# Global Stability for a Viral Infection Model with Saturated Incidence Rate

Authors: Huaqin Peng; Zhiming Guo
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

A viral infection model with saturated incidence rate and viral infection with delay is derived and analyzed; the incidence rate is assumed
to be a specific nonlinear formβxv/(1+αv). The existence and uniqueness of equilibrium are proved. The basic reproductive number R0 is given. The model
is divided into two cases: with or without delay. In each case, by constructing
Lyapunov functionals, necessary and sufficient conditions are given to ensure
the global stability of the models.


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## Body

## 1. Introduction

In recent years, study of infectious disease model has been a hot issue; the main cause of infectious disease is the virus invasion. As we know, viral cytopathicity within target cells is very common. A number of mathematical models have been used to study virus dynamics. In 1996, Nowak et al. [1] designed a simple but natural mathematical model based on ordinary differential equation. The model is as follows:
(1)dx(t)dt=λ-dx(t)-βx(t)v(t),dy(t)dt=βx(t)v(t)-ay(t),dv(t)dt=κy(t)-γv(t),
where x(t) denotes the number of uninfected cells, y(t) the numbers of infected cells, and v(t) the numbers of free viral particles at time t, respectively. In model (1), uninfected target cells are assumed to be produced at a constant rate λ and died at rate dx. Infection of target cells by in-host free viruses is assumed to occur at a bilinear rate βxv; infected cells are lost at a rate ay. Free viruses are produced by infected cells at a rate κy, in which κ is the average number of viral particles produced over the lifetime of a single infected cell. Free viral particles die at a rate γv. For model (1), Korobeinikov [2] established the condition of global stability in 2004. Some other viral dynamical models were proposed by later researchers; see for example [3–8].In [7], Wodarz and Levy pointed out that the term ay(t) in model (1) should consist of two parts: one is the natural death of infected cells, the other is viral cytopathicity. In 2012, Li et al. [4] assumed that infected cells burst and then release viral particles (i.e., viral cytopathicity occurs) after uninfected cells were infected by a constant period of time τ; that is, the time period of viral cytopathicity within target cells is τ. They incorporated the delay of viral cytopathicity within target cells and built a new model:
(2)dx(t)dt=λ-dx(t)-βx(t)v(t),dy(t)dt=βx(t)v(t)-βe-dτx(t-τ)v(t-τ)-dy(t),dv(t)dt=κβe-dτx(t-τ)v(t-τ)-γv(t).
By constructing Lyapunov functionals, necessary and sufficient conditions were obtained ensuring the global stability of the model.In models (1) and (2), the researcher studied the viral dynamics with bilinear incidence rate βxv. As we know, as the viral particles diffuse in the body, the person often takes some actions when v gets large. In order to describe the inhibitory effect from the uninfected cells when the number of viral cytopathicity is large enough, following the idea of [9], we propose an incidence rate βxv/(1+αv), where βv measures the infection force of the viral, and α reflects the level of inhibitory action.Similar to the discussions in [4], we assume that the viral cytopathicity has time delay. When the delay of viral cytopathicity within target cells is τ and the natural death rate of per target cell is d, the number of infected cells at time t(t>τ) can be represented by
(3)y(t)=∫t-τt‍βx(θ)v(θ)1+αv(θ)e-d(t-θ)dθ,fort>τ,
where e-d(t-θ) is the probability that target cells survive from time θ to time t, and (βx(θ)v(θ)/(1+αv(θ)))e-d(t-θ) is the number of target cells being infected at time θ and still surviving at time t.Differentiatingy(t) of (3), we get
(4)dy(t)dt=-d∫t-τt‍βx(θ)v(θ)1+αv(θ)e-d(t-θ)dθ+βx(t)v(t)1+αv(t)-e-dτβx(t-τ)v(t-τ)1+αv(t-τ)=βx(t)v(t)1+αv(t)-βe-dτx(t-τ)v(t-τ)1+αv(t-τ)-dy(t),
where βe-dτ(x(t-τ)v(t-τ)/(1+αv(t-τ))) is the transfer rate of the infected cells being used to produce free viruses at time t; the recruitment rate of free virus at time t is κβe-dτ(x(t-τ)v(t-τ)/(1+αv(t-τ))), in which κ is the average number of viral particles produced by an infected target cell when viral cytopathicity occurs, which implies that the recruitment of virus at time t depends on the number of target cells that were newly infected at time t-τ and still alive at time t. Therefore following the model (2), we obtain a basic viral dynamical model of delay differential equations:
(5)dx(t)dt=λ-dx(t)-βx(t)v(t)1+αv(t),dy(t)dt=βx(t)v(t)1+αv(t)-βe-dτx(t-τ)v(t-τ)1+αv(t-τ)-dy(t),dv(t)dt=κβe-dτx(t-τ)v(t-τ)1+αv(t-τ)-γv(t).Since the variabley does not appear in the first and the third equations of (5), we only focus on the following equations:
(6)dx(t)dt=λ-dx(t)-βx(t)v(t)1+αv(t),dv(t)dt=κβe-dτx(t-τ)v(t-τ)1+αv(t-τ)-γv(t),
which has the same dynamics with system (5).Letb=κe-dτ, by (6), we have
(7)dx(t)dt=λ-dx(t)-βx(t)v(t)1+αv(t),dv(t)dt=βbx(t-τ)v(t-τ)1+αv(t-τ)-γv(t),
where all the parameters are assumed to be positive.The rest of this paper is organized as follows. In the next section we will derive the infection-free equilibrium and the infection equilibrium. In Section3, we carry out a qualitative analysis of the model, and stability conditions for the infection-free equilibrium and the infection equilibrium are derived, respectively. A brief conclusion will be given in Section 4.


## 2. Positive Solutions and Equilibria

Due to the biological meaning of the components(x(t),v(t)), we consider system (7) with the following initial conditions:
(8)dx(t)dt=λ-dx(t)-βx(t)v(t)1+αv(t),dv(t)dt=βbx(t-τ)v(t-τ)1+αv(t-τ)-γv(t),x(θ)=φ1(θ)≥0,φ1(0)>0,θ∈[-τ,0],v(θ)=φ2(θ)≥0,φ2(0)>0,θ∈[-τ,0].
Equation (8) is a system of retarded differential equations in C=C([-τ,0],ℝ2). C is a Banach space of continuous mappings from [-τ,0] into ℝ2 with norm ∥ψ∥=sup⁡-τ≤θ≤0⁡|ψ(θ)| for ψ∈C. We denote
(9)C+={(φ1,φ2)∈C∣φ1(0)>0,φ2(0)>0,φ1(θ)≥0,φ2(θ)≥0,θ∈[-τ,0]}.
As usual, for any continuous function x∈C([-τ,+∞),ℝ) and any given t≥0, xt is defined as xt∈C([-τ,0]ℝ), xt(θ)=x(t+θ), for any θ∈[-τ,0].Theorem 1.
All the solutions(x(t),v(t))T of (8) under the initial conditions are positive on [0,∞).Proof.
Assume that there is at1(t1>0) such that x(t1)=0; then by x(0)>0 and the continuity of x, there is t*=inf⁡{t>0,x(t)=0}>0 such that x(t)>0 for t∈[0,t*). Then we have x′(t*)≤0. However, x′(t*)=λ>0 by the first equation of (8); this is a contradiction. Therefore x(t)>0 for all t>0.
From the second equation of (8)
(10)dv(t)dt=βbx(t-τ)v(t-τ)1+αv(t-τ)-γv(t).
Multiplying eγt in both sides of the above equation and integrating it from 0 to t, we have
(11)∫0t‍eγθdv(θ)+∫0t‍eγθγv(θ)dθ=∫0t‍eγθβbx(θ-τ)v(θ-τ)1+αv(θ-τ)dθ,eγtv(t)-v(0)=∫0t‍eγθβbx(θ-τ)v(θ-τ)1+αv(θ-τ)dθ,v(t)=[v(0)+∫0t‍eγθβbx(θ-τ)v(θ-τ)1+αv(θ-τ)dθ]e-γt.
Let η=θ-τ(12)v(t)=[v(0)+∫-τt-τ‍eγ(η+τ)βbx(η)v(η)1+αv(η)dη]e-γt.
Since x(t)≥0, v(t)≥0, and v(0)>0 for -τ≤t≤0, then v(t)>0 for 0≤t<τ.
Further, whenτ≤t<2τ, we have
(13)v(t)={v(0)+∫-τ0eγ(η+τ)βbx(η)v(η)1+αv(η)dη+∫0t-τeγ(η+τ)βbx(η)v(η)1+αv(η)dη}e-γt.
By the fact that x(η)≥0 and v(η)≥0 for -τ≤η<0, then
(14)∫-τ0‍eγ(η+τ)βbx(η)v(η)1+αv(η)dη≥0.
Also x(η)>0 and v(η)>0 for 0≤η<τ; then
(15)∫0t-τ‍eγ(η+τ)βbx(η)v(η)1+αv(η)dη>0.
Consequently, v(t)>0 for τ≤t<2τ, which implies that v(t)>0 holds true for 0≤t<2τ.
We assume that for a positive integerk, v(t)>0 for 0≤t<kτ. When kτ≤t<(k+1)τ, we have
(16)v(t)={v(0)+∫-τ(k-1)τeγ(η+τ)βbx(η)v(η)1+αv(η)dη+∫(k-1)τt-τeγ(η+τ)βbx(η)v(η)1+αv(η)dη}e-γt.
Then similar discussions show that v(t)>0 for kτ≤t<(k+1)τ. Hence, v(t)>0 for all t>0.Theorem 2.
All the solutions(x(t),v(t))T of (8) under the initial conditions are ultimately bounded.Proof.
For any solution(x(t),v(t))T of (8), define a function f(t)=bx(t-τ)+v(t). Then the derivative of f(t) is
(17)df(t)dt=bdx(t-τ)dt+dv(t)dt=bλ-bdx(t-τ)-βbx(t-τ)v(t-τ)1+αv(t-τ)+βbx(t-τ)v(t-τ)1+αv(t-τ)-γv(t)=bλ-bdx(t-τ)-γv(t)≤bλ-ρf(t),
where ρ=min⁡{d,γ}.
Integrating both sides of inequality above from0 to t, we have
(18)f(t)≤bλρ+(f(0)-bλρ)e-ρt.
It means that bxt(-τ)+vt(0)≤bλ/ρ for any t≥0 as long as bφ1(-τ)+φ2(0)≤bλ/ρ. Also,
(19)limsup⁡t→∞f(t)≤bλρ.
From the first equation of (8), we have
(20)dx(t)dt≤λ-dx(t).
Similar discussion shows that
(21)x(t)≤λd+(x(0)-λd)e-dt.
Then xt(0)≤λ/d for any t≥0 as long as φ1(0)≤λ/d. Moreover,
(22)limsup⁡t→∞x(t)≤λd.
Thus, the region
(23)Ω={(φ1,φ2)∈C+:φ1(0)≤λd,bφ1(-τ)+φ2(0)≤bλρ}
is an invariant set and an attractor of system (8) with initial condition (φ1,φ2)∈C+.In what follows, we study the existence of equilibria.We consider algebraic equations(24)λ-dx-βxv1+αv=0,βbxv1+αv-γv=0.
It is easy to see that system (24) always has an infection-free equilibrium E1(λ/d,0). To find the other equilibrium, we assume v≠0. By the second equation of (24), we have
(25)βbx1+αv=γ.Then(26)x=γ(1+αv)βb.We putx into the first equation of (24); then
(27)λ-dγ(1+αv)βb-βγ(1+αv)vβb(1+αv)=0,βbλ-dγ=(dγα+βγ)v.
Thus the positive root E2 exists if and only if βbλ-dγ>0. For system (8), define the basic reproduction number [10] as follows
(28)R0=βbλdγ.It is easy to see that(i)
IfR0≤1, then system (8) has a unique equilibrium E1(λ/d,0), which corresponds to the case that viruses die out, and it is called infection-free equilibrium.(ii)
IfR0>1, then system (8) has two equilibria, one is the infection-free equilibrium E1(λ/d,0) and the other is a positive equilibrium E2((γ+αbλ)/b(dα+β),d(R0-1)/(dα+β)).


## 3. Stability of the Equilibrium

In this section, we consider the stability of the equilibrium. There are two cases,τ=0 and τ>0.


### 3.1. Local Stability of Equilibria

First we consider the case ofτ=0. In this case system (8) is reduced to a system of ordinary differential equations. In order to examine local stability of an equilibrium, we should compute the eigenvalues of the linearized operator for system (8) at the equilibrium.By a direct computation, the Jacobian matrix is as follows:(29)(-d-βv1+αv,-βx(1+αv)2βbv1+αv,βbx(1+αv)2-γ).
Consider infection-free equilibrium E1(λ/d,0). The characteristic equation is obtained by the standard method as follows.It is obvious thatμ1=-d<0, and μ2=βbλ/d-γ=γ(R0-1) are the characteristic roots of the characteristic equation. Therefore, we have the following theorem.Theorem 3.
(i) IfR0<1, then infection-free equilibrium E1(λ/d,0) is locally asymptotically stable.
(ii) IfR0>1, then infection-free equilibrium E1(λ/d,0) is unstable.
(iii) IfR0=1, then infection-free equilibrium E1(λ/d,0) is degenerated.Now, local stability of the infection equilibriumE2((γ+αbλ)/b(dα+β),d(R0-1)/(dα+β)) is considered. As we know, infection equilibrium E2((γ+αbλ)/b(dα+β), d(R0-1)/(dα+β)) exists if and only if R0>1.Theorem 4.
IfR0>1, then the infection equilibrium E2 is locally asymptotically stable.Proof.
Setf(x,v)=βxv/(1+αv). Then system (8) at the equilibrium E2(x*,v*) has Jacobian matrix
(30)A=(-d-∂f∂x(x*,v*),-∂f∂v(x*,v*)b∂f∂x(x*,v*),b∂f∂v(x*,v*)-γ).
A direct computation shows that the characteristic equation is
(31)h(μ)=μ2+[d+γ+∂f∂x(x*,v*)-b∂f∂v(x*,v*)]μ+dγ-bd∂f∂v(x*,v*)+γ∂f∂x(x*,v*)=0.
By Hurwitz criterion, all of the eigenvalues of characteristic equation have negative real parts if and only if
(32)d+γ+∂f∂x(x*,v*)-b∂f∂v(x*,v*)>0,dγ-bd∂f∂v(x*,v*)+γ∂f∂x(x*,v*)>0.
Indeed,
(33)d+γ+∂f∂x(x*,v*)-b∂f∂v(x*,v*)=(d+γ)(1+αv*)2+βv*(1+αv*)-bβx*(1+αv*)2=(α2d+α2γ+αβ)v*2+(2αd+2γα+β)v*-bβx*+d+γ(1+αv*)2=(α2d+α2γ+αβ)v*2+(2αd+γα+β)v*+d(1+αv*)2>0,dγ-bd∂f∂v(x*,v*)+γ∂f∂x(x*,v*)=dγ(1+αv*)2-bdβx*+γβv*+γβαv*2(1+αv*)2=(α2dγ+αγβ)v*2+(dαγ+γβ)v*(1+αv*)2>0.
This implies that all the eigenvalues of characteristic equation have negative real parts. Then the infection equilibrium E2 is locally asymptotically stable. This completes the proof of theorem.Now we consider the caseτ>0. By linearizing system (8) at the infection-free equilibrium E1(λ/d,0), we obtain the characteristic equation as follows:
(34)(μ+d)(μ+γ-e-μτβbλd)=0.It is easy to see thatμ1=-d<0; hence we only need to discuss the roots of the following equation:
(35)h(μ,τ)=μ+γ-e-μτβbλd=0.Theorem 5.
Whenτ>0, then(i)
IfR0<1, then the infection-free equilibrium E1(λ/d,0) is locally asymptotically stable.(ii)
IfR0>1, then the infection-free equilibrium E1(λ/d,0) is unstable.(iii)
IfR0=1, then the infection-free equilibrium E1(λ/d,0) is degenerated.Proof.
(i) By implicit function theorem for complex variables, we know that the roots of (35) are continuous on the parameter τ.
IfR0<1, then 0 is not a root of (35) for all τ>0. Note that all complex roots of (35) must come in conjugate pairs and the root of (35) is negative for τ=0. Thus, all roots of (35) have negative real parts for small τ; that is, 0<τ=1. Suppose that there exists a positive number τ=τ0 such that (35) has a pair of purely imaginary roots λ=±ωi; here ω is a positive number. We have
(36)ωi+γ-e-ωiτ0βbλd=0.
Then
(37)βbλdcos⁡ωτ0=γ,βbλdsinωτ0=-ω.
Summing up the square of both equations in (37) we obtain
(38)ω2=β2b2λ2d2-γ2=β2b2λ2-γ2d2d2=R02d2γ2-γ2d2d2=γ2(R02-1).
When R0<1, then ω2<0. It is a contradiction with ω2>0 which leads to the nonexistence of τ0. This contradiction proves the result.
(ii) Whenμ=0, and R0>1, then
(39)h(0,τ)=γ-βbλd=γ(1-R0)<0,lim⁡μ→∞h(μ,τ)=+∞.
Therefore equation must have a positive real root for all τ>0.
(iii) IfR0=1, it is easy to know that μ=0 is a root of (35) for all τ>0, which leads to conclusion. (iii) This completes the proof of theorem.Now we consider the local stability of the infection equilibriumE2((γ+αbλ)/b(dα+β),d(R0-1)/(dα+β)). As we know, the infection equilibrium E2((γ+αbλ)/b(dα+β),d(R0-1)/(dα+β)) exists if and only if R0>1. By computation, the associated transcendental characteristic equation of (8) at E2 becomes
(40)μ2+Aμ+B-(Cμ+D)e-μτ=0,
where
(41)A=d+γ+∂f∂x(x*,v*),B=(d+∂f∂x(x*,v*))γ,C=b∂f∂v(x*,v*),D=db∂f∂v(x*,v*).Theorem 6.
Whenτ>0, if R0>1, then the infection equilibrium E2 is locally asymptotically stable.Proof.
By implicit function theorem for complex variables, we know that the root of (40) is continuous on the parameter τ. If R0>1, then all roots of (40) have negative real parts as τ=0 and (40) has no zero root for all τ>0. Thus, all roots of (40) have negative real parts for very small τ; that is, 0<τ≪1. Assume that there exists a positive τ0 such that (40) has a pair of purely imaginary roots ±ωi, ω>0. Then ω>0 must satisfy
(42)-ω2+Aωi+B-(Cωi+D)(cos⁡ωτ0-isinωτ0)=0.
Separating the real and imaginary parts, we have
(43)Cωsinωτ0+Dcos⁡ωτ0=B-ω2,Dsinωτ0-Cωcos⁡ωτ0=-Aω
which implies that
(44)ω4+(A2-2B-C2)ω2+B2-D2=0.
Direct computation shows that
(45)B2-D2=[γ(d+∂f∂x(x*,v*))]2-[db∂f∂v(x*,v*)]2=[γ∂f∂x(x*,v*)+γd-db∂f∂v(x*,v*)]×[γ∂f∂x(x*,v*)+γd+db∂f∂v(x*,v*)]>0,A2-2B-C2=d2+γ2+(∂f∂x(x*,v*))2+2d∂f∂x(x*,v*)-b2(∂f∂v(x*,v*))2=(d+∂f∂x(x*,v*))2-b2(∂f∂v(x*,v*))2+γ2=((d2+γ2)(1+αv*)4+β2(v*)2(1+αv*)2+2dβv*(1+αv*)3-b2β2(x*)2)×((1+αv*)4)-1.
Let
(46)H(x*,v*)=(d2+γ2)(1+αv*)4+β2(v*)2(1+αv*)2+2dβv*(1+αv*)3-b2β2(x*)2.
Note that 1+αv*=(βb/γ)x*,
(47)H(x*,v*)=(d2+γ2)β4b4γ4(x*)4+β2(v*)2β2b2γ2(x*)2+2dβv*β3b3γ3(x*)3-β2b2(x*)2=(x*)2[β4b2γ4(b2d2(x*)2+γ2(v*)2+2γdbv*x*)+β4b4γ2(x*)2-b2β2]=(x*)2[β4b2γ4(bdx*+γv*)2+β2b2γ2(β2b2(x*)2-γ2)]=(x*)2[β4b2γ4(bdx*+γv*)2+β2b2(α2(v*)2+2αv*)]>0.
By Hurwitz criterion, (44) has no positive roots, which implies the nonexistence of τ0. Thus all roots of (40) have negative real parts for τ>0.


### 3.2. Global Stability of Equilibria

In the section, we study the global stability of equilibria; we first consider the infection-free equilibriumE1.Theorem 7.
Whenτ=0,(i)
IfR0≤1, then infection-free equilibrium E1(λ/d,0) is globally asymptotically stable.(ii)
IfR0>1, then infection equilibrium E2(x*,v*) is globally asymptotically stable.Proof.
(i) Define a Lyapunov function as what follows(48)V(x,v)=x-λdln⁡x+vb+λdln⁡λd-λd,dV(x,v)dt|(8)=x-(λ/d)x(λ-dx-βxv1+αv)+1b(βbxv1+αv-γv)=-(dx-λ)2dx+βxv1+αv(1-dx-λdx)-γbv=-(dx-λ)2dx+(βλd(1+αv)-γb)v=-(dx-λ)2dx+γb(R0-1)v-βγdαv21+αv.
It means that dV(x,v)/dt|(8) is negative semidefinite as R0≤1. Moreover, the last equality of the above equation shows that the largest invariant set of system (8) on the region {(x,v)T∈ℝ+2:dV/dt=0} is the singleton {E1}. Therefore, the infection-free equilibrium E1 is global asymptotically stability.
(ii) We rewrite the system (8)
(49)dx(t)dt=λ-dx(t)-βx(t)v(t)1+αv(t)=Q(x,v),dv(t)dt=βbx(t)v(t)1+αv(t)-γv(t)=P(x,v).
Choose a Dulac function
(50)D(x,v)=1+αvβv.
We have
(51)∂(DP)∂v+∂(DQ)∂x=-γαβ-d1+αvβv-1<0.
Thus system (49) does not have nontrivial periodic orbits in Ω. The conclusion follows.Theorem 8.
Whenτ>0, if R0≤1, then infection-free equilibrium E1(λ/d,0) is globally asymptotically stable.Proof.
Define a functionalg:C+→ℝ as follows:
(52)g(φ1,φ2)=12[φ1(0)-λd]2+λd[φ2(0)+βb∫-τ0‍φ1(θ)φ2(θ)1+αφ2(θ)dθ].
For any t≥0, xt,vt∈C, then
(53)g(xt,vt)=b2[x(t)-λd]2+λd[v(t)+βb∫t-τt‍x(θ)v(θ)1+αv(θ)dθ],dg(xt,vt)dt=b(x(t)-λd)dx(t)dt+λddv(t)dt+βbλd[x(t)v(t)1+αv(t)-x(t-τ)v(t-τ)1+αv(t-τ)]=b[x(t)-λd][λ-dx(t)-βx(t)v(t)1+αv(t)]+λd[βbx(t)v(t)1+αv(t)-γv(t)]=-bd[x(t)-λd]2-βbx(t)v(t)1+αv(t)[x(t)-λd]+λd[βbx(t)v(t)1+αv(t)-γv(t)]=-b(d+βv(t)1+αv(t))[x(t)-λd]2+λv(t)d(βbλd(1+αv(t))-γ)=-b(d+βv(t)1+αv(t))[x(t)-λd]2+λv(t)γd(R01+αv(t)-1)=-b(d+βv(t)1+αv(t))[x(t)-λd]2+λv(t)γd(R0-1)-αv(t)R01+αv(t)≤-b(d+βv(t)1+αv(t))[x(t)-λd]2+λv(t)γd(R0-1).
Then,
(54)dg(φ1,φ2)dt|(8)=-b(d+βφ2(0)1+αφ2(0))[φ1(0)-λd]2+λφ2(0)γd(R0-1)-αφ2(0)R01+αφ2(0).
When R0≤1, we have dg(xt,vt)/dt≤0. It is easy to know, when R0≤1 the largest invariant set of system (8) on the region {(φ1,φ2)∈C+:dg(φ1,φ2)/dt=0} is the singleton {E1}. By Lassalle invariant principle for autonomous retarded differential equations [11], infection-free equilibrium E1 is globally asymptotically stable. This completes the proof.Theorem 9.
Whenτ>0, if R0>1, then the infection equilibrium E2(x*,v*) is globally asymptotically stable.Proof.
LetV(t)=v(t+τ) and X(t)=x(t); system (8) becomes
(55)dX(t)dt=λ-dX(t)-βX(t)V(t-τ)1+αV(t-τ),dV(t)dt=βbX(t)V(t-τ)1+αV(t-τ)-γV(t).
Denoteu(x)=x/(1+αx). Evaluating both sides of (55) at E2, we obtain
(56)λ=dx*+βx*u(v*),γv*=βbx*u(v*).
Define a Lyapunov functional L:C+→ℝ as follows:
(57)L(φ1,φ2)=1βu(v*)L1(φ1,φ2)+v*βbx*u(v*)L2(φ1,φ2)+L3(φ1,φ2),
where
(58)L1(φ1,φ2)=φ1(0)x*-1-ln⁡φ1(0)x*,L2(φ1,φ2)=φ2(0)v*-1-ln⁡φ2(0)v*,L3(φ1,φ2)=∫-τ0‍(φ2(θ)v*-1-ln⁡φ2(θ)v*)dθ.
Thus, L(φ1,φ2)≥0 with equality if and only if φ1(0)/x*=φ2(0)/v*=1.
For anyt≥0, Xt,Vt∈C, then
(59)L1(Xt,Vt)=X(t)x*-1-ln⁡X(t)x*,L2(Xt,Vt)=V(t)v*-1-ln⁡V(t)v*,L3(Xt,Vt)=∫-τ0‍(V(t+θ)v*-1-ln⁡V(t+θ)v*)dθ.
We calculate derivatives ofL1(Xt,Vt), L2(Xt,Vt), and L3(Xt,Vt) with respect to the system (55):
(60)dL1(Xt,Vt)dt=1x*(1-x*X)(λ-dX-βXu(V(t-τ)))=1x*(1-x*X)(dx*+βx*u(v*)-dX-βXu(V(t-τ)))=-d(X-x*)2Xx*+βu(v*)(1-x*X)(1-Xu(V(t-τ))x*u(v*))=-d(X-x*)2Xx*+βu(v*)(1-x*X+u(V(t-τ))u(v*)-Xu(V(t-τ))x*u(v*)),dL2(Xt,Vt)dt=1v*(1-v*V)(βbXu(V(t-τ))-γV)=1v*(1-v*V)(βbx*u(v*)Xu(V(t-τ))x*u(v*)-γVv*v*)=βbx*v*u(v*)(1-v*V)(Xx*u(V(t-τ))u(v*)-Vv*)=βbx*v*u(v*)×(1-v*VXx*u(V(t-τ))u(v*)+Xx*u(V(t-τ))u(v*)-Vv*),dL3(Xt,Vt)dt=∫-τ0‍ddt(V(t+θ)v*-1-ln⁡V(t+θ)v*)dθ=V(t)v*-ln⁡V(t)v*-V(t-τ)v*+ln⁡V(t-τ)v*.
We obtain
(61)dL(Xt,Vt)dt=-dβu(v*)(X-x*)2Xx*+1-x*X+u(V(t-τ))u(v*)-Xu(V(t-τ))x*u(v*)+1-v*VXx*u(V(t-τ))u(v*)+Xx*u(V(t-τ))u(v*)-Vv*+Vv*-ln⁡Vv*-V(t-τ)v*+ln⁡V(t-τ)v*=-dβu(v*)(X-x*)2Xx*+2-x*X+u(V(t-τ))u(v*)-v*VXx*u(V(t-τ))u(v*)-ln⁡Vv*-V(t-τ)v*+ln⁡V(t-τ)v*.
By adding and subtracting the quantity ln⁡((X/x*)(u(V(t-τ))/u(v*))), we have
(62)dL(Xt,Vt)dt=-dβu(v*)(X-x*)2Xx*+2-x*X+u(V(t-τ))u(v*)-v*VXx*u(V(t-τ))u(v*)-V(t-τ)v*+ln⁡V(t-τ)v*+ln⁡(Xx*v*Vu(V(t-τ))u(v*))-ln⁡(Xx*)-ln⁡(u(V(t-τ))u(v*))=-dβu(v*)(X-x*)2Xx*-(x*X-1+ln⁡Xx*)-(Xx*v*Vu(V(t-τ))u(v*)-1ii-ln⁡(Xx*v*Vu(V(t-τ))u(v*)))+u(V(t-τ))u(v*)-V(t-τ)v*+ln⁡V(t-τ)v*-ln⁡(u(V(t-τ))u(v*)).
Then,
(63)dL(φ1,φ2)dt=-dβu(v*)(φ1(0)-x*)2φ1(0)x*-(x*φ1(0)-1+ln⁡φ1(0)x*)-(φ1(0)x*v*φ2(0)u(φ2(-τ))u(v*)-1mm-ln⁡(φ1(0)x*v*φ2(0)u(φ2(-τ))u(v*)))+u(φ2(-τ))u(v*)-φ2(-τ)v*+ln⁡φ2(-τ)v*-ln⁡(u(φ2(-τ))u(v*)).
It is easy to know that  u(φ2(-τ))/u(v*)-φ2(-τ)/v*+ln⁡⁡(φ2(-τ)/v*)-ln⁡(u(φ2(-τ))/u(v*))≤0, and u(φ2(-τ))/u(v*)-φ2(-τ)/v*+ln⁡⁡(φ2(-τ)/v*)-ln⁡(u(φ2(-τ))/u(v*))=0 if and only if φ2(-τ)/v*=1. It follows that dL(φ1,φ2)/dt≤0, and dL(φ1,φ2)/dt=0 if and only if φ1(0)/x*=φ2(0)/v*=φ2(-τ)/v*=1. By classical stability theory for functional differential equations, E2 is globally asymptotically stable. This completes the proof.


## 3.1. Local Stability of Equilibria

First we consider the case ofτ=0. In this case system (8) is reduced to a system of ordinary differential equations. In order to examine local stability of an equilibrium, we should compute the eigenvalues of the linearized operator for system (8) at the equilibrium.By a direct computation, the Jacobian matrix is as follows:(29)(-d-βv1+αv,-βx(1+αv)2βbv1+αv,βbx(1+αv)2-γ).
Consider infection-free equilibrium E1(λ/d,0). The characteristic equation is obtained by the standard method as follows.It is obvious thatμ1=-d<0, and μ2=βbλ/d-γ=γ(R0-1) are the characteristic roots of the characteristic equation. Therefore, we have the following theorem.Theorem 3.
(i) IfR0<1, then infection-free equilibrium E1(λ/d,0) is locally asymptotically stable.
(ii) IfR0>1, then infection-free equilibrium E1(λ/d,0) is unstable.
(iii) IfR0=1, then infection-free equilibrium E1(λ/d,0) is degenerated.Now, local stability of the infection equilibriumE2((γ+αbλ)/b(dα+β),d(R0-1)/(dα+β)) is considered. As we know, infection equilibrium E2((γ+αbλ)/b(dα+β), d(R0-1)/(dα+β)) exists if and only if R0>1.Theorem 4.
IfR0>1, then the infection equilibrium E2 is locally asymptotically stable.Proof.
Setf(x,v)=βxv/(1+αv). Then system (8) at the equilibrium E2(x*,v*) has Jacobian matrix
(30)A=(-d-∂f∂x(x*,v*),-∂f∂v(x*,v*)b∂f∂x(x*,v*),b∂f∂v(x*,v*)-γ).
A direct computation shows that the characteristic equation is
(31)h(μ)=μ2+[d+γ+∂f∂x(x*,v*)-b∂f∂v(x*,v*)]μ+dγ-bd∂f∂v(x*,v*)+γ∂f∂x(x*,v*)=0.
By Hurwitz criterion, all of the eigenvalues of characteristic equation have negative real parts if and only if
(32)d+γ+∂f∂x(x*,v*)-b∂f∂v(x*,v*)>0,dγ-bd∂f∂v(x*,v*)+γ∂f∂x(x*,v*)>0.
Indeed,
(33)d+γ+∂f∂x(x*,v*)-b∂f∂v(x*,v*)=(d+γ)(1+αv*)2+βv*(1+αv*)-bβx*(1+αv*)2=(α2d+α2γ+αβ)v*2+(2αd+2γα+β)v*-bβx*+d+γ(1+αv*)2=(α2d+α2γ+αβ)v*2+(2αd+γα+β)v*+d(1+αv*)2>0,dγ-bd∂f∂v(x*,v*)+γ∂f∂x(x*,v*)=dγ(1+αv*)2-bdβx*+γβv*+γβαv*2(1+αv*)2=(α2dγ+αγβ)v*2+(dαγ+γβ)v*(1+αv*)2>0.
This implies that all the eigenvalues of characteristic equation have negative real parts. Then the infection equilibrium E2 is locally asymptotically stable. This completes the proof of theorem.Now we consider the caseτ>0. By linearizing system (8) at the infection-free equilibrium E1(λ/d,0), we obtain the characteristic equation as follows:
(34)(μ+d)(μ+γ-e-μτβbλd)=0.It is easy to see thatμ1=-d<0; hence we only need to discuss the roots of the following equation:
(35)h(μ,τ)=μ+γ-e-μτβbλd=0.Theorem 5.
Whenτ>0, then(i)
IfR0<1, then the infection-free equilibrium E1(λ/d,0) is locally asymptotically stable.(ii)
IfR0>1, then the infection-free equilibrium E1(λ/d,0) is unstable.(iii)
IfR0=1, then the infection-free equilibrium E1(λ/d,0) is degenerated.Proof.
(i) By implicit function theorem for complex variables, we know that the roots of (35) are continuous on the parameter τ.
IfR0<1, then 0 is not a root of (35) for all τ>0. Note that all complex roots of (35) must come in conjugate pairs and the root of (35) is negative for τ=0. Thus, all roots of (35) have negative real parts for small τ; that is, 0<τ=1. Suppose that there exists a positive number τ=τ0 such that (35) has a pair of purely imaginary roots λ=±ωi; here ω is a positive number. We have
(36)ωi+γ-e-ωiτ0βbλd=0.
Then
(37)βbλdcos⁡ωτ0=γ,βbλdsinωτ0=-ω.
Summing up the square of both equations in (37) we obtain
(38)ω2=β2b2λ2d2-γ2=β2b2λ2-γ2d2d2=R02d2γ2-γ2d2d2=γ2(R02-1).
When R0<1, then ω2<0. It is a contradiction with ω2>0 which leads to the nonexistence of τ0. This contradiction proves the result.
(ii) Whenμ=0, and R0>1, then
(39)h(0,τ)=γ-βbλd=γ(1-R0)<0,lim⁡μ→∞h(μ,τ)=+∞.
Therefore equation must have a positive real root for all τ>0.
(iii) IfR0=1, it is easy to know that μ=0 is a root of (35) for all τ>0, which leads to conclusion. (iii) This completes the proof of theorem.Now we consider the local stability of the infection equilibriumE2((γ+αbλ)/b(dα+β),d(R0-1)/(dα+β)). As we know, the infection equilibrium E2((γ+αbλ)/b(dα+β),d(R0-1)/(dα+β)) exists if and only if R0>1. By computation, the associated transcendental characteristic equation of (8) at E2 becomes
(40)μ2+Aμ+B-(Cμ+D)e-μτ=0,
where
(41)A=d+γ+∂f∂x(x*,v*),B=(d+∂f∂x(x*,v*))γ,C=b∂f∂v(x*,v*),D=db∂f∂v(x*,v*).Theorem 6.
Whenτ>0, if R0>1, then the infection equilibrium E2 is locally asymptotically stable.Proof.
By implicit function theorem for complex variables, we know that the root of (40) is continuous on the parameter τ. If R0>1, then all roots of (40) have negative real parts as τ=0 and (40) has no zero root for all τ>0. Thus, all roots of (40) have negative real parts for very small τ; that is, 0<τ≪1. Assume that there exists a positive τ0 such that (40) has a pair of purely imaginary roots ±ωi, ω>0. Then ω>0 must satisfy
(42)-ω2+Aωi+B-(Cωi+D)(cos⁡ωτ0-isinωτ0)=0.
Separating the real and imaginary parts, we have
(43)Cωsinωτ0+Dcos⁡ωτ0=B-ω2,Dsinωτ0-Cωcos⁡ωτ0=-Aω
which implies that
(44)ω4+(A2-2B-C2)ω2+B2-D2=0.
Direct computation shows that
(45)B2-D2=[γ(d+∂f∂x(x*,v*))]2-[db∂f∂v(x*,v*)]2=[γ∂f∂x(x*,v*)+γd-db∂f∂v(x*,v*)]×[γ∂f∂x(x*,v*)+γd+db∂f∂v(x*,v*)]>0,A2-2B-C2=d2+γ2+(∂f∂x(x*,v*))2+2d∂f∂x(x*,v*)-b2(∂f∂v(x*,v*))2=(d+∂f∂x(x*,v*))2-b2(∂f∂v(x*,v*))2+γ2=((d2+γ2)(1+αv*)4+β2(v*)2(1+αv*)2+2dβv*(1+αv*)3-b2β2(x*)2)×((1+αv*)4)-1.
Let
(46)H(x*,v*)=(d2+γ2)(1+αv*)4+β2(v*)2(1+αv*)2+2dβv*(1+αv*)3-b2β2(x*)2.
Note that 1+αv*=(βb/γ)x*,
(47)H(x*,v*)=(d2+γ2)β4b4γ4(x*)4+β2(v*)2β2b2γ2(x*)2+2dβv*β3b3γ3(x*)3-β2b2(x*)2=(x*)2[β4b2γ4(b2d2(x*)2+γ2(v*)2+2γdbv*x*)+β4b4γ2(x*)2-b2β2]=(x*)2[β4b2γ4(bdx*+γv*)2+β2b2γ2(β2b2(x*)2-γ2)]=(x*)2[β4b2γ4(bdx*+γv*)2+β2b2(α2(v*)2+2αv*)]>0.
By Hurwitz criterion, (44) has no positive roots, which implies the nonexistence of τ0. Thus all roots of (40) have negative real parts for τ>0.


## 3.2. Global Stability of Equilibria

In the section, we study the global stability of equilibria; we first consider the infection-free equilibriumE1.Theorem 7.
Whenτ=0,(i)
IfR0≤1, then infection-free equilibrium E1(λ/d,0) is globally asymptotically stable.(ii)
IfR0>1, then infection equilibrium E2(x*,v*) is globally asymptotically stable.Proof.
(i) Define a Lyapunov function as what follows(48)V(x,v)=x-λdln⁡x+vb+λdln⁡λd-λd,dV(x,v)dt|(8)=x-(λ/d)x(λ-dx-βxv1+αv)+1b(βbxv1+αv-γv)=-(dx-λ)2dx+βxv1+αv(1-dx-λdx)-γbv=-(dx-λ)2dx+(βλd(1+αv)-γb)v=-(dx-λ)2dx+γb(R0-1)v-βγdαv21+αv.
It means that dV(x,v)/dt|(8) is negative semidefinite as R0≤1. Moreover, the last equality of the above equation shows that the largest invariant set of system (8) on the region {(x,v)T∈ℝ+2:dV/dt=0} is the singleton {E1}. Therefore, the infection-free equilibrium E1 is global asymptotically stability.
(ii) We rewrite the system (8)
(49)dx(t)dt=λ-dx(t)-βx(t)v(t)1+αv(t)=Q(x,v),dv(t)dt=βbx(t)v(t)1+αv(t)-γv(t)=P(x,v).
Choose a Dulac function
(50)D(x,v)=1+αvβv.
We have
(51)∂(DP)∂v+∂(DQ)∂x=-γαβ-d1+αvβv-1<0.
Thus system (49) does not have nontrivial periodic orbits in Ω. The conclusion follows.Theorem 8.
Whenτ>0, if R0≤1, then infection-free equilibrium E1(λ/d,0) is globally asymptotically stable.Proof.
Define a functionalg:C+→ℝ as follows:
(52)g(φ1,φ2)=12[φ1(0)-λd]2+λd[φ2(0)+βb∫-τ0‍φ1(θ)φ2(θ)1+αφ2(θ)dθ].
For any t≥0, xt,vt∈C, then
(53)g(xt,vt)=b2[x(t)-λd]2+λd[v(t)+βb∫t-τt‍x(θ)v(θ)1+αv(θ)dθ],dg(xt,vt)dt=b(x(t)-λd)dx(t)dt+λddv(t)dt+βbλd[x(t)v(t)1+αv(t)-x(t-τ)v(t-τ)1+αv(t-τ)]=b[x(t)-λd][λ-dx(t)-βx(t)v(t)1+αv(t)]+λd[βbx(t)v(t)1+αv(t)-γv(t)]=-bd[x(t)-λd]2-βbx(t)v(t)1+αv(t)[x(t)-λd]+λd[βbx(t)v(t)1+αv(t)-γv(t)]=-b(d+βv(t)1+αv(t))[x(t)-λd]2+λv(t)d(βbλd(1+αv(t))-γ)=-b(d+βv(t)1+αv(t))[x(t)-λd]2+λv(t)γd(R01+αv(t)-1)=-b(d+βv(t)1+αv(t))[x(t)-λd]2+λv(t)γd(R0-1)-αv(t)R01+αv(t)≤-b(d+βv(t)1+αv(t))[x(t)-λd]2+λv(t)γd(R0-1).
Then,
(54)dg(φ1,φ2)dt|(8)=-b(d+βφ2(0)1+αφ2(0))[φ1(0)-λd]2+λφ2(0)γd(R0-1)-αφ2(0)R01+αφ2(0).
When R0≤1, we have dg(xt,vt)/dt≤0. It is easy to know, when R0≤1 the largest invariant set of system (8) on the region {(φ1,φ2)∈C+:dg(φ1,φ2)/dt=0} is the singleton {E1}. By Lassalle invariant principle for autonomous retarded differential equations [11], infection-free equilibrium E1 is globally asymptotically stable. This completes the proof.Theorem 9.
Whenτ>0, if R0>1, then the infection equilibrium E2(x*,v*) is globally asymptotically stable.Proof.
LetV(t)=v(t+τ) and X(t)=x(t); system (8) becomes
(55)dX(t)dt=λ-dX(t)-βX(t)V(t-τ)1+αV(t-τ),dV(t)dt=βbX(t)V(t-τ)1+αV(t-τ)-γV(t).
Denoteu(x)=x/(1+αx). Evaluating both sides of (55) at E2, we obtain
(56)λ=dx*+βx*u(v*),γv*=βbx*u(v*).
Define a Lyapunov functional L:C+→ℝ as follows:
(57)L(φ1,φ2)=1βu(v*)L1(φ1,φ2)+v*βbx*u(v*)L2(φ1,φ2)+L3(φ1,φ2),
where
(58)L1(φ1,φ2)=φ1(0)x*-1-ln⁡φ1(0)x*,L2(φ1,φ2)=φ2(0)v*-1-ln⁡φ2(0)v*,L3(φ1,φ2)=∫-τ0‍(φ2(θ)v*-1-ln⁡φ2(θ)v*)dθ.
Thus, L(φ1,φ2)≥0 with equality if and only if φ1(0)/x*=φ2(0)/v*=1.
For anyt≥0, Xt,Vt∈C, then
(59)L1(Xt,Vt)=X(t)x*-1-ln⁡X(t)x*,L2(Xt,Vt)=V(t)v*-1-ln⁡V(t)v*,L3(Xt,Vt)=∫-τ0‍(V(t+θ)v*-1-ln⁡V(t+θ)v*)dθ.
We calculate derivatives ofL1(Xt,Vt), L2(Xt,Vt), and L3(Xt,Vt) with respect to the system (55):
(60)dL1(Xt,Vt)dt=1x*(1-x*X)(λ-dX-βXu(V(t-τ)))=1x*(1-x*X)(dx*+βx*u(v*)-dX-βXu(V(t-τ)))=-d(X-x*)2Xx*+βu(v*)(1-x*X)(1-Xu(V(t-τ))x*u(v*))=-d(X-x*)2Xx*+βu(v*)(1-x*X+u(V(t-τ))u(v*)-Xu(V(t-τ))x*u(v*)),dL2(Xt,Vt)dt=1v*(1-v*V)(βbXu(V(t-τ))-γV)=1v*(1-v*V)(βbx*u(v*)Xu(V(t-τ))x*u(v*)-γVv*v*)=βbx*v*u(v*)(1-v*V)(Xx*u(V(t-τ))u(v*)-Vv*)=βbx*v*u(v*)×(1-v*VXx*u(V(t-τ))u(v*)+Xx*u(V(t-τ))u(v*)-Vv*),dL3(Xt,Vt)dt=∫-τ0‍ddt(V(t+θ)v*-1-ln⁡V(t+θ)v*)dθ=V(t)v*-ln⁡V(t)v*-V(t-τ)v*+ln⁡V(t-τ)v*.
We obtain
(61)dL(Xt,Vt)dt=-dβu(v*)(X-x*)2Xx*+1-x*X+u(V(t-τ))u(v*)-Xu(V(t-τ))x*u(v*)+1-v*VXx*u(V(t-τ))u(v*)+Xx*u(V(t-τ))u(v*)-Vv*+Vv*-ln⁡Vv*-V(t-τ)v*+ln⁡V(t-τ)v*=-dβu(v*)(X-x*)2Xx*+2-x*X+u(V(t-τ))u(v*)-v*VXx*u(V(t-τ))u(v*)-ln⁡Vv*-V(t-τ)v*+ln⁡V(t-τ)v*.
By adding and subtracting the quantity ln⁡((X/x*)(u(V(t-τ))/u(v*))), we have
(62)dL(Xt,Vt)dt=-dβu(v*)(X-x*)2Xx*+2-x*X+u(V(t-τ))u(v*)-v*VXx*u(V(t-τ))u(v*)-V(t-τ)v*+ln⁡V(t-τ)v*+ln⁡(Xx*v*Vu(V(t-τ))u(v*))-ln⁡(Xx*)-ln⁡(u(V(t-τ))u(v*))=-dβu(v*)(X-x*)2Xx*-(x*X-1+ln⁡Xx*)-(Xx*v*Vu(V(t-τ))u(v*)-1ii-ln⁡(Xx*v*Vu(V(t-τ))u(v*)))+u(V(t-τ))u(v*)-V(t-τ)v*+ln⁡V(t-τ)v*-ln⁡(u(V(t-τ))u(v*)).
Then,
(63)dL(φ1,φ2)dt=-dβu(v*)(φ1(0)-x*)2φ1(0)x*-(x*φ1(0)-1+ln⁡φ1(0)x*)-(φ1(0)x*v*φ2(0)u(φ2(-τ))u(v*)-1mm-ln⁡(φ1(0)x*v*φ2(0)u(φ2(-τ))u(v*)))+u(φ2(-τ))u(v*)-φ2(-τ)v*+ln⁡φ2(-τ)v*-ln⁡(u(φ2(-τ))u(v*)).
It is easy to know that  u(φ2(-τ))/u(v*)-φ2(-τ)/v*+ln⁡⁡(φ2(-τ)/v*)-ln⁡(u(φ2(-τ))/u(v*))≤0, and u(φ2(-τ))/u(v*)-φ2(-τ)/v*+ln⁡⁡(φ2(-τ)/v*)-ln⁡(u(φ2(-τ))/u(v*))=0 if and only if φ2(-τ)/v*=1. It follows that dL(φ1,φ2)/dt≤0, and dL(φ1,φ2)/dt=0 if and only if φ1(0)/x*=φ2(0)/v*=φ2(-τ)/v*=1. By classical stability theory for functional differential equations, E2 is globally asymptotically stable. This completes the proof.


## 4. Conclusion

The viral infection model addressed in this paper has saturated incidence rate and viral infection with delay. The basic reproductive numberR0 is given. When R0<1, for the model with or without delay time, the infection-free equilibrium is globally asymptotically stable, which implies that the viral infection goes extinct eventually; when R0>1, the infection equilibrium is globally asymptotically stable, which implies that the viral infection persists in the body of the host.



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*Source: 187897-2014-04-06.xml*