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

Authors: Shuo Liu; Lina Ma; Jianquan Li; Qingbo Zhao
Journal: Journal of Applied Mathematics
 (2013)

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## 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.


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## Body

## 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:
(1)x′=λ-dx-βxv,y′=βxv-ay,v′=kay-γv,
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 Section2, 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

LetQ(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:
(2)w(t)=w0(t)+∫0t‍βx(θ)v(θ)Q(t-θ)e-d(t-θ)dθ,
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:
(3)x′=λ-dx-βxv,w(t)=w0(t)+∫0t‍βx(θ)v(θ)Q(t-θ)e-d(t-θ)dθ,y′=F(x,y,w)-ay,v′=kay-γv,
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, functionQ(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.WhenQ(t)=exp(-εt), (2) becomes
(4)w(t)=w0e-(d+ε)t+∫0t‍βx(θ)v(θ)e-(d+ε)(t-θ)dθ,
where w0 is the concentration of the latently infected cells at t=0, then we have
(5)w′(t)=βx(t)v(t)-(d+ε)w(t).
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
(6)x′=λ-dx-βxv,w′=βxv-(d+ε)w,y′=εw-ay,v′=kay-γv.WhenQ(t) is a step function, that is,
(7)Q(t)={1fort∈[0,τ),0fort∈[τ,+∞),w0(t)=0 for t≥τ. Thus, for t≥τ the integral equation (2) becomes
(8)w(t)=∫t-τt‍βx(θ)v(θ)e-d(t-θ)dθ.
It is equivalent to the following delay differential equation:
(9)w′(t)=βx(t)v(t)-βe-dτx(t-τ)v(t-τ)-dw(t),
with w(0)=∫-τ0‍βx(θ)v(θ)edθdθ, 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
(10)x′=λ-dx-βxv,w′=βxv-βe-dτx(t-τ)v(t-τ)-dw,y′=βe-dτx(t-τ)v(t-τ)-ay,v′=kay-γv.
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:
(11)x1′=λ1-d1x1-β1x1v,w1(t)=w10(t)+∫0t‍β1x1(θ)v(θ)Q1(t-θ)e-d1(t-θ)dθ,y1′=F1(x1,y1,w1)-a1y1,x2′=λ2-d2x2-β2x2v,w2(t)=w20(t)+∫0t‍β2x2(θ)v(θ)Q2(t-θ)e-d2(t-θ)dθ,y2′=F2(x2,y2,w2)-a2y2,v′=k1a1y1+k2a2y2-γv.Similarly, when the probability functions in (11) are exponential  function, model (11) can become
(12)x1′=λ1-d1x1-β1x1v,w1′=β1x1v-(d1+ε1)w1,y1′=ε1w1-a1y1,x2′=λ2-d2x2-β2x2v,w2′=β2x2v-(d2+ε2)w2,y2′=ε2w2-a2y2,v′=k1a1y1+k2a2y2-γv.
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
(13)x1′(t)=λ1-d1x1(t)-β1x1(t)v(t),w1′(t)=β1x1(t)v(t)-β1e-d1τ1x1(t-τ1)×v(t-τ1)-d1w1(t),y1′(t)=β1e-d1τ1x1(t-τ1)v(t-τ1)-a1y1(t),x2′(t)=λ2-d2x2(t)-β2x2(t)v(t),w2′(t)=β2x2(t)v(t)-β2e-d2τ2x2(t-τ)×v(t-τ2)-d2w2(t),y2′=β2e-d2τ2x2(t-τ2)v(t-τ2)-a2y2(t),v′=k1a1y1(t)+k2a2y2(t)-γv(t).
For system (13), variables w1 and w2 do not appear in the equations of xi, yi (i=1,2), and v, then denoting b1=e-d1τ1, and b2=e-d2τ2, gives a subsystem of (13) as follows:
(14)x1′(t)=λ1-d1x1(t)-β1x1(t)v(t),y1′(t)=β1b1x1(t-τ1)v(t-τ1)-a1y1(t),x2′(t)=λ2-d2x2(t)-β2x2(t)v(t),y2′=β2b2x2(t-τ2)v(t-τ2)-a2y2(t),v′=k1a1y1(t)+k2a2y2(t)-γv(t).
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];ℝ+5). From the biological meaning, the initial conditions for system (14) are given as follows:
(15)x1(θ)=ϕ1(θ),y1(θ)=ϕ2(θ),x2(θ)=ϕ3(θ),y2(θ)=ϕ4(θ),v(θ)=ϕ5(θ),θ∈[-τ,0],
where (ϕ1(θ),ϕ2(θ),ϕ3(θ),ϕ4(θ),ϕ5(θ))T∈C+ and ϕi(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 functionalL01=b1x1(t)+y1(t+τ1) then from the first two equations of (14), we have
(16)L01′=b1λ1-[b1d1x1(t)+a1y1(t+τ1)]≤b1λ1-ρ1L01,
where ρ1=min{d1,a1}. It follows that limsupt→+∞[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, fort>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
(17)v′(t)<k1a1(b1λ1ρ1+ε)+k2a2(b2λ2ρ2+ε)-γv(t),
for t>T. It implies that
(18)limsupt→+∞v(t)≤1γ[k1a1(b1λ1ρ1+ε)+k2a2(b2λ2ρ2+ε)].
Summarizing the above inference, Theorem1 holds.Since the positive numberε in the proof of Theorem 1 is arbitrary, we can know that the set
(19)Ω={biλiρixT(t)∈C+:bixi(t)+yi(t+τi)≤biλiρi,i=1,2,v(t)≤ρ-}
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*(x1*,y1*,x2*,y2*,v*)(v*>0) is determined by the following equations:
(20)λ1-d1x1-β1x1v=0,β1b1x1v-a1y1=0,λ2-d2x2-β2x2v=0,β2b2x2v-a2y2=0,k1a1y1+k2a2y2-γv=0.From the first and third equations of (20), we have
(21)x1=λ1d1+β1v,x2=λ2d2+β2v,
respectively. Substituting them into the second and fourth equations of (20) yields
(22)y1=β1b1λ1va1(d1+β1v),y2=β2b2λ2va2(d2+β2v),
respectively. When v≠0, substituting the above y1 and y2 into the last equation of (20) gives
(23)k1β1b1λ1d1+β1v+k2β2b2λ2d2+β2v=γ.Since the function ofv 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(24)R0=(k1β1b1λ1d1+k2β2b2λ2d2)1γ;
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*(x1*,y1*,x2*,y2*,v*), where
(25)x1*=λ1d1+β1v*,x2*=λ2d2+β2v*,y1*=β1b1λ1v*a1(d1+β1v*),y2*=β2b2λ2v*a2(d2+β2v*),
and v* is determined by (23).Note thatR0 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 equilibriumE*, we first introduce an inequality as lemma, which was proved in [18].Lemma 3.
Forn positive numbers ci (i=1,2,…,n), the inequality
(26)n-c1-c2-⋯-cn+ln(c1c2⋯cn)≤0
is true, and the equality holds if and only if c1=c2=⋯=cn=1.Theorem 4.
WhenR0≤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 equilibriumE0 of (14), we define a Lyapunov function:
(27)L11=k1b1(x1-x10-x10lnx1x10)+k1y1+k2b2(x2-x20-x20lnx2x20)+k2y2+v,
then
(28)dL11dt=k1b1(x1-x10)(λ1x1-d1-β1v)+k1[β1b1x(t-τ1)v(t-τ1)-a1y1]+k2b2(x2-x20)(λ2x2-d2-β2v)+k2[β2b2x(t-τ2)v(t-τ2)-a2y2]+k1a1y1+k2a2y2-γv=k1b1(x1-x10)[λ1(1x1-1x10)-β1v]+k1[β1b1x(t-τ1)v(t-τ1)-a1y1]+k2b2(x2-x20)[λ2(1x2-1x20)-β2v]+k2[β2b2x(t-τ2)v(t-τ2)-a2y2]+k1a1y1+k2a2y2-γv=k1b1λ1(2-x1x10-x10x1)+k2b2λ2(2-x2x20-x20x2)+k1b1β1[x1(t-τ1)v(t-τ1)-x1(t)v(t)]+k2b2β2[x2(t-τ2)v(t-τ2)-x2(t)v(t)]+(k1b1β1x10+k2b2β2x20-γ)v.
Let L1=L11+k1b1β1∫t-τ1t‍x1(θ)v(θ)dθ+k2b2β2∫t-τ2t‍x2(θ)v(θ)dθ; then
(29)dL1dt=k1b1λ1(2-x1x10-x10x1)+k2b2λ2(2-x2x20-x20x2)+γ(R0-1)v,
where R0=(k1b1β1x10+k2b2β2x20)/γ is used.
It is easy to see that2≤xi/xi0+xi0/xi (i=1,2), then dL1/dt≤0 as R0≤1. Note that, for R0<1, dL1/dt=0 if and only if xi=xi0 (i=1,2) and v=0; for R0=1, dL1/dt=0 if and only if xi=xi0 (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 equilibriumE*(x1*,y1*,x2*,y2*,v*), define the following Lyapunov functions and functionals:
(30)L21=k1b1(x1-x1*-x1*lnx1x1*)+k1(y1-y1*-y1*lny1y1*),L22=k2b2(x2-x2*-x2*lnx2x2*)+k2(y2-y2*-y2*lny2y2*),L23=(v-v*-v*lnvv*),L24=k1b1β1x1*v*×∫t-τ1t‍[x1(θ)v(θ)x1*v*-1-lnx1(θ)v(θ)x1*v*]dθ+k2b2β2x2*v*×∫t-τ2t‍[x2(θ)v(θ)x2*v*-1-lnx2(θ)v(θ)x2*v*]dθ.
According to (20), we have
(31)d1=λ1x1*-β1v*,a1=β1b1x1*v*y1*,d2=λ2x2*-β2v*,a2=β2b2x2*v*y2*,γ=k1a1y1*v*+k2a2y2*v*.
Then, system (14) can be rewritten as
(32)x1′(t)=x1(t)[λ1(1x1(t)-1x1*)-β1(v(t)-v*)],y1′(t)=β1b1y1(t)[x1(t-τ1)v(t-τ1)y1(t)-x1*v*y1*],x2′(t)=x2(t)[λ2(1x2(t)-1x2*)-β2(v(t)-v*)],y2′(t)=β2b2y2(t)[x2(t-τ2)v(t-τ2)y2(t)-x2*v*y2*],v′(t)=v(t)[k1a1(y1(t)v(t)-y1*v*)+k2a2(y2(t)v(t)-y2*v*)].
By usingλ1=d1x1*+β1x1*v* and λ2=d2x2*+β2x2*v*, direct computations show that
(33)L21′=k1b1d1x1*(2-x1*x1-x1x1*)+k1b1β1x1*v*(2-x1*x1-x1vx1*v*+vv*)+k1b1β1x1*v*[x1(t-τ1)v(t-τ1)x1*v*-y1y1*-x1(t-τ1)y1*v(t-τ1)x1*y1v*],L22′=k2b2d2x2*(2-x2*x2-x2x2*)+k2b2β2x2*v*(2-x2*x2-x2vx2*v*+vv*)+k2b2β2x2*v*[x2(t-τ2)v(t-τ2)x2*v*-y2y2*-x2(t-τ2)y2*v(t-τ2)x2*y2v*],L23′=k1a1y1*(y1y1*-y1v*y1*v-vv*+1)+k2a2y2*(y2y2*-y2v*y2*v-vv*+1),L24′=k1b1β1x1*v*[x1vx1*v*-x1(t-τ1)v(t-τ1)x1*v*+lnx1(t-τ1)v(t-τ1)x1(t)v(t)]+k2b2β2x2*v*[x2vx2*v*-x2(t-τ2)v(t-τ2)x2*v*+lnx2(t-τ2)v(t-τ2)x2(t)v(t)].
Let L2=L21+L22+L23+L24, then
(34)L2′=k1b1d1x1*(2-x1*x1-x1x1*)+k2b2d2x2*(2-x2*x2-x2x2*)+k1b1β1x1*v*[3-x1*x1(t)-x1(t-τ1)y1*v(t-τ1)x1*y1v*-y1(t)v*y1*v(t)+lnx1(t-τ1)v(t-τ1)x1(t)v(t)]+k2b2β2x2*v*[3-x2*x2(t)-x2(t-τ2)y2*v(t-τ2)x2*y2v*-y2(t)v*y2*v(t)+lnx2(t-τ2)v(t-τ2)x2(t)v(t)].
According to the relationship between the arithmetic and the associated geometric means and Lemma3, dL2/dt≤0 and dL2/dt=0 if and only if x1=x1*, x2=x2*, y1/y1* = y2/y2* = 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
(35)R01(O)=βλkεdγ(d+ε),R02(O)=[β1λ1k1ε1d1(d1+ε1)+β2λ2k2ε2d2(d2+ε2)]1γ,
respectively [6, 7]. For delay differential systems (10) and (13) (or (14)), the basic reproduction numbers are
(36)R01(D)=βλke-dτdγ,R02(D)=[β1λ1k1e-d1τ1d1+β2λ2k2e-d2τ2d2]1γ,
respectively, where R01(D) was obtained in [17]; R02(D) 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
(37)R01(O)=βλkdγ(1+dτ),R02(O)=[β1λ1k1d1(1+d1τ1)+β2λ2k2d2(1+d2τ2)]1γ,
respectively.From the inequality1/(1+u)>e-u for u>0, it follows that
(38)R01(O)>R01(D),R02(O)>R02(D).
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.



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