Analysis of a HBV Model with Diffusion and Time Delay

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.

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: 𝑢̇(𝑡)(𝑡)(𝑡)𝑣(𝑡),𝑤̇(𝑡)(𝑡)𝑣(𝑡)(𝑡),𝑣̇(𝑡)(𝑡)(𝑡),=𝑠−𝜇𝑢−𝛽𝑢=𝛽𝑢−𝑎𝑤=𝑘𝑤−𝑑𝑣 (1.1) 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: 𝑥̇(𝑡)(𝑡)−(𝑡)𝑣(𝑡)=𝜆−𝑑𝑥𝛽𝑥𝑥(𝑡)(𝑡),𝑦̇(𝑡)=+𝑦𝛽e^−𝑚𝜏𝑥(𝑡−𝜏)𝑣(𝑡−𝜏)𝑥()()(𝑡),𝑣̇(𝑡)(𝑡)(𝑡).𝑡−𝜏+𝑦𝑡−𝜏−𝑎𝑦=𝑘𝑦−𝑢𝑣 (1.2) 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: 𝜕𝑢()−()𝑣()𝜕𝑡=𝐿−𝑑𝑢𝑥,𝑡𝛽𝑢𝑥,𝑡𝑥,𝑡𝑢()(),𝑥,𝑡+𝑤𝑥,𝑡𝜕𝑤=𝜕𝑡𝛽e^−𝑚𝜏𝑢(𝑥,𝑡−𝜏)𝑣(𝑥,𝑡−𝜏)𝑢()()(),𝑥,𝑡−𝜏+𝑤𝑥,𝑡−𝜏−𝑎𝑤𝑥,𝑡𝜕𝑣()(),𝜕𝑡=𝐷Δ𝑣+𝑘𝑤𝑥,𝑡−𝑝𝑣𝑥,𝑡 (1.3) for 𝑡>0, 𝑥∈Ω, with homogeneous Neumann boundary conditions 𝜕𝑣𝜕𝜂=0, (1.4) and initial conditions 𝑢(𝑥,𝜃)=𝜙₁(𝑥,𝜃)≥0,𝑤(𝑥,𝜃)=𝜙₂()𝑣()𝑥,𝜃≥0,𝑥,𝜃=𝜙₃(𝑥,𝜃)≥0,𝜃∈[−𝜏,0],𝑥∈Ω. (1.5) In the previous problem Ω is a bounded domain in ℝ^𝑛 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 Section 3 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 functions 𝑈=(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: 𝜕𝑢𝑢()−𝛽𝑢()𝑣()𝜕𝑡≥𝐿−𝑑𝑥,𝑡𝑥,𝑡𝑥,𝑡𝑢()+𝑤(),𝜕𝑤𝑥,𝑡𝑥,𝑡≥𝜕𝑡𝛽e^−𝑚𝜏𝑢(𝑥,𝑡−𝜏)𝑣(𝑥,𝑡−𝜏)𝑢()+𝑤()𝑤(),𝜕𝑣𝑥,𝑡−𝜏𝑥,𝑡−𝜏−𝑎𝑥,𝑡𝑤()𝑣(),𝜕𝑢𝜕𝑡≥𝐷Δ𝑣+𝑘𝑥,𝑡−𝑝𝑥,𝑡𝑢()−𝛽𝑢()𝑣()𝜕𝑡≤𝐿−𝑑𝑥,𝑡𝑥,𝑡𝑥,𝑡𝑢()+𝑤(),𝜕𝑤𝑥,𝑡𝑥,𝑡≤𝜕𝑡𝛽e^−𝑚𝜏𝑢(𝑥,𝑡−𝜏)𝑣(𝑥,𝑡−𝜏)𝑢()+𝑤()𝑤(),𝜕𝑣𝑥,𝑡−𝜏𝑥,𝑡−𝜏−𝑎𝑥,𝑡𝑤()𝑣(),𝜕𝑡≤𝐷Δ𝑣+𝑘𝑥,𝑡−𝑝𝑥,𝑡 (2.1) for (𝑥,𝑡)∈Ω×(0,∞), and 𝜕𝑢𝜕𝑢𝜕𝜂≤0≤,𝜕𝑤𝜕𝜂𝜕𝑤𝜕𝜂≤0≤,𝜕𝑣𝜕𝜂𝜕𝑣𝜕𝜂≤0≤()(),𝑢()𝜕𝜂𝑥,𝑡∈𝜕Ω×0,∞𝑥,𝑡≤𝜙₁(𝑥,𝑡)≤𝑢(𝑥,𝑡),𝑤(𝑥,𝑡)≤𝜙₂()≤𝑤(),𝑣()𝑥,𝑡𝑥,𝑡𝑥,𝑡≤𝜙₃(𝑥,𝑡)≤𝑣(𝑥,𝑡),(𝑥,𝑡)∈Ω×[−𝜏,0]. (2.2)

The following lemma then follows from Theorem 3.4 developed by Redlinger [5].

Lemma 2.2. Let 𝑈 and 𝑈 be a pair of coupled upper and lower solutions for problem (1.3)–(1.5) and suppose that the initial functions 𝜙_𝑖 (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 that 0=(0,0,0) and 𝐾=(𝐾₁,𝐾₂,𝐾₃) are a pair of coupled lower-upper solutions to problem (1.3)–(1.5), where 𝐾₁𝐿=max𝑑,sup_{−𝜏≤𝜃≤0}𝜙₁(⋅,𝜃)_{𝐶(Ω,𝑅)},𝐾₂=max𝑘𝛽e^−𝑚𝜏𝐾₁𝑎𝑝,sup_{−𝜏≤𝜃≤0}𝜙₂(⋅,𝜃)_{𝐶(Ω,𝑅)},𝐾₃𝑘=max²𝛽e^−𝑚𝜏𝐾₁𝑎𝑝²,sup_{−𝜏≤𝜃≤0}𝜙₃(⋅,𝜃)_{𝐶(Ω,𝑅)}. (2.3) 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 𝜙_𝑖(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 𝑢^∗(𝜏)=𝐿e^−𝑚𝜏𝑑e^−𝑚𝜏+𝑎𝑅₀−1,𝑤^∗(𝜏)=𝐿e^−𝑚𝜏𝑅₀−1𝑑e^−𝑚𝜏+𝑎𝑅₀,𝑣−1_∗(𝜏)=𝐿𝑘e^−𝑚𝜏𝑅₀−1𝑝𝑑e^−𝑚𝜏+𝑎𝑅₀.−1 (3.1)

Let 0=𝜇₁<𝜇₂<⋯ be the eigenvalues of the operator −Δ on Ω with the homogeneous Neumann boundary conditions, and let 𝐸(𝜇_𝑖) be the eigenspace corresponding to 𝜇_𝑖 in 𝐶¹(Ω).

Let 𝕏=[𝐶¹(Ω)]³, let {𝜙_𝑖𝑗;𝑗=1,2,…,dim𝐸(𝜇_𝑖)} be an orthonormal basis of 𝐸(𝜇_𝑖), and let 𝕏_𝑖𝑗={𝑐𝜙_𝑖𝑗∣𝑐∈ℝ³}, then 𝕏=^∞_𝑖=0𝕏_𝑖,𝕏_𝑖=^{dim𝐸(𝑝𝑖)}_𝑗=1𝕏_𝑖𝑗. (3.2) Let 𝒟=diag(0,0,𝐷), 𝑍=𝑍(𝑥,𝑡)=(𝑈(𝑥,𝑡),𝑊(𝑥,𝑡),𝑉(𝑥,𝑡)), ℒ𝑍=𝒟Δ𝑍+𝐽_𝐸^∗𝑍+𝐽_𝜏𝐸^∗𝑍(𝑡−𝜏), where 𝐽_𝐸^∗=𝛽−𝑑−𝑤𝑣(𝑢+𝑤)²𝛽𝑢𝑣(𝑢+𝑤)²−𝛽𝑢𝑢+𝑤,𝐽0−𝑎00𝑘−𝑝_𝜏=000𝛽e^−𝑚𝜏𝑣𝑤(𝑢+𝑤)²−𝛽e^−𝑚𝜏𝑢𝑣(𝑢+𝑤)²𝛽e^−𝑚𝜏𝑢𝑢+𝑤,000 (3.3) 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 𝑍_𝑡=ℒ𝑍. For each 𝑖≥1,𝕏_𝑖 is invariant under the operator ℒ, and 𝜆 is an eigenvalue of the matrix −𝜇_𝑖𝒟+𝐽_𝐸^∗+𝐽_𝜏𝐸^∗ for some 𝑖≥1, then, there is an eigenvector in 𝕏_𝑖.

The characteristic equation on the equilibrium 𝐸₁ is (𝜆+𝑑)𝜆²+𝑎₁𝜆+𝑎₀+𝑏₀(𝜏)e^−𝜆𝜏=0, (3.4) where 𝑎₀=𝑎𝑝+𝜇_𝐼𝐷,𝑎₁=𝑎+𝑝+𝜇_𝑖𝐷,𝑏₀(𝜏)=−𝑘𝛽e^−𝑚𝜏. (3.5) The characteristic equation has the negative root 𝜆=−𝑑. All other roots of (3.4) are given by the transcendental equation 𝜆²+𝑎₁𝜆+𝑎₀+𝑏₀(𝜏)e^−𝜆𝜏=0. (3.6) Let 𝑓(𝜆)=𝜆²+𝑎₁𝜆+𝑎₀+𝑏₀(𝜏)e^−𝜆𝜏, (3.7) if 𝑅₀>1, note that for 𝜆 real and 𝑖=1 (in this case 𝜇₁=0), 𝑓(0)=𝑎₀+𝑏₀=𝑎𝑝−𝑘𝛽e^−𝑚𝜏<0,lim_𝜆→∞𝑓(𝜆)=+∞. (3.8) 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.

If 𝑅₀<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.

For 𝜏>0 if 𝑖𝜔(𝜔>0) is a solution of (3.6), separating in real and imaginary parts, we obtain −𝜔²+𝑎₀=−𝑏₀(𝜏)(),𝑎cos𝜔𝜏₁𝜔=𝑏₀(𝜏)sin(𝜔𝜏). (3.9) Squaring and adding the above equations and taking 𝑧=𝜔² we obtain 𝑧²+𝑎²₁−2𝑎₀𝑧+𝑎²₀−𝑏₀(𝜏)²=0, (3.10) where 𝑎²₁−2𝑎₀=𝑎²+𝑝+𝜇_𝑖𝐷²𝑎>0,₂₀−𝑏²₀(𝜏)=𝑎𝑝+𝜇_𝑖𝐷−𝑘𝛽e^−𝑚𝜏𝑎𝑝+𝜇_𝑖𝐷+𝑘𝛽e^−𝑚𝜏>0, (3.11) 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 𝜆³+𝑎₂(𝜏)𝜆²+𝑎₁𝜆+𝑎₀(𝜏)+𝑏₂(𝜏)𝜆²+𝑏₁(𝜏)𝜆+𝑏₀(𝜏)e^−𝜆𝜏=0, (3.12) where 𝑎₂(𝜏)=𝑎+𝑑+𝑝+𝜇_𝑖𝐷+𝛽𝑣^∗𝑤^∗(𝑢^∗+𝑤^∗)²,𝑎₁=𝑎𝑝+𝜇_𝑖𝐷+𝑎+𝑝+𝜇_𝑖𝐷𝑎𝑑,₀(𝜏)=𝑎𝑝+𝜇_𝑖𝐷𝑑+𝛽𝑤^∗𝑣^∗(𝑢^∗+𝑤^∗)²,𝑏₂(𝜏)=𝛽e^−𝑚𝜏𝑢^∗𝑣^∗(𝑢^∗+𝑤^∗)^∗,𝑏₁(𝜏)=𝑑+𝑝+𝜇_𝑖𝐷𝛽e^−𝑚𝜏𝑢^∗𝑤^∗(𝑢^∗+𝑤^∗)²−𝑘𝛽e^−𝑚𝜏𝑢^∗𝑢^∗+𝑤^∗,𝑏₀(𝜏)=𝑑𝑝+𝜇_𝑖𝐷𝛽e^−𝑚𝜏𝑢^∗𝑣^∗(𝑢^∗+𝑤^∗)²−𝑑𝑘𝛽e^−𝑚𝜏𝑢^∗𝑢^∗+𝑤^∗, (3.13) when 𝜏=0 becomes 𝜆³+𝑎₂(0)+𝑏₂(0)𝜆²+𝑎₁+𝑏₁(0)𝜆+𝑎₀(0)+𝑏₀(0)=0. (3.14) Note that 𝑎₂(0)+𝑏₂(0)>0; adding 𝑎₀(0)+𝑏₀(0) and replacing 𝑢^∗(0) and 𝑤^∗(0) we obtain 𝑎₀(0)+𝑏₀(0)=𝑎𝑝+𝜇_𝑖𝐷𝛽𝑤^∗(0)𝑣^∗(0)(𝑢^∗(0)+𝑤^∗(0))²+𝑑𝑝+𝜇_𝑖𝐷𝛽𝑢^∗(0)𝑣^∗(0)(𝑢^∗(0)+𝑤^∗(0))²𝑎>0,₂(0)+𝑏₂(0)𝑎₁+𝑏₁(0)−𝑎₀(0)+𝑏₀(0)=𝑑𝑎+𝑝+𝜇_𝑖𝐷+𝑢^∗(0)+𝑤^∗(0)𝑝+𝜇_𝑖𝐷𝛽𝑣^∗(0)(𝑢^∗(0)+𝑤^∗(0))²×𝑎+𝑑+𝑝+𝜇_𝑖𝐷+𝑢^∗(0)+𝑤^∗(0)𝛽𝑣^∗(𝑢^∗(0)+𝑤^∗(0))²+𝜆𝛽𝑣^∗(0)(𝑢^∗(0)+𝑤^∗(0))²𝑎+𝑑+𝑢^∗(0)+𝑤^∗(0)𝛽𝑣^∗(0)(𝑢^∗(0)+𝑤^∗(0))²>0. (3.15) By the Routh-Hurwitz criteria, all roots have negative real parts if 𝑅₀>1.

For the case 𝜏>0 we look for solutions 𝜆=𝑖𝜔(𝜔>0) for (3.12), separating real and imaginary parts, it follows that 𝜔³−𝑎₁𝜔=𝑏₂(𝜏)𝜔²−𝑏₀(𝜏)sin(𝜔𝜏)+𝑏₁(𝜏)(),𝑎𝜔cos𝜔𝜏₂(𝜏)𝜔²−𝑎₀(𝜏)=−𝑏₂(𝜏)𝜔²−𝑏₀(𝜏)cos(𝜔𝜏)+𝑏₁(𝜏)𝜔sin(𝜔𝜏). (3.16) Squaring and adding the two equations, we derive that 𝜔⁶+𝐶₁𝜔⁴+𝐶₂𝜔²+𝐶₃=0, (3.17) where 𝐶₁=𝑝+𝜇_𝑖𝐷²+𝑑+𝛽𝑤^∗(𝜏)𝑣^∗(𝜏)(𝑢^∗(𝜏)+𝑤^∗(𝜏))²²+𝑎𝑢^∗(𝜏)𝑢^∗(𝜏)+𝑤^∗(𝜏)𝑎+𝛽e^−𝑚𝜏𝑢^∗(𝜏)𝑣^∗(𝜏)(𝑢^∗(𝜏)+𝑤^∗(𝜏))²𝐶>0,₂=𝑎²𝑑²+𝑝+𝜇_𝑖𝐷²𝑤^∗(𝜏)𝑢^∗(𝜏)+2𝑤^∗(𝜏)(𝑢^∗(𝜏)+𝑤^∗(𝜏))²+𝑎²𝛽𝑤^∗(𝜏)𝑣^∗(𝜏)(𝑢^∗(𝜏)+𝑤^∗(𝜏))²2𝑑+𝛽𝑤^∗(𝜏)𝑣^∗(𝜏)(𝑢^∗(𝜏)+𝑤^∗(𝜏))²+𝑝+𝜇_𝑖𝐷𝑑+𝛽𝑤^∗(𝜏)+𝑣^∗(𝜏)(𝑢^∗(𝜏)+𝑤^∗(𝜏))²𝐶>0,₃=𝑎𝑝+𝜇_𝑖𝐷²𝛽𝑣^∗(𝜏)𝑑e^−𝑚𝜏𝑢^∗(𝜏)+𝑎𝑤^∗(𝜏)(𝑢^∗(𝜏)+𝑤^∗(𝜏))³𝑑2𝑥^∗(𝜏)+𝑤^∗(𝜏)+𝛽𝑤^∗(𝜏)𝑣^∗(𝜏)𝑢^∗(𝜏)+𝑤^∗(𝜏)>0, (3.18) implying that (3.17) has no positive roots 𝑧=𝜔².

Theorem 3.1. If 𝑅₀<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: 𝑢̇₁(𝑡)=𝑎₁𝛽e^−𝑚𝜏𝑢₂(𝑡−𝜏)𝑎₁+𝑢₁(𝑡−𝜏)−𝑎𝑢₁(𝑡),𝑢̇₂(𝑡)=𝑘𝑢₁(𝑡)−𝑝𝑢₂(𝑡), (4.1) with initial conditions 𝑢_𝑖(𝑠)=𝜙_𝑖(𝑠)≥0,𝑠∈[−𝜏,0),𝜙_𝑖(0)>0,𝜙_𝑖∈𝐶[−𝜏,0],ℝ₊. (4.2) 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^∗(𝑢^∗₁,𝑢^∗₂) where 𝑢^∗₁=𝑎₁𝑘𝛽e^−𝑚𝜏−𝑎𝑝𝑎𝑝,𝑢^∗₂=𝑎₁𝑘𝑘𝛽e^−𝑚𝜏−𝑎𝑝𝑎𝑝², (4.3) and according to [2], for system (4.1), one has the following.

Lemma 4.1. If 𝑘𝛽e^−𝑚𝜏>𝑎𝑝, then the positive equilibrium 𝐴^∗(𝑢^∗₁,𝑢^∗₂) is globally stable.
If 𝑘𝛽e^−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 𝜙_𝑖(𝑥,0)≢0(𝑖=1,2,3). If 𝑅₀>1 and(H1)𝑑𝑝e^−𝑚𝜏>𝑘𝛽e^−𝑚𝜏−𝑎𝑝, then lim_𝑡→∞(𝑢(𝑥,𝑡),𝑤(𝑥,𝑡),𝑣(𝑥,𝑡))=𝑢^∗,𝑤^∗,𝑣^∗uniformlyfor𝑥∈Ω, (4.4) 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 𝜙_𝑖≢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 𝑢=limsup_𝑡→∞max_𝑥∈Ω𝑢(𝑥,𝑡),𝑢=liminf_𝑡→∞min_𝑥∈Ω𝑢(𝑥,𝑡),𝑤=limsup_𝑡→∞max_𝑥∈Ω𝑤(𝑥,𝑡),𝑤=liminf_𝑡→∞min_𝑥∈Ω𝑤(𝑥,𝑡),𝑣=limsup_𝑡→∞max_𝑥∈Ω𝑣(𝑥,𝑡),𝑣=liminf_𝑡→∞min_𝑥∈Ω𝑣(𝑥,𝑡). (4.5) First we look for upper solutions for the system (1.3). Let (𝑢¹(𝑥,𝑡),𝑤¹(𝑥,𝑡),𝑣¹(𝑥,𝑡)) be a solution for the following problem: 𝜕𝑢⁽¹⁾𝜕𝑡=𝐿−𝑑𝑢⁽¹⁾()𝜕𝑥,𝑡,𝑡>0,𝑥∈Ω,𝑤⁽¹⁾=𝜕𝑡𝛽e^−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)=𝜕𝜕𝑡𝑤⁽¹⁾(𝑥,𝑡)=𝜕𝜕𝑡𝑣⁽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. (4.6) 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 0≤𝑢(𝑥,𝑡)≤𝑢⁽¹⁾(𝑥,𝑡),0≤𝑤(𝑥,𝑡)≤𝑤⁽¹⁾(𝑥,𝑡),0≤𝑣(𝑥,𝑡)≤𝑣⁽¹⁾(𝑥,𝑡). (4.7) From the first equation of (4.6) lim_𝑡→∞𝑢⁽¹⁾()=𝐿𝑥,𝑡𝑑=𝑀^𝑢₁. (4.8) Hence, by comparison, for all 𝜖>0 sufficiently small, there exists 𝑡₁>0 such that if 𝑡>𝑡₁max_𝑥∈Ω𝑢⁽¹⁾(𝑥,𝑡)≤𝑀^𝑢₁+𝜖, (4.9) since 𝜖 is arbitrary and sufficiently small we can conclude that 𝑢=limsup_𝑡→∞max_𝑥∈Ω𝑢(𝑥,𝑡)≤𝑀^𝑢₁. (4.10) Now consider the problem related with the second and third equations of (4.6) 𝜕𝜔₂⁽¹⁾=𝜕𝑡𝛽e^−𝑚𝜏𝑀^𝑢₁+𝜖𝜔₃⁽¹⁾(𝑥,𝑡−𝜏)𝑀^𝑢₁+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>𝑡₁𝜔,𝑥∈𝜕Ω,₂₍₁₎(𝑥,𝑡)=𝑤(𝑥,𝑡),𝜔₃⁽¹⁾(𝑥,𝑡)=𝑣(𝑥,𝑡),𝑡∈−𝜏,𝑡₁,𝑥∈Ω. (4.11) Consider the solution for 𝑢̇₂=𝛽e^−𝑚𝜏𝑀^𝑢₁+𝜖𝑢₃(𝑡−𝜏)𝑀^𝑢₁+𝜖+𝑢₂(𝑡−𝜏)−𝑎𝑢₂,𝑡>𝑡₁,𝑢̇₃=𝑘𝑢₂−𝑝𝑢₃,𝑡>𝑡₁,𝑢₂(𝑡)=max_𝑥∈Ω𝑤(𝑥,𝑡),𝑢₃(𝑡)=max_𝑥∈Ω𝑣(𝑥,𝑡),𝑡∈−𝜏,𝑡₁. (4.12) 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 lim_𝑡→∞𝑢₂(𝑡)=𝑘𝛽e^−𝑚𝜏−𝑎𝑝𝑀^𝑢₁+𝜖,𝑎𝑝lim_𝑡→∞𝑢₃(𝑡)=𝑘𝑘𝛽e^−𝑚𝜏−𝑎𝑝𝑀^𝑢₁+𝜖𝑎𝑝². (4.13) Hence, for all 𝜖>0 sufficiently small, by comparison there exists a 𝑡₂>𝑡₁ such that if 𝑡>𝑡₂max_𝑥∈Ω𝑤⁽¹⁾(𝑥,𝑡)<𝑀^𝑤₁+𝜖,max_𝑥∈Ω𝑣⁽¹⁾(𝑥,𝑡)<𝑀^𝑣₁+𝜖, (4.14) where 𝑀^𝑤₁=𝑘𝛽e^−𝑚𝜏−𝑎𝑝𝑀^𝑢₁𝑎𝑝,𝑀^𝑣₁=𝑘𝑘𝛽e^−𝑚𝜏−𝑎𝑝𝑀^𝑢₁𝑎𝑝². (4.15) Since 𝜖>0 is arbitrary and sufficiently small, we conclude that 𝑤=limsup_𝑡→∞max_𝑥∈Ω𝑤(𝑥,𝑡)≤𝑀^𝑤₁,𝑣=limsup_𝑡→∞max_𝑥∈Ω𝑣(𝑥,𝑡)≤𝑀^𝑣₁. (4.16) Now for lower solutions, let (𝑢⁽¹⁾(𝑥,𝑡),𝑤⁽¹⁾(𝑥,𝑡),𝑣⁽¹⁾(𝑥,𝑡)) be the solution for the following problem: 𝜕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⁾=𝜕𝑡𝛽e^−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>𝑡₂𝑢,𝑥∈𝜕Ω,₍₁₎()=1𝑥,𝑡2𝑢(𝑥,𝑡),𝑤⁽¹⁾()=1𝑥,𝑡2𝑤(),𝑣𝑥,𝑡₍₁₎()=1𝑥,𝑡2𝑣(𝑥,𝑡),𝑡∈−𝜏,𝑡₂,𝑥∈Ω. (4.17) 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 𝜕𝑢⁽¹⁾𝜕𝑡≥𝐿−𝑑𝑢⁽¹⁾(𝑥,𝑡)−𝛽𝑀^𝑣₁+𝜖,𝑡>𝑡₂,𝑥∈Ω. (4.18) By comparing the above equation with the following problem: 𝜕𝜔₁⁽¹⁾𝜕𝑡=𝐿−𝑑𝜔₁⁽¹⁾(𝑥,𝑡)−𝛽𝑀^𝑣₁+𝜖,𝑡>𝑡₂,𝑥∈Ω,𝜕𝜔₁𝜔𝜂=0,𝑡>𝑡₂,𝑥∈𝜕Ω,𝜔₁𝑥,𝑡₂=12𝑢𝑥,𝑡₂,𝑥∈Ω, (4.19) we obtain lim_𝑡→∞𝜔₁()=𝑀𝑥,𝑡𝐿−𝛽^𝑣₁+𝜖𝑑, (4.20) 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, min_𝑥∈Ω𝑢⁽¹⁾(𝑥,𝑡)≥𝑁^𝑢₁−𝜖, (4.21) where 𝑁^𝑢₁=𝐿−𝛽𝑀^𝑣₁𝑑. (4.22) Since 𝜖>0 is arbitrary sufficiently small, by comparison we conclude that 𝑢=liminf_𝑡→∞min_𝑥∈Ω𝑢(𝑥,𝑡)≥𝑁^𝑢₁=𝐿−𝛽𝑀^𝑣₁𝑑. (4.23) Now consider the following problem related with the second and third equations of (4.17): 𝜕𝜔₂=𝜕𝑡𝛽e^−𝑚𝜏𝑁^𝑢₁−𝜖𝜔₃(𝑥,𝑡−𝜏)𝑁^𝑢₁−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>𝑡₃𝜔,𝑥∈𝜕Ω,₂()=1𝑥,𝑡2𝑤(𝑥,𝑡),𝜔₃()=1𝑥,𝑡2𝑣(𝑥,𝑡),𝑡∈−𝜏,𝑡₃,𝑥∈Ω. (4.24) Now let us consider the solution for the problem 𝑢̇₂(𝑡)=𝛽e^−𝑚𝜏𝑁^𝑢₁−𝜖𝑢₃(𝑡−𝜏)𝑁^𝑢₁−𝜖+𝑢₂(𝑡−𝜏)−𝑎𝑢₂(𝑡),𝑡>𝑡₃,𝑢̇₃(𝑡)=𝑘𝑢₂(𝑡)−𝑝𝑢₃(𝑡),𝑡>𝑡₃,𝑢₂(𝑡)=12min_𝑥∈Ω𝑤(𝑥,𝑡),𝑢₃(𝑡)=12min_𝑥∈Ω𝑣(𝑥,𝑡),𝑡∈−𝜏,𝑡₃,𝑡>𝑡₃, (4.25) and according to Lemma 4.1lim_𝑡→∞𝑢₂(𝑡)=𝑘𝛽e^−𝑚𝜏−𝑎𝑝𝑁^𝑢₁−𝜖,𝑎𝑝lim_𝑡→∞𝑢₃(𝑡)=𝑘𝑘𝛽e^−𝑚𝜏−𝑎𝑝𝑁^𝑢₁−𝜖𝑎𝑝². (4.26) Hence, for all 𝜖>0 sufficiently small, by comparison there exists a 𝑡₄>𝑡₃ such that if 𝑡>𝑡₄min_𝑥∈Ω𝑤⁽¹⁾(𝑥,𝑡)>𝑁^𝑤₁−𝜖,min_𝑥∈Ω𝑣⁽¹⁾(𝑥,𝑡)>𝑁^𝑣₁−𝜖, (4.27) where 𝑁^𝑤₁=𝑘𝛽e^−𝑚𝜏−𝑎𝑝𝑁^𝑢₁𝑎𝑝,𝑁^𝑣₁=𝑘𝑘𝛽e^−𝑚𝜏−𝑎𝑝𝑁^𝑢₁𝑎𝑝². (4.28) Since 𝜖>0 is arbitrary and sufficiently small, we conclude that 𝑤=liminf_𝑡→∞min_𝑥∈Ω𝑤(𝑥,𝑡)≥𝑁^𝑤₁,𝑣=liminf_𝑡→∞min_𝑥∈Ω𝑣(𝑥,𝑡)≥𝑁^𝑣₁. (4.29) Now we look for the closest upper and lower solutions. Let (𝑢⁽²⁾,𝑤⁽²⁾,𝑣⁽²⁾) be a solution for the problem 𝜕𝑢⁽²⁾𝜕𝑡=𝐿−𝑑𝑢⁽²⁾()−𝛽𝑥,𝑡𝑢⁽²⁾(𝑥,𝑡)𝑣⁽¹⁾(𝑥,𝑡)𝑢⁽¹⁾+𝑤⁽¹⁾(𝑥,𝑡),𝑡>𝑡₄𝜕,𝑥∈Ω,𝑤⁽²⁾=𝜕𝑡𝛽e^−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⁾=𝜕𝜕𝜂𝑤⁽²⁾=𝜕𝜕𝜂𝑣⁽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. (4.30) For all 𝜖>0 sufficiently small it follows form the first equation of (4.30) and the inequalities (4.27) and (4.14) that 𝜕𝑢⁽²⁾𝜕𝑡≤𝐿−𝑑𝑢⁽²⁾()−𝛽𝑥,𝑡𝑢⁽²⁾(𝑥,𝑡)𝑁^𝑣₁−𝜖𝑀^𝑢₁+𝜖+𝑀^𝑤₁+𝜖,𝑡>𝑡₄,𝑥∈Ω. (4.31) Let 𝜔₁⁽²⁾(𝑥,𝑡) be the solution for the following problem: 𝜕𝜔₁⁽²⁾𝜕𝑡=𝐿−𝑑𝜔₁⁽²⁾(𝑥,𝑡)−𝛽𝜔₁⁽²⁾(𝑥,𝑡)𝑁^𝑣₁−𝜖𝑀^𝑢₁+𝜖+𝑀^𝑤₁+𝜖,𝑡>𝑡₄,𝑥∈Ω,𝜕𝜔₁⁽²⁾𝜕𝜂=0,𝑡>𝑡₄𝜔,𝑥∈𝜕Ω,₁₍₂₎𝑥,𝑡₄=𝑢𝑥,𝑡₄,𝑥∈Ω, (4.32) it follows that lim_𝑡→∞𝜔₁⁽²⁾()=𝐿𝑀𝑥,𝑡^𝑢1+𝑀^𝑤₁+2𝜖𝑑𝑀^𝑢₁+𝑀^𝑤₁+2𝜖+𝛽𝑁^𝑣₁.−𝜖 (4.33) 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>𝑡₅max_𝑥∈Ω𝑢⁽²⁾(𝑥,𝑡)=𝑀^𝑢₂+𝜖, (4.34) where 𝑀^𝑢₂=𝐿𝑀^𝑢₁+𝑀^𝑤₁𝑑𝑀^𝑢₁+𝑀^𝑤₁+𝛽𝑁^𝑣₁. (4.35) Since (4.34) is valid for 𝜖>0 arbitrary and sufficiently small, by comparison we conclude that 𝑢=limsup_𝑡→∞max_𝑥∈Ω𝑢(𝑥,𝑡)≤𝑀^𝑢₂. (4.36) Now consider the following problem related with the second and third equations of (4.30): 𝜕𝜔₂⁽²⁾=𝜕𝑡𝛽e^−𝑚𝜏𝑀^𝑢₂+𝜖𝜔₃⁽²⁾(𝑥,𝑡)𝑀^𝑢₂+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>𝑡₅𝜔,𝑥∈𝜕Ω,₂₍₂₎(𝑥,𝑡)=𝑤(𝑥,𝑡),𝜔₃⁽²⁾(𝑥,𝑡)=𝑣(𝑥,𝑡),𝑡∈−𝜏,𝑡₅,𝑥∈Ω. (4.37) Let (𝑢₂(𝑡),𝑢₃(𝑡)) be the positive solution to the following problem: 𝑢̇₂(𝑡)=𝛽e^−𝑚𝜏𝑀^𝑢₂+𝜖𝑢₃(𝑡−𝜏)𝑀^𝑢₂+𝜖+𝑢₂(𝑡−𝜏)−𝑎𝑢₂(𝑡),𝑡>𝑡₅,𝑢̇₃(𝑡)=𝑘𝑢₂(𝑡)−𝑝𝑢₃(𝑡),𝑡>𝑡₅,𝑢₂(𝑡)=max_𝑥∈Ω𝑤(𝑥,𝑡),𝑢₃(𝑡)=max_𝑥∈Ω𝑣(𝑥,𝑡),𝑡∈−𝜏,𝑡₅. (4.38) Then by Lemma 4.1 and the previous system we have lim𝑢₂(𝑡)=𝑘𝛽e^−𝑚𝜏−𝑎𝑝𝑀^𝑢₂+𝜖,𝑎𝑝lim𝑢₃(𝑡)=𝑘𝑘𝛽e^−𝑚𝜏𝑀^𝑢₂+𝜖𝑎𝑝². (4.39) Hence for all 𝜖>0 sufficiently small, by comparison there is a 𝑡₆>𝑡₅ such that if 𝑡>𝑡₆max_𝑥∈Ω𝑤(𝑥,𝑡)<𝑀^𝑤₂+𝜖,max_𝑥∈Ω𝑣(𝑥,𝑡)<𝑀^𝑣₂+𝜖, (4.40) where 𝑀^𝑤₂=𝑘𝛽e^−𝑚𝜏−𝑎𝑝𝑀^𝑢₂,𝑀𝑎𝑝_𝑣2=𝑘𝑘𝛽e^−𝑚𝜏𝑀^𝑢₂𝑎𝑝². (4.41) Since 𝜖>0 is arbitrary and sufficiently small, we conclude that 𝑤=limsup_𝑡→∞max_𝑥∈Ω𝑤(𝑥,𝑡)≤𝑀^𝑤₂,𝑣=limsup_𝑡→∞max_𝑥∈Ω𝑣(𝑥,𝑡)≤𝑀^𝑣₂. (4.42) Let (𝑢⁽²⁾,𝑤⁽²⁾,𝑣⁽²⁾) be a solution for the following problem: 𝜕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⁾=𝜕𝑡𝛽e^−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>𝑡₆𝑢,𝑥∈Ω,₍₂₎()=1𝑥,𝑡2𝑢(𝑥,𝑡),𝑤⁽²⁾()=1𝑥,𝑡2𝑤(),𝑤𝑥,𝑡₍₂₎()=1𝑥,𝑡2𝑣(𝑥,𝑡),𝑡∈−𝜏,𝑡₆,𝑥∈Ω. (4.43) 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∈Ω𝑢⁽²⁾(𝑥,𝑡)≤𝑢(𝑥,𝑡)≤𝑢⁽²⁾(𝑥,𝑡),𝑤⁽²⁾(𝑥,𝑡)≤𝑤(𝑥,𝑡)≤𝑤⁽²⁾(),𝑣𝑥,𝑡₍₂₎(𝑥,𝑡)≤𝑣(𝑥,𝑡)≤𝑣⁽²⁾(𝑥,𝑡). (4.44) For all 𝜖>0 sufficiently small, it follow from the first equation of (4.43), and the inequalities 𝜕𝑢⁽²⁾𝜕𝑡≥𝐿−𝑑𝑢⁽²⁾(𝑥,𝑡)−𝛽𝑢⁽²⁾(𝑥,𝑡)𝑀^𝑣₁+𝜖𝑁^𝑢₁−𝜖+𝑁^𝑤₁.−𝜖 (4.45) 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 𝜕𝑣₁⁽²⁾𝜕𝑡=𝐿−𝑑𝑣₁⁽²⁾(𝑥,𝑡)−𝛽𝑣₁⁽²⁾(𝑥,𝑡)𝑀^𝑣₁+𝜖𝑁^𝑢₁−𝜖+𝑁^𝑤₁,−𝜖𝜕𝑣₁⁽²⁾𝜕𝜂=0,𝑡>𝑡₆,𝑥∈𝜕Ω, (4.46) which has satisfies lim_𝑡→∞𝑣₁⁽²⁾()=𝐿𝑁𝑥,𝑡^𝑢1−𝜖+𝑁^𝑤₁−𝜖𝑑𝑁^𝑢₁−𝜖+𝑁^𝑤₁−𝜖+𝛽𝑀^𝑣₁.+𝜖 (4.47) Hence for all 𝜖>0 sufficiently small, by comparison, there is a 𝑡₇>𝑡₆ such that if 𝑡>𝑡₇min𝑢⁽²⁾(𝑥,𝑡)≥𝑁^𝑢₂−𝜖, (4.48) with 𝑁^𝑢₂=𝐿𝑁^𝑢₁+𝑁^𝑤₁𝑑𝑁^𝑢₁+𝑁^𝑤₁+𝛽𝑀^𝑣₁. (4.49) Since this holds true for arbitrary 𝜖>0 sufficiently small, by comparison we conclude that 𝑢=liminf_𝑡→∞min_𝑥∈Ω𝑢(𝑥,𝑡)≥𝑁^𝑢₂. (4.50) Now consider the following problem: 𝜕𝑣₂⁽²⁾=𝜕𝑡𝛽e^−𝑚𝜏𝑁^𝑢₂−𝜖𝑣₃⁽²⁾(𝑥,𝑡−𝜏)𝑁^𝑢₂−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>𝑡₇𝑣,𝑥∈Ω,₂₍₂₎()=1𝑥,𝑡2𝑤(𝑥,𝑡),𝑣₃⁽²⁾()=1𝑥,𝑡2𝑣(𝑥,𝑡),𝑡∈−𝜏,𝑡₇,𝑥∈Ω. (4.51) Let (𝑢₂(𝑡),𝑢₃(𝑡)) be the positive solution for the following problem: 𝑣̇₂(𝑡)=𝛽e^−𝑚𝜏𝑁^𝑢₂−𝜖𝑢₂(𝑡−𝜏)𝑁^𝑢₂−𝜖+𝑢₂(𝑡−𝜏)−𝑎𝑢₂(𝑡),𝑡>𝑡₇,𝑣̇₃(𝑡)=𝑘𝑣₂(𝑡)−𝑝𝑣₃(𝑡),𝑡>𝑡₇,𝑣₂(𝑡)=12min_𝑥∈Ω,𝑣₃(𝑡)=12min_𝑥∈Ω𝑣(𝑥,𝑡),𝑡∈−𝜏,𝑡₇. (4.52) By Lemma 4.1 it follows lim_𝑡→∞𝑣₂(𝑡)=𝑁^𝑢₂−𝜖𝑘𝛽e^−𝑚𝜏−𝑎𝑝𝑎𝑝,lim_𝑡→∞𝑣₃(𝑡)=𝑘𝑁^𝑢₂−𝜖𝑘𝛽e^−𝑚𝜏−𝑎𝑝𝑎𝑝², (4.53) hence, for all 𝜖>0 sufficiently small, by comparison there exists a 𝑡₈>𝑡₇ such that if 𝑡>𝑡₈, min_𝑥∈Ω𝑤⁽²⁾(𝑥,𝑡)>𝑁^𝑤₂−𝜖,min_𝑥∈Ω𝑣⁽²⁾(𝑥,𝑡)>𝑁^𝑣₂−𝜖, (4.54) where 𝑁^𝑤₂=𝑁^𝑢₂𝑘𝛽e^−𝑚𝜏−𝑎𝑝𝑎𝑝,𝑁^𝑣₂=𝑘𝑁^𝑢₂𝑘𝛽e^−𝑚𝜏−𝑎𝑝𝑎𝑝². (4.55) Since 𝜖>0 is arbitrary and sufficiently small, we conclude that 𝑤=liminf_𝑡→∞min_𝑥∈Ω𝑤(𝑥,𝑡)≥𝑁^𝑤₂,𝑣=liminf_𝑡→∞min_𝑥∈Ω𝑣(𝑥,𝑡)≥𝑁^𝑣₂, (4.56) continuing this process, we derive six sequences 𝑀^𝑢_𝑛, 𝑀^𝑤_𝑛, 𝑀^𝑣_𝑛, 𝑁^𝑢_𝑛, 𝑁^𝑤_𝑛, and 𝑁^𝑣_𝑛(𝑛=1,2,…) such that, for 𝑛≥2, 𝑀^𝑢_𝑛=𝐿𝑀^𝑢_𝑛−1+𝑀^𝑤_𝑛−1𝑑𝑀^𝑢_𝑛−1+𝑀^𝑤_𝑛−1+𝛽𝑁^𝑣_𝑛−1,𝑀^𝑤_𝑛=𝑘𝛽e^−𝑚𝜏−𝑎𝑝𝑀^𝑢_𝑛,𝑀𝑎𝑝_𝑣𝑛=𝑘𝑘𝛽e^−𝑚𝜏−𝑎𝑝𝑀^𝑢_𝑛𝑎𝑝²,𝑁^𝑢_𝑛=𝐿𝑁^𝑢_𝑛−1+𝑁^𝑤_𝑛−1𝑑𝑁^𝑢_𝑛−1+𝑁^𝑤_𝑛−1+𝛽𝑀^𝑣_𝑛,𝑁^𝑤_𝑛=𝑘𝛽e^−𝑚𝜏−𝑎𝑝𝑁^𝑢_𝑛,𝑁𝑎𝑝_𝑣𝑛=𝑘𝑘𝛽e^−𝑚𝜏−𝑎𝑝𝑁^𝑢_𝑛𝑎𝑝². (4.57) It is readily seen that 𝑁^𝑢_𝑛≤𝑢≤𝑢≤𝑀^𝑢_𝑛,𝑁^𝑤_𝑛≤𝑤≤𝑤≤𝑀^𝑤_𝑛,𝑁^𝑣_𝑛≤𝑣≤𝑣≤𝑀^𝑣_𝑛. (4.58) The sequences 𝑀^𝑢_𝑛, 𝑀^𝑤_𝑛, and 𝑀^𝑣_𝑛 are nonincreasing and the sequences 𝑁^𝑢_𝑛, 𝑁^𝑤_𝑛, and 𝑁^𝑣_𝑛 are nondecreasing.

To prove the monotonicity of 𝑁^𝑢_𝑛 and 𝑀^𝑢_𝑛, we follow the ideas of Uh Zapata et al. [6]; consider 𝑅₀=𝛽𝑘e^−𝑚𝜏/𝑎𝑝 and the following expressions for 𝑁^𝑢, 𝑁^𝑤, 𝑁^𝑣, 𝑀^𝑢, 𝑀^𝑤, and 𝑀^𝑣𝑀^𝑢_𝑛=𝐿𝑀^𝑢_𝑛−1+𝑀^𝑤_𝑛−1𝑑𝑀^𝑢_𝑛−1+𝑀^𝑤_𝑛−1+𝛽𝑁_𝑛−1,𝑀^𝑤_𝑛=𝑅₀−1𝑀^𝑢_𝑛,𝑀^𝑣_𝑛=𝑘𝑝𝑅₀−1𝑀^𝑢_𝑛,𝑁^𝑢_𝑛=𝐿𝑁^𝑢_𝑛−1+𝑁^𝑤_𝑛−1𝑑𝑁^𝑢_𝑛−1+𝑁^𝑤_𝑛−1+𝛽𝑀^𝑣_𝑛,𝑁^𝑤_𝑛=𝑅₀−1𝑁^𝑢_𝑛,𝑁^𝑣_𝑛=𝑘𝑝𝑅₀−1𝑁^𝑢_𝑛. (4.59) We prove the result by induction so we first show that 𝑀^𝑢₂−𝑀^𝑢₁≤0, 𝑀^𝑢₂−𝑀^𝑢₁=𝐿e^−𝑚𝜏𝑀^𝑢₁𝑑e𝑀^𝑢₁+𝑎𝑅₀−1𝑁^𝑢₁−𝐿𝑑≤𝐿𝑑−𝐿𝑑=0 (4.60) and 𝑁^𝑢₂−𝑁^𝑢₁>0, 𝑁^𝑢₂−𝑁^𝑢₁=𝐿𝑅₀𝑁₁𝑑𝑅₀𝑁^𝑢₁+𝛽ℎ/𝑝𝑅₀−1𝑀^𝑢₂−𝑁₁=𝑁^𝑢₁𝐿𝑅₀−𝑑𝑅₀𝑁^𝑢₁−𝛽ℎ/𝑝𝑅₀−1𝑀^𝑢₂𝑑𝑅₀𝑁^𝑢₁+𝛽ℎ/𝑝𝑅₀−1𝑀^𝑢₂=𝑁^𝑢₁𝑑𝑅₀𝑁^𝑢₁+𝛽ℎ/𝑝𝑅₀−1𝑀^𝑢₂𝐿𝑅₀−𝐿𝑅₀+𝛽𝑅₀𝑀^𝑣₁−ℎ𝛽𝑝𝑅₀−1𝑀^𝑢₂=𝑁^𝑢₁𝑑𝑅₀𝑁^𝑢₁+𝛽ℎ/𝑝𝑅₀−1𝑀^𝑢₂𝛽ℎ𝑝𝑅₀𝑅₀−1𝑀^𝑢₁−ℎ𝛽𝑝𝑅₀−1𝑀^𝑢₂=𝛽ℎ𝑅₀−1𝑁^𝑢₁𝑝𝑑𝑅₀𝑁^𝑢₁+𝛽ℎ/𝑝𝑅₀−1𝑀^𝑢₂𝑅₀𝑀^𝑢₁−𝑀^𝑢₂>𝛽ℎ𝑅₀−1𝑁^𝑢₁𝑝𝑑𝑅₀𝑁^𝑢₁+𝛽ℎ/𝑝𝑅₀−1𝑀^𝑢₂𝑅₀𝑀^𝑢₁−𝑀^𝑢₂>0. (4.61) 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 𝑀^𝑢_𝑛≤𝑀^𝑢_𝑛−1 and 𝑁^𝑢_𝑛≥𝑁^𝑢_𝑛−1, 𝑀^𝑢_𝑛+1=𝐿𝑅₀𝑀^𝑢_𝑛𝑑𝑅₀𝑀^𝑢_𝑛+𝛽𝑁^𝑢_𝑛=𝐿𝑅₀𝑀^𝑢_𝑛𝑑𝑅₀𝑀^𝑢_𝑛+𝛽ℎ/𝑝𝑅₀−1𝑁^𝑢_𝑛. (4.62) The last function is increasing and by the induction hypothesis we have 𝑀^𝑢_𝑛+1≤𝐿𝑅₀𝑀^𝑢_𝑛−1𝑑𝑅₀𝑀^𝑢_𝑛−1+𝛽ℎ/𝑝𝑅₀−1𝑁^𝑢_𝑛≤𝐿𝑅₀𝑀^𝑢_𝑛−1𝑑𝑅₀𝑀^𝑢_𝑛−1+𝛽ℎ/𝑝𝑅₀−1𝑁^𝑢_𝑛=𝑀^𝑢_𝑛, (4.63) therefore the sequence 𝑀^𝑢_𝑛 is nonincreasing. For the sequence 𝑁^𝑢_𝑛 we use similar ideas and the behaviour of the sequence 𝑀^𝑢_𝑛 just proved. One has 𝑁^𝑢_𝑛+1=𝐿𝑅₀𝑁^𝑢_𝑛𝐿𝑅₀𝑁^𝑢_𝑛+𝛽𝑀^𝑣_𝑛+1=𝐿𝑅₀𝑁^𝑢_𝑛𝐿𝑅₀𝑁^𝑢_𝑛+𝛽ℎ/𝑝𝑅₀−1𝑀^𝑢_𝑛+1. (4.64) The last function is increasing and by the induction hypothesis we have 𝑁^𝑢_𝑛+1≥𝐿𝑅₀𝑁_𝑛−1𝐿𝑅₀𝑁_𝑛−1+𝛽ℎ/𝑝𝑅₀−1𝑀^𝑢_𝑛+1≥𝐿𝑅₀𝑁_𝑛−1𝐿𝑅₀𝑁_𝑛−1+𝛽ℎ/𝑝𝑅₀−1𝑀^𝑢_𝑛=𝑁^𝑢_𝑛, (4.65) therefore the sequence 𝑁^𝑢_𝑛 is nondecreasing. The behaviour for the sequences 𝑁^𝑤_𝑛, 𝑁^𝑣_𝑛, 𝑀^𝑤_𝑛, and 𝑀^𝑣_𝑛 follows from the nonincreasing sequence 𝑀^𝑢_𝑛 and the nondecreasing sequence 𝑁^𝑢_𝑛.

Hence, the limit of each sequence in 𝑁^𝑢_𝑛, 𝑁^𝑤_𝑛, 𝑁^𝑣_𝑛, 𝑀^𝑢_𝑛, 𝑀^𝑤_𝑛, and 𝑀^𝑣_𝑛 exists. Denote 𝑥=lim_𝑛→∞𝑀^𝑢_𝑛,𝑥=lim_𝑛→∞𝑁^𝑢_𝑛,𝑦=lim_𝑛→∞𝑀^𝑤_𝑛,𝑦=lim_𝑛→∞𝑁^𝑤_𝑛,𝑧=lim_𝑛→∞𝑀^𝑣_𝑛,𝑧=lim_𝑛→∞𝑁^𝑣_𝑛. (4.66) We therefore obtain from (4.57) and (4.66) that 𝑥−𝑥𝑑𝑘𝛽e^−𝑚𝜏−𝑎𝑝𝑘𝛽𝑘𝛽e^−𝑚𝜏−𝑎𝑝𝑎𝑝²=0. (4.67) Noting that (H1) holds and 𝑅₀>1, it follows that 𝑑𝑘𝛽e^−𝑚𝜏>𝑎𝑝𝑘𝛽𝑘𝛽e^−𝑚𝜏−𝑎𝑝𝑎𝑝², (4.68) 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 equilibrium

Theorem 4.3. If 𝑅₀<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 𝜙_𝑖(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: 𝜕𝑢⁽¹⁾𝜕𝑡=𝐿−𝑑𝑢⁽¹⁾(𝑥,𝑡),𝜕𝑤⁽¹⁾=𝜕𝑡𝛽e^−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). (4.69) Therefore for 𝑡>0, 𝑥∈Ω we have 0≤𝑢(𝑥,𝑡)≤𝑢⁽¹⁾(𝑥,𝑡),0≤𝑤(𝑥,𝑡)≤𝑤⁽¹⁾(𝑥,𝑡),0≤𝑣(𝑥,𝑡)≤𝑣⁽¹⁾(𝑥,𝑡). (4.70) We derive from the first equation that lim_𝑡→∞𝑢⁽¹⁾()=𝐿𝑥,𝑡𝑑, (4.71) so we can conclude limsup_𝑡→∞()≤𝐿max𝑢𝑥,𝑡𝑑. (4.72) 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: 𝜕𝜔₂⁽¹⁾=𝜕𝑡𝛽e^{(−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>𝑡₁𝜔;𝑥∈𝜕Ω,₂₍₁₎(𝑥,𝑡)=𝑤(𝑥,𝑡),𝜔₃⁽¹⁾(𝑥,𝑡)=𝑣(𝑥,𝑡),𝑡∈−𝜏,𝑡₁,𝑥∈Ω. (4.73) Consider (𝑢₂(𝑡),𝑢₃(𝑡)) as the solution for 𝑢̇₂=𝛽e^−𝑚𝜏(𝐿+𝜖)𝑢₃(𝑡−𝜏)𝐿/𝑑+𝜖+𝑢₂(𝑡−𝜏)−𝑎𝑢₂(𝑡),𝑢̇₃=𝑘𝑢₂(𝑡)−𝑝𝑢₃(𝑡),𝑢₂(𝑡)=max_𝑥∈Ω𝑤(𝑥,𝑡),𝑢₃(𝑡)=max_𝑥∈Ω𝑣(𝑥,𝑡),𝑡∈−𝜏,𝑡₁. (4.74) Then with 𝑅₀<1 according to Lemma 4.1 we have that lim_𝑡→∞𝑢₂(𝑡)=0,lim_𝑡→∞𝑢₃(𝑡)=0. (4.75) By comparison, it follows that lim_𝑡→∞𝑤(𝑡)=0,lim_𝑡→∞𝑣(𝑡)=0 (4.76) 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 Theorem 4.2  𝑢(𝑥,𝑡) is an upper solution for the following problem: 𝜕𝜔₁⁽¹⁾=𝐿−𝑑𝜔₁⁽¹⁾−𝛽𝜔₁⁽¹⁾𝜖,𝜕𝜔₁⁽¹⁾𝜕𝜂=0,𝑡≥𝑡₂,𝑥∈Ω,𝜔₁⁽¹⁾𝑥,𝑡₂=12𝑢𝑥,𝑡₂,𝑥∈Ω, (4.77) from the above equation we have that lim_𝑡→∞𝜔₁⁽¹⁾()=𝐿𝑥,𝑡𝑑+𝛽𝜖 (4.78) uniformly for 𝑥∈Ω. Since this holds for arbitrary 𝜖>0 sufficiently small, by comparison we conclude that liminf_𝑡→∞()≥𝐿min𝑢𝑥,𝑡𝑑, (4.79) which together with (4.72) gives lim_𝑡→∞𝑢()=𝐿𝑥,𝑡𝑑 (4.80) 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 parameters 𝐿=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 Figure 2 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 parameters 𝐿=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 Figure 3 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^∗.

(a) 𝐷=0.01,𝜏=4

(b) 𝐷=0.01,𝜏=0.9

(c) 𝐷=1,𝜏=4

(d) 𝐷=1,𝜏=0.9

Figure 3 

In this case the parameters are 𝐿=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=0.9

(b) 𝐷=0.01,𝜏=4

(c) 𝐷=1,𝜏=0.9

(d) 𝐷=1,𝜏=4

Figure 4 

In this case the parameter are 𝐿=27, 𝑑=3.78−2, 𝑎=3.38𝑑, 𝑝=0.67, 𝛽=1.45−4, 𝑘=5.183 and 𝑚=0.2.

Acknowledgments

This paper was supported by UADY under Grant UADY-CA-34 and Mexican SNI under Grants 15284 and 33365.