# Analysis of a HBV Model with Diffusion and Time Delay

Authors: Noé Chan Chí; Eric ÁvilaVales; Gerardo García Almeida
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

This paper discussed a hepatitis B virus infection with delay, spatial diffusion, and standard incidence function. The local stability of equilibrium is obtained via characteristic equations. By using comparison arguments, it is proved that if the basic reproduction number is less than unity, the infection-free equilibrium is globally asymptotically stable. If the basic reproductive number is greater than unity, by means of an iteration technique, sufficiently conditions are obtained for the global asymptotic stability of the infected steady state. Numerical simulations are carried out to illustrate our findings.


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## Body

## 1. Introduction

Human infection with hepatitis B virus (HBV) is a major global health problem. Between 300 and 400 million people are chronically infected worldwide. The virus is contracted through contact with blood or other fluids from the body, which could lead to develop viral persistence in the individual in the absence of strong antibody or some immune depression. Mathematical models have the potential to improve the understanding of the dynamics of this disease; one of the earliest models is referred to as the basic virus infection model, introduced by Nowak et al. [1]. They proposed a basic mathematical model for uninfected susceptible host cells (hepatocytes), u, infected host cells, w, and free virus particles, v, as follows:
(1.1)u˙(t)=s-μu(t)-βu(t)v(t),w˙(t)=βu(t)v(t)-aw(t),v˙(t)=kw(t)-dv(t),
where hepatocytes are produced at a rate s, uninfected cells die at rate μ and become and infected at rate βu(t)v(t), infected hepatocytes are produced at rate βu(t)v(t) and die at rate aw(t). Free viruses are produced from infected cells at rate kw(t) and are removed at rate dv(t). It is assumed that all parameters are positive constants. Previous models assume that the infectious process is instantaneous; that is, in the very moment that the virus enters an uninfected cell, this one starts to produce virus particles; we know that this is not biologically reasonable. Thus, models with delays have been considered; in [2], the authors studied the following hepatitis B virus infection model with a time delay:
(1.2)x˙(t)=λ-dx(t)-βx(t)v(t)x(t)+y(t),y˙(t)=βe-mτx(t-τ)v(t-τ)x(t-τ)+y(t-τ)-ay(t),v˙(t)=ky(t)-uv(t).
The authors gave results about local and global stability of feasible equilibria.For HBV infection, susceptible host cells and infected cells are hepatocytes and cannot move under normal conditions, but viruses move freely in liver [3]; therefore, the authors introduce an HBV model with diffusion and delay. Xu and Ma [4] considered also a diffusion model with delay but instead of bilinear response of the infection rate, they considered saturation response.In this work motivated by the work of Xu and Ma, we study the following model:(1.3)∂u∂t=L-du(x,t)-βu(x,t)v(x,t)u(x,t)+w(x,t),∂w∂t=βe-mτu(x,t-τ)v(x,t-τ)u(x,t-τ)+w(x,t-τ)-aw(x,t),∂v∂t=DΔv+kw(x,t)-pv(x,t),
for t>0, x∈Ω, with homogeneous Neumann boundary conditions
(1.4)∂v∂η=0,
and initial conditions
(1.5)u(x,θ)=ϕ1(x,θ)≥0,w(x,θ)=ϕ2(x,θ)≥0,v(x,θ)=ϕ3(x,θ)≥0,θ∈[-τ,0],x∈Ω¯.
In the previous problem Ω is a bounded domain in ℝn with smooth boundary ∂Ω, ∂/∂η denotes the outward normal derivative on ∂Ω.This paper is ordered as follows. In the next section we present a result about the existence, uniqueness, and positivity. In Section3 we discuss the local stability of each of the feasible equilibria of system (1.3), by analyzing the corresponding characteristic equations. In Section 4, by using comparison arguments and an iterative technique, we establish sufficient conditions for the global stability of the equilibria of system (1.3). In Section 5 numerical simulations are carried out to illustrate our principal results and we compare the effect of the diffusion and the delay on the system (1.3).


## 2. Preliminaries

Consider problem (1.3)–(1.5) and the following definitions.Definition 2.1.
A pair of functionsU~=(u~,w~,v~), U^=(u^,w^,v^) in C([0,∞)×Ω-)∩C(1,2)((0,∞)×Ω) are called coupled upper and lower solutions to system (1.3)–(1.5) if u^≤u~, w^≤w~, v^≤v~ in Ω-×[-τ,∞) and the following differential inequalities hold:
(2.1)∂u~∂t≥L-du~(x,t)-βu~(x,t)v^(x,t)u~(x,t)+w^(x,t),∂w~∂t≥βe-mτu~(x,t-τ)v~(x,t-τ)u~(x,t-τ)+w~(x,t-τ)-aw~(x,t),∂v~∂t≥DΔv~+kw~(x,t)-pv~(x,t),∂u^∂t≤L-du~(x,t)-βu~(x,t)v^(x,t)u~(x,t)+w^(x,t),∂w^∂t≤βe-mτu^(x,t-τ)v^(x,t-τ)u^(x,t-τ)+w^(x,t-τ)-aw^(x,t),∂v^∂t≤DΔv^+kw^(x,t)-pv^(x,t),
for (x,t)∈Ω×(0,∞), and
(2.2)∂u^∂η≤0≤∂u~∂η,∂w^∂η≤0≤∂w~∂η,∂v^∂η≤0≤∂v~∂η(x,t)∈∂Ω×(0,∞),u^(x,t)≤ϕ1(x,t)≤u~(x,t),w^(x,t)≤ϕ2(x,t)≤w~(x,t),v^(x,t)≤ϕ3(x,t)≤v~(x,t),(x,t)∈Ω-×[-τ,0].The following lemma then follows from Theorem 3.4 developed by Redlinger [5].Lemma 2.2.
LetU~ and U^ be a pair of coupled upper and lower solutions for problem (1.3)–(1.5) and suppose that the initial functions ϕi (i=1,2,3) are Hölder continuous in [-τ,0]×Ω. Then problem (1.3)–(1.5) has exactly one regular solution U(x,t)=(u(x,t),w(x,t),v(x,t)) satisfying U^≤U≤U~ in Ω-×[-τ,∞).It is not hard to see that0=(0,0,0) and K=(K1,K2,K3) are a pair of coupled lower-upper solutions to problem (1.3)–(1.5), where
(2.3)K1=max{Ld,sup-τ≤θ≤0∥ϕ1(·,θ)∥C(Ω-,R)},K2=max{kβe-mτK1ap,sup-τ≤θ≤0∥ϕ2(·,θ)∥C(Ω-,R)},K3=max{k2βe-mτK1ap2,sup-τ≤θ≤0∥ϕ3(·,θ)∥C(Ω-,R)}.
Hence, 0≤u(x,t)≤K1, 0≤w(x,t)≤K2, 0≤v(x,t)≤K3 for (x,t)∈Ω-×[-τ,∞), and also, by the maximum principle, if ϕi(x,0)≢0(i=1,2,3), we have u(x,t)>0, w(x,t)>0, v(x,t)>0 for all t>0, x∈Ω-.


## 3. Local Stability

System (1.3) has the equilibrium E1(L/d,0,0). Let R0=βke-mτ/ap>1 then system (1.3) has a unique infected steady state E2(u*(τ),w*(τ),v*(τ)); the previous notation is because the equilibrium involves τ and we use this as the parameter for the stability analysis, where
(3.1)u*(τ)=Le-mτde-mτ+a(R0-1),w*(τ)=Le-mτ(R0-1)de-mτ+a(R0-1),v*(τ)=Lke-mτ(R0-1)p[de-mτ+a(R0-1)].Let0=μ1<μ2<⋯ be the eigenvalues of the operator -Δ on Ω with the homogeneous Neumann boundary conditions, and let E(μi) be the eigenspace corresponding to μi in C1(Ω).Let𝕏=[C1(Ω)]3, let {ϕij;j=1,2,…,dimE(μi)} be an orthonormal basis of E(μi), and let 𝕏ij={cϕij∣c∈ℝ3}, then
(3.2)𝕏=⨁i=0∞𝕏i,𝕏i=⨁j=1dimE(pi)𝕏ij.
Let 𝒟=diag(0,0,D), Z=Z(x,t)=(U(x,t),W(x,t),V(x,t)), ℒZ=𝒟ΔZ+JE*Z+JτE*Z(t-τ), where
(3.3)JE*=(-d-βw-v-(u-+w-)2βu-v-(u-+w-)2-βu-u-+w-0-a00k-p),Jτ=(000βe-mτv-w-(u-+w-)2-βe-mτu-v-(u-+w-)2βe-mτu-u-+w-000),
and E*(u-,w-,v-) represents any feasible steady state of the system (1.3). The linearization of system (1.3) at E* is of the form Zt=ℒZ. For each i≥1,𝕏i is invariant under the operator ℒ, and λ is an eigenvalue of the matrix -μi𝒟+JE*+JτE* for some i≥1, then, there is an eigenvector in 𝕏i.The characteristic equation on the equilibriumE1 is
(3.4)(λ+d)(λ2+a1λ+a0+b0(τ)e-λτ)=0,
where
(3.5)a0=a(p+μID),a1=a+p+μiD,b0(τ)=-kβe-mτ.
The characteristic equation has the negative root λ=-d. All other roots of (3.4) are given by the transcendental equation
(3.6)λ2+a1λ+a0+b0(τ)e-λτ=0.
Let
(3.7)f(λ)=λ2+a1λ+a0+b0(τ)e-λτ,
if R0>1, note that for λ real and i=1 (in this case μ1=0),
(3.8)f(0)=a0+b0=ap-kβe-mτ<0,limλ→∞f(λ)=+∞.
Hence, (3.7) has a positive root. Therefore, there is a characteristic root λ with positive real part in the spectrum of ℒ. Accordingly, if R0>1, the disease-free steady state E1(λ/d,0,0) is unstable.IfR0<1, when τ=0 the coefficients of (3.7) are a1 and a0+b0(0), and under the hypothesis R0<1 the coefficients are positive and according to the criterion of Routh-Hurwitz, the equilibrium E1(λ/d,0,0) is locally asymptotically stable.Forτ>0 if iω(ω>0) is a solution of (3.6), separating in real and imaginary parts, we obtain
(3.9)-ω2+a0=-b0(τ)cos(ωτ),a1ω=b0(τ)sin(ωτ).
Squaring and adding the above equations and taking z=ω2 we obtain
(3.10)z2+(a12-2a0)z+a02-b0(τ)2=0,
where
(3.11)a12-2a0=a2+(p+μiD)2>0,a02-b02(τ)=(a(p+μiD)-kβe-mτ)(a(p+μiD)+kβe-mτ)>0,
the last inequality is true because R0<1. Therefore there is no positive root z=ω2 of (3.10). In conclusion if R0<1 the equilibrium E1(λ/d,0,0) is locally asymptotically stable.The characteristic equation of system (1.3) at the endemic equilibrium E2(u*,w*,v*) is of the form
(3.12)λ3+a2(τ)λ2+a1λ+a0(τ)+(b2(τ)λ2+b1(τ)λ+b0(τ))e-λτ=0,
where
(3.13)a2(τ)=a+d+p+μiD+βv*w*(u*+w*)2,a1=a(p+μiD)+(a+p+μiD)d,a0(τ)=a(p+μiD)[d+βw*v*(u*+w*)2],b2(τ)=βe-mτu*v*(u*+w*)*,b1(τ)=(d+p+μiD)βe-mτu*w*(u*+w*)2-kβe-mτu*u*+w*,b0(τ)=d(p+μiD)βe-mτu*v*(u*+w*)2-dkβe-mτu*u*+w*,
when τ=0 becomes
(3.14)λ3+(a2(0)+b2(0))λ2+(a1+b1(0))λ+(a0(0)+b0(0))=0.
Note that a2(0)+b2(0)>0; adding a0(0)+b0(0) and replacing u*(0) and w*(0) we obtain
(3.15)a0(0)+b0(0)=a(p+μiD)βw*(0)v*(0)(u*(0)+w*(0))2+d(p+μiD)βu*(0)v*(0)(u*(0)+w*(0))2>0,(a2(0)+b2(0))(a1+b1(0))-a0(0)+b0(0)=(d(a+p+μiD)+(u*(0)+w*(0))(p+μiD)βv*(0)(u*(0)+w*(0))2)×(a+d+p+μiD+(u*(0)+w*(0))βv*(u*(0)+w*(0))2)+λβv*(0)(u*(0)+w*(0))2(a+d+(u*(0)+w*(0))βv*(0)(u*(0)+w*(0))2)>0.
By the Routh-Hurwitz criteria, all roots have negative real parts if R0>1.For the caseτ>0 we look for solutions λ=iω(ω>0) for (3.12), separating real and imaginary parts, it follows that
(3.16)ω3-a1ω=(b2(τ)ω2-b0(τ))sin(ωτ)+b1(τ)ωcos(ωτ),a2(τ)ω2-a0(τ)=-(b2(τ)ω2-b0(τ))cos(ωτ)+b1(τ)ωsin(ωτ).
Squaring and adding the two equations, we derive that
(3.17)ω6+C1ω4+C2ω2+C3=0,
where
(3.18)C1=(p+μiD)2+(d+βw*(τ)v*(τ)(u*(τ)+w*(τ))2)2+au*(τ)u*(τ)+w*(τ)(a+βe-mτu*(τ)v*(τ)(u*(τ)+w*(τ))2)>0,C2=a2(d2+(p+μiD)2)w*(τ)(u*(τ)+2w*(τ))(u*(τ)+w*(τ))2+a2βw*(τ)v*(τ)(u*(τ)+w*(τ))2(2d+βw*(τ)v*(τ)(u*(τ)+w*(τ))2)+(p+μiD)(d+βw*(τ)+v*(τ)(u*(τ)+w*(τ))2)>0,C3=a(p+μiD)2βv*(τ)(de-mτu*(τ)+aw*(τ))(u*(τ)+w*(τ))3(d(2x*(τ)+w*(τ)+βw*(τ)v*(τ)u*(τ)+w*(τ)))>0,
implying that (3.17) has no positive roots z=ω2.Theorem 3.1.
IfR0<1 the disease-free equilibrium is locally asymptotically stable; if R0>1 it is unstable and the endemic equilibrium exists and it is locally asymptotically stable.


## 4. Global Stability

We will discuss in this section the global stability of the infected steady state and the disease-free equilibrium. The technique of proof is to use comparison arguments and to successively modify the coupled lower-upper solutions pairs.Consider the following delay system:(4.1)u˙1(t)=a1βe-mτu2(t-τ)a1+u1(t-τ)-au1(t),u˙2(t)=ku1(t)-pu2(t),
with initial conditions
(4.2)ui(s)=ϕi(s)≥0,s∈[-τ,0),ϕi(0)>0,ϕi∈C([-τ,0],ℝ+).
System (4.1) always have the trivial equilibrium A0(0,0). If kβe-mτ>ap, then system (4.1) has a unique positive equilibrium A*(u1*,u2*) where
(4.3)u1*=a1(kβe-mτ-ap)ap,u2*=a1k(kβe-mτ-ap)ap2,
and according to [2], for system (4.1), one has the following.Lemma 4.1.
Ifkβe-mτ>ap, then the positive equilibrium A*(u1*,u2*) is globally stable.
Ifkβe-mτ<ap, then the equilibrium A0(0,0) is globally stable.Now we stablish and prove our result about global stability.Theorem 4.2.
Let(u(x,t),w(x,t),v(x,t)) be a solution to problem (1.3)–(1.5), let ϕi(x,0)≢0(i=1,2,3). If R0>1 and(H1)
dpe-mτ>kβe-mτ-ap,then
(4.4)limt→∞(u(x,t),w(x,t),v(x,t))=(u*,w*,v*)uniformly  forx∈Ω,
that is, the infected steady state E* is globally asymptotically stable.Proof.
Let(u(x,t),w(x,t),v(x,t)) be a solution to problem (1.3)–(1.5), let ϕi≢0(i=1,2,3). We have u(x,t)>0, w(x,t)>0, and v(x,t)>0 for all x∈Ω-, t>0. Denote
(4.5)u¯=limsupt→∞maxx∈Ω-u(x,t),u_=liminft→∞minx∈Ω-u(x,t),w¯=limsupt→∞maxx∈Ω-w(x,t),w_=liminft→∞minx∈Ω-w(x,t),v¯=limsupt→∞maxx∈Ω-v(x,t),v_=liminft→∞minx∈Ω-v(x,t).
First we look for upper solutions for the system (1.3). Let (u¯1(x,t),w¯1(x,t),v¯1(x,t)) be a solution for the following problem:
(4.6)∂u¯(1)∂t=L-du¯(1)(x,t),t>0,x∈Ω,∂w¯(1)∂t=βe-mτu¯(1)(x,t-τ)v¯(1)(x,t-τ)u¯(1)(x,t-τ)+w¯(1)(x,t-τ)-aw¯(1)(x,t),t>0,x∈Ω,∂v¯(1)∂t=DΔv¯(1)(x,t)+kw¯(1)(x,t)-pv¯(1)(x,t),t>0,x∈Ω,∂u¯(1)(x,t)∂t=∂w¯(1)(x,t)∂t=∂v¯(1)(x,t)∂t=0,t>0,x∈∂Ω,u¯(1)(x,t)=u(x,t),v¯(1)(x,t)=v(x,t),w¯(1)(x,t)=w(x,t),t∈[-τ,0],x∈Ω-.
We note that the solution of this system is an upper solution of system (1.3)–(1.5). For t>0, x∈Ω- we have
(4.7)0≤u(x,t)≤u¯(1)(x,t),0≤w(x,t)≤w¯(1)(x,t),0≤v(x,t)≤v¯(1)(x,t).
From the first equation of (4.6)
(4.8)limt→∞u¯(1)(x,t)=Ld=M1u.
Hence, by comparison, for all ϵ>0 sufficiently small, there exists t1>0 such that if t>t1(4.9)maxx∈Ω-u(1)(x,t)≤M1u+ϵ,
since ϵ is arbitrary and sufficiently small we can conclude that
(4.10)u¯=limsupt→∞maxx∈Ω-u(x,t)≤M1u.
Now consider the problem related with the second and third equations of (4.6)
(4.11)∂ω2(1)∂t=βe-mτ(M1u+ϵ)ω3(1)(x,t-τ)M1u+ϵ+ω2(1)(x,t-τ)-aω2(1)(x,t),t>t1,x∈Ω,∂ω3(1)∂t=DΔω3(1)(x,t)+kω2(1)(x,t)-pω3(1),t>t1,x∈Ω,∂ω2(1)∂η=∂ω3(1)∂η=0,t>t1,x∈∂Ω,ω2(1)(x,t)=w(x,t),ω3(1)(x,t)=v(x,t),t∈[-τ,t1],x∈Ω-.
Consider the solution for
(4.12)u˙2=βe-mτ(M1u+ϵ)u3(t-τ)M1u+ϵ+u2(t-τ)-au2,t>t1,u˙3=ku2-pu3,t>t1,u2(t)=maxx∈Ω-w(x,t),u3(t)=maxx∈Ω-v(x,t),t∈[-τ,t1].
Note that (u1(t),u2(t)) is an upper solution for system (4.11), and using the assumption that R0>1, by Lemma 4.1, it follows from (4.12) that
(4.13)limt→∞u2(t)=(kβe-mτ-ap)(M1u+ϵ)ap,limt→∞u3(t)=k(kβe-mτ-ap)(M1u+ϵ)ap2.
Hence, for all ϵ>0 sufficiently small, by comparison there exists a t2>t1 such that if t>t2(4.14)maxx∈Ω-w¯(1)(x,t)<M1w+ϵ,maxx∈Ω-v¯(1)(x,t)<M1v+ϵ,
where
(4.15)M1w=(kβe-mτ-ap)M1uap,M1v=k(kβe-mτ-ap)M1uap2.
Since ϵ>0 is arbitrary and sufficiently small, we conclude that
(4.16)w¯=limsupt→∞maxx∈Ω-w(x,t)≤M1w,v¯=limsupt→∞maxx∈Ω-v(x,t)≤M1v.
Now for lower solutions, let (u_(1)(x,t),w_(1)(x,t),v_(1)(x,t)) be the solution for the following problem:
(4.17)∂u_(1)∂t=L-du_(1)(x,t)-βu_(1)(x,t)v¯(1)(x,t)u_(1)(x,t)+w¯(1)(x,t),t>t2,x∈Ω,∂w_(1)∂t=βe-mτu_(1)(x,t)v_(1)(x,t)u_(1)(x,t)+w_(1)(x,t)-aw_(1)(x,t),t>t2,x∈Ω,∂v_(1)∂t=DΔv_(1)(x,t)+kw_(1)(x,t)-pv_(1)(x,t),t>t2,x∈Ω,∂u_(1)∂η=∂w_(1)∂η=∂v_(1)∂η=0,t>t2,x∈∂Ω,u_(1)(x,t)=12u(x,t),w_(1)(x,t)=12w(x,t),v_(1)(x,t)=12v(x,t),t∈[-τ,t2],x∈Ω-.
Note that the solution of (4.17) is a lower solution to (1.3)–(1.5). For all ϵ>0 sufficiently small, from the first equation of (4.17) and (4.16) it follows
(4.18)∂u_(1)∂t≥L-du_(1)(x,t)-β(M1v+ϵ),t>t2,x∈Ω.
By comparing the above equation with the following problem:
(4.19)∂ω1(1)∂t=L-dω1(1)(x,t)-β(M1v+ϵ),t>t2,x∈Ω,∂ω1ωη=0,t>t2,x∈∂Ω,ω1(x,t2)=12u(x,t2),x∈Ω-,
we obtain
(4.20)limt→∞ω1(x,t)=L-β(M1v+ϵ)d,
so u_(1)(x,t)≥ω1(x,t), t>t2, and x∈Ω. Hence, for all ϵ>0 sufficiently small, there is a t3>t2 such that if t>t3,
(4.21)minx∈Ω-u_(1)(x,t)≥N1u-ϵ,
where
(4.22)N1u=L-βM1vd.
Since  ϵ>0 is arbitrary sufficiently small, by comparison we conclude that
(4.23)u_=liminft→∞minx∈Ω-u(x,t)≥N1u=L-βM1vd.
Now consider the following problem related with the second and third equations of (4.17):
(4.24)∂ω2∂t=βe-mτ(N1u-ϵ)ω3(x,t-τ)N1u-ϵ+ω2(x,t-τ)-aω2(x,t),t>t3,x∈Ω,∂ω3∂t=DΔω3(x,t)+kω2(x,t)-pω3(x,t),t>t3,x∈Ω,∂ω2∂η=∂ω3∂η=0,t>t3,x∈∂Ω,ω2(x,t)=12w(x,t),ω3(x,t)=12v(x,t),t∈[-τ,t3],x∈Ω-.
Now let us consider the solution for the problem
(4.25)u˙2(t)=βe-mτ(N1u-ϵ)u3(t-τ)N1u-ϵ+u2(t-τ)-au2(t),t>t3,u˙3(t)=ku2(t)-pu3(t),t>t3,u2(t)=12minx∈Ω-w(x,t),u3(t)=12minx∈Ω-v(x,t),t∈[-τ,t3],t>t3,
and according to Lemma 4.1(4.26)limt→∞u2(t)=(kβe-mτ-ap)(N1u-ϵ)ap,limt→∞u3(t)=k(kβe-mτ-ap)(N1u-ϵ)ap2.
Hence, for all ϵ>0 sufficiently small, by comparison there exists a t4>t3 such that if t>t4(4.27)minx∈Ω-w_(1)(x,t)>N1w-ϵ,minx∈Ω-v_(1)(x,t)>N1v-ϵ,
where
(4.28)N1w=(kβe-mτ-ap)N1uap,N1v=k(kβe-mτ-ap)N1uap2.
Since ϵ>0 is arbitrary and sufficiently small, we conclude that
(4.29)w_=liminft→∞minx∈Ω-w(x,t)≥N1w,v_=liminft→∞minx∈Ω-v(x,t)≥N1v.
Now we look for the closest upper and lower solutions. Let (u¯(2),w¯(2),v¯(2)) be a solution for the problem
(4.30)∂u¯(2)∂t=L-du¯(2)(x,t)-βu¯(2)(x,t)v_(1)(x,t)u¯(1)+w¯(1)(x,t),t>t4,x∈Ω,∂w¯(2)∂t=βe-mτu¯(2)(x,t-τ)v¯(2)(x,t)u¯(2)(x,t-τ)+w¯(2)(x,t-τ)-aw¯(2)(x,t),t>t4,x∈Ω,∂v¯(2)∂t=DΔv¯(x,t)+kw¯(2)(x,t)-pv¯(2)(x,t),t>t4,x∈Ω,∂u¯(2)∂η=∂w¯(2)∂η=∂v¯(2)∂η=0,t>t4,x∈∂Ω,u¯(2)(x,t)=u(x,t),w¯(2)(x,t)=w(x,t),v¯(2)(x,t)=v(x,t),t∈[-τ,t4],x∈Ω-.
For all ϵ>0 sufficiently small it follows form the first equation of (4.30) and the inequalities (4.27) and (4.14) that
(4.31)∂u¯(2)∂t≤L-du¯(2)(x,t)-βu¯(2)(x,t)(N1v-ϵ)M1u+ϵ+M1w+ϵ,t>t4,x∈Ω.
Let ω1(2)(x,t) be the solution for the following problem:
(4.32)∂ω1(2)∂t=L-dω1(2)(x,t)-βω1(2)(x,t)(N1v-ϵ)M1u+ϵ+M1w+ϵ,t>t4,x∈Ω,∂ω1(2)∂η=0,t>t4,x∈∂Ω,ω1(2)(x,t4)=u(x,t4),x∈Ω-,
it follows that
(4.33)limt→∞ω1(2)(x,t)=L(M1u+M1w+2ϵ)d(M1u+M1w+2ϵ)+β(N1v-ϵ).
By comparison we have that u¯(2)≤ω1(2), t>t4, and x∈Ω-. Hence, for all ϵ>0 sufficiently small, by comparison, there is a t5>t4 such that if t>t5(4.34)maxx∈Ω-u¯(2)(x,t)=M2u+ϵ,
where
(4.35)M2u=L(M1u+M1w)d(M1u+M1w)+β(N1v).
Since (4.34) is valid for ϵ>0 arbitrary and sufficiently small, by comparison we conclude that
(4.36)u¯=limsupt→∞maxx∈Ω-u(x,t)≤M2u.
Now consider the following problem related with the second and third equations of (4.30):
(4.37)∂ω2(2)∂t=βe-mτ(M2u+ϵ)ω3(2)(x,t)M2u+ω2(2)(x,t-τ)-aω2(2)(x,t),t>t5,x∈Ω,∂ω3(2)∂t=DΔω3(2)(x,t)+kω2(2)(x,t)-pω3(2)(x,t),t>t5,x∈Ω,∂ω2(2)∂η=∂ω3(2)∂η=0,t>t5,x∈∂Ω,ω2(2)(x,t)=w(x,t),ω3(2)(x,t)=v(x,t),t∈[-τ,t5],x∈Ω-.
Let (u2(t),u3(t)) be the positive solution to the following problem:
(4.38)u˙2(t)=βe-mτ(M2u+ϵ)u3(t-τ)M2u+ϵ+u2(t-τ)-au2(t),t>t5,u˙3(t)=ku2(t)-pu3(t),t>t5,u2(t)=maxx∈Ω-w(x,t),u3(t)=maxx∈Ω-v(x,t),t∈[-τ,t5].
Then by Lemma 4.1 and the previous system we have
(4.39)limu2(t)=(kβe-mτ-ap)(M2u+ϵ)ap,limu3(t)=k(kβe-mτ)(M2u+ϵ)ap2.
Hence for all ϵ>0 sufficiently small, by comparison there is a t6>t5 such that if t>t6(4.40)maxx∈Ω-w¯(x,t)<M2w+ϵ,maxx∈Ω-v¯(x,t)<M2v+ϵ,
where
(4.41)M2w=(kβe-mτ-ap)M2uap,M2v=k(kβe-mτ)M2uap2.
Since ϵ>0 is arbitrary and sufficiently small, we conclude that
(4.42)w¯=limsupt→∞maxx∈Ω-w(x,t)≤M2w,v¯=limsupt→∞maxx∈Ω-v(x,t)≤M2v.
Let (u_(2),w_(2),v_(2)) be a solution for the following problem:
(4.43)∂u_(2)∂t=L-du_(2)(x,t)-βu_(2)(x,t)v¯(1)(x,t)u_(1)(x,t)+w_(1)(x,t),t>t6,x∈Ω,∂w_(2)∂t=βe-mτu_(2)(x,t-τ)v_(2)(x,t-τ)u_(2)(x,t-τ)+w_(2)(x,t-τ)-aw_(2)(x,t)t>t6,x∈Ω,∂v_(2)∂t=DΔv_(2)(x,t)+kw_(2)-pv_(2)(x,t)t>t6,x∈Ω,u_(2)(x,t)=12u(x,t),w_(2)(x,t)=12w(x,t),w_(2)(x,t)=12v(x,t),t∈[-τ,t6],x∈Ω-.
Then (u_(2),w_(2),v_(2)) and (u¯(2),w¯(2),v¯(2)) are a pair of coupled lower and upper solutions to system (1.3)–(1.5). Hence we have that for t≥t6,  x∈Ω-(4.44)u_(2)(x,t)≤u(x,t)≤u¯(2)(x,t),w_(2)(x,t)≤w(x,t)≤w¯(2)(x,t),v_(2)(x,t)≤v(x,t)≤v¯(2)(x,t).
For all ϵ>0 sufficiently small, it follow from the first equation of (4.43), and the inequalities
(4.45)∂u_(2)∂t≥L-du_(2)(x,t)-βu_(2)(x,t)(M1v+ϵ)N1u-ϵ+N1w-ϵ.
By comparison we have that u_(2)(x,t)≥v1(2)(x,t), t>t6, and x∈Ω- where v1(2) is the solution to problem
(4.46)∂v1(2)∂t=L-dv1(2)(x,t)-βv1(2)(x,t)(M1v+ϵ)N1u-ϵ+N1w-ϵ,∂v1(2)∂η=0,t>t6,x∈∂Ω,
which has satisfies
(4.47)limt→∞v1(2)(x,t)=L(N1u-ϵ+N1w-ϵ)d(N1u-ϵ+N1w-ϵ)+β(M1v+ϵ).
Hence for all ϵ>0 sufficiently small, by comparison, there is a t7>t6 such that if t>t7(4.48)minu_(2)(x,t)≥N2u-ϵ,
with
(4.49)N2u=L(N1u+N1w)d(N1u+N1w)+βM1v.
Since this holds true for arbitrary ϵ>0 sufficiently small, by comparison we conclude that
(4.50)u_=liminft→∞minx∈Ω-u(x,t)≥N2u.
Now consider the following problem:
(4.51)∂v2(2)∂t=βe-mτ(N2u-ϵ)v3(2)(x,t-τ)N2u-ϵ+v2(2)(x,t-τ)-av2(2)(x,t),t>t7,x∈Ω,∂v3(2)∂t=DΔv3(2)(x,t)+kv2(2)-pv3(2)(x,t),t>t7,x∈Ω,∂v2(2)∂η=∂v3(2)∂η=0,t>t7,x∈Ω,v2(2)(x,t)=12w(x,t),v3(2)(x,t)=12v(x,t),t∈[-τ,t7],x∈Ω-.
Let (u2(t),u3(t)) be the positive solution for the following problem:
(4.52)v˙2(t)=βe-mτ(N2u-ϵ)u2(t-τ)N2u-ϵ+u2(t-τ)-au2(t),t>t7,v˙3(t)=kv2(t)-pv3(t),t>t7,v2(t)=12minx∈Ω-,v3(t)=12minx∈Ω-v(x,t),t∈[-τ,t7].
By Lemma 4.1 it follows
(4.53)limt→∞v2(t)=(N2u-ϵ)(kβe-mτ-ap)ap,limt→∞v3(t)=k(N2u-ϵ)(kβe-mτ-ap)ap2,
hence, for all ϵ>0 sufficiently small, by comparison there exists a t8>t7 such that if t>t8,
(4.54)minx∈Ω-w_(2)(x,t)>N2w-ϵ,minx∈Ω-v_(2)(x,t)>N2v-ϵ,
where
(4.55)N2w=N2u(kβe-mτ-ap)ap,N2v=kN2u(kβe-mτ-ap)ap2.
Since ϵ>0 is arbitrary and sufficiently small, we conclude that
(4.56)w_=liminft→∞minx∈Ω-w(x,t)≥N2w,v_=liminft→∞minx∈Ω-v(x,t)≥N2v,
continuing this process, we derive six sequences Mnu, Mnw, Mnv, Nnu, Nnw, and Nnv(n=1,2,…) such that, for n≥2,
(4.57)Mnu=L(Mn-1u+Mn-1w)d(Mn-1u+Mn-1w)+βNn-1v,Mnw=(kβe-mτ-ap)Mnuap,Mnv=k(kβe-mτ-ap)Mnuap2,Nnu=L(Nn-1u+Nn-1w)d(Nn-1u+Nn-1w)+βMnv,Nnw=(kβe-mτ-ap)Nnuap,Nnv=k(kβe-mτ-ap)Nnuap2.
It is readily seen that
(4.58)Nnu≤u_≤u¯≤Mnu,Nnw≤w_≤w¯≤Mnw,Nnv≤v_≤v¯≤Mnv.
The sequences Mnu, Mnw, and Mnv are nonincreasing and the sequences Nnu, Nnw, and Nnv are nondecreasing.To prove the monotonicity ofNnu and Mnu, we follow the ideas of Uh Zapata et al. [6]; consider R0=βke-mτ/ap and the following expressions for Nu, Nw, Nv, Mu, Mw, and Mv(4.59)Mnu=L(Mn-1u+Mn-1w)d(Mn-1u+Mn-1w)+βNn-1,Mnw=(R0-1)Mnu,Mnv=kp(R0-1)Mnu,Nnu=L(Nn-1u+Nn-1w)d(Nn-1u+Nn-1w)+βMnv,Nnw=(R0-1)Nnu,Nnv=kp(R0-1)Nnu.
We prove the result by induction so we first show that M2u-M1u≤0,
(4.60)M2u-M1u=Le-mτM1udeM1u+a(R0-1)N1u-Ld≤Ld-Ld=0
and N2u-N1u>0,
(4.61)N2u-N1u=LR0N1dR0N1u+(βh/p)(R0-1)M2u-N1=N1u(LR0-dR0N1u-(βh/p)(R0-1)M2udR0N1u+(βh/p)(R0-1)M2u)=N1udR0N1u+(βh/p)(R0-1)M2u(LR0-LR0+βR0M1v-hβp(R0-1)M2u)=N1udR0N1u+(βh/p)(R0-1)M2u(βhpR0(R0-1)M1u-hβp(R0-1)M2u)=βh(R0-1)N1up(dR0N1u+(βh/p)(R0-1)M2u)(R0M1u-M2u)>βh(R0-1)N1up(dR0N1u+(βh/p)(R0-1)M2u)(R0M1u-M2u)>0.
For the next step consider the function f(x)=ax/(bx+c), with a, b, and c positive, which is monotone increasing. The induction hypothesis is Mnu≤Mn-1u and Nnu≥Nn-1u,
(4.62)Mn+1u=LR0MnudR0Mnu+βNnu=LR0MnudR0Mnu+(βh/p)(R0-1)Nnu.
The last function is increasing and by the induction hypothesis we have
(4.63)Mn+1u≤LR0Mn-1udR0Mn-1u+(βh/p)(R0-1)Nnu≤LR0Mn-1udR0Mn-1u+(βh/p)(R0-1)Nnu=Mnu,
therefore the sequence Mnu is nonincreasing. For the sequence Nnu we use similar ideas and the behaviour of the sequence Mnu just proved. One has
(4.64)Nn+1u=LR0NnuLR0Nnu+βMn+1v=LR0NnuLR0Nnu+(βh/p)(R0-1)Mn+1u.
The last function is increasing and by the induction hypothesis we have
(4.65)Nn+1u≥LR0Nn-1LR0Nn-1+(βh/p)(R0-1)Mn+1u≥LR0Nn-1LR0Nn-1+(βh/p)(R0-1)Mnu=Nnu,
therefore the sequence Nnu is nondecreasing. The behaviour for the sequences Nnw, Nnv, Mnw, and Mnv follows from the nonincreasing sequence Mnu and the nondecreasing sequence Nnu.Hence, the limit of each sequence inNnu, Nnw, Nnv, Mnu, Mnw, and Mnv exists. Denote
(4.66)x¯=limn→∞Mnu,x_=limn→∞Nnu,y¯=limn→∞Mnw,y_=limn→∞Nnw,z¯=limn→∞Mnv,z_=limn→∞Nnv.
We therefore obtain from (4.57) and (4.66) that
(4.67)(x¯-x_)[dkβe-mτap-kβ(kβe-mτ)-apap2]=0.
Noting that (H1) holds and R0>1, it follows that
(4.68)dkβe-mτap>kβ(kβe-mτ-ap)ap2,
which together with the previous equation yields x¯=x_. We therefore derive from (4.57) y (4.66) that y¯=y_, z¯=z_. Noting that if R0>1, by Theorem 3.1, the virus-infected equilibrium E2 is locally asymptotically stable, and if in addition (H1) holds, we conclude that E2 is globally asymptotically stable.Now we prove the global stability for the disease-free equilibriumTheorem 4.3.
IfR0<1 the disease-free equilibrium E1(L/d,0,0) of (1.3) is globally asymptotically stable.Proof.
Let(u(x,t),w(x,t),v(x,t)) be a solution to problem with ϕi(x,0)≠0, (i=1,2,3). We have u(x,t)>0, w(x,t)>0, and v(x,t)>0 for all x∈Ω-. Let (u(1)(x,t),w(1)(x,t),v(1)(x,t)) be a solution to the following problem:
(4.69)∂u(1)∂t=L-du(1)(x,t),∂w(1)∂t=βe-mτu(1)(x,t-τ)v(1)(x,t-τ)u(1)(x,t-τ)+w(1)(x,t-τ)-aw(1)(x,t),∂v(1)∂t=DΔv(1)(x,t)+kw(1)(x,t)-pv(1)(x,t).
Therefore for t>0, x∈Ω- we have
(4.70)0≤u(x,t)≤u(1)(x,t),0≤w(x,t)≤w(1)(x,t),0≤v(x,t)≤v(1)(x,t).
We derive from the first equation that
(4.71)limt→∞u(1)(x,t)=Ld,
so we can conclude
(4.72)limsupt→∞maxu(x,t)≤Ld.
Hence, for ϵ>0 sufficiently small, there exists a t1 such that u(1)(x,t)≤L/d+ϵ for all x∈Ω- and t≥t1. Hence, (w(x,t),v(x,t)) is a lower solution to the following problem:
(4.73)∂ω2(1)∂t=βe(-mτ)(L/d+ϵ)v(1)(x,t-τ)L/d+ϵ+ω2(1)(x,t-τ)-aω2(1)(x,t),∂ω3(1)∂t=DΔω3(1)+kω2(1)(x,t)-pω3(1)(x,t),∂ω2(1)∂η=∂ω3(1)∂η=0,t>t1;x∈∂Ω^,ω2(1)(x,t)=w(x,t),ω3(1)(x,t)=v(x,t),t∈[-τ,t1],x∈Ω-.
Consider (u2(t),u3(t)) as the solution for
(4.74)u˙2=βe-mτ(L+ϵ)u3(t-τ)L/d+ϵ+u2(t-τ)-au2(t),u˙3=ku2(t)-pu3(t),u2(t)=maxx∈Ω-w(x,t),u3(t)=maxx∈Ω-v(x,t),t∈[-τ,t1].
Then with R0<1 according to Lemma 4.1 we have that
(4.75)limt→∞u2(t)=0,limt→∞u3(t)=0.
By comparison, it follows that
(4.76)limt→∞w(t)=0,limt→∞v(t)=0
uniformly for x∈Ω-. Hence, for ϵ>0 sufficiently small, by comparison there is a t2≥t1 such that if t≥t2, w(x,t)<ϵ, v(x,t)<ϵ for all x∈Ω- and t≥t2.
As in the proof of Theorem4.2  u(x,t) is an upper solution for the following problem:
(4.77)∂ω1(1)=L-dω1(1)-βω1(1)ϵ,∂ω1(1)∂η=0,t≥t2,x∈Ω-,ω1(1)(x,t2)=12u(x,t2),x∈Ω-,
from the above equation we have that
(4.78)limt→∞ω1(1)(x,t)=Ld+βϵ
uniformly for x∈Ω-. Since this holds for arbitrary ϵ>0 sufficiently small, by comparison we conclude that
(4.79)liminft→∞minu(x,t)≥Ld,
which together with (4.72) gives
(4.80)limt→∞u(x,t)=Ld
uniformly for x∈Ω-. We already have by Theorem 3.1 that the disease-free equilibrium E1 is locally asymptotically stable. And now we have proved that it is also globally asymptotically stable.


## 5. Numerical Simulations

In this section we illustrate some numerical solutions for systems (1.3). In the numerical simulation display in Figure 1 we illustrate the stability for the disease-free equilibrium according to Theorem 3.1. In this case the basic reproductive number is R0=0.039426. In the graphics we see how the level of uninfected cells increases from the initial condition and the number of infected cells and virus in the body goes to zero.Figure 1
Simulations with parametersL=211, d=3.78-2, a=3.38d, p=0.67, β=1.45-6, k=5.183, m=0.2, and D=0.5. R0=0.039426.In Figure2 consider the case R0>1 in this case we consider a bigger rate of infection for the cells in the graphics we see how the number of infected cells and viruses increases when the time passes, and when the number of susceptible cells decreases the number of virus also decreases to the value v*.Figure 2
Simulations with parametersL=211, d=3.78-2, a=3.38d, p=0.67, β=1.45-4, k=5.18×103, m=0.2, and D=0.1. R0=3.9426.Now in Figure3 we just show the level of virus in different for different values of the diffusion constant D and the delay τ. We see that a bigger delay increases the time needed for the virus to reach the value v*, meanwhile a mayor value for the constant D just affects the levels of the virus according to the space and does not affect significantly the time needed for the virus to reach v*. In Figure 4 we consider a lower value for λ, which is the uninfected cell production rate. In this case we see how the time to reach the value v* of the endemic equilibrium is lower and again the diffusion rate has no significant effect on the time; its effect is on the level of virus in the system. The delay is what really affect the time to reach the value v*.In this case the parameters areL=211, d=3.78-2, a=3.38d, p=0.67, β=1.45-4, k=5.183 and m=0.2.
(a)
D
=
0.01
,
τ
=
4
(b)
D
=
0.01
,
τ
=
0.9
(c)
D
=
1
,
τ
=
4
(d)
D
=
1
,
τ
=
0.9In this case the parameter areL=27, d=3.78-2, a=3.38d, p=0.67, β=1.45-4, k=5.183 and m=0.2.
(a)
D=0.01,τ=0.9
(b)
D
=
0.01
,
τ
=
4
(c)
D
=
1
,
τ
=
0.9
(d)
D
=
1
,
τ
=
4



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*Source: 578561-2012-11-05.xml*