∗₁𝑦₁𝑦^∗₁−𝑦₁𝑣^∗𝑦^∗₁𝑣−𝑣𝑣^∗+1+𝑘₂𝑎₂𝑦^∗₂𝑦₂𝑦^∗₂−𝑦₂𝑣^∗𝑦^∗₂𝑣−𝑣𝑣^∗,𝐿+1_′24=𝑘₁𝑏₁𝛽₁𝑥^∗₁𝑣^∗𝑥₁𝑣𝑥^∗₁𝑣^∗−𝑥₁𝑡−𝜏₁𝑣𝑡−𝜏₁𝑥^∗₁𝑣^∗𝑥+ln¹𝑡−𝜏₁𝑣𝑡−𝜏₁𝑥₁(𝑡)𝑣(𝑡)+𝑘₂𝑏₂𝛽₂𝑥^∗₂𝑣^∗𝑥₂𝑣𝑥^∗₂𝑣^∗−𝑥₂𝑡−𝜏₂𝑣𝑡−𝜏₂𝑥^∗₂𝑣^∗𝑥+ln²𝑡−𝜏₂𝑣𝑡−𝜏₂𝑥₂(𝑡)𝑣(𝑡). (33) Let 𝐿₂=𝐿₂₁+𝐿₂₂+𝐿₂₃+𝐿₂₄, then 𝐿^′₂=𝑘₁𝑏₁𝑑₁𝑥^∗₁𝑥2−^∗1𝑥₁−𝑥₁𝑥^∗₁+𝑘₂𝑏₂𝑑₂𝑥^∗₂𝑥2−^∗2𝑥₂−𝑥₂𝑥^∗₂+𝑘₁𝑏₁𝛽₁𝑥^∗₁𝑣^∗𝑥3−^∗1𝑥₁(𝑡)−𝑥₁𝑡−𝜏₁𝑦^∗₁𝑣𝑡−𝜏₁𝑥^∗₁𝑦₁𝑣^∗−𝑦₁(𝑡)𝑣^∗𝑦^∗₁𝑣(𝑡)𝑥+ln¹𝑡−𝜏₁𝑣𝑡−𝜏₁𝑥₁(𝑡)𝑣(𝑡)+𝑘₂𝑏₂𝛽₂𝑥^∗₂𝑣^∗𝑥3−^∗2𝑥₂(𝑡)−𝑥₂𝑡−𝜏₂𝑦^∗₂𝑣𝑡−𝜏₂𝑥^∗₂𝑦₂𝑣^∗−𝑦₂(𝑡)𝑣^∗𝑦^∗₂𝑣(𝑡)𝑥+ln²𝑡−𝜏₂𝑣𝑡−𝜏₂𝑥₂(𝑡)𝑣(𝑡). (34)
According to the relationship between the arithmetic and the associated geometric means and Lemma 3, 𝑑𝐿₂/𝑑𝑡≤0 and 𝑑𝐿₂/𝑑𝑡=0 if and only if 𝑥₁=𝑥^∗₁, 𝑥₂=𝑥^∗₂, 𝑦₁/𝑦^∗₁ = 𝑦₂/𝑦^∗₂ = 𝑣(𝑡)/𝑣^∗ = 𝑣(𝑡−𝜏₁)/𝑣^∗ = 𝑣(𝑡−𝜏₂)/𝑣^∗. It is easy to see that the largest invariant set of (14) on the set {(𝑥₁,𝑦₁,𝑥₂,𝑦₂,𝑣)∈Ω∶𝑑𝐿₂/𝑑𝑡=0} is the singleton {𝐸^∗}. Since any solution of (14) is bounded, it follows from the Lyapunov-LaSalle Invariance Principle for functional differential equations that the infection equilibrium 𝐸^∗ is globally stable on the set Ω when 𝑅₀>1 [19].

4. Discussion

In this paper, we first present the probability function describing the latency of the infected cells such that some models of viral infection with latent stage are unified. When the function is an exponential one, the associated model is a system of ordinary differential equations; when the function is a step function, the associated one is a delay differential system. Both of the two types of models have the similar dynamical behaviors. That is, when the basic reproduction number is less than or equal to unity, the infection-free equilibrium is globally stable, which implies that the in-host free virus will be cleared out finally; when the basic reproduction number is greater than unity, the infection equilibrium is globally stable, which implies that the viral infection will be chronic and persist in-host [6, 7, 17]. But there is a difference between the basic reproduction numbers for the two types of models.

In fact, for ordinary differential systems (6) and (12), the basic reproduction numbers are 𝑅^(𝑂)₀₁=𝛽𝜆𝑘𝜀(),𝑅𝑑𝛾𝑑+𝜀_(𝑂)02=𝛽₁𝜆₁𝑘₁𝜀₁𝑑₁𝑑₁+𝜀₁+𝛽₂𝜆₂𝑘₂𝜀₂𝑑₂𝑑₂+𝜀₂1𝛾, (35) respectively [6, 7]. For delay differential systems (10) and (13) (or (14)), the basic reproduction numbers are 𝑅^(𝐷)₀₁=𝛽𝜆𝑘𝑒^−𝑑𝜏,𝑅𝑑𝛾_(𝐷)02=𝛽₁𝜆₁𝑘₁𝑒^{−𝑑1𝜏1}𝑑₁+𝛽₂𝜆₂𝑘₂𝑒^{−𝑑2𝜏2}𝑑₂1𝛾, (36) respectively, where 𝑅^(𝐷)₀₁ was obtained in [17]; 𝑅^(𝐷)₀₂ is 𝑅₀ in this paper.

According to the definition of probability function of staying in the latent stage, for the two common types of probability functions, exponential function and step function, we assume that the associated average latent periods of infected cells are equal, that is, 𝜏=1/𝜀. Then, the basic reproduction numbers of ordinary differential systems (6) and (12) can be rewritten by 𝑅^(𝑂)₀₁=𝛽𝜆𝑘(),𝑅𝑑𝛾1+𝑑𝜏_(𝑂)02=𝛽₁𝜆₁𝑘₁𝑑₁1+𝑑₁𝜏₁+𝛽₂𝜆₂𝑘₂𝑑₂1+𝑑₂𝜏₂1𝛾, (37) respectively.

From the inequality 1/(1+𝑢)>𝑒^−𝑢 for 𝑢>0, it follows that 𝑅^(𝑂)₀₁>𝑅^(𝐷)₀₁,𝑅^(𝑂)₀₂>𝑅^(𝐷)₀₂. (38) They imply that for certain models of viral infection, the basic reproduction number of ordinary differential system may be greater than that of delay differential system. It is well known that the basic reproduction number usually determines dynamics of models of viral infection. Therefore, the difference between the basic reproduction numbers of the different types of models reminds us that it is necessary to elect an appropriate type of probability function, when predicting the final outcome of viral infection in-host.

Acknowledgments

The authors would like to thank the referees for their valuable comments. This work was partially supported by the National Natural Science Foundation of China (nos. 11071256, 11171267, and 11371369).