# Spatial and Temporal Dynamics of a Viral Infection Model with Two Nonlocal Effects

Authors: Xiaoyan Wang; Yuming Chen; Junyuan Yang
Journal: Complexity
 (2019)

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## Abstract

We propose and study a viral infection model with two nonlocal effects and a general incidence rate. First, the semigroup theory and the classical renewal process are adopted to compute the basic reproduction numberR0 as the spectral radius of the next-generation operator. It is shown that R0 equals the principal eigenvalue of a linear operator associated with a positive eigenfunction. Then we obtain the existence of endemic steady states by Shauder fixed point theorem. A threshold dynamics is established by the approach of Lyapunov functionals. Roughly speaking, if R0<1, then the virus-free steady state is globally asymptotically stable; if R0>1, then the endemic steady state is globally attractive under some additional conditions on the incidence rate. Finally, the theoretical results are illustrated by numerical simulations based on a backward Euler method.


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## Body

## 1. Introduction

Viruses are very tiny germs. Their presence in the body causes not only familiar infectious diseases (such as the common cold, flu, and warts) but also severe illnesses (such as HIV/AIDS [1–3], hepatitis B [4–7], hepatitis C [6, 7], and human T cell leukemia [8]). Because of the long infectious periods and difficulties in treating them, viral infections have been regarded as serious health problems and have brought heavy economic burden worldwide. The mechanism on a viral infection is quite complicated. Roughly, viruses invade target cells (healthy cells) and replicate in them and then replicated viruses are released. Mathematical modeling has been a very important and efficient way to better understand the evolution of viral infections and to evaluate antiviral drug therapies. In a typical compartmental viral infection model, there are three compartments for uninfected target cells (T), infected cells (I), and free virions (v).In recent years, spatial-structured models have played a crucial role in exploring viral dynamics. In most of the study (see, e.g., [5, 9–11]), uninfected cells and infected cells are assumed to be motionless, while virions can move freely. Obtained results include the existence of travelling waves [5] and asymptotical behavior [10, 11]. To the best of our knowledge, the spatial domain Ω is either one-dimensional or infinitely dimensional. Usually a Laplacian operator with a diffusion coefficient is used to describe the random diffusion of each virion in the adjacent habitat (position). The diffusion term follows Fick’s law and this leads to systems coupling two ordinary differential equations with one parabolic equation.Such mentioned diffusion is assumed such that the diffusive habitat is small. In fact, the motions of virions are always free and thus the limitations for the short diffusion are not reasonable. Thus, nonlocal diffusion accounting for the long-range diffusion effect has been proposed and extensively investigated. The convolution diffusion operator takes the form(1)Lv≔d∫ΩJx-yvy-vxdy,x,y∈Ω,where v∈B and B is a suitable Banach space. This means that a virion at position y can affect another virion at position x at the probability of J(x-y). In [12], Garcia-Melian and Rossi assumed that J(x-y) represents the probability of skipping from location y to x. Then virions arrive at location x from other places at rate ∫ΩJ(x-y)v(y)dy. Based on peculiar features of the nonlocal diffusion operators, models in ecology [13–15], in epidemiology [16–19], and even in materials science [20, 21] have been investigated.It is well known that incidence rates play a key role in understanding intrinsic mechanisms of viral infections. Though commonly used, the bilinear incidence (or called mass action) may not completely capture the viral dynamics. To overcome this deficiency, several forms of incidence rates have been proposed, which include the saturated incidence rateβTv/1+αv(α≥0) [22], Crowley-Martin functional response βTv/(1+γT)(1+αv)(α,γ≥0) [23], and a general form of f(T,v) [8, 24].Motivated by the aforementioned works, we propose a nonlocal diffusive viral infection model with a general incidence rate. To build the model, we still assume that there are three compartments involved in the viral infection for uninfected target cells, infected cells, and free virions. Their densities at timet and position x∈Ω are denoted by T(t,x), I(t,x), and v(t,x), respectively. Here Ω⊂Rn is compact and connected with a smooth boundary and satisfies Ω=int⁡(Ω)¯. The model to be studied is (2)∂Tt,x∂t=λx-μxTt,x-∫Ωfx,y,Tt,x,vt,ydy,x∈Ω,∂It,x∂t=∫Ωfx,y,Tt,x,vt,ydy-δxIt,x,x∈Ω,∂vt,x∂t=∫Ωθx,yvt,ydy-γxvt,x+pxIt,x-cxvt,x,x∈Ωwith the initial condition(3)T0,x=T0x>0,I0,x=I0x≥0,v0,x=v0x≥0forx∈Ω.Here λ(x) is the created rate of uninfected cells at x; μ(x), δ(x), and c(x) are, respectively, the death rates of uninfected cells, infected cells, and virions at x; ∫Ωf(x,y,T(t,x),v(t,y))dy denotes the rate of new infections at time t and location x; ∫Ωθ(x,y)v(t,y)dy represents the transfer rate of virions from positions in Ω∖x to position x with a kernel function θ(x,y) while -γ(x)v(t,x) is the total transfer rate of virions from location x to all the other locations. Note that if we take the kernel function θ(x,y)=J(x-y), then system (2) becomes a classical within-host model with the nonlocal diffusion.The main contribution of this paper has three aspects. Firstly, it is easy to see that the equation forI has an integral term and thus the solution of system (2) lacks strong regularity. This implies that the semiflow generated by system (2) is not compact. To overcome this difficulty, we use Arzelà-Ascoli Theorem [25] to establish the asymptotic smoothness of the semiflow, which enables us to pass the dissipativity of system (2) from L1(Ω;R+) to C(Ω;R+). Secondly, the nonlocal diffusion further weakens the regularity of the solution and this requires nonroutine methods to deal with compactness and nonsupporting of the next generation operator. Finally, the nonlocal diffusion term enhances the difficulty in proving the global stability of the steady states. Inspired by the work of Thieme [26, Section 11], in order to construct a suitable Lyapunov functional, a nonnegative Borel measurable function should be picked to balance the nonlocal term. We explicitly identify such a Borel measurable function as v∗(x) instead of in an abstract form.The organization of this paper is as follows. Section2 gives the existence, uniqueness, and nonnegativity of solutions to system (2). Section 3 shows that the solution semiflow is asymptotically smooth by applying Arzelà-Ascoli Theorem. Section 4 focuses on the basic reproduction number R0 of system (2) defined as the spectral radius of the next-generation operator R(x) and the relationship between R0 and the spectral bound of a linear operator. Section 5 is devoted to the existence of endemic steady states by employing Shauder fixed theorem. We discuss the stability of the virus-free steady state in Section 6. The global behavior of system (2) including uniform persistence and global stability of endemic steady states is established in Sections 7 and 8. In Section 9, we carry out numerical experiments to validate the theoretical results. Section 10 concludes the paper with a succinct discussion.


## 2. Preliminaries

LetX=C(Ω;R) equip with the supremum norm ·X(4)ϕX=supx∈Ωϕx,ϕ∈X.Moreover, for ϕ∈X, we denote ϕ¯=supx∈Ωϕ(x) and ϕ_=infx∈Ωϕ(x). Let Y=X3 and define(5)ψY=ψ1X2+ψ2X2+ψ3X2forψ=ψ1,ψ2,ψ3∈Y.Then Y is also a Banach space equipped with the norm ·Y. Clearly, X+=C(Ω;R+) and Y+=X+3 are positive cones of X and Y, respectively.To study the asymptotic dynamics of system (2), we make the following assumptions.Assumption 1.
(i)
λ ( · ), μ(·), δ(·), p(·), and c(·)∈X are all strictly positive. (ii)
f ∈ C ( Ω 2 × R + 2 , R ) is continuously differentiable with respect to the third and fourth variables and also satisfies the following:
(ii-1)
f ( x , y , 0 , v ) = f ( x , y , T , 0 ) = 0 for all (x,y,T,v)∈Ω2×R+2; (ii-2)
for all(x,y)∈Ω2, f(x,y,T,v) is increasing in both T and v; (ii-3)
for all(x,y,T)∈Ω2×R+, the function f(x,y,T,v)/v is decreasing associated with the variable v on (0,∞); (ii-4)
for anyc>0 there exists some Lc>0 such that(6)fx,y,T,v-fx,y,T¯,v¯≤LcT-T¯+v-v¯forT,T¯,v,v¯∈0,c. (iii)
θ ∈ C ( Ω 2 , R + ) and satisfies θ(x,x)>0. Moreover, it is irreducible and symmetrical for all x,y∈Ω. (iv)
γ ∈ X + and θ satisfy the balance condition(7)∫Ωθx,ydy=γxfor  allx∈Ω.To continue the discussion, we define a linear operatorA and a nonlinear operator F on Y by(8)Aψx=-μxψ1x,-δxψ2x,∫Ωθx,yψ3ydy-cx+γxψ3xand(9)Fψx=λx-∫Ωfx,y,ψ1x,ψ3ydy,∫Ωfx,y,ψ1x,ψ3ydy,pxψ2xfor ψ∈Y and x∈Ω, respectively. Here f is extended to Ω2×R2 through continuity.Lemma 2.
The operatorA defined by (8) generates a uniformly continuous semigroup {etA}t≥0 on Y. Furthermore, eAtY+⊂Y+ for all t∈R+.Proof.
We decompose the operatorA as A=A1+A2, where(10)A1ψx=-μxψ1x,-δxψ2x,-cx+γxψ3xand(11)A2ψx=0,0,∫Ωθx,yψ3ydyfor ψ∈Y and x∈Ω. We readily see that A1 generates a strongly continuous and positive semigroup {eA1t}t≥0 with(12)eA1tψx=e-μxtψ1x,e-δxtψ2x,e-cx+γxtψ3xfor ψ∈Y and x∈Ω. Moreover, it follows from Assumption 1 that the operator A2 is bounded. Then [27, Corollary VI 1.11] and [28, Theorem 1.2] combined ensure that the operator A generates a positively continuous semigourp {eAt}t≥0. This completes the proof.The following proposition gives the existence and uniqueness of solutions of system (2).Proposition 3.
For allu0∈Y, system (2) admits a unique classical solution u(·,u0)∈C([0,Tmax),Y)∩C1((0,Tmax),Y) with Tmax>0. Moreover, the solution has the following properties: (i)
IfTmax<∞ then limt→Tmax-u(t,u0)Y=∞.(ii)
LetT∈(0,Tmax). If {un}⊂Y with un→u0 in Y as n→∞, then(13)limn→∞supt∈0,Tut,un-ut,u0Y=0.(iii)
T(t,x)>0 for t∈(0,Tmax) and x∈Ω.(iv)
Ifu0∈Y+, then u(t,u0)∈Y+ for all t∈(0,Tmax).Proof.
[29, Lemma 3.1], together with Assumption 1, ensures that the operator F defined by (9) is continuously Fréchet differentiable on Y. Therefore,(14)ut,u0x=eAtu0x+∫0teAt-sFus,xdsforx∈Ω.Then from [30, Proposition 4.16] and [28, Theorems 1.2–1.5 in Chapter 6] assertions (i) and (ii) follow immediately.
We use by way of contradiction to prove (iii). We claim that there existst0∈(0,Tmax) and x0∈Ω such that T(t0,x0)=0. Notice that T(t,x0)>0 for all small enough t. Hence, by the continuous dependence of the solution on the initial values, define(15)t^=inf⁡t∈0,Tmax∣Tt,x0=0.Thus t^∈(0,Tmax) and T(t^,x0)=0 with ∂T(t^,x0)/∂t≤0. But then(16)∂Tt^,x0∂t=λx0>0is a contradiction. This proves assertion (iii).
In order to establish (iv), note that for each positiver there exists a sufficiently large κr such that(17)Fut,·+κrut,·>0fort∈R+andu∈CR+,Y+∩B0,r,where B(0,r) is an open ball in Y with the center at (0,0,0) and radius r. We rewrite system (2) in the form of(18)dut,·dt=Aκrut,·+Fκrut,·,Aκr=A-κId where Id is an indentity operator on X and Fκr(u)=F(u)+κru. Applying the method of variation of constant, we have(19)ut,x=eAκrtu0x+∫0tFκrus,xeAκrt-sds.Hence, for (t,u0)∈(0,Tmax)×Y+, u(t,u0)∈Y+.In fact, as the following result shows, solutions exist globally with initial values inY+.Proposition 4.
Letu0∈Y+. Then the solution of (2) with the initial value u0 exists on R+.Proof.
LetJ=[0,Tmax) be the maximal existence interval of the solution through u0. Define Z(t,x)=T(t,x)+I(t,x), for (t,x)∈(J×Ω). Then, for any t∈J, first, we add the first two equations of (2) to obtain (20)∂Zt,x∂t=λx-μxTt,x-δxIt,x≤λx-νxZt,x,where ν(x)=min⁡μx,δx. It follows that (21)Zt,x≤e-tνxZ0,x+λxνx1-e-tνx≤M1x,where M1(x)=Z(0,x)+λ(x)/ν(x). Next we integrate the third equation of system (2) on the set Ω to obtain (22)d∫Ωvt,xdxdt=∫Ω∫Ωθx,yvt,ydydx-∫Ωγxvt,xdx+∫ΩpxIt,xdx-∫Ωcxvt,xdx=∫Ω∫Ωθx,yvt,ydxdyinterchange  order  of  integration-∫Ωγxvt,xdx+∫ΩpxIt,xdx-∫Ωcxvt,xdx=∫Ω∫Ωθy,xvt,ydxdy-∫Ωγxvt,xdxbyAssumption1+∫ΩpxIt,xdx-∫Ωcxvt,xdx=∫Ωγyvt,ydy-∫Ωγxvt,xdx+∫ΩpxIt,xdx-∫Ωcxvt,xdx=∫ΩpxIt,xdx-∫Ωcxvt,xdx≤∫ΩpxM1xdx-c_∫Ωvt,xdx.Solving this differential inequality, we get (23)∫Ωvt,xdx≤e-c_t∫Ωv0,xdx+∫ΩpxM1xdxc_1-e-c_t≤∫Ωv0,xdx+∫ΩpxM1xdxc_≔M2for t∈J. Lastly, it also follows from the last equation of (2) that(24)∂vt,·∂t≤θ¯M2+p¯M1¯-γ_+c_vt,·,which gives(25)vt,·≤v0,·e-γ_+c_t+θ¯M2+p¯M1¯γ_+c_1-e-γ_+c_tfort∈J.
In summary, we have obtained that(26)ut,·Y≤2M1¯2+v0,·X+θ¯M2+p¯M1¯γ_+c_2fort∈J.By (i) of Proposition 3, we get Tmax=∞ and this completes the proof.By Propositions3 and 4, a solution semiflow Φ:R+×Y+→Y+ is defined by(27)Φt,ψ=Tt,·,It,·,vt,·for (t,ψ)∈R+×Y+. Moreover, the existence of the solution of (2) is indeed global and the semiflow Φ is bounded and dissipative. Define(28)Γ=ψ∈Y+0<ψ1x+ψ2x≤λxmin⁡μx,δxforx∈Ωand∫Ωψ3xdx≤∫Ωpxλx/min⁡μx,δxdxc_.Then it follows easily from the proof of Proposition 4 (actually using the same differential inequalities and the resulting inequalities in the proof) that Γ is positively invariant and attracts all the bounded subsets of (2) in Y+. Therefore, to study the asymptotic behavior of (2), we only need to focus on solutions with initial values in Γ.


## 3. The Asymptotic Smoothness

A semiflowΦ isasymptotically smooth if there exists a nonempty compact set attracting each forward invariant bounded closed set (see [31, Definition 2.25]), in other words, if the semiflow is asymptotically compact on every forward invariant bounded closed set.Theorem 5.
The semiflow{Φ}t≥0 defined by (27) is asymptotically compact and hence it is asymptotically smooth.Proof.
The proof is inspired by the ideal in [26, Section 4]. Assume that B is any forward invariant bounded subset of Y+. We show that Φ is asymptotically compact on B [32, pp. 28]. This needs to show that a sequence of solutions un=(Sn,In,vn) is equi-bounded. That is, there exists a sequence {tn} with tn→∞ as n→∞ such that {un(tn)} has a convergent subsequence in Y+. Let us consider the translated solutions u¯n=(T¯n,I¯n,v¯n)=un(tn+·). Then (29)∂T¯nt,x∂t=λx-∫Ωfx,y,T¯nt,x,v¯nt,ydy-μxT¯nt,x,(30)∂I¯nt,x∂t=∫Ωfx,y,T¯nt,x,v¯nt,ydy-δxI¯nt,x,(31)∂v¯nt,x∂t=∫Ωθx,yv¯nt,ydy-γx+cxv¯nt,x+pxI¯nt,x.By the Arzelà-Ascoli theorem, it suffices to show that {T¯n(0,·)}, {I¯n(0,·)}, and {v¯n(0,·)} are equi-continuous. Then, for each x,x~∈Ω, we obtain(32)∂T¯nt,x-T¯nt,x~2∂t=2T¯nt,x-T¯nt,x~λx-λx~-μxT¯nt,x+μx~T¯nt,x~-∫Ωfx,y,T¯nt,x,v¯nt,y-fx~,y,T¯nt,x~,v¯nt,ydy=2T¯nt,x-T¯nt,x~λx-λx~-2μxT¯nt,x-T¯nt,x~2-2T¯nt,x-T¯nt,x~μx-μx~T¯nt,x~-2∫ΩT¯nt,x-T¯nt,x~fx,y,T¯nt,x,v¯nt,y-fx~,y,T¯nt,x,v¯nt,ydy-2∫ΩT¯nt,x-T¯nt,x~fx~,y,T¯nt,x,v¯nt,y-fx~,y,T¯nt,x~,v¯nt,ydy≤λx~-λx2ζ2+3ζ2T¯nt,x-T¯nt,x~2-2μxT¯nt,x-T¯nt,x~2+μx-μx~2T¯nt,x~2ζ2+1ζ2∫Ωfx~,y,T¯nt,x,v¯nt,y-fx~,y,T¯nt,x~,v¯nt,y2dy.Here we have used the fact that f is increasing with respect to T and ab≤[(ζa)2+(b/ζ)2]/2, where ζ>0 is small enough and satisfies(33)3ζ2-2μx<-ζforx∈Ω.Integrating the inequality (32) from -tn to t≥-tn, we obtain (34)T¯nt,x-T¯nt,x~2≤Tn0,x-Tn0,x~2e-ζt+tn+λx-λx~2ζ3+μx-μx~2ζ3sup-tn≤l≤tT¯n2l,x+1ζ3sup-tn≤l≤t∫Ω(fx~,y,T¯nl,x,v¯nl,y-fx~,y,T¯nl,x~,v¯nl,y2dy.By Assumption 1, for each bounded set Ω2×[0,c]2(c>0), f is uniformly continuous. The boundedness of f, together with e-ζtn→0 as n→∞, implies that (T¯n(t,x)-T¯n(t,x~))2→0 as x~→x. This limitation is uniform for both n∈N and t in compact subsets of R. The equi-continuity of {T¯n(t,·):n∈N,t∈[-T,T]} holds immediately for any T>0.
Similarly, fort≥-tn, we can get(35)I¯nt,x-I¯nt,x~2≤In0,x-In0,x~2e-ζt+tn+δx-δx~2ζ3sup-tn≤r≤tI¯n2r,x+1ζ2∫-tnte-δxt-s∫Ωfx,y,T¯ns,x,v¯ns,y-fx~,y,T¯ns,x~,v¯ns,y2dydsArguing directly, we claim that {I¯n(t^,·)} is not equi-continuous for some t^≥0. Then we can pick up a sequence {x~n}n such that x~n→x as n→∞. Moreover, we choose a subsequence of {I¯n} satisfying(36)limsupn→∞I¯nt^,x-I¯nt^,x~n2>0.Since, for each t, {T¯n(t,·)} is equi-continuous, Tn(t,x~n)→Tn(t,x) as n→∞. Applying Fatou’s lemma and Assumption 1, we get from (35) that(37)limsupn→∞I¯nt^,x-I¯nt^,x~n2≤1ζ2∫-∞t^eζtdtlimsupn→∞∫Ωfx,y,T¯nt,x,v¯nt,y-fx~n,y,T¯nt,x~n,v¯nt,y2dydsis a contradiction. This proves that {I¯n(t,·)}n:n∈N,t∈[-T,T]} is also equi-continuous for any T>0.
Finally, again similarly as before, we can obtain(38)v¯n0,x-v¯n0,x~2≤vn0,x-vn0,x~2e-ζt+tn+px-px~2ζ3sup-tn≤l≤tI¯n2l,x+γx+cx-γx~-cx~2ζ3sup-tn≤l≤tv¯n2l,x+∫Ωθx,y-θx~,ydysup-tn≤l≤tv¯nl,x.By Assumption 1 and supx~∈Ωvn(0,x~)<∞, we have (v¯n(0,x)-v¯n(0,x~))2→0 as x~→x uniformly for n∈N. This proves the equi-continuity of {v¯n(0,·)} and the hence the proof is complete.


## 4. The Basic Reproduction Number

This section is conducted for estimation of the basic reproduction number, which is defined as the expected numbers of secondary cases created by a typical infected individual among a completely susceptible population. Clearly, system (2) has a virus-free steady state E0=(T0(x),0,0), where T0(x)=λ(x)/μ(x) for x∈Ω. Linearize (2) around E0 in the disease invasion phase to obtain (39)∂It,x∂t=∫Ωfvx,y,T0x,0vt,ydy-δxIt,x,∂vt,x∂t=∫Ωθx,yvt,ydy+pxIt,x-cx+γxvt,x.Solving them, we have(40)It,x=I0xe-δxt+∫0te-δxt-s∫Ωfvx,y,T0x,0vs,ydydsand(41)vt,x=v0xe-cx+γxt+∫0te-cx+γxt-s∫Ωθx,yvs,ydy+pxIs,xds,respectively. Plugging (40) into (41) yields (42)vt,x=v0xe-cx+γxt+∫0te-cx+γxt-s∫Ωθx,yvs,ydyds+px∫0te-cx+γxt-sI0xe-δxsds+px∫0te-cx+γxt-s∫0se-δxs-τ∫Ωfvx,y,T0x,0vτ,ydydτds.Using change of variables gives us (43)vt,x=v0xe-cx+γxt+px∫0te-cx+γxt-sI0xe-δxsds+∫0te-cx+γxs∫Ωθx,yvt-s,ydyds+px∫0te-cx+γxs∫τte-δxτ-s∫Ωfvx,y,T0x,0vt-τ,ydydτds.Therefore, following the approach of Diekmann et al. [33], we define thenext-generation operatorR:X→X by(44)Rϕx=∫Ω∫0∞e-cx+γxτθx,yϕydτdy+∫Ω∫0∞∫τ∞pxe-cx+γxτe-δxs-τfvx,y,T0x,0ϕydsdτdy=∫Ωθx,yϕydycx+γx+pxδxγx+cx∫Ωfvx,y,T0x,0ϕydyfor ϕ∈X and x∈Ω. Based on Assumption 1, the operator R is well defined, continuous, and positive. In the following, we show that R is compact and nonsupporting. Nonsupporting means that, for any v∈X+∖0 and ξ∈(X∗)+∖0, there exists positive N0=N0(v,ξ) such that ∫Ω(RN0v)(y)ξ(y)dy>0.Proposition 6.
Suppose that Assumption1 holds. Then the next-generation operator R defined by (44) is compact and nonsupporting.Proof.
LetB⊂X+ be a bounded set. It follows from Assumption 1 that R(B) is bounded. To show that R is compact, it suffices to show that R(B) is equi-continuous by Arzelà-Ascoli theorem. For x, x~∈Ω, and ϕ∈B, (45)Rϕx-Rϕx~=pxδxcx+γx-px~δx~cx~+γx~∫Ωfvx,y,T0x,0ϕydy+px~δx~cx~+γx~∫Ωfvx,y,T0x,0-fvx~,y,T0x,0ϕydy+px~δx~cx~+γx~∫Ωfvx~,y,T0x,0-fvx~,y,T0x~,0ϕydy+∫Ωθx,y-θx~,yϕydyγx+cx+1γx+cx-1γx~+cx~∫Ωθx~,yϕydy.Note that the uniform continuity by Assumption 1 and ϕX≤M, where M is a positive constant. We can easily see that R(B) is equi-continuous.
Now, we show thatR is nonsupporting. For any ϕ∈X+∖{0}, by Proposition 3 and the monotonicity property of f, we have(46)Rϕx≥∫Ωθx,yϕydyc¯+γ¯,x∈Ω.From the assumption on θ, it follows that (Rϕ)(x)>0 for x∈Ω. Then, for any ξ∈(X∗)+∖0,, we have ∫Ω(Rϕ)(x)ξ(x)dx>0 (with N0=N0(ϕ,ξ)=1). This proves that R is nonsupporting.From the definition of the basic reproduction number by Diekmann [33], such value is defined by(47)R0≔rRwhere r(·) represents the spectral radius of an operator. Hence, R(·) can be considered as a next-generation operator in [33–35].Now, in order to clarify the relationship between the next-generation operator and a linear operator, we defineL:X→X:(48)Lϕx=pxδx∫Ωfvx,y,T0x,0ϕydy+∫Ωθx,yϕydy-cx+γxϕxfor ϕ∈X and x∈Ω. From definitions of operators R and L, we immediately see that L=(c+γ)(R-Id) or R=Id+1/c+γL, where Id is the identity operator on X.Theorem 7.
r ( R ) > 1, r(R)<1, and r(R)=1 if and only if s(L)>0, s(L)<0, and s(L)=0, respectively, where s(L)=sup⁡Re⁡ξ:ξ∈σL denotes the spectral bound of L.Proof.
First, suppose thatr(R)=1. It follows from [36, Proposition 4.4] that there exists a positive function ϕ∈X such that Rϕ=ϕ or (R-Id)ϕ=0. This implies that (c+γ)(R-Id)ϕ=Lϕ=0. Therefore, s(L)=0. On the other hand, suppose that s(L)=0. Then, with the help of the irreducibility of θ and [37, Theorem 2.2], we conclude that there exists an eigenfunction ϕ>0 with respect to s(L)=0; namely, Lϕ=0. It follows that Rϕ=ϕ+1/c+γLϕ=ϕ, which gives r(R)=1. This proves theta r(R)=1 if and only if s(L)=0.
Now, we only show thatr(R)>1 if and only if s(R)>0 as the proof of r(R)<1 if and only if s(L)<0 is similar. On one hand, let r(R)>1 hold. Then Proposition 6 implies that there exists a positive eigenfunction associated with r(R)>1; that is, Rϕ=ϕ+1/c+γLϕ=r(R)ϕ. It follows that Lϕ=(c+γ)(r(R)-1)ϕ. Based on Assumption 1, we know that (c+γ)(r(R)-1)>0. It follows that s(L)>0. On the other hand, let s(L)>0. Following the above approach, we obtain the existence of a positive eigenfunction ϕ with respect to the eigenvalue s(L)>0; that is, Lϕ=(c+γ)(R-Id)ϕ=s(L)ϕ. Then Rϕ=(1/c+γs(L)+1)ϕ>ϕ, which implies that r(R)>1. This completes the proof.


## 5. Existence of Endemic Steady States

This section is conducted for the existence of endemic steady states of (2). Let E∗=(T∗(·),I∗(·),v∗(·))∈Y+ be a feasible steady state and then it satisfies(49)λx-∫Ωfx,y,T∗x,v∗ydy-μxT∗x=0,∫Ωfx,y,T∗x,v∗ydy-δxI∗x=0,∫Ωθx,yv∗ydy-cx+γxv∗x+pxI∗x=0.We apply Shauder fixed point theorem to find solutions of (49). To overcome the difficulty caused by the nonlinear function f, we make the following modifications on f. For n∈N, define(50)fnx,y,T,v=fx,y,T,v∧nforx,y,T,v∈Ω2×R+2,where v∧n=min⁡v,n. We focus on the following perturbation systems:(51)λx-∫Ωfnx,y,T∗x,v∗ydy-μxT∗x=0,∫Ωfnx,y,T∗x,v∗ydy-δxI∗x=0,∫Ωθx,yv∗ydy-cx+γxv∗x-ϵn+pxI∗x=0,where {ϵn} is a bounded and decreasing sequence in R+ with ϵn→0 as n→∞.Define(52)gnT,v=λx-∫Ωfnx,y,Tx,vydy-μxTx.Since f is, and so is fn, increasing in T, for each x∈Ω and v∈X+ there exists a unique Gn(v)(x)∈[0,T0(x)] such that(53)λx-∫Ωfnx,y,Gnvx,vydy-μxGnvx=0.Lemma 8 ([26, Lemma 7.3]).
For fixedn∈N, Gn(v) is continuous with respect to x, uniformly for v in bounded subsets of X+.Theorem 9.
Suppose thatR0=r(R)>1 holds. Then system (2) admits at least one feasible endemic steady state E∗=(T∗,I∗,v∗)∈Y+ with v∗∈X+∖0.Proof.
Solving the second equation of (51) yields I∗(x)=∫Ωfn(x,y,T∗(x),v∗(y))dy/δ(x). Then substitute this expression of I∗(x) into the third equation of (51) to get(54)∫Ωθx,yv∗ydy-cx+γxv∗x-ϵn+px∫Ωfnx,y,T∗x,v∗ydyδx=0.Therefore, we can define a map Fn(n∈N):X+→X+ by(55)Fnvx=px∫Ωfx,y,Gnvx,min⁡vy,ndyδxcx+γx+∫Ωθx,ymin⁡vy,ndycx+γx+ϵnfor v∈X+ and x∈Ω. Clearly, Fn is continuous and nonincreasing by Assumption 1 and Lemma 8. Furthermore, for all v∈X+,(56)Fnvx≤n∫Ωθx,ydycx+γx+px∫Ωfx,y,Gnvx,ndyδxcx+γx+ϵ1forv∈X+andx∈Ω.Let Bn=v∈X+:v¯≤rn be a closed nonnegative ball with the radius rn≫1 and the center at 0. If we pick up large enough rn, then Fn maps Bn into itself. Since Bn is closed and convex, applying Shauder fixed theorem gives the existence of some v∈Bn such that Fn(v)=v. Note that v≠0. Then T=Gn(v), I=∫Ωf(x,y,Gn(v)(x),v(y))dy/δ(x), and v satisfy(57)λx-∫Ωfx,y,Tx,vydy-μxTx=0,∫Ωfx,y,Tx,vydy-δxIx=0,∫Ωθx,ymin⁡vy,ndy-cx+γxvx-ϵn+pxIx=0.Adding the first two equations of (57) gives(58)Zx≤λxνx,where Z(x)=T(x)+I(x) and ν=min⁡μx,δx. Integrating the last equation of (57) on Ω with respect to x, together with ∫Ωθx,ymin⁡vy,ndy≤∫Ωθ(x,y)v(y)dy, we have (59)∫Ωcxvxdx≤∫ΩpxIxdx+ϵn∫Ωcx+γxdx.Consequently, {∫Ωv(x)dx} is bounded. Thus {v} is uniformly bounded. Furthermore, we can apply the similar argument as that in the proof of Proposition 6 to show that {v} is equi-continuous. Thus it is precompact in X+. Then we can choose a subsequence, say itself, such that (T,I,v)→(T∗,I∗,v∗) uniformly in x∈Ω as n→∞. Letting n→∞ in (57), we see that (T∗,I∗,v∗) is a steady state of system (2).
Now, we show that(T∗,I∗,v∗) is an endemic steady state by showing that v∗≠0. By way of contradiction, assume that v∗≡0, which implies that I(x)→0, and T(x)→T0(x) uniformly for x∈Ω as n→∞ and v→0. For large enough n, we know that v(x)≤n for all x∈Ω. Define(60)GT,vx=∫Ωθx,yvydycx+γx+px∫Ωfx,y,Tx,vydyδxcx+γxforx∈Ω.Then v=F(v)=G(T,v)+ϵn. It follows that(61)vvX=GT,v-RvvX+RvvX+ϵnvX.Note that (62)GT,v-RvXvX=supx∈Ωpxδxcx+γx∫Ωfx,y,Tx,vyvy-fvx,y,T0x,0vyvXdy≤supx∈Ωpxδxcx+γx∫Ωfx,y,Tx,vyvy-fvx,y,Tx,0dy+supx∈Ω⁡pxδxcx+γx∫Ωfvx,y,Tx,0-fvx,y,T0x,0dy.Lemma 8, together with Assumption 1, implies that the above inequality converges to zero as n→∞. From the compactness of R and boundedness of ϵn/vX, we can pick up a subsequence, again say itself, such that(63)RϵnvX→LandϵnvX→ξasn→∞for some ξ≥0 and L∈X+. Set un=v/vX. Then un→u as n→∞ in X. Proposition 6 ensures that u∈X+. By Krein-Rutman theorem, there exists some u∗∈X∗∖0 and u∗≥0 such that R∗u∗=R0u∗. Hence(64)∫Ωuxdu∗x=∫ΩRuxdu∗x+ξu∗Ω≥∫ΩuxdR∗u∗x=R0∫Ωuxdu∗x.This leads to a contradiction with R0>1.


## 6. Stability of the Virus-Free Steady StateE0

The objective of this section is to establish the stability of the virus-free steady stateE0=(T0,0,0).Lemma 10.
IfT0(x)≤T0(x), then T(t,x)≤T0(x) for all (t,x)∈R+×Ω.Proof.
Definew(t)=T(t,x)-T0(x). Noting that λ(x)=μ(x)T0(x), we get(65)∂wt,x∂t=-μx+∫Ωfx,y,Tt,x,vt,ydywt,x-T0x∫Ωfx,y,Tt,x,vt,ydy.Solving this equation, we have (66)wt,x=w0xe-∫0tμx+∫Ωfx,y,Ts,x,vs,ydyds-T0x∫0te-∫stμ+∫Ωfx,y,Tξ,x,vξ,ydydξ∫Ωfx,y,Ts,x,vs,ydyds.Because f is nonnegative on Ω2×R+2 by Assumption 1(ii) and w0(x)=T0(x)-T0(x)≤0, we conclude that, for all (t,x)∈(R+×Ω), w(t,x)=T(t,x)-T0(x)≤0.DefineΦ^:D(Φ^)→X+2 by(67)Φ^ψIψvx=∫Ωfvx,y,T0x,0ψvydy-δxψIx∫Ωθx,yψvydy-cx+γxψvx+pxψIx,where (ψI,ψv)∈D(Φ^) and x∈Ω. We can separate Φ^ into two operators F^ and A^ defined by(68)F^ψIψvx=0∫Ωθx,yψvydy+pxψIxand(69)A^ψIψvx=∫Ωfvx,y,T0x,0ψvydy-δxψIx-cx+γxψvx,respectively. Denote A^=(A^I,A^v)T. Observe that A^v actually acts on ψv by(70)A^vψv=-cx+γxψv,ψv∈X,x∈Ω.Lemma 11.
If the problem(71)A^ψ=ηF^ψ,ψ∈X2has a positive eigenvalue η0 with a positive eigenfunction ψ, then(72)R0=r-F^A^-1=r-A^-1F^=1η0.Proof.
Letη∈ρ(A^), where ρ(A^) represents the resolvent set of the operator A^. For any ϕ∈X2, let ψ∈D(A^) such that(73)ψ=ηId-A^-1ϕ.Then(74)ψI=1η+δϕI+∫Ωfvx,y,T0x,0ψvydyand(75)ψv=ϕvη+c+γ.Let Z be a positive semigroup generated by the operator A^. Then(76)ηId-A^-1ϕ=∫0∞e-ηtZt,ϕdt,ϕ∈X2.Without loss of the generality, letting η=0 and ϕI=0 gives -A^-1ϕ=∫0∞Z(t,ϕ)dt for each ϕ∈X2. By the definition of F^, we have (77)F^-A^-1ϕx=0px∫Ωfvx,y,T0x,0ϕvt,ydyδxcx+γx+∫Ωθx,yϕvydycx+γx=0Rϕvx.Let ψ~≔-A^-1ϕ. Then the above equality can be rewritten:(78)F^ψ~x=0-RA^vψ~vx.This means that(79)0-A^vψ~vx=R-1F^ψ~x.Following the approach in [38], we derive that the eigenvalue problem (80)A^xψ=ηF^xψ,x∈Ωhas a positive eigenvalue η0 with a positive eigenvector ψ(x) for x∈Ω. Since ψ~ is positive, it follows that η0=1/R0.Theorem 12.
Suppose thatR0<1. Then the virus-free steady state E0=(T0(x),0,0) is locally asymptotically stable.Proof.
Linearizing system (2) around the virus-free steady state E0, we have(81)∂Tt,x∂t=-μxTt,x-∫Ωfvx,y,T0x,0vt,ydy,∂It,x∂t=∫Ωfvx,y,T0x,0vt,ydy-δxIt,x,∂vt,x∂t=∫Ωθx,yvt,ydy-cx+γxvt,x+pxIt,x.Let T(t,x)=ψT(x)eηt, I(t,x)=ψI(x)eηt, and v(t,x)=ψv(x)eηt be a solution of system (81). After substitution, we arrive at(82)ηψTx=-μxψTx-∫Ωfvx,y,T0x,0ψvydy,ηψIx=∫Ωfvx,y,T0x,0ψvydy-δxψIx,ηψvx=∫Ωθx,yψvydy-cx+γxψvx+pxψIx.Observe that the first equation of system (82) decouples with the other two equations. Furthermore, the last two equations can be rewritten in the form(83)ηId-A^ψx=F^ψxwhere F^ and A^ are defined by (68) and (69), respectively. By [Theorem 2.2, [39]], the eigenvalue problem (83) has a principal eigenvalue η0 with a unique positive eigenfunction. Lemma 11 implies that η0<0 if R0<1. Otherwise, (ψI,ψv)=(0,0) and then it follows from the first equation of (82) that η=0. Therefore, local stability of the virus-free steady state E0 follows immediately.Next, we show the global attractivity of the virus-free steady stateE0. To establish this result, we need the following Volterra-type:(84)gz=z-1-ln⁡z,z>0.g has the property as follows: for all z>0  g(z)≥0 and the quality holds if and only if z=1. This function has been used very often to construct Lyapunov functionals (see, e.g., Yang et al. [40], Kuniya and Wang [29], and McCluskey [41] and the references therein).Theorem 13.
Suppose thatR0<1 and u∈Γ. The virus-free steady state E0 is globally asymptotically stable.Proof.
For(t,x)∈R+×Ω, we consider the following Lyapunov functional:(85)V^ut,x=ΔxVTt,x+VIt,x+Vvt,x,where VI(t,x)=I(t,x), Vv(t,x)=v(t,x), VT(,x)=gT(t,x)/T0(x), and Δ(x) is a weighted positive function to be determined later. Based on Lemma 10, V^ is well defined.
Taking the derivative ofVT(t,x) along solutions of (2) with respect to t, one arrives at (86)∂VTt,x∂t2=1T0x1-T0xTt,x∂Tt,x∂t=1T0x1-T0xTt,xλx-∫Ωf(x,y,Tt,x,vt,ydy-μTt,x=1T0x1-T0xTt,xμT0x-Tt,x-∫Ωf(x,y,Tt,x,vt,ydy=1T0x-μTt,x-T0x2Tt,x-∫Ωfx,y,Tt,x,vt,ydy+T0xTt,x∫Ωfx,y,Tt,x,vt,ydy.Similarly, we get(87)∂VIt,x∂t2=∫Ωfx,y,Tt,x,vt,ydy-δxIt,xand(88)∂Vvt,x∂t2=∫Ωθx,yvt,ydy+pxIt,x-cx+γxvt,x.Therefore, we can pick up Δ(x)=T0(x) and take a Lyapunov functional in the form of(89)Vut=∫ΩpxψvxδxT0xVTt,x+VIt,x+Vvt,xdx,where ψv defined in (82) is a positive eigenvector function associated with eigenvalue η0. Then (90)dVutdt2=∫ΩpxψvxδxT0x∂VTt,x∂t+∂VIt,x∂t+∂Vvt,x∂tdx≤∫Ωψvxpxδx∫Ωfx,y,T0x,0vt,ydy+∫Ωθx,yvt,ydy-γx+cxvt,xdx=∫ΩLψvxvt,xdx=η0∫Ωψvxvt,xdx,where L is defined by (48). It follows from Theorem 7 that dV[u](t)/dt≤0 provided that R0<1. The equality holds if and only if v(t,x)=0 and T(t,x)=T0(x) for each x∈Ω. Therefore, the largest invariant set in M=u∈Γ∣V˙t=0 is {E0}. We employ the LaSalle invariant principle to conclude that E0 is globally asymptotically stable.


## 7. Uniform Persistence

Denote(91)Γ0=ψ∈Γ∣ψ3≢0,∂Γ0=ψ∈Γ∣ψ3=0.Let(92)M∂=ψ∈∂Γ0∣Φt,ψ∈∂Γ0fort∈R+,And let O+(ϕ) be the positive orbit {Φtϕ}t≥0. From Theorem 13, we readily have the following result.Lemma 14.
For everyu0∈M∂, ω(Φ(t,u0))=E0, where ω(Φ(t,u0)) represents the omega limit set of O+(ϕ).The semiflow{Φt}t≥0 is said to be persistent associated with (Γ0,∂Γ0) if there exists an ϵ such that(93)liminft→∞Tt,·_≥ϵ,liminft→∞It,·_≥ϵ,liminft→∞vt,·_≥ϵfor any solution u(t)=(T(t,·),I(t,·),v(t,·)) with u(0)∈Γ0.Lemma 15.
Letu(t)=(T(t,·),I(t,·),v(t,·)) be any solution of system (2) with the initial value u0∈Γ0. Then u(t,·)>0 for all t>0.Proof.
Proposition3(iii) has already asserted that T(t,·)>0 for t>0. Now we show that I(t,·)>0 and v(t,·)>0 for all t>0. Moreover, since u0=(ψ1,ψ2,ψ3)∈Γ0, we have ψ3≠0. Then a similar argument to that for the proof of Proposition 3(iii) can also give v(t,·)>0 for t>0. Now, from the second equation of (2), it follows that, for t>0,(94)It,·≥e-δxt∫0t∫Ωf(·,y,Ts,·,vs,ydyds>0.Lemma15 implies that Γ0 is a positive invariant of (2). Let Φ|Γ0 be the restricted semiflow of Φ on R+×Γ0.Lemma 16.
Suppose thatR0>1. Then system (2) is uniformly weakly persistent. That is to say that there exists a positive value ϵ such that, for all ψ∈Γ0,(95)limsupt→∞Φt,ψ-E0∞≥ϵ.Proof.
For anyϵ>0, define F^ϵ by(96)F^ϵψIψvx=∫Ωfx,y,T0x-ϵ,ϵϵψvydy0.Clearly, F^ϵ→F^ as ϵ→0+. Note that s(Φ^)=r(F^-A^)>0 since R0>1. Then, for small enough ϵ0, ηψ=(F^ϵ0-A^)ψ has a positive principle eigenvalue ηϵ0>0 with an associated eigenvector function ψϵ0>0.
By way of contradiction, assume that the conclusion fails. Then, forϵ0 chosen above, there exists ψε0∈Γ0 (will be denoted by ψ for simplicity of notation) such that(97)limsupt→∞Φt,ψ-E0Y<ϵ02.Then, with a possible shift in time t and by Lemma 15, we can assume that, for t∈R+,(98)0<vt,x≤ϵ0andT0x-ϵ0≤Tt,x≤T0x+ϵ0.Then, by the monotonicity of f in T and v, we get(99)∂It,x∂t≥∫Ωfx,y,T0x-ϵ0,vt,ydy-δxIt,x.From Assumption 1 (ii-3), we obtain(100)∂It,x∂t≥∫Ωfx,y,T0x-ϵ0,ϵ0ϵ0vt,ydy-δxIt,x.Let u~=(I~,v~) be the solution of the following auxiliary system:(101)∂I~t,x∂t=∫Ωfx,y,T0x-ϵ0,ϵ0ϵ0v~t,ydy-δxI~t,x,x∈Ω,∂v~t,x∂t=∫Ωθx,yv~t,ydy-γx+cxv~t,x+pxI~t,x,x∈Ωwith ψ~=(ψ~(0,·),v~(0,·))=(ψ2,ψ3). Then we have u(t,·)≥u~(t,·) for t∈R+. Let ξ>0 such that ψ2≥ξψIϵ0 and ψ3≥ξψvϵ0. By comparison principle, we get(102)ut,·≥u~t,·≥ξeηϵ0tψ~·for t∈R+. This contradicts with the fact that (I(t,·),v(t,·)) is bounded and hence the proof is complete.With the help of Lemmas15 and 16, we can establish the uniform persistence.Theorem 17.
IfR0>1 and u0∈Γ0, then system (2) is uniformly strongly persistent and hence the deduced semiflow Φ|Γ0 has a global attractor A in Γ0.Proof.
Defineρ:Y+→R+ by(103)ρψ=ψ3_forψ=ψ1,ψ2,ψ3∈Y+.Clearly, ρ-1(R+)⊂Γ0. It follows from Lemma 15 that ρ has the property that ρ(Φ(t,ψ))>0 for t>0 if ρ(ψ)>0 or ψ∈Γ0 with ρ(ψ)=0. Lemma 14 ensures that, for any ψ∈M∂, we have limt→∞Φ(t,ψ)=E0 and hence there is no cycle in ∂Γ0 from E0 to itself. Moreover, Lemma 16 tells us that WS(E0)∩Γ0=∅, where WS(E0) is the stable manifold of E0. Applying [42, Theorem 3] gives a positive number ϵv such that(104)minψ∈ωϕρψ>ϵvfor  allϕ∈Γ0,that is(105)liminft→∞vt,·_>ϵ~v.
Sincef is continuously differentiable and system (2) is dissipative, there exists a positive constant M such that(106)∂Tt,·∂t≥λ·-μ·+MTt,·.Using comparison principle again, one admits(107)liminft→∞Tt,·_≥λ_μ¯+M≔ϵT.Then, it follows from the second equation of system (2) that(108)∂It,x∂t≥∫Ωfx,y,ϵT,ϵvdy-δxIt,x.Therefore,(109)liminft→∞It,·_≥supx∈Ω⁡∫Ωfx,y,ϵT,ϵvdyδ¯≔ϵI.Letting ϵ={ϵT,ϵI,ϵ~v} completes the proof.


## 8. Global Attractivity of Endemic Steady State

This section is devoted to the uniqueness and the global stability of endemic steady states underR0>1 and some additional condition on f. The approach is Lyapunov functional method. Recall that the semiflow Φ on Y+ is bounded. Then the following result, which follows directly from this and Theorem 17, is helpful to construct the Lyapunov functional.Lemma 18.
Suppose thatu(t,x)=(T(t,x),I(t,x),v(t,x))∈Γ0 is a solution and E∗=(T∗,I∗,v∗) is an endemic steady state of system (2). Then there exist two positive numbers δ1 and δ2 such that(110)δ1≤Tt,xT∗x,It,xI∗x,vt,xv∗x≤δ2fort>0andx∈Ω.In order to establish the global attractivity of an endemic steady state, we impose the following hypothesis.Assumption 19.
Suppose that an endemic steady stateE∗=(T∗,I∗,v∗) of system (2) satisfies(111)T∗xT-fx,y,T∗x,v∗yfx,y,T,vT∗xT-fx,y,T∗x,v∗yvfx,y,T,vv∗y≤0for all x, y∈Ω and T, I, v>0.Assumption19 is obviously satisfied if(112)fx,y,Tt,x,vt,x=βxTt,xvpt,xor(113)fx,y,Tt,x,vt,x=βxTt,xvpt,x1+vpt,x,where p∈(0,1].Theorem 20.
Suppose thatR0>1 and u0∈Γ0. Under Assumption 19 and f(x,y,T(x),v(y))=f(T(x),v(x)), the endemic steady state E∗ of system (2) is globally attractive.Proof.
Consider the Lyapunov functional defined by(114)Vut=∫ΩGxUt,xdx,where G(x) is a strictly positive function to be defined later and(115)Ut,x=VTt,x+VIt,x+δxpxVvt,xwith (116)VTt,x=T∗xgTt,xT∗x,VIt,x=I∗xgIt,xI∗x,Vvt,x=v∗xgvt,xv∗x,and g is defined as in the proof of Theorem 13. Lemmas 18 and 15 ensure that Vj(t,x) (j=T, I, v) are well defined for (t,x)∈(R+×Ω).
DifferentiatingVT(t,x) along trajectories of system (2) yields (117)∂VTt,x∂t2=1-T∗xTt,x∂Tt,x∂t=1-T∗xTt,xλx-μxTt,x-∫Ωfx,y,Tt,x,vt,ydy=μxT∗x2-T∗xTt,x-Tt,xT∗x+1-T∗xTt,x∫Ωfx,y,T∗x,v∗ydy-∫Ωfx,y,Tt,x,vt,ydy≤∫Ωfx,y,T∗x,v∗y1-T∗xTt,x-fx,y,Tt,x,vt,yfx,y,T∗x,v∗y+T∗xfx,y,Tt,x,vt,yTt,xfx,y,T∗x,v∗ydy.Similarly, we get(118)∂VIt,x∂t2=1-I∗xIt,x∂It,x∂t=1-I∗xIt,x∫Ωfx,y,Tt,x,vt,ydy-δxIt,x=∫Ωfx,y,T∗x,v∗y1+fx,y,Tt,x,vt,yfx,y,T∗x,v∗y-It,xI∗x-I∗xIt,xfx,y,Tt,x,vt,yfx,y,T∗x,v∗ydyand(119)∂Vvt,x∂t2=1-v∗xvt,x∂vt,x∂t=1-v∗xvt,x∫Ωθx,yvt,ydy+pxIt,x-cx+γxvt,x=pxI∗x1-vt,xv∗x+It,xI∗x-It,xI∗xv∗xvt,x+∫Ωθx,yv∗y1+vt,yv∗y-v∗xvt,yvt,xv∗y-vt,xv∗xdy.Remember that I∗(x)=∫Ωf(x,y,T∗(x),v∗(y))dy/δ(x). Then we substitute (117)–(119) into U(t,x) to obtain(120)∂Ut,x∂t2≤∫Ωfx,y,T∗x,v∗y3-T∗xTt,x+T∗xfx,y,Tt,x,vt,yTt,xfx,y,T∗x,v∗y-I∗xfx,y,Tt,x,vt,yIt,xfx,y,T∗x,v∗y-vt,xv∗x-It,xI∗xv∗xvt,xdy+δxpx∫Ωθx,yv∗y1+vt,yv∗y-v∗xvt,yvt,xv∗y-vt,xv∗xdy.We will rearrange the terms in (120) as follows.
First note that(121)3-T∗xTt,x+T∗xfx,y,Tt,x,vt,yTt,xfx,y,T∗x,v∗y-I∗xfx,y,Tt,x,vt,yIt,xfx,y,T∗x,v∗y-vt,xv∗x-v∗xvt,yvt,xv∗y=-gT∗xTt,x+gI∗xfx,y,Tt,x,vt,yIt,xfx,y,T∗x,v∗y+gIt,xv∗xI∗xvt,x+T∗xfx,y,Tt,x,vt,yTt,xfx,y,T∗x,v∗y-vt,xv∗x+ln⁡Tt,xfx,y,T∗x,v∗yvt,xT∗xfx,y,Tt,x,vt,yv∗x.Now the last term of (121) can be rewritten as(122)T∗xfx,y,Tt,x,vt,yTt,xfx,y,T∗x,v∗y-vt,xv∗x+ln⁡Tt,xfx,y,T∗x,v∗yvt,xT∗xfx,y,Tt,x,vt,yv∗x=-gTt,xfx,y,T∗x,v∗yvt,yT∗xfx,y,Tt,x,vt,yv∗y+ln⁡vt,xv∗x-vt,xv∗x-ln⁡vt,yv∗y+vt,yv∗y+Tt,xfx,y,T∗x,v∗yvt,yT∗xfx,y,Tt,x,vt,yv∗y-1+T∗xfx,y,Tt,x,vt,yTt,xfx,y,T∗x,v∗y-vt,yv∗y.Furthermore, the last term of equation (122) can be rearranged to be(123)Tt,xfx,y,T∗x,v∗yvt,yT∗xfx,y,Tt,x,vt,yv∗y-1+T∗xfx,y,Tt,x,vt,yTt,xfx,y,T∗x,v∗y-vt,yv∗y=T∗xfx,y,Tt,x,vt,yTt,xfx,y,T∗x,v∗y-1+Tt,xfx,y,T∗x,v∗yvt,yT∗xfx,y,Tt,x,vt,yv∗y×1-T∗xfx,y,Tt,x,vt,yTt,xfx,y,T∗x,v∗y=T∗xfx,y,Tt,x,vt,yTt,xfx,y,T∗x,v∗y-11-Tt,xfx,y,T∗x,v∗yvt,yT∗xfx,y,Tt,x,vt,yv∗y=f(x,y,Tt,x,vt,yfx,y,T∗x,v∗yTt,xT∗xT∗xTt,x-fx,y,T∗x,v∗yfx,y,Tt,x,vt,y×T∗xTt,x-fx,y,T∗x,v∗yvt,yfx,y,Tt,x,vt,yv∗y.Next we can simplify the factor in the last term of (120) as (124)∫Ωθx,yv∗y1+vt,yv∗y-v∗xvt,yvt,xv∗y-vt,xv∗xdy=-∫Ωθx,yv∗ygv∗xvt,yvt,xv∗ydy+∫Ωθx,yv∗yvt,yv∗y-ln⁡vt,yv∗y-vt,xv∗x+lnvt,xv∗xdyIn summary, we have found(125)∂Wt,x∂t2≤-∫Ωfx,y,T∗x,v∗ygT∗xTt,x+gI∗xfx,y,Tt,x,vt,yIt,xfx,y,T∗x,v∗y+gIt,xv∗xI∗xvt,x+gTt,xfx,y,T∗x,v∗yvt,yT∗xfx,y,Tt,x,vt,yv∗ydy+∫Ωfx,y,Tt,x,vt,yTt,xT∗xT∗xTt,x-fx,y,T∗x,v∗yfx,y,Tt,x,vt,y×T∗xTt,x-fx,y,T∗x,v∗yvt,yfx,y,Tt,x,vt,yv∗ydy+∫Ωfx,y,T∗x,v∗y+δxpxθx,yvt,yv∗y-ln⁡vt,yv∗y-vt,xv∗x+ln⁡vt,xv∗xdy-δxpx∫Ωθx,yv∗ygv∗xvt,yvt,xv∗ydy.If f(x,y,T(x),v(y)=f(T(x),v(x)), then the last but two term of (125) is equal to zero. Following [26, Proposition 11.1], we catch v∗(x) as a positive Borel function on Ω such that (126)∫Ω∫Ωθx,yv∗xv∗yvt,yv∗y-ln⁡vt,yv∗y-vt,xv∗x+ln⁡vt,xv∗xdydx=∫Ω∫Ωθx,yv∗xv∗yvt,yv∗y-ln⁡vt,yv∗y-vt,xv∗x+ln⁡vt,xv∗xdxdy=∫Ω∫Ωθy,xv∗yv∗xvt,xv∗x-ln⁡vt,xv∗x-vt,yv∗y+ln⁡vt,yv∗ydydx=∫Ω∫Ωθx,yv∗xv∗yvt,xv∗x-ln⁡vt,xv∗x-vt,yv∗y+ln⁡vt,yv∗ydydx.Therefore,(127)2∫Ωθx,yv∗xv∗yvt,yv∗y-ln⁡vt,yv∗y-vt,xv∗x+ln⁡vt,xv∗xdydx=0.Setting G(x)=2p(x)v∗(x)/δ(x) and plugging (127) into (125), we arrive at (128)dVtdt2≤-2∫Ω∫Ωpxfx,y,T∗x,v∗yv∗xδxgT∗xTt,x+gI∗xf(x,y,Tt,x,vt,yIt,xfx,y,T∗x,v∗y+gIt,xv∗xI∗xvt,x+gTt,xfx,y,T∗x,v∗yvt,yT∗xf(x,y,Tt,x,vt,yv∗ydydx+2∫Ω∫Ωfx,y,Tt,x,vt,ypxTt,xδxT∗xv∗xT∗xTt,x-fx,y,T∗x,v∗yf(x,y,Tt,x,vt,y×T∗xTt,x-fx,y,T∗x,v∗yvt,yf(x,y,Tt,x,vt,yv∗ydydx-2∫Ω∫Ωθx,yv∗xv∗ygv∗xvt,yvt,xv∗ydydx.By virtue of g and Assumption 19, dV(t)/dt(2)≤0 and the equality holds if and only if(129)Tt,x=T∗x,fx,y,T∗x,v∗yfx,y,Tt,x,vt,y=1,It,xv∗xI∗xvt,x=1.Replacing T(t,x) in the first equation of system (2) by T∗(x), we have(130)0=λx-∫Ωfx,y,T∗x,vt,ydy-μxT∗x.From Assumption 1, it is easy to see that v(t,x)=v∗(x) for all (t,x)∈R+×Ω. Using the second equation of system (2), we conclude that the largest invariant set M={T·,I·,v·∈Γ0∣dVt/dt(2)=0}={E∗}. Consequently, it follows from LaSalle Invariance Principle that the endemic steady state E∗ is globally attractive.Remark 21.
The existence and attractivity of endemic steady states (see Theorems9 and 20) indicate that system (2) has a unique endemic steady state if the conditions of Theorem 20 hold.


## 9. Numerical Simulations

In this section, we perform numerical experiments to illustrate our main theoretical results and compare the effects of diffusion rate and incidence. For this purpose, we takeΩ=[-1,1]⊂R and use the initial values in [29].(131)T0,x=0.99cos⁡π2x,I0,x=0.01cos⁡π2x,v0,x=10-6cos⁡π2x.Moreover, we fix the following parameters:(132)λx=2.0,μx=0.5,δx=1,px=2,γx=10,cx=0.5.The incidence rate takes the form (133)fx,y,T,v=βxTt,xvt,y1+αvt,x,where(134)βx=β~1+0.1cos⁡5πx,x∈-1,1with β~ varying. For the nonlocal effect, we set(135)θx,y=1.5dx-y2,where d is the diffusive coefficient. Firstly, we set β~=1.1, d=10-5, and α=0.5. Then R0≈0.9219<1. Figures 1(a) and 1(b) show that the density of free viruses approaches zero as t goes to infinity.Figure 1
The evolution of the density of free viruses with parameters listed in the text. ((a) and (b)) The variation of the free virus density withβ~=1.1. ((c) and (d)) The variation of the free virus density with β~=1.5.
(a) (b) (c) (d)Now, we enlargeβ~ to β~=1.5 and get R0≈1.257>1. In view of Theorem 20, the endemic steady state converges to a spatially positive endemic steady state (see Figures 1(c) and 1(d)).Secondly, Figure2(a) shows that increasing the diffusion rate enhances the infected risk and increases the final infected size. This means that controlling virions diffusion beats the viral replication. Figure 2(b) reflects that viral infection models with bilinear incidence rate have higher infected risk than ones with saturating incidence rates.Figure 2
The evolution of the cumulative density of free viruses with parameters as those for Figure1 except d and α. (a) d∈[5.1:5:25.1] and α=0.5; (b) α∈[0:5:20] and d=10-5.
(a) (b)


## 10. Discussion

In this paper, we firstly proposed a within-host viral infection model with nonlocal diffusion and nonlocal transmission. The model can be considered as a spatial generalization of that proposed by Nowark and Bangham [43], a continuous spatial model of Funk et al. [9]. We derived the next-generation operator R and built the relationship between the basic reproduction number R0=r(R) and the spectral bound of the operator L (see Theorem 7). The asymptotic smoothness of the semiflow Φ was established by Arzelà-Ascoli theorem in Section 3. The threshold dynamics of system (2) has been established by constructing suitable Lyapunov functionals. Biologically, whether or not the viral infection outbreaks is determined by the basic reproduction number R0.Compared with other within-host models with diffusion (discrete style [9] and continuous style (Laplace operator) [5]), our model can be considered as a generalization of the model proposed by Zhao and Ruan [18], where for the incidence they took the particular form(136)∫Ωfx,y,Tt,x,vt,ydy=βxTt,xvt,x.In [18], they adopted the semigroup method to investigate the asymptotical behavior of system (2). In this paper, we used functional analysis method together with the Lyapunov functional method to study the asymptotical stability of the system. We believe that the method used here can also be applicable to or be generalized to deal with other nonscalar systems with nonlocal diffusions.



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