# Modeling Cell-to-Cell Spread of HIV-1 with Nonlocal Infections

Authors: Xiaoting Fan; Yi Song; Wencai Zhao
Journal: Complexity
 (2018)

---

## Abstract

This paper is devoted to develop a nonlocal and time-delayed reaction-diffusion model for HIV infection within host cell-to-cell viral transmissions. In a bounded spatial domain, we study threshold dynamics in terms of basic reproduction numberR0 for the heterogeneous model. Our results show that if R0<1, the infection-free steady state is globally attractive, implying infection becomes extinct, while if R0>1, virus will persist in the host environment.


---

## Body

## 1. Introduction and Model Derivation

Infectious diseases have been threatening human health; infectious diseases such as smallpox, cholera, and acquired immune deficiency syndrome (AIDS) have brought great disaster to the national economy of a country and people’s livelihood. In order to control the spread of infectious diseases, researchers have proposed a great deal of mathematical models to study the evolution behavior of infectious diseases from different views [1–7]. Mathematical models have been confirmed to be an effective and valuable approach to understand the dynamical behavior of infectious diseases [8–11]. Many diseases are caused by a virus, for example, human immunodeficiency virus (HIV) can destroy the immune cells, reduce human immunity, and ultimately lead to AIDS. A lot of mathematical models have been built to explain the virus infection process from different views [12–15]. In recent years, the virus infection dynamical models incorporating spatial dispersion [16, 17] have received widely attentions. For example, under spatial domain equipped with suitable boundary conditions, K. Wang and W. Wang [18] established the existence of travelling wave solutions via the geometric singular perturbation method and the spatial domain is assumed to be one dimensional, that is, x,t∈−∞,∞×0,+∞. And then, Brauner et al. [19] extended the works [18] to a two-dimensional square domain 0,l×0,l with periodic boundary conditions by introducing a new parameter λ0 which is the largest eigenvalue of some Sturm-Liouville problem and taking the heterogeneous reproductive ratio into account. Recently, Wang et al. [20] noticed that a realistic domain should be bounded but is typically not a square, under no-flux boundary condition (homogenous Neumann boundary condition) in a general bounded domain Ω⊂Rn with smooth boundary ∂Ω, and threshold dynamics of the model were investigated by appealing to the theory of uniform persistence and the comparison theorem. Some further developments have been performed on the virus infection dynamical models with diffusion term (see, for example, Hattaf and Yousfi [21], Wang and Ma [22–24], Lai and Zou [25], Wang and Xu [26], and Wang et al. [27]).It is widely known that virus-to-cell transmission and direct cell-to-cell transmission are two predominant infection modes in within-host environment (see, for example, [28, 29]). The virus-to-cell transmission has iterative process: the binding of virus to a receptor on the surface of CD4+ T cells, the fusion of virus with host cells and then the release of genetic materials, the transcription of genetic materials in an infected cell, the assembling of virus inside the infected cells, and budding through the membrane of the infected cells [30]. The cell-to-cell transmission process is due to that a large number of viruses can be transferred from infected cells to uninfected cells through the formation of virally induced structures termed virological synapses [31]. In order to examine the effects of both diffusion and spatial heterogeneity, Wang et al. [32] proposed the following virus infection dynamical model incorporating cell-to-cell transmission under the homogeneous Neumann boundary condition:
(1)∂Ux,t∂t=ξx−dUxUx,t−β1xUx,tωx,t−β2xUx,tVx,t,∂Vx,t∂t=β1xUx,tωx,t+β2xUx,tVx,t−dVxVx,t,∂ωx,t∂t=DωΔω+kxVx,t−dωxωx,t.In model (1), ξx is the production rate of newly produced uninfected cells. dUx, dVx, and dUx represent the death rates of uninfected cells, infected cells, and free viruses, respectively. β1x denotes the transmission coefficient for virus-to-cell infection. β2x represents the transmission coefficient for cell-to-cell infection. kx denotes the production rate of virus due to the lysis of infected cells. Dω is the diffusion coefficient and Δ is Laplacian. For model (1), the virus dynamics are fully determined by an important parameter which is called basic reproduction number R0, in the following sense: if R0<1, the infection-free steady state is globally asymptotically stable, while if R0>1, the model is uniformly persistent and the infection steady state is globally asymptotically stable.The mobility of cells in the latent period will result in a delay term with spatial averaging on the spatial habitatΩ [33]. To formulate this process with the latency properly, we introduce an infection age variable a1. The infected cells can be divided into two epidemiological categories: latently infected cells (I=Ix,t) and actively infected cells (V=Vx,t). Let Mt,a1,x represent the density of cells with infection age a1 at time t and habitat x. From the biological perspective, we assume that all populations remain confined to the region Ω for all time and subject to no-flux boundary condition for Mt,a1,x:
(2)Da1,x∇Mt,a1,x·ν=0,where ν is the outward normal to the smooth boundary ∂Ω. We adopt the standard model on describing age-structured population with spatial diffusion [34]. Then
(3)∂Mt,a1,x∂t+∂Mt,a1,x∂a1=∇·Da1,x∇Mt,a1,x−dVxMt,a1,x.In (3), ∇Mt,a1,x is the gradient of Mt,a1,x with respect to the spatial variable x and ∇·Da1,x∇Mt,a1,x represents the divergence of Da1,x∇Mt,a1,x. Da1,x represents the diffusion rate of cells at age a1 and habitat x. dVx is the natural death rate which is independent of the infection age a1. Let τ be the average incubation period. Then
(4)Ix,t=∫0τMt,a1,xda1,x,t=∫τ∞Mt,a1,xda1.In order to make the model mathematically tractable yet without losing the main characteristics, we make some assumptions for functionDa1,x as follows:
(5)Da1,x=DIx,fort≥0,a1∈0,τandx∈Ω,DVx,fort≥0,a1∈τ,∞andx∈Ω.Integrating both sides of (3) from 0 to τ and from τ to ∞, it then follows that
(6)∂Ix,t∂t=∇·DIx∇Ix,t−dVxIx,t−Mt,τ,x+Mt,0,x,and
(7)∂Vx,t∂t=∇·DVx∇Vx,t−dVxVx,t+Mt,τ,x−Mt,∞,x.For biological reasons, we assume thatMt,∞,x=0 (see, for example, [35]), since the recruitment of newly infected cells Mt,0,x is divided into two parts: the contact of uninfected cells and virus and the contact of uninfected cells and infected cells. Thus, adopting Beddington-DeAngelis functional response which was firstly proposed by [36, 37] leads to the following condition:
(8)Mt,0,x=β1xUx,tωx,t1+a1xωx,t+b1xUx,t+β2xUx,tVx,t1+a2xVx,t+b2xUx,t,where a1x and a2x are the measure of virus interference during infection and b1x and b2x determine how fast the infection rate approaches its saturation value (see, for example, [36, 37]).In the following, we determineMt,τ,x by the method of characteristics. Let vr,a1,x=Ma1+r,a1,x with r≥0. For a1∈0,τ, it then follows that
(9)∂vr,a1,x∂a1=∂Mr,a1,x∂t+∂Mr,a1,x∂a1t=a1+r=∇·DIx∇Ma1+r,a1,x−dVxMa1+r,a1,x=∇·DIx∇vr,a1,x−dVxvr,a1,x,vr,0,x=β1xUr,xωr,x1+a1xωr,x+b1xUr,x+β2xUr,xVr,x1+a2xVr,x+b2xUr,x.Regardingr as a parameter and solving the above equation, we obtain that
(10)vr,a1,x=∫ΩΓa1,x,yβ1yUy,rωy,r1+a1yωy,r+b1yUy,rdy+∫ΩΓa1,x,yβ2yUy,rωy,r1+a2yωy,r+b2yUy,rdy,where Γa1,x,y is the fundamental solution associated with the partial differential operator ∇·DI·∇−dV·, which satisfies ∫ΩΓa1,x,ydy=1, ∀x∈Ω. Γa1,x,y=Γa1,y,x, and Γa1,x,y>0 ∀x∈Ω, x≠y (see, [38], Chapter 1).SinceMt,τ,x=vt−τ,τ,x, ∀t≥τ, we get the formula for Mt,τ,x for a1=τ:
(11)Mt,τ,x=∫ΩΓτ,x,yβ1yUy,t−τωy,t−τ1+a1yωy,t−τ+b1yUy,t−τdy+∫ΩΓτ,x,yβ2yUy,t−τVy,t−τ1+a2yVy,t−τ+b2yUy,t−τdy.It then follows that(12)∂Ix,t∂t=∇·DIx∇Ix,t−dVxIx,t−∫ΩΓτ,x,yβ1yUy,t−τωy,t−τ1+a1yωy,t−τ+b1yUy,t−τdy−∫ΩΓτ,x,yβ2yUy,t−τVy,t−τ1+a2yVy,t−τ+b2yUy,t−τdy+β1xUx,tωx,t1+a1xωx,t+b1xUx,t+β2xUx,tVx,t1+a2xVx,t+b2xUx,t,and
(13)∂Vx,t∂t=∇·DVx∇Vx,t−dVxVx,t+∫ΩΓτ,x,yβ1yUy,t−τωy,t−τ1+a1yωy,t−τ+b1yUy,t−τdy+∫ΩΓτ,x,yβ2yUy,t−τVy,t−τ1+a2yVy,t−τ+by2Uy,t−τdy.With all these assumptions, the HIV infection dynamics modeling cell-to-cell with nonlocal infections can be described by the following nonlocal and time-delayed reaction-diffusion equations:(14)∂Ux,t∂t=∇·DUx∇Ux,t+ξx−dUxUx,t−β1xUx,tωx,t1+a1xωx,t+b1xUx,t−β2xUx,tVx,t1+a2xVx,t+b2xUx,t,∂Ix,t∂t=∇·DIx∇Ix,t−dVxIx,t−∫ΩΓτ,x,yβ1yUy,t−τωy,t−τ1+a1yωy,t−τ+b1yUy,t−τdy−∫ΩΓτ,x,yβ2yUy,t−τVy,t−τ1+a2yVy,t−τ+b2yUy,t−τdy+β1xUx,tωx,t1+a1xωx,t+b1xUx,t+β2xUx,tVx,t1+a2xVx,t+b2xUx,t,∂Vx,t∂t=∇·DVx∇Vx,t−dVxVx,t+∫ΩΓτ,x,yβ1yUy,t−τωy,t−τ1+a1yωy,t−τ+b1yUy,t−τdy+∫ΩΓτ,x,yβ2yUy,t−τVy,t−τ1+a2yVy,t−τ+b2yUy,t−τdy,∂ωx,t∂t=∇·Dωx∇ωx,t+kxVx,t−dωxωx,t.SinceIx,t is decoupled from other equations for model (14), we consider the following dynamical model:
(15)∂Ux,t∂t=∇·DUx∇Ux,t+ξx−dUxUx,t−β1xUx,tωx,t1+a1xωx,t+b1xUx,t−β2xUx,tVx,t1+a2xVx,t+b2xUx,t,∂Vx,t∂t=∇·DVx∇Vx,t−dVxVx,t+∫ΩΓτ,x,yβ1yUy,t−τωy,t−τ1+a1yωy,t−τ+b1yUy,t−τdy+∫ΩΓτ,x,yβ2yUy,t−τVy,t−τ1+a2yVy,t−τ+b2yUy,t−τdy,∂ωx,t∂t=∇·Dωx∇ωx,t+kxVx,t−dωxωx,t,for x∈Ω, t>0. We consider a closed environment in the sense that the fluxes for each of these subpopulations are aero, and hence we propose no-flux condition on the boundary:
(16)DUx∇Ux,t·ν=DVx∇Vx,t·ν=Dωx∇ωx,t·ν=0,x∈∂Ω,t>0,and initial conditions
(17)Ux,θ=ϕ1x,θ≥0,Vx,θ=ϕ2x,θ≥0,ωx,θ=ϕ3x,θ≥0,θ∈−τ,0.We now briefly outline the plan of this paper. In Section2, we establish basic reproduction number R0. In Section 3, we study threshold dynamics in terms of basic reproduction number for the heterogeneous model. At last, we give discussions and biological implications in Section 4.


## 2. Basic Reproduction Number

LetX≔CΩ¯,ℝ3 be the Banach space with the supremum norm ∥·∥X by ϕX=supx∈Ω¯∣ϕx∣=supx∈Ω¯ϕ12x+ϕ22x+ϕ32x, ϕ=ϕ1,ϕ2,ϕ3T, where ∣·∣ denotes the Euclidean norm in ℝ3 and T denotes the transpose of the vector, and X+≔CΩ¯,ℝ+3. We easily get that X+ is a closed cone of X.Defineℂ=C−τ,0,X with the norm ϕ=supθ∈−τ,0ϕθX and ℂ+=C−τ,0,X+, then ℂ,ℂ+ is an ordered Banach space. For any given continuous function u:−τ,σ→X for σ>0, define ut∈ℂ+ by utθ=ut+θ, θ∈−τ,0. Let ℤ≔ℂΩ¯,ℝ and ℤ+≔ℂΩ¯,ℝ+. Suppose that M1t, M2t, and M3t are the strongly continuous semigroups associated with ∇·DU·∇−dU·, ∇·DV·∇−dV·, and ∇·Dω·∇−dω· subject to homogeneous Neumann boundary conditions, respectively, that is,
(18)ℳitϕx=∫ΩΓix,y,tϕydy,for any ϕ∈ℤ, t≥0, and Mi0ϕ=ϕ, i=1,2,3. Γ1,Γ2, and Γ3 are the Green functions associated with ∇·DU·∇−dU·, ∇·DV·∇−dV·, and ∇·Dω·∇−dω·, respectively, subject to homogenous Neumann boundary conditions. According to Section 7.1 and Corollary 7.2.3 in [39], it follows that Mit:ℤ→ℤ,i=1,2,3 is strongly positive and compact for each t>0. We can also obtain that Kt=M1t,M2t,M3t:X→X, t≥0, is a strongly continuous semigroup. Let Ai:DAi→ℤ be the generator of M1t,M2t,M3t, respectively i=1,2,3. Then, we get that Kt:X→X is a semigroup generated by the operator A=A1,A2,A3 defined on DA=DA1×DA2×DA3.In view of Lemma 2.2 in [35], model (15) exists a unique infection-free steady state E0=U^x,0,0. Linearizing model (15) at the infection-free steady state E0, we get the following linearized system:
(19)∂v1∂t=∇·DUx∇v1x,t−dUxv1x,t−β1xU^x1+b1xU^xv3x,t−β2xU^x1+b2xU^xv2x,t,∂v2∂t=∇·DVx∇v2x,t+∫ΩΓτ,x,yβ1yU^y1+b1yU^yv3t−τ,ydy+∫ΩΓτ,x,yβ2yU^y1+b2yU^yv2t−τ,ydy−dVxv2x,t,∂v3∂t=∇·Dωx∇v3x,t+kxv2x,t−dωxv3x,t,satisfying the following boundary conditions
(20)DUx∇v1x,t⋅ν=DVx∇v2x,t⋅ν=Dωx∇v3x,t⋅ν=0,∀x∈∂Ω,t>0.We consider the following model:(21)∂v1∂t=∇·DUx∇v1x,t−dUxv1x,t−β1xU^x1+b1xU^xv3x,t−β2xU^x1+b2xU^xv2x,t,∂v2∂t=∇·DVx∇v2x,t+∫ΩΓτ,x,yβ1yU^y1+b1yU^yv3t,ydy+∫ΩΓτ,x,yβ2yU^y1+b2yU^yv2t,ydy−dVxv2x,t,∂v3∂t=∇·Dωx∇v3x,t+kxv2x,t−dωxv3x,t.Substitutingv2x,t=eλtφ1x and v3x,t=eλtφ2x into v2 and v3 of model (21) which is called the infectious compartments results in the following eigenvalue problem:
(22)λφ1x=∇⋅DVx∇φ1x+∫ΩΓτ,x,yβ1yU^y1+b1yU^yφ2ydy+∫ΩΓτ,x,yβ2yU^y1+b2yU^yφ1ydy−dVxφ1x,λφ2x=∇⋅Dωx∇φ2x+kxφ1x−dωxφ2x,Dvx∇φ1x⋅ν=Dωx∇φ2x⋅ν=0,∀x∈∂Ω,t>0,φ=φ1,φ2∈ℤ×ℤ.By the similar argument to Theorem 7.6.1 in [39], it then follows that eigenvalue problem (22) has a principal eigenvalue λ0U^x with a positive eigenfunction.We further consider the nonlocal eigenvalue problem associated with(23)λφ1x=∇⋅DVx∇φ1x+e−λτ∫ΩΓτ,x,yβ1yU^y1+b1yU^yφ2ydy+e−λτ∫ΩΓτ,x,yβ2yU^y1+b2yU^yφ1ydy−dVxφ1x,λφ2x=∇⋅Dωx∇φ2x+kxφ1x−dωxφ2x,Dvx∇φ1x⋅ν=Dωx∇φ2x⋅ν=0,∀x∈∂Ω,t>0,φ=φ1,φ2∈ℤ×ℤ.From Theorem 2.2 in [40], we get some useful information eigenvalue problem (23).Lemma 2.1.
The eigenvalue problem (23) has a principal eigenvalue λ¯0U^x with a strictly positive eigenfunction, and for any τ≥0, λ¯0U^x has the same sign as λ0U^x.We adopt the same ideas as in [41–44] to define basic reproduction number of model (15). For this purpose, we define the positive linear operator as follows:
(24)ℋφx=ℋ1φx,ℋ2φx,∀ϕ=φ1,φ2∈ℤ×ℤ,x∈Ω¯,where
(25)ℋ1φx=∫ΩΓτ,x,yβ1yU^y1+b1yU^yφ2ydy+∫ΩΓτ,x,yβ2yU^y1+b2yU^yφ1ydy,ℋ2φx=kxφ1x.In order to compute basic reproduction numberℛ0, we assume that both infected cells and virus are near the infection-free steady state E0 and introduce initial infected cells and virus at time t=0, where the distribution of initial infected cells and virus is described by φ1x,φ2x. From model (15) and as time evolves, those distributions can reach
(26)ℳtφx=ℳ2tφ1x,ℳ3tφ2x,at time t. Thus, the total distribution of new infected cells is
(27)∫0+∞ℋ1ℳtφdtx=∫τ+∞ℋ1ℳt−τφdtx=∫0+∞∫ΩΓτ,x,yβ1yU^y1+b1yU^yℳ3tφ2ydyd+∫0+∞∫ΩΓτ,x,yβ2yU^y1+b2yU^yℳ2tφ1ydydt.Similarly, the total distribution of new virus can be described as(28)∫0+∞ℋ2ℳtφdtx=∫0+∞kxℳ2tφ1xdt.It then follows that(29)ℒφ=∫0+∞ℋℳtφdt=ℋ∫0+∞ℳtφdt,where ℒ is the next infection operator (see, for example, [41–44]). It maps the initial distribution of infected cells and virus to the total distribution of new infected cells and virus produced during the infection period.By [41–44], we define the spectral radius of ℒ as the basic reproduction number of model (15), namely,
(30)ℛ0=rℒ.In view of the general results developed in [42], we have the following observation.Lemma 2.2.
ℛ0−1 has the same sign as λ0U^x.


## 3. Threshold Dynamics in a Bounded Spatial Domain

In this section, we study threshold dynamics in terms ofℛ0 for model (15), assuming a bounded spatial domain Ω⊂ℝn. We assume that all the parameters ξx, βx, ax, bx, dUx, dVx, dωx, and kx are spatially dependent. Assume these functions are positive, continuous, and bounded in the domain Ω¯. Let
(31)ξ¯=maxx∈Ω¯ξx>0,β¯=maxx∈Ω¯βx>0,k¯=maxx∈Ω¯kx>0,a¯=minx∈Ω¯ax>0,b¯=minx∈Ω¯bx>0,d¯U=minx∈Ω¯dUx>0,d¯V=minx∈Ω¯dVx>0,d¯ω=minx∈Ω¯dωx>0.For anyϕ=ϕ1,ϕ2,ϕ3∈ℂ+ and x∈Ω¯, we define ℱ=ℱ1,ℱ2,ℱ3:ℂ+→X by
(32)ℱ1ϕx=ξx−β1xϕ10,xϕ30,x1+a1xϕ30,x+b1xϕ10,x−β2xϕ10,xϕ20,x1+a2xϕ20,x+b2xϕ10,x,ℱ2ϕx=∫ΩΓτ,x,yβ1yϕ1−τ,yϕ3−τ,y1+a1xϕ3−τ,y+b1xϕ1−τ,ydy+∫ΩΓτ,x,yβ2yϕ1−τ,yϕ2−τ,y1+a2xϕ2−τ,y+b2xϕ1−τ,ydy,ℱ3ϕx=kxϕ20,x.Obviously,ℱ is locally Lipschitz. Then, model (15) can be rewritten as the following abstract functional differential equation:
(33)dudt=Au+ℱut,t>0,u0=ϕ∈ℂ+,where u=U,V,ω.Lemma 3.1.
For any givenϕ∈ℂ+, there exists a unique mild solution ut,·,ϕ of model (15) defined on its maximal interval of existence 0,tϕ with u0=ϕ, where tϕ≤∞. Further, ut,·,ϕ∈ℂ+, ∀t∈0,tϕ, and ut,·,ϕ are a classical solution of model (15), ∀t>τ.Proof 1.
For anyϕ=ϕ1,ϕ2,ϕ3∈ℂ+ and for sufficiently small k′≥0, we obtain that
(34)ϕ+kℱϕx=ϕ1+k′ξx−k′β1xϕ10,xϕ30,x1+a1xϕ30,x+b1xϕ10,x−k′β2xϕ10,xϕ20,x1+a2xϕ20,x+b2xϕ10,xϕ2+∫ΩΓτ,x,yk′β1yϕ1−τ,yϕ3−τ,y1+a1yϕ3−τ,y+b1yϕ1−τ,ydy+∫ΩΓτ,x,yk′β2yϕ1−τ,yϕ2−τ,y1+a2yϕ2−τ,y+b2yϕ1−τ,ydyϕ3+k′kxϕ20,x≥1−k′β¯1a¯1−k′β¯2a¯2ϕ1ϕ2ϕ3≥000.
Thus,ϕ+k′ℱϕx∈ℂ+. This implies that limk′→0+1/k′distϕ+k′ℱϕx,ℂ+=0, for all ϕ∈ℂ+. From Corollary 4 in [45] (see, also [46], Corollary 8.1.3), we obtain the conclusion stated in Lemma 3.1. The proof is completed.We are now in the position to study the well-posedness of model (15) in the sense of the following theorem.Theorem 3.1.
For anyϕ=ϕ1,ϕ2,ϕ3∈ℂ+, model (15) has a unique solution ut,·,ϕ on 0,∞ with u0=ϕ. Moreover, the solution semiflow Φt=ut,·,ϕ:ℂ+→ℂ+ has a compact global attractor for t≥0.Proof 2.
It is easy to see that model (15) defines a semiflow Φt≔ut·:ℂ+→ℂ+ by
(35)Φtϕθ,x=ut+θ,x,ϕ,∀θ∈−τ,0,x∈Ω¯,t>0.
For any fixedϕ=ϕ1,ϕ2,ϕ3∈ℂ+→ℂ+, from the first equation of model (15), we obtain that
(36)∂Ux,t∂t≤∇·DUx∇Ux,t+ξ¯−d¯UUx,t.
From the comparison principle, we easily get that there existst1ϕ>0 such that Ux,t≤2ξ¯/d¯U≕ℬ1 for t>t1. From the second equation of model (15), we have that
(37)∂Vx,t∂t≤∇·DVx∇Vx,t+β¯1ℬ1a¯1+β¯2ℬ1a¯2−d¯VVx,t.
There existst2ϕ>0 such that Vx,t≤2β¯1ℬ1/a¯1d¯V+2β¯2ℬ1/a¯2d¯V≕ℬ2 for t>t2. From the third equation of model (15), we get
(38)∂ωx,t∂t≤∇·Dωx∇ωx,t+k¯ℬ2−d¯ωωx,t.
There also existst3ϕ>0 such that ωx,t≤2k¯ℬ2/d¯ω≕ℬ3 for t>t3.
Consequently, the existence of solutionsut,·,ϕ of model (15) claimed in Lemma 3.1 is indeed global (i.e., tϕ=∞). The solution semiflow Φt:ℂ+→ℂ+ is point dissipative. Moreover, by Theorem 2.2.6 in [46], we have that Φt:ℂ+→ℂ+ is compact for any t>τ. Thus, from Theorem 3.4.8 in [47], we know that Φt:ℂ+→ℂ+ has a compact global attractor in ℂ+ for t≥0. We complete the proof.The following results will play an important role in establishing uniform persistence of model (15).Lemma 3.2.
Letut,·,ϕ be the solution of model (15) with u0,·,ϕ=ϕ∈ℂ+, then we have the following:
(i) If there exists somet0>0 such that Vt0,·,ϕ≢0 and ωt0,·,ϕ≢0, then
(39)Vt,·,ϕ>0,ωt,·,ϕ>0,for all t>t0, x∈Ω¯.
(ii) It holds thatUt,·,ϕ>0 for any t>0, x∈Ω¯, and
(40)liminft→∞Ut,·,ϕ≥ξ¯dU+β¯1/a¯1+β¯2/a¯2,uniformly for x∈Ω¯.Proof 3.
From the second and third equations of model (15), it follows that
(41)∂Vx,t∂t≥∇·DVx∇Vx,t−dVxVx,t,∂ωx,t∂t≥∇·Dωx∇ωx,t−dωxωx,t,with the following boundary conditions
(42)∂Vx,t∂ν=∂ωx,t∂ν=0,x∈∂Ω.
IfVt0,·,ϕ≢0 and ωt0,·,ϕ≢0 for some t0≥0, it follows from the strong maximum principle (see, for example, [48] p. 172, Theorem 4) and the Hopf boundary lemma (see, for example, [48] p. 170, Theorem 3) that Vt,·,ϕ>0, Mt,·,ϕ>0, and ωt,·,ϕ>0 for t>t0, x∈Ω¯, that is, the conclusion i holds.
From the first equation of model (15), we get
(43)∂Ux,t∂t≥D0ΔU+ξ¯−dU+β¯1a¯1+β¯2a¯2Ux,t.
Letvx,t,ϕ be the solution of
(44)∂vx,t∂t=D0Δv+ξ¯−dU+β¯1a¯1+β¯2a¯2vx,t,∂v∂ν=0,x∈∂Ω,v0,x=ϕ0,x.
It then follows thatUt,x,ϕ≥vt,x,ϕ>0 for all t>0 and x∈Ω¯. It follows from the standard parabolic comparison principle that liminft→∞Ut,·,ϕ≥ξ¯/dU+β¯1/a¯1+β¯2/a¯2 uniformly for x∈Ω¯. The proof is completed.The following theorem indicates thatℛ0 is a threshold quantity for virus extinction or persistence.Theorem 3.2.
Supposeux,t,ϕ is the solution of model (15) with u0=ϕ∈ℂ+.. Then, the following statements hold:
(i)
Ifℛ0<1, then the infection-free steady state E0 of model (15) is globally attractive.
(ii)
Ifℛ0>1, then model (15) has at least one coexistence steady state and there exists ς>0 such that any nonnegative solution ut,x,ϕ with ϕ20≢0 and ϕ30≢0, we get
(45)liminft→∞Vx,t≥ς,liminft→∞ωx,t≥ς,uniformly for all x∈Ω¯.Proof 4.
From Lemma2.2, we easily obtain that λ0U^0<0 when ℛ0<1. Since limε→0λ0U^0+ε=λ0U^0<0, there exists ε0>0 sufficiently small such that λ0U^0+ε0<0. For fixed ε0>0, there exists T2>0 such that Ux,t≤U^0+ε0 for all t≥T2, x∈Ω¯. Therefore, for all t≥T2, from the second and third equations of model (15), it follows that
(46)∂V∂t≤∇·DVx∇Vx,t+∫ΩΓτ,x,yβ1yU^+ε01+b1yU^+ε0ωy,t−τdy+∫ΩΓτ,x,yβ2yU^+ε01+b2yU^+ε0Vy,t−τdy−dVxVx,t,∂ω∂t≤∇·Dωx∇ωx,t+kxVx,t−dωxωx,t.
According to Lemma2.1, we get λ¯0U^+ε0<0, and there exists a positive eigenfunction ψ0 corresponding to λ¯0U^+ε0<0. It then obtains the following linear system:
(47)∂u2∂t=∇·DVx∇u2x,t+∫ΩΓτ,x,yβ1yU^+ε01+b1yU^+ε0u3y,t−τdy+∫ΩΓτ,x,yβ2yU^+ε01+b2yU^+ε0u2y,t−τdy−dVxu2,∂u3∂t=∇·Dωx∇u3x,t+kxu2−dωxu3,with the following boundary conditions
(48)∂u2∂ν=∂u3∂ν=0,x∈Ω¯,admitting a solution ut,x=eλ¯0U^0+ε0tψ0x. Then, for any given ϕ∈ℂ+, there exists α′>0 such that
(49)Vt,·,ϕ,ωt,·,ϕ≤α'ut,·,t∈T2−τ,T2.
By the comparison principle, it follows that(50)Vt,x,ϕ,ωt,x,ϕ≤α′eλ0U^0+ε0tψ0x,∀t≥T2.
Thus,limt→+∞Vt,x,ϕ,ωt,x,ϕ=0 uniformly for x∈Ω¯. Hence, by [35], it then follows that limt→+∞Ut,x,ϕ=U^0x uniformly for ∀x∈Ω¯, which ends the proof of Part (i).Forℛ0>1, we employ the persistence theory in [49] to study uniform persistence of model (15). Define
(51)X0=ϕ=ϕ1,ϕ2,ϕ3∈ℂ+:ϕ20≢0andϕ30≢0.Obviously, we have that(52)∂X0≔ℂ+\X0=ϕ∈ℂ+:ϕ20≡0orϕ30≡0.In view of Lemma3.2, it then follows that X0 is positively invariant for the solution semiflow Φt. Define
(53)W∂≔ϕ∈∂X0:Φtϕ∈∂X0,∀t≥0.Letωϕ be the omega limit set of the orbit of Φt through ϕ∈ℂ+, and set W1≔U^x,0,0.Claim 1.
∪ϕ∈W∂ωϕ=W1.
For any givenϕ∈W∂, it follows that utϕ∈∂X0 for all t≥0. Since utϕ=ut,·,ϕ, we easily have that for each t≥0 either Vt,·,ϕ≡0 or ωt,·,ϕ≡0. For the case where Vt,·,ϕ≡0, ∀t≥0, combined with the third equation of model (15), it can be obtained that limt→+∞ωt,·,ϕ=0 uniformly for x∈Ω¯. Further, from the first equation of model (15) and the theory of asymptotically autonomous semiflows [50], it then follows that limt→+∞Ut,·,ϕ=U^x uniformly for x∈Ω¯. In the case where Vt0,·,ϕ≢0 for some t0>0, by parabolic maximum principle, we have that Vt,·,ϕ>0 for all t>t0 and x∈Ω¯. Therefore, we get ωt,·,ϕ≡0 for all t≥t0. From the third equation of model (15), we obtain that Vt,·,ϕ≡0 for all t≥t0, which is a contradiction. Hence, we get ∪ϕ∈W∂ωϕ=W1. We complete the proof of Claim 1.Claim 2.
Similar to the proof of Lemma 3.3 in [17], we easily get that E0 is a uniform weak repeller in the sense that
(54)limsupt→∞∥ut,·,ϕ−U^x,0,0∥ℂ+≥χ.
We define a continuous functionp:ℂ+→ℝ+ by
(55)pϕ=minminx∈Ω¯ϕ20x,minx∈Ω¯ϕ30x,∀ϕ∈ℂ+.
We easily verify thatp−10,+∞⊂X0. By Lemma 3.3, p has the property that if either pϕ=0 and ϕ∈X0 or pϕ>0, then pΦtϕ>0 for all t>0. Consequently, we have that p is a generalized distance function for the semiflow Φt:ℂ+→ℂ+ (see [49]). From the above claims, it follows that any forward orbit of Φt in W∂ converges to W1 which is isolated in ℂ+, and WsW1∩X0=0, where WsW1 is the stable set of W1 (see, for example, [49]). Further, we find that no subset of W1 forms a cycle in ∂X0 from E0 to E0. By Theorem 3 in [49], we prove that there exists ς>0 such that minpψ:ψ∈ωϕ>ς for any ϕ∈X0. In view of Theorem 1.3.6 in [51], we obtain that model (15) has at least one coexistence steady state if ℛ0>1. The proof is completed.


## 4. Discussions and Biological Implications

This paper is devoted to develop a nonlocal and time-delayed reaction-diffusion model for HIV infection within host cell-to-cell viral transmissions. For model (15), the virus dynamics are fully determined by an important parameter which is called basic reproduction number ℛ0, in the following sense: if ℛ0<1, the infection-free steady state E0 is globally attractive, implying infection becomes extinct; while if ℛ0>1, the virus will persist in the host environment.



---

*Source: 2139290-2018-08-01.xml*