# Spatiotemporal Dynamics of a Delayed Viral Infection Model with Two Modes of Transmission and Immune Impairment

Authors: Hui Miao; Meiyan Jiao
Journal: Discrete Dynamics in Nature and Society
 (2022)

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## Abstract

In this paper, the dynamical behaviors for a diffusive virus infection model with two general nonlinear incidence functions, CTL immune impairment and discrete delays, are investigated. Cell-to-cell transmission plays a more important role in virus spreading in addition to virus-to-cell infection. Nonnegativity and boundcedness properties of the solutions are proven to show the well-posedness of the model. Five reproduction numbers are calculated and verified to be thresholds determining the global properties of the virus model. A significant impact of cell-to-cell transmission is that they increase the basic reproduction number. If one neglects either the cell-to-cell transmission or virus-to-cell infection, the basic reproduction number of the model is underevaluated. Numerical simulations are carried out to illustrate the dynamical behavior of the model.


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## Body

## 1. Introduction

Mathematical modeling has been developed to explore the pathogenesis and dynamical behaviors of the within-host virus infection process [1–32]. During virus infections, our immunity system protects us against pathogens. Some virus models have been proposed to describe the dynamics with cellular immune response [5–7, 18, 21, 27, 30, 33] or with humoral immune response [8–11, 31, 32, 34]. Therefore, an effective means to prevent viral infection needs both cellular and humoral immune responses at the same time. In addition, virus models that take into account the time delay not only can explain biological phenomena but also can exhibit much more complicated dynamical behavior [2, 5–7, 10–12, 14, 15, 19–21, 23, 27, 30–32]. In [12], Yan et al. developed the following viral infection model as follows:(1)dTtdt=s−dTt−βTtVt,dItdt=βe−m1τTt−τVt−τ−aIt−pItZt,dVtdt=kIt−μVt−qVtWt,dWtdt=δVtWt−hWt,dZtdt=cItZt−bZt,where the uninfected cells Tt are produced at a rate s, die at a rate d, and become infected at a rate β. τ≥0 accounts for the time between viral entry into a target cell and the production of new viral particles. Infected cells, It, die at a rate a and are killed by the CTL immune cells at a rate p. Free viruses Vt are produced from infected cells at a rate k and are removed at a rate μ, q is the antibody cells neutralize rate. CTL immune cells Zt expand in response to a viral antigen derived from infected cells at a rate c, and decay at a rate b. Antibody cells, Wt, are the production of new B cells by antigenic stimulation at a rate δ and h denotes the death rate of the antibody response.The influence of spatial structure is significant in the study of virus dynamics. In 2007, Wang and Wang [13] proposed a spatial HBV model. Based on the model in [13], a delayed HBV model considering diffusion and saturation incidence rate is studied by Xu and Ma in [14]. Subsequently, Yang and Wei [15] further analyzed the dynamics of a spatially viral model with time delay. Since then, several viral models incorporating delay and diffusion have been studied [13–17, 19, 20]. In fact, the process of virus transmission and the interaction between virus and the immune response is often local [35–37]. Meanwhile, it is assumed that the infection process in model (1) is a virus-to-cell infection. However, cell-to-cell transmission seems to be a more efficient method than virus-to-cell infection [38–42]. Recent studies in [22–26, 43, 44] formulated virus models considering two modes of transmission. Thus, it is essential to study a diffusive viral model incorporating cell-to-cell transmission.Furthermore, most viral infection models assume that the presence of antigen can only simulate immunity and ignore immune impairment. In certain circumstances, some pathogens can also suppress immune responses or even destroy immunity, especially when the load of pathogens is too high [27]. There has been much interest in mathematical modeling of viral dynamics within-host. Some study the CTL immune impairment [28–30] and others contain the humoral immune impairment [31, 32, 34]. Motivated by the works of Lin et al. [10], Yan and Wang [12], McCluskey and Yang [20], and Shu et al. [26], we propose a delayed virus infection model with spatial diffusion, two modes of transmission, and CTL immune impairment as follows:(2)∂T∂t=d1△T+s−dTx,t−fTx,t,Ix,t,Vx,tVx,t−gTx,t,Ix,tIx,t,∂I∂t=d2△I+e−m1τ1fTx,t−τ1,Ix,t−τ1,Vx,t−τ1Vx,t−τ1+gTx,t−τ1,Ix,t−τ1Ix,t−τ1−aIx,t−pIx,tZx,t,∂V∂t=d3△V+ke−m2τ2Ix,t−τ2−μVx,t−qVx,tWx,t,∂W∂t=d4△W+δVx,tWx,t−hWx,t,∂Z∂t=d5△Z+cIx,tZx,t−bZx,t−mIx,tZx,t,for t>0, x∈Ω. m is the immune impairment rate. Δ is the Laplacian operator. dii=1,2,3,4,5 are the diffusion coefficients. τ1 represents the time needed for infected cells to produce viruses after viral entry. m1 is the death rate for infected but not yet virus-producing cells. Therefore, the probability of surviving from time t−τ1 to time t is e−m1τ1 [6, 7, 10, 12, 20]. τ2 represents the time necessary for the newly produced viruses to become mature and then produce infectious particles. The average lifetime of an immature virus is given by 1/m2 and the probability of survival of immature viruses is given by e−m2τ2 [2]. And the other parameters are the same meaning as the model (1). Further, fT,I,V and g(T, I) satisfy the following assumptions:H1fT,I,V is continuously differentiable; f0,I,V=0 for all I≥0 and V≥0; ∂fT,I,V/∂T>0,∂fT,I,V/∂I≤0, and ∂fT,I,V/∂V≤0 for all T≥0,I≥0, and V≥0.H2g0,I=0 for all I≥0; ∂gT,I/∂T≥0 and ∂gT,I/∂I≤0 for all T≥0 and I≥0.For the biological interpretations of the above two assumptions, we refer the reader to the works [16, 42, 43].The model (2) we considered in this paper satisfies the following initial conditions:(3)Tx,θ=ϕ1x,θ≥0,Ix,θ=ϕ2x,θ≥0,Vx,θ=ϕ3x,θ≥0,Wx,θ=ϕ4x,θ≥0,Zx,θ=ϕ5x,θ≥0,x∈Ω¯,θ∈−τ,0.And the Neumann boundary conditions are(4)∂T∂n⟶=∂I∂n⟶=∂V∂n⟶=∂W∂n⟶=∂Z∂n⟶=0,t>0,x∈∂Ω,where τ=maxτ1,τ2, and ϕix,θi=1,2,3,4,5 is Hölder continuous in Ω¯×−τ,0, we give out a bounded domain Ω in ℝn and its boundary ∂Ω is smooth, and the outward normal derivative on ∂Ω is denoted by ∂/∂n⟶.In this paper, our purpose is to investigate the global dynamics of model (2). The organization of this paper is as follows: in the next section, the reproduction ratios are calculated and the existence of feasible equilibria is established. In Section 3, under the additional assumption H3, the global asymptotic stabilities of each of the feasible equilibria are investigated. In Section 4, we present numerical simulations to illustrate the theoretical results. In the last section, a brief summary is given to conclude this work.


## 2. Positivity, Boundedness, and Equilibrium

In this section, we establish the existence, uniqueness, positivity, and boundedness of solutions for model (2)–(4). Further, we determine the basic reproduction numbers and equilibria of model (2).LetC=C−τ,0,X be a Banach space composed with continuous functions from −τ,0 into X and its norm is ϕ=maxθ∈−τ,0ϕθX. X is also a Banach space CΩ¯,R5, which denotes the space of continuous functions from the topological space Ω¯ into the space R5. Further, we recognize ϕ∈C as a function from Ω¯×−τ,0 into R5 and we define ϕx,s=ϕsx. For any continuous function ω⋅:−τ,b⟶X for b>0, we further define ωt∈C by ωts=ωt+s,s∈−τ,0. It is natural to know that t⟶ωt is a continuous function from 0,b to C.Theorem 1.
For any given initial conditionϕ∈C satisfying (3), model (2)–(4) has a unique nonnegative solution. When d1=d2=d5, and d3=d4, this solution is defined on 0,+∞ and remains nonnegative and bounded for all t≥0.Proof:.
For anyϕ=ϕ1,ϕ2,ϕ3,ϕ4,ϕ5T∈C and x∈Ω¯, F=F1,F2,F3,F4,F5:C⟶X is defined by(5)F1ϕx=s−dϕ1x,0−fϕ1x,0,ϕ2x,0,ϕ3x,0ϕ3x,0−gϕ1x,0ϕ2x,0ϕ2x,0,F2ϕx=e−m1τ1fϕ1x,−τ1,ϕ2x,−τ1,ϕ3x,−τ1ϕ3x,−τ1+gϕ1x,−τ1,ϕ2x,−τ1ϕ2x,−τ1−aϕ2x,0−pϕ2x,0ϕ5x,0,F3ϕx=ke−m2τ2ϕ2x,−τ2−μϕ3x,0−qϕ3x,0ϕ4x,0,F4ϕx=δϕ3x,0ϕ4x,0−hϕ4x,0,F5ϕx=cϕ2x,0ϕ5x,0−bϕ5x,0−mϕ2x,0ϕ5x,0.
Then, we rewrite model (2)–(4) as an abstract function differential equation as(6)ω′t=Aω+Fωt,t>0,ω0=ϕ∈X,where ω=T,I,V,W,ZT,ϕ=ϕ1,ϕ2,ϕ3,ϕ4,ϕ5T, and Aω=d1△T,d2△I,d3△V,d4△W,d5△ZT. It is easy to see that F is locally Lipschitz in X. From Wu [45], we derive that model (6) has a unique local solution on 0,Tmax, where Tmax denotes the maximal existence time for the solution of model (6). Further, we obtain Tx,t≥0,Ix,t≥0,Vx,t≥0,Wx,t≥0, and Zx,t≥0 from the fact that 0 is a subsolution of each equation of model (2).
Next, we continue to prove the boundedness of solutions. Let us denoteN1x,t=e−m1τ1Tx,t−τ1+Ix,t+p/c−mZx,t, then we have(7)∂N1x,t∂t=d1e−m1τ1△T+d2△I+pc−md5△Z+se−m1τ1−de−m1τ1Tt−τ1−aI−pbc−mZx,t≤d1△N1+se−m1τ1−l1N1x,t,where d1=d2=d5,l1=mind,a,b. Therefore,(8)N1x,t≤maxse−m1τ1l1,maxx∈Ω¯e−m1τ1ϕ1x,τ1+ϕ2x,0+pc−mϕ5x,0.
Which yields thatT,I, and Z are bounded for ∀x,t∈Ω¯×0,Tmax.
Let us denoteN2x,t=Vx,t+q/δWx,t. Similarly, we can derive that(9)∂N2x,t∂t=d3△V+qδd4△W+ke−m2τ2Ix,t−τ2−μV−qhδWx,t≤d3△N2+ke−m2τ2ξ−l2N2x,t,where d3=d4,ξ=maxse−m1τ1/l1,maxx∈Ω¯N1x,0,l2=minμ,h. Thus,(10)N2x,t≤maxke−m2τ2ξl2,maxx∈Ω¯ϕ3x,0+qδϕ4x,0.
From the above analysis, we proved thatTx,t,Ix,t,Vx,t,Wx,t, and Zx,t are bounded on Ω¯×0,Tmax. Furthermore, the standard theory for semilinear parabolic systems [45] follow that Tmax=+∞.
Now, we discuss the existence of equilibria of model (2). It is easy to know that any equilibrium E=T,I,V,W,Z of model (2) satisfies(11)s−dTx−fTx,Ix,VxVx−gTx,IxIx=0,e−m1τ1fTx,Ix,VxVx+gTx,IxIx−aIx−pIxZx=0,ke−m2τ2Ix−μVx−qVxWx=0,δVxWx−hWx=0,c−mIxZx−bZx=0.
Obviously,E0=T0,0,0,0,0 with T0=s/d is always an infection-free equilibrium. Let us define(12)R0=ke−m1τ1−m2τ2fs/d,0,0+μe−m1τ1gs/d,0aμ.
Which is the basic reproductive number of virus infections for model (2). If Z=0 and W=0, then we obtain I=s−dT/aem1τ1,V=kλ−dT/aμem1τ1+m2τ2, and(13)ke−m1τ1−m2τ2fT,s−dTaem1τ1,kλ−dTaμem1τ1+m2τ2+μe−m1τ1gT,s−dTaem1τ1=aμ.
SinceI≥0, we have T≤s/d. Let us denote(14)F1T=ke−m1τ1−m2τ2fT,s−dTaem1τ1,kλ−dTaμem1τ1+m2τ2+μe−m1τ1gT,s−dTaem1τ1−aμ.
We haveF10=−aμ<0,F1s/d=aμR0−1 and(15)F1′T=ke−m1τ1−m2τ2∂f∂T−de−m1τ1a∂f∂I−kde−m1τ1−m2τ2aμ∂f∂V+μe−m1τ1∂g∂T−de−m1τ1a∂g∂I>0.
By the conditionH1, we know that F1T is strictly monotonically increasing with respect to T. When R0>1, there exists a unique T1∈0,s/d such that F1T1=0. Thus, we obtain a unique immune-free equilibrium E1=T1,I1,V1,0,0 with T1∈0,s/d, I1=s−dT1/aem1τ1, and V1=kλ−dT1/aμem1τ1+m2τ2.
IfZ=0 and W≠0, we have I=s−dT/aem1τ1 and V=h/δ. From the first and second equations of (11), we have(16)ahfT,s−dTaem1τ1,hδ+δs−dTe−m1τ1gT,s−dTaem1τ1=aδs−dT.
SinceW=kδe−m1τ1−m2τ2s−dT−aμh/aqh≥0, we get T≤s/d−aμhem1τ1+m2τ2/kδd. Let us denote(17)F2T=ahfT,s−dTaem1τ1,hδ+δs−dTe−m1τ1gT,s−dTaem1τ1−aδs−dT.
We haveF20=−aδs<0 and(18)F2′T=ah∂f∂T−de−m1τ1a∂f∂I+δs−dTe−m1τ1∂g∂T−de−m1τ1a∂g∂I+dδa−e−m1τ1gT,s−dTaem1τ1>0.
Let us define(19)R1=δhV1.
As for the antibody response reproductive number, which represents the average number of B cells activated by viruses when virus infection is successful and cell response has not been established.
IfR1>1, then V1>h/δ, T1<s/d−aμhem1τ1+m2τ2/kδd, and(20)F2sd−aμhem1τ1+m2τ2kδd=ahem1τ1+m2τ2kF1sd−aμhem1τ1+m2τ2kδd<ahem1τ1+m2τ2kF1T1=0.
Thus, ifR1>1, there exists a unique infection equilibrium with only antibody response E2=T2,I2,V2,W2,0 with T2∈0,s/d−aμhem1τ1+m2τ2/kδd,I2=s−dT2/aem1τ1,V2=h/δ, and W2=kδe−m1τ1−m2τ2s−dT2−aμh/aqh.
IfW=0 and Z≠0, we have I=b/c−m and V=kbe−m2τ2/μc−m. From (11), we obtain ke−m2τ2fT,b/c−m,kbe−m2τ2/μc−m+μgT,b/c−m=μc−m/bs−dT. As Z=c−me−m1τ1s−dT−ab/pb≥0, then T≤s/d−abem1τ1/c−md. Let us denote(21)F3T=ke−m2τ2fT,bc−m,kbe−m2τ2μc−m+μgT,bc−m−μc−mbs−dT.
We haveF30=−μc−ms/b<0 and F3′T=∂f/∂T+c−md/kbe−m2τ2>0.
Let us define the CTL immune response reproductive number, which denotes the average number of T cells activated by infectious cells when virus infection is successful and humoral immune response has not been established.(22)R2=c−mbI1.
We see thatR2>1 is equivalent to I1>b/c−m,T2<s/d−ab/c−mde−m1τ1, and(23)F3sd−abc−mde−m1τ1=em1τ1F1sd−abc−mde−m1τ1<em1τ1F1T1=0.
Hence,R2>1, there exists a unique infection equilibrium with only CTL response E3=T3,I3,V3,0,Z3 with T3∈0,s/d−ab/c−mde−m1τ1,I3=b/c−m,V3=kbe−m2τ2/μc−m, and Z3=c−me−m1τ1s−dT3−ab/pb.
IfW≠0 and Z≠0, we have I=b/c−m and V=h/δ. From the first equation of (11), we have h/δfT,b/c−m,h/δ+b/c−mgT,b/c−m=s−dT. Since Z=c−me−m1τ1s−dT−ab/pb≥0, we deduce that T≤s/d−ab/c−mde−m1τ1. Let us denote(24)F4T=h/δfT,b/c−m,h/δ+b/c−mgT,b/c−m−s−dT.
We haveF40=−s<0, F4′T=h/δ∂f/∂T+b/c−m∂g/∂T+d>0 and F4s/d=h/δfs/d,b/c−m,h/δ+b/c−mgs/d,b/c−m. Hence, there exists a unique T4∈0,s/d such that F4T4=0.
Let us define the antibody immune competitive reproductive number and the CTL immune competitive reproductive number, respectively. From the fourth and the fifth equations of (11), we obtain that W4=μ/qkbδe−m2τ2/μhc−m−1=μ/qR3−1 and(25)Z4=ap⋅e−m1τ1fT4,I4,V4hc−m/δb+gT4,I4a−1=apR4−1,R3=δhV3,R4=c−mbI2.
IfR3>1 and R4>1, model (2) has a unique infection equilibrium with CTL and antibody responses E4=T4,I4,V4,W4,Z4 with T4∈0,s/d, I4=b/c−m,V4=h/δ, W4=μ/qR3−1 and Z4=a/pR4−1.


## 3. Stability Analysis

To state the global stability, we need an additional assumption as follows:(26)H31−fT,I,VfT,Ii,VifT,Ii,VifT,I,V−VVi≤0,1−fTi,Ii,VigT,IfT,Ii,VigTi,IifT,Ii,VigTi,IifTi,Ii,VigT,I−IIi≤0,For allT,I,V>0, where Ii and Vi are the components of equilibrium Eii=1,2,3,4.


### 3.1. Stability of EquilibriumE0

Theorem 2.
IfR0≤1, then the infection-free equilibrium E0 is globally asymptotically stable.Proof:.
Let us define a Lyapunov functional(27)L1t=∫Ωem1τ1I+fT0,0,0μV+qfT0,0,0μδW+pem1τ1c−mZ+∫t−τ1tfTx,θ,Ix,θ,Vx,θVx,θ+gTx,θ,Ix,θIx,θdθ+ke−m2τ2fT0,0,0μ∫t−τ2tIx,θdθdx.
Calculating the time derivative ofL1t along the solution and using T0=s/d, we obtain(28)dL1tdt=∫Ωd2em1τ1△I+d3fT0,0,0μ△V+d4qfT0,0,0μδ△W+pd5em1τ1c−m△Z+fT,I,V−fT0,0,0V+aem1τ1e−m1τ1agT,I+ke−m1τ1−m2τ2fT0,0,0aμ−1I−qhfT0,0,0μδW−pbem1τ1fT0,0,0c−mZdx.
From the first equation of model (2), we obtain ∂T/∂t≤d1△T+s−dTx,t. By the comparison principle of partial functional differential equations [45], we get(29)Tx,t≤maxx∈Ω¯ϕ1x,0e−dt+sd1−e−dt.
For allx,t∈Ω¯×−τ,0. Hence, limsupt⟶+∞Tx,t≤T0. Without loss of generality, we can assume Tx,t≤T0 for all x,t∈Ω¯×−τ,0. Furthermore, assumptions H1-H2 imply that fT,I,V≤fT0,0,0 and gT,I≤gT0,0 for any x,t∈Ω¯×−τ,0. Thus, together with the explicit formula of R0 given in (12), we further can get(30)dL1tdt≤∫Ωd2em1τ1△I+d3fT0,0,0μ△V+d4qfT0,0,0μδ△W+pd5em1τ1c−m△Z+fT,I,V−fT0,0,0V+aem1τ1R0−1I−qhfT0,0,0μδW−pbem1τ1fT0,0,0c−mZdx.
Using the divergence theorem and the homogeneous Neumann boundary conditions, we get(31)∫Ω△Idx=∫∂Ω∂I∂n→dx=0,∫Ω△Vdx=∫∂Ω∂V∂n→dx=0,∫Ω△Wdx=∫∂Ω∂W∂n→dx=0,∫Ω△Zdx=∫∂Ω∂Z∂n→dx=0.
Thus,(32)dL1tdt≤∫ΩfT,I,V−fT0,0,0V+aem1τ1R0−1I−qhfT0,0,0μδW−pbem1τ1fT0,0,0c−mZdx≤∫Ωaem1τ1R0−1I−qhfT0,0,0μδW−pbem1τ1fT0,0,0c−mZdx
Therefore,dL1t/dt≤0 with equality holding if and only if T=T0, I=0, W=0 and Z=0. Let M be the largest invariant set of T,I,V,W,Z∈R+5:dL1t/dt=0, we have M=E0. From LaSalle’s invariance principle [45], E0 is globally asymptotically stable when R0≤1.


### 3.2. Stability of EquilibriumE1

Theorem 3.
Let us assume thatH3 for i=1 holds. If R0>1, R1≤1, and R2≤1, then the immune-free equilibrium E1 is globally asymptotically stable.Proof:.
LetGξ=ξ−1−lnξ. Clearly, Gξ≥0 for ξ>0 and G1=0. Let us define a Lyapunov functional L2t=∫ΩU1x,t+U2x,tdx, where(33)U1x,t=T−T1−∫t−τ1tfT1,I1,V1fX,I1,V1dX+em1τ1I1GII1+fT1,I1,V1V1em2τ2kI1V1GVV1+qfT1,I1,V1V1δke−m2τ2I1W+pem1τ1c−mZ.
And(34)U2x,t=fT1,I1,V1V1∫t−τ1tGfTx,θ,Ix,θ,Vx,θVx,θfT1,I1,V1V1dθ+gT1,I1I1∫t−τ1tGgTx,θ,Ix,θIx,θgT1,I1I1dθ+fT1,I1,V1V1∫t−τ2tGIx,θI1dθ.
By using(35)s=dT1+fT1,I1,V1V1+gT1,I1I1,e−m1τ1fT1,I1,V1V1+gT1,I1I1=aI1,e−m2τ2kI1=μV1.
Calculating the time derivative ofU1x,t and U2x,t along the solution, we obtain(36)∂U1x,t∂t+∂U2x,t∂t=dT1−dT1−fT1,I1,V1fT,I1,V1+d11−fT1,I1,V1fT,I1,V1△T+d2em1τ11−I1I△I+d3fT1,I1,V1V1em2τ2kI11−V1V△V+d4qfT1,I1,V1V1δke−m2τ2I1△W+d5pem1τ1c−m△Z+fT1,I1,V1V1−1−VV1+fT,I,VVfT,I1,V1V1+fT,I1,V1fT,I,V+gT1,I1I1−1−II1+fT1,I1,V1gT,IIfT,I1,V1gT1,I1I1+fT,I1,V1gT1,I1fT1,I1,V1gT,I−fT1,I1,V1V1GfT1,I1,V1fT,I1,V1+GfT,I1,V1fT,I,V+GIt−τ2V1I1V+GfTt−τ1,It−τ1,Vt−τ1Vt−τ1I1fT1,I1,V1V1I−gT1,I1I1GfT1,I1,V1fT,I1,V1+GgTt−τ1,It−τ1It−τ1gT1,I1I+GfT,I1,V1gT1,I1fT1,I1,V1gT,I+pem1τ1I1−bc−mZ+qfT1,I1,V1V1ke−m2τ2I1V1−hδW.
Meanwhile, from the homogeneous Neumann boundary conditions, we get by divergence theorem(37)∫Ω△Tdx=∫∂Ω∂T∂n→dx=0,∫Ω△TfT,I1,V1dx=∫Ω∇T2f2T,I1,V1dx,∫Ω△IIdx=∫Ω∇I2I2dx,∫Ω△VVdx=∫Ω∇V2V2dx.
It follows fromH1 and H3 that(38)1−TT11−fT1,I1,V1fT,I1,V1≤0,−1−VV1+fT,I,VVfT,I1,V1V1+fT,I1,V1fT,I,V=1−fT,I,VfT,I1,V1fT,I1,V1fT,I,V−VV1≤0,−1−II1+fT1,I1,V1gT,IIfT,I1,V1gT1,I1I1+fT,I1,V1gT1,I1fT1,I1,V1gT,I=1−fT1,I1,V1gT,IfT,I1,V1gT1,I1fT,I1,V1gT1,I1fT1,I1,V1gT,I−II1≤0.
Thus, we have(39)dL2tdt=∫ΩdT11−TT11−fT1,I1,V1fT,I1,V1+fT1,I1,V1V11−fT,I,VfT,I1,V1fT,I1,V1fT,I,V−VV1+gT1,I1I11−fT1,I1,V1gT,IfT,I1,V1gT1,I1fT,I1,V1gT1,I1fT1,I1,V1gT,I−II1−fT1,I1,V1V1GfT1,I1,V1fT,I1,V1+GfT,I1,V1fT,I,V+GIt−τ2V1I1V+GfTt−τ1,It−τ1,Vt−τ1Vt−τ1I1fT1,I1,V1V1I−gT1,I1I1GfT1,I1,V1fT,I1,V1+GfT,I1,V1gT1,I1fT1,I1,V1gT,I+GgTt−τ1,It−τ1It−τ1gT1,I1I+pem1τ1I1−bc−mZ+qfT1,I1,V1V1ke−m2τ2I1V1−hδWdx−d1fT1,I1,V1∫Ω∇T2f2T,I,Vdx−d2em1τ1I1∫Ω∇I2I2dx−d3fT1,I1,V1V1em2τ2kI1∫Ω∇V2V2dx.
Therefore, we havedL2t/dt≤0 with equality holding if and only if Tt=T1,It=I1,Vt=V1, Wt=0, and Zt=0. From LaSalle’s invariance principle [45], E1 is globally asymptotically stable when R0>1, R1≤1, and R2≤1.


### 3.3. Stability of EquilibriumE2

Theorem 4.
Let us assume thatH3 for i=2 holds. If R0>1, R1>1, and R3≤1, then the antibody response equilibrium E2 is globally asymptotically stable.Proof:.
Let us define a Lyapunov functional(40)L3t=∫ΩT−T2−∫T2TfT2,I2,V2fX,I2,V2dX+em1τ1I2GII2+pem1τ1c−mZ+fT2,I2,V2V2em2τ2kI1V2GVV2+qfT2,I2,V2V2δke−m2τ2I2W2GWW2+fT2,I2,V2V2∫t−τ1tGfTx,θ,Ix,θ,Vx,θVx,θfT2,I2,V2V2dθ+gT2,I2I2∫t−τ1tGgTx,θ,Ix,θIx,θgT2,I2I2dθ+fT2,I2,V2V2∫t−τ2tGIx,θI2dθdx.
By using(41)s=dT2+fT2,I2,V2V2+gT2,I2I2,V2=hδ,e−m1τ1fT2,I2,V2V2+gT2,I2I2=aI2,e−m2τ2kI2=μ+qW2V2.
Calculating the time derivative ofL3t along the solution, we obtain(42)dL3tdt=∫ΩdT21−TT21−fT2,I2,V2fT,I2,V2+fT2,I2,V2V21−fT,I,VfT,I2,V2fT,I2,V2fT,I,V−VV2+gT2,I2I21−fT2,I2,V2gT,IfT,I2,V2gT2,I2fT,I2,V2gT2,I2fT2,I2,V2gT,I−II2−fT2,I2,V2V2GfT2,I2,V2fT,I2,V2+GfT,I2,V2fT,I,V+GIt−τ2V2I2V+GfTt−τ1,It−τ1,Vt−τ1Vt−τ1I2fT2,I2,V2V2I−gT2,I2I2GfT2,I2,V2fT,I2,V2+GgTt−τ1,It−τ1It−τ1gT2,I2I+GfT,I2,V2gT2,I2fT2,I2,V2gT,I+pem1τ1I2−bc−mZdx−d1fT2,I2,V2∫Ω∇T2f2T,I,Vdx−d2em1τ1I2∫Ω∇I2I2dx−d3fT2,I2,V2V2em2τ2kI2∫Ω∇V2V2dx−d4qfT2,I2,V2V2δke−m2τ2I2∫Ω∇W2W2dx.
Therefore, we havedL3t/dt≤0 with equality holding if and only if Tt=T2,It=I2,Vt=V2, Zt=0, and Wt=W2. From LaSalle’s invariance principle [45], E2 is globally asymptotically stable when R0>1, R1≤1, and R3≤1.


### 3.4. Stability of EquilibriumE3

Theorem 5.
Let us assume thatH3 for i=3 holds. If R0>1, R2>1, and R4≤1, then the infection equilibrium E3 with only CTL response is globally asymptotically stable.Proof:.
Let us define a Lyapunov functional(43)L4t=∫ΩT−T3−∫T3TfT3,I3,V3fX,I3,V3dX+em1τ1I3GII3+pem1τ1c−mZ3GZZ3+fT3,I3,V3V3em2τ2kI3V3GVV3+qfT3,I3,V3V3δke−m2τ2I3W+fT3,I3,V3V3∫t−τ1tGfTx,θ,Ix,θ,Vx,θVx,θfT3,I3,V3V3dθ+gT3,I3I3∫t−τ1tGgTx,θ,Ix,θIx,θgT3,I3I3dθ+fT3,I3,V3V3∫t−τ2tGIx,θI3dθdx.
By using(44)s=dT3+fT3,I3,V3V3+gT3,I3I3,e−m2τ2kI3=μV3,e−m1τ1fT3,I3,V3V3+gT3,I3I3=a+pZ3I3,I3=bc−m.
Calculating the time derivative ofL4t along the solution, we obtain(45)dL4tdt=∫ΩdT31−TT31−fT3,I3,V3fT,I3,V3+fT3,I3,V3V31−fT,I,VfT,I3,V3fT,I3,V3fT,I,V−VV3+gT3,I3I31−fT3,I3,V3gT,IfT,I3,V3gT3,I3fT,I3,V3gT3,I3fT3,I3,V3gT,I−II3−fT3,I3,V3V3GfT3,I3,V3fT,I3,V3+GfT,I3,V3fT,I,V+GIt−τ2V3I3V+GfTt−τ1,It−τ1,Vt−τ1Vt−τ1I3fT3,I3,V3V3I−gT3,I3I3GfT3,I3,V3fT,I3,V3+GgTt−τ1,It−τ1It−τ1gT3,I3I+GfT,I3,V3gT3,I3fT3,I3,V3gT,I+qfT3,I3,V3V3ke−m2τ2I3V3−hδWdx−d1fT3,I3,V3∫Ω∇T2f2T,I,Vdx−d2em1τ1I3∫Ω∇I2I2dx−d3fT3,I3,V3V3em2τ2kI3∫Ω∇V2V2dx−d5pem1τ1Z3c−m∫Ω∇Z2Z2dx.
Therefore, we havedL4t/dt≤0 with equality holding if and only if Tt=T3,It=I3,Vt=V3,Zt=Z3, and Wt=0. From LaSalle’s invariance principle [45], E3 is globally asymptotically stable when R0>1,R2>1, and R4≤1.


### 3.5. Stability of EquilibriumE4

Theorem 6.
Let us assume thatH3 for i=4 holds. If R0>1, R1>1, R3>1, and R4>1, then the infection equilibrium E4 with CTL and antibody responses is globally asymptotically stableProof:.
Let us define a Lyapunov functional(46)L5t=∫ΩT−T4−∫T4TfT4,I4,V4fX,I4,V4dX+em1τ1I4GII4+pem1τ1c−mZ4GZZ4+fT4,I4,V4V4em2τ2kI4V4GVV4+qfT4,I4,V4V4δke−m2τ2I4W4GWW4+fT4,I4,V4V4∫t−τ1tGfTx,θ,Ix,θ,Vx,θVx,θfT4,I4,V4V4dθ+gT4,I4I4∫t−τ1tGgTx,θ,Ix,θIx,θgT4,I4I4dθ+fT4,I4,V4V4∫t−τ2tGIx,θI4dθdx.
Using the above similar method, we obtain(47)dL5tdt=∫ΩdT41−TT41−fT4,I4,V4fT,I4,V4+fT4,I4,V4V41−fT,I,VfT,I4,V4fT,I4,V4fT,I,V−VV4+gT4,I4I41−fT4,I4,V4gT,IfT,I4,V4gT4,I4fT,I4,V4gT4,I4fT4,I4,V4gT,I−II4−fT4,I4,V4V4GfT4,I4,V4fT,I4,V4+GfT,I4,V4fT,I,V+GIt−τ2V4I4V+GfTt−τ1,It−τ1,Vt−τ1Vt−τ1I4fT4,I4,V4V4I−gT4,I4I4GfT4,I4,V4fT,I4,V4+GfT,I4,V4gT4,I4fT4,I4,V4gT,I+GgTt−τ1,It−τ1It−τ1gT4,I4Idx−d1fT4,I4,V4∫Ω∇T2f2T,I,Vdx−d2em1τ1I4∫Ω∇I2I2dx−d3fT4,I4,V4V4em2τ2kI4∫Ω∇V2V2dx−d5pem1τ1Z4c−m∫Ω∇Z2Z2dx−d4qfT4,I4,V4V4em2τ2W4δkI4∫Ω∇W2W2dx.
Therefore, we havedL5t/dt≤0 with equality holding if and only if T=T4,I=I4,andV=V4. From LaSalle’s invariance [45], E4 is globally asymptotically stable when R0>1,R1>1,R3>1, and R4>1.


## 3.1. Stability of EquilibriumE0

Theorem 2.
IfR0≤1, then the infection-free equilibrium E0 is globally asymptotically stable.Proof:.
Let us define a Lyapunov functional(27)L1t=∫Ωem1τ1I+fT0,0,0μV+qfT0,0,0μδW+pem1τ1c−mZ+∫t−τ1tfTx,θ,Ix,θ,Vx,θVx,θ+gTx,θ,Ix,θIx,θdθ+ke−m2τ2fT0,0,0μ∫t−τ2tIx,θdθdx.
Calculating the time derivative ofL1t along the solution and using T0=s/d, we obtain(28)dL1tdt=∫Ωd2em1τ1△I+d3fT0,0,0μ△V+d4qfT0,0,0μδ△W+pd5em1τ1c−m△Z+fT,I,V−fT0,0,0V+aem1τ1e−m1τ1agT,I+ke−m1τ1−m2τ2fT0,0,0aμ−1I−qhfT0,0,0μδW−pbem1τ1fT0,0,0c−mZdx.
From the first equation of model (2), we obtain ∂T/∂t≤d1△T+s−dTx,t. By the comparison principle of partial functional differential equations [45], we get(29)Tx,t≤maxx∈Ω¯ϕ1x,0e−dt+sd1−e−dt.
For allx,t∈Ω¯×−τ,0. Hence, limsupt⟶+∞Tx,t≤T0. Without loss of generality, we can assume Tx,t≤T0 for all x,t∈Ω¯×−τ,0. Furthermore, assumptions H1-H2 imply that fT,I,V≤fT0,0,0 and gT,I≤gT0,0 for any x,t∈Ω¯×−τ,0. Thus, together with the explicit formula of R0 given in (12), we further can get(30)dL1tdt≤∫Ωd2em1τ1△I+d3fT0,0,0μ△V+d4qfT0,0,0μδ△W+pd5em1τ1c−m△Z+fT,I,V−fT0,0,0V+aem1τ1R0−1I−qhfT0,0,0μδW−pbem1τ1fT0,0,0c−mZdx.
Using the divergence theorem and the homogeneous Neumann boundary conditions, we get(31)∫Ω△Idx=∫∂Ω∂I∂n→dx=0,∫Ω△Vdx=∫∂Ω∂V∂n→dx=0,∫Ω△Wdx=∫∂Ω∂W∂n→dx=0,∫Ω△Zdx=∫∂Ω∂Z∂n→dx=0.
Thus,(32)dL1tdt≤∫ΩfT,I,V−fT0,0,0V+aem1τ1R0−1I−qhfT0,0,0μδW−pbem1τ1fT0,0,0c−mZdx≤∫Ωaem1τ1R0−1I−qhfT0,0,0μδW−pbem1τ1fT0,0,0c−mZdx
Therefore,dL1t/dt≤0 with equality holding if and only if T=T0, I=0, W=0 and Z=0. Let M be the largest invariant set of T,I,V,W,Z∈R+5:dL1t/dt=0, we have M=E0. From LaSalle’s invariance principle [45], E0 is globally asymptotically stable when R0≤1.


## 3.2. Stability of EquilibriumE1

Theorem 3.
Let us assume thatH3 for i=1 holds. If R0>1, R1≤1, and R2≤1, then the immune-free equilibrium E1 is globally asymptotically stable.Proof:.
LetGξ=ξ−1−lnξ. Clearly, Gξ≥0 for ξ>0 and G1=0. Let us define a Lyapunov functional L2t=∫ΩU1x,t+U2x,tdx, where(33)U1x,t=T−T1−∫t−τ1tfT1,I1,V1fX,I1,V1dX+em1τ1I1GII1+fT1,I1,V1V1em2τ2kI1V1GVV1+qfT1,I1,V1V1δke−m2τ2I1W+pem1τ1c−mZ.
And(34)U2x,t=fT1,I1,V1V1∫t−τ1tGfTx,θ,Ix,θ,Vx,θVx,θfT1,I1,V1V1dθ+gT1,I1I1∫t−τ1tGgTx,θ,Ix,θIx,θgT1,I1I1dθ+fT1,I1,V1V1∫t−τ2tGIx,θI1dθ.
By using(35)s=dT1+fT1,I1,V1V1+gT1,I1I1,e−m1τ1fT1,I1,V1V1+gT1,I1I1=aI1,e−m2τ2kI1=μV1.
Calculating the time derivative ofU1x,t and U2x,t along the solution, we obtain(36)∂U1x,t∂t+∂U2x,t∂t=dT1−dT1−fT1,I1,V1fT,I1,V1+d11−fT1,I1,V1fT,I1,V1△T+d2em1τ11−I1I△I+d3fT1,I1,V1V1em2τ2kI11−V1V△V+d4qfT1,I1,V1V1δke−m2τ2I1△W+d5pem1τ1c−m△Z+fT1,I1,V1V1−1−VV1+fT,I,VVfT,I1,V1V1+fT,I1,V1fT,I,V+gT1,I1I1−1−II1+fT1,I1,V1gT,IIfT,I1,V1gT1,I1I1+fT,I1,V1gT1,I1fT1,I1,V1gT,I−fT1,I1,V1V1GfT1,I1,V1fT,I1,V1+GfT,I1,V1fT,I,V+GIt−τ2V1I1V+GfTt−τ1,It−τ1,Vt−τ1Vt−τ1I1fT1,I1,V1V1I−gT1,I1I1GfT1,I1,V1fT,I1,V1+GgTt−τ1,It−τ1It−τ1gT1,I1I+GfT,I1,V1gT1,I1fT1,I1,V1gT,I+pem1τ1I1−bc−mZ+qfT1,I1,V1V1ke−m2τ2I1V1−hδW.
Meanwhile, from the homogeneous Neumann boundary conditions, we get by divergence theorem(37)∫Ω△Tdx=∫∂Ω∂T∂n→dx=0,∫Ω△TfT,I1,V1dx=∫Ω∇T2f2T,I1,V1dx,∫Ω△IIdx=∫Ω∇I2I2dx,∫Ω△VVdx=∫Ω∇V2V2dx.
It follows fromH1 and H3 that(38)1−TT11−fT1,I1,V1fT,I1,V1≤0,−1−VV1+fT,I,VVfT,I1,V1V1+fT,I1,V1fT,I,V=1−fT,I,VfT,I1,V1fT,I1,V1fT,I,V−VV1≤0,−1−II1+fT1,I1,V1gT,IIfT,I1,V1gT1,I1I1+fT,I1,V1gT1,I1fT1,I1,V1gT,I=1−fT1,I1,V1gT,IfT,I1,V1gT1,I1fT,I1,V1gT1,I1fT1,I1,V1gT,I−II1≤0.
Thus, we have(39)dL2tdt=∫ΩdT11−TT11−fT1,I1,V1fT,I1,V1+fT1,I1,V1V11−fT,I,VfT,I1,V1fT,I1,V1fT,I,V−VV1+gT1,I1I11−fT1,I1,V1gT,IfT,I1,V1gT1,I1fT,I1,V1gT1,I1fT1,I1,V1gT,I−II1−fT1,I1,V1V1GfT1,I1,V1fT,I1,V1+GfT,I1,V1fT,I,V+GIt−τ2V1I1V+GfTt−τ1,It−τ1,Vt−τ1Vt−τ1I1fT1,I1,V1V1I−gT1,I1I1GfT1,I1,V1fT,I1,V1+GfT,I1,V1gT1,I1fT1,I1,V1gT,I+GgTt−τ1,It−τ1It−τ1gT1,I1I+pem1τ1I1−bc−mZ+qfT1,I1,V1V1ke−m2τ2I1V1−hδWdx−d1fT1,I1,V1∫Ω∇T2f2T,I,Vdx−d2em1τ1I1∫Ω∇I2I2dx−d3fT1,I1,V1V1em2τ2kI1∫Ω∇V2V2dx.
Therefore, we havedL2t/dt≤0 with equality holding if and only if Tt=T1,It=I1,Vt=V1, Wt=0, and Zt=0. From LaSalle’s invariance principle [45], E1 is globally asymptotically stable when R0>1, R1≤1, and R2≤1.


## 3.3. Stability of EquilibriumE2

Theorem 4.
Let us assume thatH3 for i=2 holds. If R0>1, R1>1, and R3≤1, then the antibody response equilibrium E2 is globally asymptotically stable.Proof:.
Let us define a Lyapunov functional(40)L3t=∫ΩT−T2−∫T2TfT2,I2,V2fX,I2,V2dX+em1τ1I2GII2+pem1τ1c−mZ+fT2,I2,V2V2em2τ2kI1V2GVV2+qfT2,I2,V2V2δke−m2τ2I2W2GWW2+fT2,I2,V2V2∫t−τ1tGfTx,θ,Ix,θ,Vx,θVx,θfT2,I2,V2V2dθ+gT2,I2I2∫t−τ1tGgTx,θ,Ix,θIx,θgT2,I2I2dθ+fT2,I2,V2V2∫t−τ2tGIx,θI2dθdx.
By using(41)s=dT2+fT2,I2,V2V2+gT2,I2I2,V2=hδ,e−m1τ1fT2,I2,V2V2+gT2,I2I2=aI2,e−m2τ2kI2=μ+qW2V2.
Calculating the time derivative ofL3t along the solution, we obtain(42)dL3tdt=∫ΩdT21−TT21−fT2,I2,V2fT,I2,V2+fT2,I2,V2V21−fT,I,VfT,I2,V2fT,I2,V2fT,I,V−VV2+gT2,I2I21−fT2,I2,V2gT,IfT,I2,V2gT2,I2fT,I2,V2gT2,I2fT2,I2,V2gT,I−II2−fT2,I2,V2V2GfT2,I2,V2fT,I2,V2+GfT,I2,V2fT,I,V+GIt−τ2V2I2V+GfTt−τ1,It−τ1,Vt−τ1Vt−τ1I2fT2,I2,V2V2I−gT2,I2I2GfT2,I2,V2fT,I2,V2+GgTt−τ1,It−τ1It−τ1gT2,I2I+GfT,I2,V2gT2,I2fT2,I2,V2gT,I+pem1τ1I2−bc−mZdx−d1fT2,I2,V2∫Ω∇T2f2T,I,Vdx−d2em1τ1I2∫Ω∇I2I2dx−d3fT2,I2,V2V2em2τ2kI2∫Ω∇V2V2dx−d4qfT2,I2,V2V2δke−m2τ2I2∫Ω∇W2W2dx.
Therefore, we havedL3t/dt≤0 with equality holding if and only if Tt=T2,It=I2,Vt=V2, Zt=0, and Wt=W2. From LaSalle’s invariance principle [45], E2 is globally asymptotically stable when R0>1, R1≤1, and R3≤1.


## 3.4. Stability of EquilibriumE3

Theorem 5.
Let us assume thatH3 for i=3 holds. If R0>1, R2>1, and R4≤1, then the infection equilibrium E3 with only CTL response is globally asymptotically stable.Proof:.
Let us define a Lyapunov functional(43)L4t=∫ΩT−T3−∫T3TfT3,I3,V3fX,I3,V3dX+em1τ1I3GII3+pem1τ1c−mZ3GZZ3+fT3,I3,V3V3em2τ2kI3V3GVV3+qfT3,I3,V3V3δke−m2τ2I3W+fT3,I3,V3V3∫t−τ1tGfTx,θ,Ix,θ,Vx,θVx,θfT3,I3,V3V3dθ+gT3,I3I3∫t−τ1tGgTx,θ,Ix,θIx,θgT3,I3I3dθ+fT3,I3,V3V3∫t−τ2tGIx,θI3dθdx.
By using(44)s=dT3+fT3,I3,V3V3+gT3,I3I3,e−m2τ2kI3=μV3,e−m1τ1fT3,I3,V3V3+gT3,I3I3=a+pZ3I3,I3=bc−m.
Calculating the time derivative ofL4t along the solution, we obtain(45)dL4tdt=∫ΩdT31−TT31−fT3,I3,V3fT,I3,V3+fT3,I3,V3V31−fT,I,VfT,I3,V3fT,I3,V3fT,I,V−VV3+gT3,I3I31−fT3,I3,V3gT,IfT,I3,V3gT3,I3fT,I3,V3gT3,I3fT3,I3,V3gT,I−II3−fT3,I3,V3V3GfT3,I3,V3fT,I3,V3+GfT,I3,V3fT,I,V+GIt−τ2V3I3V+GfTt−τ1,It−τ1,Vt−τ1Vt−τ1I3fT3,I3,V3V3I−gT3,I3I3GfT3,I3,V3fT,I3,V3+GgTt−τ1,It−τ1It−τ1gT3,I3I+GfT,I3,V3gT3,I3fT3,I3,V3gT,I+qfT3,I3,V3V3ke−m2τ2I3V3−hδWdx−d1fT3,I3,V3∫Ω∇T2f2T,I,Vdx−d2em1τ1I3∫Ω∇I2I2dx−d3fT3,I3,V3V3em2τ2kI3∫Ω∇V2V2dx−d5pem1τ1Z3c−m∫Ω∇Z2Z2dx.
Therefore, we havedL4t/dt≤0 with equality holding if and only if Tt=T3,It=I3,Vt=V3,Zt=Z3, and Wt=0. From LaSalle’s invariance principle [45], E3 is globally asymptotically stable when R0>1,R2>1, and R4≤1.


## 3.5. Stability of EquilibriumE4

Theorem 6.
Let us assume thatH3 for i=4 holds. If R0>1, R1>1, R3>1, and R4>1, then the infection equilibrium E4 with CTL and antibody responses is globally asymptotically stableProof:.
Let us define a Lyapunov functional(46)L5t=∫ΩT−T4−∫T4TfT4,I4,V4fX,I4,V4dX+em1τ1I4GII4+pem1τ1c−mZ4GZZ4+fT4,I4,V4V4em2τ2kI4V4GVV4+qfT4,I4,V4V4δke−m2τ2I4W4GWW4+fT4,I4,V4V4∫t−τ1tGfTx,θ,Ix,θ,Vx,θVx,θfT4,I4,V4V4dθ+gT4,I4I4∫t−τ1tGgTx,θ,Ix,θIx,θgT4,I4I4dθ+fT4,I4,V4V4∫t−τ2tGIx,θI4dθdx.
Using the above similar method, we obtain(47)dL5tdt=∫ΩdT41−TT41−fT4,I4,V4fT,I4,V4+fT4,I4,V4V41−fT,I,VfT,I4,V4fT,I4,V4fT,I,V−VV4+gT4,I4I41−fT4,I4,V4gT,IfT,I4,V4gT4,I4fT,I4,V4gT4,I4fT4,I4,V4gT,I−II4−fT4,I4,V4V4GfT4,I4,V4fT,I4,V4+GfT,I4,V4fT,I,V+GIt−τ2V4I4V+GfTt−τ1,It−τ1,Vt−τ1Vt−τ1I4fT4,I4,V4V4I−gT4,I4I4GfT4,I4,V4fT,I4,V4+GfT,I4,V4gT4,I4fT4,I4,V4gT,I+GgTt−τ1,It−τ1It−τ1gT4,I4Idx−d1fT4,I4,V4∫Ω∇T2f2T,I,Vdx−d2em1τ1I4∫Ω∇I2I2dx−d3fT4,I4,V4V4em2τ2kI4∫Ω∇V2V2dx−d5pem1τ1Z4c−m∫Ω∇Z2Z2dx−d4qfT4,I4,V4V4em2τ2W4δkI4∫Ω∇W2W2dx.
Therefore, we havedL5t/dt≤0 with equality holding if and only if T=T4,I=I4,andV=V4. From LaSalle’s invariance [45], E4 is globally asymptotically stable when R0>1,R1>1,R3>1, and R4>1.


## 4. Numerical Simulations

In this section, we present several numerical examples to illustrate the results obtained in Section3. We will use the finite difference scheme that is proposed in [46, 47] for the delayed reaction-diffusion epidemic models. For convenience, we consider model (2) under the one-dimensional spatial domain Ω=0,1. The homogeneous Neumann boundary conditions and the initial conditions are given by(48)∂T∂n→=∂I∂n→=∂V∂n→=∂W∂n→=∂Z∂n→=0,t>0,x=0,1,And(49)Tx,θ=400,Ix,θ=1,Vx,θ=0.1,Wx,θ=2,Zx,θ=0.001,for 0≤x≤1 and −τ≤θ≤0.In model (2), we choose the nonlinear incidences fT,I,VV=β1TV/1+α1V,gT,II=β2TI/1+α2I. Furthermore, β1, β2, c, m, δ, h, a, k, and b are chosen as free parameters. All other parameter values are the same as in Table 1.Table 1
List of parameters.
ParameterDefinitionValueSourcesProduction rate of uninfected cells10μl−1day−1[18]dDeath rate of uninfected cells0.01day−1[18]pCTL effectiveness1μlday−1[5, 18]α1Saturation coefficient0.01Assumedα2Saturation coefficient0.01AssumedμClearance rate of free virus3day−1[1, 5]qNeutralizing rate of humoral1μlday−1[1, 21]m1Death rate for infected cells duringt−τ1,t0.01Assumedm2Death rate for new virus duringt−τ2,t0.01AssumedIn the following Figures1–5, (a), (b), (c), (d), and (e) are Figure 2 denoted Figure 3 time-series Figure 4 figures of Tx,t, Ix,t, Vx,t, Wx,t, and Zx,t.Figure 1
Takingβ1=0.001, β2=0.001, c=0.01, m=0.001, δ=0.5, h=1.5, a=1.5, k=0.4, and b=2, we have R0=0.6797<1, E0=1000,0,0,0,0 is asymptotically stable.
(a)(b)(c)(d)(e)Figure 2
Takingβ1=0.15, β2=0.01, c=0.01, m=0.001, δ=0.5, h=1.5, a=0.5, k=0.4, and b=0.2, we have R0=52.5251>1,R1=0.7495<1, and R2=0.7977<1, E1=20.4000,17.7278,2.2484,0,0 is asymptotically stable.
(a)(b)(c)(d)(e)Figure 3
Takingβ1=0.35, β2=0.01, c=0.1, m=0.01, δ=1.5, h=1, a=0.5, k=0.4, and b=2, we have R0=98.4295>1,R1=3.3053>1, and R3=0.7818<1, E2=39.9500,17.3738,0.6667,6.9159,0 is asymptotically stable.
(a)(b)(c)(d)(e)Figure 4
Takingβ1=0.45, β2=0.15, c=0.1, m=0.001, δ=1.5, h=1, a=0.5, k=0.4, and b=0.15, we have R0=374.7362>1,R2=14.1326>1, and R4=0.2883<1, E3=31.2314,1.5152,0.1922,0,6.5663 is asymptotically stable.
(a)(b)(c)(d)(e)Figure 5
Takingβ1=0.45, β2=0.01, c=0.16, m=0.01, δ=0.1, h=0.01, a=0.5, k=4, and b=0.15, we have R0=1050.9>1,R3=18.6953>1, and R4=12.6831>1, E4=154.1877,1.0000,0.1000,35.0492,8.8477 is asymptotically stable.
(a)(b)(c)(d)(e)


## 5. Discussion

In this paper, we have discussed a generalized delayed viral infection model (2) with diffusion, CTL immune impairment, and two modes of transmission, which are virus-to-cell infection fT,I,VV and cell-to-cell transmission gT,II. Under assumptions H1-H3 on two nonlinear incidence functions, we have shown that the global stabilities of the model (2) is determined by five threshold numbers that are viral infection R0, antibody immune response R1, CTL response R2, humoral immune competition R3, and CTL immune competition R4.From the point of biology, we have shown whenR0≤1, the infection-free equilibrium E0 is globally asymptotically stable; when R0>1, R1≤1, and R2≤1, the immune-free equilibrium E1 is globally asymptotically stable; when R0>1, R1>1, and R3≤1, the antibody response equilibrium E2 is globally asymptotically stable; when R0>1, R2>1, and R4≤1, the CTL response equilibrium E3 is globally asymptotically stable; and when R3>1 and R4>1, the interior equilibrium E4 is globally asymptotically stable. Since R0,R1,R2,R3, and R4 is independent of the diffusion coefficients d1,d2,d3,d4, and d5, the results reveal that the spatial diffusion of the cells and virus, the intracellular delay, and virus replication delay has no impact on the global stability of the equilibria E0,E1,E2,E3, and E4, subject to the Neumann boundary conditions.From the expression of the basic reproduction numberR0 , we can see that R0 could be underevaluated without considering either the virus-to-cell infection or cell-to-cell transmission. Therefore, it is not enough to eliminate the disease by reducing the basic reproduction number of virus-to-cell infection due to the existence of cell-to-cell transmission. Furthermore, it is interesting to improve the model (2) by incorporating logistic growth terms for the uninfected cells and infected cells. Meanwhile, we do not consider the immune response delay an issue. On the other hand, model (2) can be extended to describe the spatiotemporal dynamics with two classes of target cells, CD4+ T cells and macrophages. We leave these points as possible future works.



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