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Research article Special Issues

Dynamics of a reaction-diffusion SIRI model with relapse and free boundary

  • This paper is concerned with the free boundary problem for a reaction-diffusion SIRI model with relapse and bilinear incidence rate. After studying the (global) existence and uniqueness of solutions, we provide some sufficient conditions on the disease spreading-vanishing dichotomies for both cases with and without relapse.

    Citation: Qian Ding, Yunfeng Liu, Yuming Chen, Zhiming Guo. Dynamics of a reaction-diffusion SIRI model with relapse and free boundary[J]. Mathematical Biosciences and Engineering, 2020, 17(2): 1659-1676. doi: 10.3934/mbe.2020087

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  • This paper is concerned with the free boundary problem for a reaction-diffusion SIRI model with relapse and bilinear incidence rate. After studying the (global) existence and uniqueness of solutions, we provide some sufficient conditions on the disease spreading-vanishing dichotomies for both cases with and without relapse.


    Though medicine and living conditions have been constantly improving, infectious diseases are still a global concern. Mathematical modeling can not only enhance our understanding of the transmission mechanisms underlying them but also help us assess the efficacy of control strategies. Among the deterministic models described by ordinary differential equations are compartmental models. One of the basic models is the Kermack-McKendric model,

    {dSdt=βSI,dIdt=βSIγI,dRdt=γI,

    where S, I, and R are the densities (or numbers) of susceptible, infectious, and recovered individuals, respectively; β is the transmission rate while γ is the recovery rate. The incidence rate is the bilinear one, βSI. To better reflect the actual biology of a given disease, the above model has been significantly modified.

    In this paper, we consider the factor of relapse. For certain diseases such as herpes, tuberculosis, simplex virus type 2 (a human disease transmitted by close physical or sexual contacts), recovered individuals may experience relapse, which means that they can revert to the infectious class with the reactivation of a latent infection. For example, this feature of recurrence for tuberculosis is often due to incomplete treatment. Tudor [1] was the first to study relapse, who built the so-called SIRI model. In this model, the bilinear incidence rate is used. Tudor investigated the existence and local stability of equilibria. Later on, Moreira and Wang [2] modified this model with an incidence rate depending on the size of the susceptible population. By means of an elementary analysis of Liénard's equation and Lyapunov's direct method, they established sufficient conditions on the global asymptotic stability of the disease-free and endemic equilibria.

    In the above mentioned studies on relapse, the population size is constant. In particular, there are no disease-induced deaths. Thus, in 2013, Vargas-De-León [3] proposed two epidemiological models with relapse and disease-induced deaths. One of them is the following one with the bilinear incidence rate,

    {dSdt=ΛβSIμS,dIdt=βSI(α+γ+μ)I+ηR,dRdt=γI(μ+η)R, (1.1)

    where Λ represents the recruitment rate, β is the transmission rate, μ is the natural death rate, α is the disease-induced death rate, γ is the recovery rate, and η is the relapse rate. All the parameters are positive. They constructed suitable Lyapunov functions to obtain threshold dynamics determined by the basic reproduction number R0. If R0<1, the disease-free equilibrium is globally asymptotically stable and hence the disease dies out. On the other hand, if R0>1, the endemic equilibrium is globally asymptotically stable and hence the disease remains endemic. For more works on SIRI models described by ordinary differential equations, we refer to [4,5,6] and references therein.

    Note that, due to mobility, the distribution of individuals in an area is not even. Modeling this phenomenon often results in reaction-diffusion equations. Consequently, inspired by [3], we have formulated a diffusive epidemic model with relapse and bilinear incidence as follows,

    {St(x,t)=dΔS(x,t)+ΛβS(x,t)I(x,t)μS(x,t),t>0,xΩ,It(x,t)=dΔI(x,t)+βS(x,t)I(x,t)(α+γ+μ)I(x,t)+ηR(x,t),t>0,xΩ,Rt(x,t)=dΔR(x,t)+γI(x,t)(μ+η)R(x,t),t>0,xΩ,S(x,0)=S0(x)0,I(x,0)=I0(x),0,R(x,0)=R0(x)0,x¯Ω;Sn(x,t)=In(x,t)=Rn(x,t)=0,t>0,xΩ. (1.2)

    Here S(x,t), I(x,t), and R(x,t) are the densities of susceptible, infective, and recovered individuals at time t and position xΩ, respectively; Ω is a bounded domain in Rn with a smooth boundary Ω; Δ is the usual Laplacian operator; n is the outward normal derivative to Ω; d is the diffusion rate which represents the ability of random mobility of individuals; and the meanings of the other parameters are the same as those in (1.1). Note that the Neumann boundary conditions imply that individuals cannot move across the boundary Ω.

    It should be pointed out that any solution of (1.2) is always positive for any time t>0 no matter what the nonnegative nontrivial initial condition is. Thus the disease spreads to the whole area immediately, even though the infectious are confined to a quite small part of the habitat at the beginning. This does not agree with the observed fact that diseases always spread gradually. To compensate for the gradual disease spreading progress, a better modeling technique is to introduce free boundary.

    The equation governing the free boundary, h(t)=μIx(h(t),t), is a special case of the well-known Stefan condition, which has been established in [7] for diffusive populations and used in the modeling of a number of applied problems. For example, it was used to describe the melting of ice in contact with water [8] and to model oxygen in muscles [9] as well as wound healing [10]. There is a vast literature on Stefan problems. Some important recent theoretical advances can be found in [11]. As a typical case, in 2013, Kim et al. [12] studied a diffusive SIR epidemic model in a radially symmetric domain with free boundary. They provided sufficient conditions on disease vanishing and spreading.

    Motivated by the above discussion, in this paper, we investigate the behavior of nonnegative solutions (S(x,t),I(x,t),R(x,t);h(t)) of the following reaction-diffusion SIRI epidemic with free boundary,

    {St(x,t)=dSxx(x,t)+ΛβS(x,t)I(x,t)δS(x,t),x>0,t>0,It(x,t)=dIxx(x,t)+βS(x,t)I(x,t)(α+γ+δ)I(x,t)+ηR(x,t),0<x<h(t),t>0,Rt(x,t)=dRxx(x,t)+γI(x,t)(δ+η)R(x,t),0<x<h(t),t>0,Sx(0,t)=Ix(0,t)=Rx(0,t)=0,t>0,I(x,t)=R(x,t)=0,xh(t),t>0,h(t)=μIx(h(t),t),t>0,h(0)=h0,S(x,0)=S0(x)0,I(x,0)=I0(x)0,R(x,0)=R0(x)0,x0, (1.3)

    where x=h(t) is the moving boundary to be determined, μ represents the moving rate of the free boundary, δ is the natural death rate, and the meanings of the rest parameters are the same as those in model (1.2). All parameters are assumed to be positive. The nonnegative initial functions S0, I0 and R0 satisfy

    {S0C2([0,+)),I0,R0C2([0,h0]),I0(x)=R0(x)=0 for x[h0,+) and I0(x)>0 for x[0,h0). (1.4)

    In reality, I0(x)=0 for x[h0,+) and I00 on [0,h0). Since for t>0, the solution though the initial condition (S0,I0,R0;h0) with such an I0 satisfy I(x,t)>0 on [0,h(t)) and I(x,t)=0 for x[h(t),+). Thus, without loss of generality, we make the assumption (1.4). Biologically, model (1.3) means that beyond the free boundary x=h(t), there are only susceptible individuals. We will also consider the case without relapse, that is, η=0. In this case, (1.3) reduces to

    {St(x,t)=dSxx(x,t)+ΛβS(x,t)I(x,t)δS(x,t),x>0,t>0,It(x,t)=dIxx(x,t)+βS(x,t)I(x,t)(α+γ+δ)I(x,t),0<x<h(t),t>0,Rt(x,t)=dRxx(x,t)+γI(x,t)δR(x,t),0<x<h(t),t>0,Sx(0,t)=Ix(0,t)=Rx(0,t)=0,t>0,I(x,t)=R(x,t)=0,xh(t),t>0,h(t)=μIx(t,h(t)),t>0,h(0)=h0,S(x,0)=S0(x)0,I(x,0)=I0(x)0,R(x,0)=R0(x)0,x0. (1.5)

    The remainder of this paper is organized as follows. In Section 2, we prove some general results on the existence and uniqueness of solutions to (1.3)–(1.4). In particular, solutions are global. Then, in Section 3, we provide some sufficient conditions on disease spreading and vanishing. For (1.5), the disease will die out either if the basic reproduction number R0<1 or if R0>1 and the initial infected area, boundary moving rate, and initial value of infected individuals are sufficiently small; while the disease will spread to the whole area if R0>1 and either the initial infected area is suitably large or the diffusion rate is suitably small. For (1.3), when the basic reproduction number ˜R01, the disease will disappear, whereas when ˜R0>R0>1 and the initial infected area is suitably large, the disease will successfully spread. The paper ends with a brief conclusion and discussion.

    First, we state the result on the local existence of solutions to (1.3)–(1.4), which can be proved with some modifications of the arguments in [10] and [13]. Hence we omit the proof to avoid repetition.

    Theorem 2.1. For any given (S0,I0,R0) satisfying (1.4) and any r(0,1), there is a T>0 such that problem (1.3) admits a unique bounded solution

    (S,I,R;h)C1+r,(1+r)2(DT)×[C1+r,(1+r)2(DT)]2×C1+r2([0,T]);

    moreover,

    SC1+r,(1+r)2(DT)+IC1+r,(1+r)2(DT)+RC1+r,(1+r)2(DT)+hC1+r2([0,T])C,

    where DT={(x,t)R2:x[0,+),t[0,T]} and DT={(x,t)R2:x[0,h(t)],t[0,T]}. Here C and T only depend on h0, r, S0C2([0,+)), I0C2([0,h0]), and R0C2([0,h0]).

    Next we make some preparations to show the global existence of solutions.

    Lemma 2.1. Problem (1.3)–(1.4) admits a unique and uniformly bounded solution (S,I,R;h) on (0,T0) for some T0(0,+], that is, there exists a constant M independent of T0 such that

    0<S(x,t)Mfor0x<+,t(0,T0).0<I(x,t),R(x,t)Mfor0x<h(t),t(0,T0).

    Proof. As long as the solution exists, it is easy to see that S0, I0, and R0 on [0,+)×[0,T0]. By applying the strong maximum principle to the equations on {(x,t):x[0,h(t)],t[0,T0]}, we immediately obtain

    S(x,t)>0 for 0x<+0<t<T0I(x,t),R(x,t)>0 for 0x<h(t)0<t<T0.

    It remains to prove the uniform boundedness of the solution (S(x,t),I(x,t),R(x,t);h(t)). For this purpose, define

    U(x,t)=S(x,t)+I(x,t)+R(x,t),0x<+,t(0,T0).

    A direct calculation gives

    dUdt=dSxx+dIxx+dRxx+ΛδS(δ+α)IδR=dUxx+Λδ(S+I+R)αIdUxx+ΛδU,

    which gives U(x,t)max{U0,Λδ}M, where

    U0=S(x,0)+I(x,0)+R(x,0). 

    Now the required result follows immediately.

    Finally, we show that the free boundary of (1.3)–(1.4) is strictly monotonically increasing.

    Lemma 2.2. Let (S,I,R;h) be a solution to problem (1.3)–(1.4) defined for t(0,T0) for some T0(0,+]. Then there exists a constant C1 independent of T0 such that

    0<h(t)C1 fort(0,T0).

    Proof. Using the strong maximum principle and Hopf boundary lemma to the equation of I, we can obtain Ix(h(t),t)<0 for t(0,T0). This, combined with the Stefan condition h(t)=μIx(h(t),t), gives h(t)>0 for t(0,T0).

    In order to get a bound for h(t), we denote

    ΩN:={(x,t):h(t)N1<x<h(t),0<t<T0},

    and construct an auxiliary function

    ωN(x,t):=M[2N(h(t)x)N2(h(t)x)2].

    We will choose N so that ωN(x,t)I(x,t) holds over ΩN.

    Clearly, for (x,t)ΩN,

    (ωN)t=2MNh(t)[1N(h(t)x)]0,(ωN)xx=2MN2,βSI(α+γ+δ)I+ηRβM2+ηM.

    Therefore, if N2βM+η2d then

    (ωN)td(ωN)xx2dMN2βM2+ηM.

    On the other hand, we have the boundary condition

    ωN(h(t)N1,t)=MI(h(t)N1,t),ωN(h(t),t)=0=I(h(t),t).

    To employ the maximum principle to (ωNI) over ΩN to deduce that I(x,t)ωN(x,t), we only have to find some N independent of T0 such that I0(x)ωN(x,0) for x[h0N1,h0]. It would then follow that

    Ix(h(t),t)(ωN)x(h(t),t)=2NM,h(t)=μIx(h(t),t)2μNM.

    Note that

    I0(x)=I0(x)I0(h0)=h(t)xI0(s)ds(h0x)I0C[0,h0]

    and

    ωN(x,0):=M[2N(h0x)N2(h0x)2]MN(h0x),x[h0N1,h0].

    It suffices to have

    (h0x)I0C[0,h0]MN(h0x).

    Thus choosing

    N:=max{βM+η2d,I0C([0,h0])M}

    completes the proof.

    By a similar argument as the one in [12,13], we can have the following result.

    Theorem 2.2. The solution of problem (1.3)–(1.4) exists and is unique for all t(0,+).

    This section is devoted to the spreading-vanishing dichotomy. We distinguish two cases, η=0 and η>0. We start with a sufficient condition on disease vanishing, which will be used in the coming discussion.

    It follows from Lemma 2.2 that if x=h(t) is monotonically increasing, then h:=limth(t)(h0,+] is well defined.

    Lemma 3.1. If h<+, then limt+I(,t)C([0,h(t)])=0. Moreover, limt+R(,t)C([0,h(t)])=0 and limt+S(x,t)=Λδ uniformly in any bounded subset of [0,+).

    Proof. Define

    s=h0xh(t),u(s,t)=S(x,t),v(s,t)=I(x,t),w(s,t)=R(x,t).

    Then it is easy to see that

    It=vth(t)h(t)svs,Ix=h0h(t)vs,Ixx=h20h2(t)vss.

    It follows that v(s,t) satisfies

    {vth(t)h(t)svsdh20h2(t)vss=v[βu(α+δ+γ)]+ηw,0<s<h0,t>0,vs(0,t)=v(h0,t)=0,t>0,v(s,0)=I0(s)0,0sh0.

    This means that the transformation changes the free boundary x=h(t) into the fixed line s=h0 and hence we have an initial boundary value problem over a fixed area s<h0.

    Since h0h(t)<h<+, the differential operator is uniformly parabolic. With the bounds in Lemma 2.1 and Lemma 2.2, there exist positive constants M1 and M2 such that

    v(βu(α+μ+γ))+ηwLM1andh(t)h(t)sLM2.

    Applying the standard Lp theory and the Sobolev embedding theorem [14], we obtain that

    vC1+α,1+α2([0,h0]×[0,+))M3

    for some constant M3 depending on α, h0, M1, M2, and I0C2[0,h0]. It follows that there exists a constant ˜C depending on α, h0, (S0,I0,R0), and h such that

    hC1+α2([0,+))˜C. (3.1)

    Assume lim supt+I(,t)C([0,h(t)])=ϖ>0 by contradiction. Then there exists a sequence {(xk,tk)} in [0,h)×(0,+) such that I(xk,tk)ϖ2 for all kN and tk+ as k+. Since I(h(t),t)=0 and since (3.1) indicates that Ix(h(t),t) is uniformly bounded for t[0,+), there exists σ>0 such that xkh(tk)σ for all k1. Then there is a subsequence of {xk} which converges to x0[0,hσ]. Without loss of generality, we assume xkx0 as k+. Correspondingly,

    sk:=h0xkh(tk)s0:=h0x0h<h0.

    Define Sk(x,t)=S(x,tk+t), Ik(x,t)=I(x,tk+t), and Rk(x,t)=R(x,tk+t) for (x,t)(0,h(tk+t))×(tk,+). It follows from the parabolic regularity that {(Sk,Ik,Rk)} has a subsequence {(Ski,Iki,Rki)} such that (Ski,Iki,Rki)(˜S,˜I,˜R) as i+. Since hC1+α2([0,+))˜C, h(t)>0, and h(t)h<+, it is necessary that h(t)0 as t+. Hence (˜S,˜I,˜R) satisfies

    {˜Std1˜Sxx=Λβ˜S˜Iδ˜S,0<x<h,t(,+),˜Itd2˜Ixx=β˜S˜I(α+γ+δ)˜I+η˜R,0<x<h,t(,+),˜Rtd3˜Rxx=γ˜I(δ+η)˜R,0<x<h,t(,+).

    Since ˜I(x0,0)ϖ2, the maximum principle implies that ˜I>0 on [0,h)×(,+). Thus we can apply the Hopf lemma to conclude that σ0:=˜Is(h0,0)<0. It follows that

    vx(h(tki),tki)=Iki(h0,0)sh0h(tki)σ02h0h<0

    for all large i. Hence h(tki)μσ02h0h>0 for all large i, which contradicts with h(t)0 as t+. This proves limt+I(,t)C([0,h(t)])=0.

    Using a simple comparison argument, we can deduce that limt+R(,t)C([0,h(t)])=0 and limt+S(x,t)=Λδ uniformly in any bounded subset of [0,+). In fact, for any ε>0, there exists a T00 such that I(x,t)ε for tT0. Then, for tT0, we have

    StdSxx+Λ(βε+δ)S(x,t)

    and

    RtdRxx+γε(δ+η)R(x,t).

    It follows that

    lim inft+S(x,t)Λβε+δuniformlyinanyboundedsubsetof[0,+)

    and

    lim supt+R(,,t)C([0,h(t)])γεδ+η.

    As ε is arbitrarily, letting ε0+ gives us

    lim inft+S(x,t)Λδuniformlyinanyboundedsubsetof[0,+)

    and

    lim supt+R(,,t)C([0,h(t)])0.

    This immediately gives limt+R(,,t)C([0,h(t)])=0. Moreover, for t0, we have

    StdSxx+ΛδS(x,t).

    Then S(x,t)ˉS(t) for x(0,+) and t(0,+), where

    ˉS(t):=Λδ+(ˉS(0)Λδ)eδt

    is the solution of the problem

    dˉS(t)dt=ΛδˉS(t),t>0;ˉS(0)=max{Λδ,S0}.

    Since limt+ˉS(t)=Λδ, we deduce that

    lim supt+S(x,t)limt+ˉS(t)=Λδuniformly forx[0,+).

    Therefore, we have limt+S(x,t)=Λδ uniformly in any bounded subset of [0,+).

    Consider the following eigenvalue problem,

    {dϕxx+βΛδϕ(α+γ+δ)ϕ+λϕ=0,x(0,h0),ϕx(0)=0,ϕ(h0)=0. (3.2)

    It admits a principal eigenvalue λ1, where

    λ1=dπ24h20βΛδ+(α+γ+δ).

    The basic reproduction number of (1.5) denoted by R0 is given by

    R0=βΛδ(γ+α+δ).

    With the assistance of the expression of R0, we can rewrite the expression of λ1 as

    λ1=dπ24h20βΛδ+(α+γ+δ)=dπ24h20(11R0)βΛδ.

    It follows that λ1>0 either if R01 or if R0>1 and h0<dδπ24βΛ(11R0).

    We first give some sufficient conditions on disease vanishing.

    Theorem 3.2. If R0<1, then limt+I(,t)C([0,h(t)])=0 and limt+R(,t)C([0,h(t)])=0. Moreover, limt+S(x,t)=Λδ uniformly in any bounded subset of [0,+).

    Proof. From the proof of Lemma 3.1, we have obtained that

    lim supt+S(x,t)Λδuniformlyforx[0,+).

    Since R0<1, there exists T0 such that S(x,t)Λδ1+R02R0 on [0,+)×(T0,+). Then I(x,t) satisfies

    {It(x,t)dIxx+[βΛδ1+R02R0(α+γ+δ)]I(x,t),0<x<h(t),t>T0,Ix(0,t)=0,I(h(t),t)=0,t>T0,I(x,T0)>0,0xh(T0).

    We know that I(x,t)ˉI(x,t) for (x,t){(x,t):x[0,h(t)],t(T0,+)}, where ˉI(x,t) satisfies

    {ˉIt(x,t)=dˉIxx+[βΛδ1+R02R0(α+γ+δ)]ˉI(x,t),0<x<h(t),t>T0,ˉIx(0,t)=ˉI(h(t),t)=0,t>T0,ˉI(x,T0)I(,T0)>0,0xh(T0).

    Since βΛδ1+R02R0(α+γ+δ)=(α+γ+δ)(R01)2<0, we have limt+ˉI(,t)C[0,h(t)]=0. Then it follows from I(x,t)ˉI(x,t) that I(,t)C[0,h(t)]0 as t+. The remaining part follows from Lemma 3.1.

    Theorem 3.2. Suppose R0>1. Then h<+ for given initial condition (S0,I0,R0;h0) satisfying h0min{d16k0,d16γ} and μd8K, where k0=βMαγδ>0, M=max{S0,Λδ}, and K=43max{I0,R0}.

    Proof. Since R0>1, one can easily see that k0>0. Inspired by [13], we define

    ˉS(x,t)=M,ˉI(x,t)={KeθtV(xˉh(t)),0xˉh(t),0,x>ˉh(t),ˉR(x,t)={KeθtV(xˉh(t)),0xˉh(t),0,x>ˉh(t),V(y)=1y2,0y1,ˉh(t)=2h0(2eθt),t0,

    where θ is a constant to be determined later. In the following, we show that (ˉS,ˉI,ˉR;ˉh) is an upper solution to (1.5).

    For 0<x<ˉh(t) and t>0, direct computations yield

    ˉStdˉSxx=0ΛδˉS,ˉItdˉIxx(βˉSαγδ)ˉI=ˉItdˉIxxk0ˉI=Keθt[θVxˉhˉh2Vdˉh2Vk0V]Keθt[d8h20θk0],ˉRtdˉRxx(γˉIδˉR)Keθt[d8h20θγ],ˉh(t)=2h0θeθt,μˉIx(ˉh(t),t)=2Kμˉh1(t)eθt.

    Moreover,

    ˉS(x,0)S0(x),ˉI(x,0)=K(1x24h20)34Kfor x[0,h0]ˉR(x,0)=K(1x24h20)34Kfor x[0,h0].

    Choose θ=d16h20. Noting ˉh(t)4h0, we have

    {ˉStdˉSxxΛδˉS,x>0,t>0,ˉItdˉIxxβˉSˉI(α+γ+δ)ˉI,0<x<ˉh(t),t>0,ˉRtdˉRxxαˉIδˉR,0<x<ˉh(t),t>0,ˉSx(0,t)0,ˉIx(0,t)0,ˉRx(0,t)0,t>0,ˉI(x,t)=ˉR(x,t)=0,xˉh(t),0<tT,ˉh(t)μˉIx(ˉh(t),t),ˉh(0)=2h0h0,t>0,ˉS(x,0)S0(x),ˉI(x,0)I0(x),ˉR(x,0)R0(x),0xh0.

    This verifies that (ˉS,ˉI,ˉR;ˉh) is an upper solution to (1.5). Then we can apply a result similar as [12,Lemma 4.1] (which can be proved in the same manner as [13,Lemma 5.6]) to conclude that h(t)ˉh(t) for t>0. This implies that hlimt+ˉh(t)=4h0<+.

    Theorem 3.3. Assume that R0>1. For given initial condition (S0,I0,R0;h0), we have h<+ provided that h0<h:=min{dπ24[βN(α+γ+δ)],dγ4γ} and both I0 and R0 are sufficiently small (which is specified in the proof), where N=max{Λδ,S0}.

    Proof. Note that h is well defined since R0>1. As in the proof of Theorem 3.2, we will construct a suitable upper solution to (1.5). Since h0<h, there exists ε1>0 such that h0<dπ24[β(N+ε1)(α+γ+δ)]. Then the principal eigenvalue of the eigenvalue problem

    {dϕxx+β(N+ε1)ϕ(α+γ+δ)ϕ+λϕ=0,0<x<h0ϕx(0)=ϕ(h0)=0.

    is

    ˜λ1=dπ24h20β(N+ε1)+α+δ+γ>0

    and it is has a normalized positive eigenfunction ˜ϕ on (0,h0). Moreover, ˜ϕx<0 on (0,h0]. Choose ε2(0,γ) such that

    ˜λ1>[β(N+ε1)+ε2](1+ε2)2β(N+ε1)>0.

    Recall that lim supt+S(t,x)Λδ uniformly for x[0,+). Thus there exists a T0>0 such that 0<S(x,t)(N+ε1) in [0,+)×[T0,+). As in [15], we define

    ϑ(t)=h0(1+ε2ε22eε2t),ˉS(x,t)=(N+ε1),tT0,ˉI(x,t)={ιeε2t˜ϕ(xh0ϑ(t)),0xϑ(t),tT0,0,x>ϑ(t),tT0,ˉR(x,t)={ιeε2tV(xϑ(t)),0xϑ(t),tT0,0,x>ϑ(t),tT0.V(y)=1y2,0y1,

    where ι is a positive number to be determined later. As ˜ϕ(h0)=0, it follows that ˉI(ϑ(t),t)=0 for tT0, which implies that the function ˉI(x,t) is continuous on [0,+)×[0,+). Similarly, as V(1)=0, we know that ˉR is also continuous on [0,+)×[0,+). Detailed calculations yield ˉStdˉSxx=0ΛδˉS and, for 0xϑ(t),

    ˉItdˉIxxβˉSˉI+(α+γ+δ)ˉI=ιeε2t[ε2˜ϕxh0ϑ(t)ϑ2(t)˜ϕxdh20ϑ2(t)˜ϕxxβ(N+ε1)˜ϕ+(α+γ+δ)˜ϕ]=ιeε2t{ε2˜ϕxh0ϑ(t)ϑ2(t)˜ϕxh20ϑ2(t)[β(N+ε1)˜ϕ+(α+γ+δ)˜ϕ˜λ1˜ϕ]β(N+ε1)˜ϕ+(α+γ+δ)˜ϕ}=ιeε2t[ε2˜ϕxh0ϑ(t)ϑ2(t)˜ϕx+(h20ϑ2(t)1)β(N+ε1)˜ϕ+(1h20ϑ2(t))(α+γ+δ)˜ϕ+h20ϑ2(t)˜λ1˜ϕ]˜ϕιeε2t{ε2+h20ϑ2(t)[β(N+ε1)+˜λ1]β(N+ε1)}˜ϕιeε2t{ε2+h20h20(1+ε2)2[β(N+ε1)+˜λ1]β(N+ε1)}˜ϕιeε2t{ε2+1(1+ε2)2[β(N+ε1)+˜λ1]β(N+ε1)}.

    Here we have used the fact that ˜ϕx<0 for x(0,h0]. It follows that ˉItdˉIxxβˉSˉI+(α+γ+δ)ˉI0. On the other hand, as h0<h, we can obtain

    ˉRtdˉRxxγˉI+δˉRιeε2t(ε2γ+d8h20)ιeε2t(2γ+d8h20)0.

    Moreover,

    μˉIx(ϑ(t),t)=μιeε2t˜ϕx(h0)h0ϑ(t).

    If we choose 0<ιε22h0(1+ε22)/2μ˜ϕx(h0), then

    ϑ(t)μˉIx(ϑ(t),t)

    since ˜ϕx(h0)<0. Obviously, ˉS(x,0)S0. If I0ιϕ(x1+ε22) and R0V(xh0(1+ε22)) for x[0,h0], then I0(x)ˉI(x,0) and R0(x)ˉR(x,0) for x>0. This proves that (ˉS,ˉI,ˉR;ϑ(t)) is an upper solution of (1.5). Thus, similalrly as in the proof of Theorem 3.2, we can get h(t)ϑ(t), which yields h<limt+ϑ(t)=h0(1+ε2)<+. This completes the proof.

    We provide a sufficient condition on disease spreading to conclude this subsection.

    Theorem 3.4. If R0>1 and h0>h:=dδπ24βΛ(11R0), then h=+.

    Proof. By way of contradiction, we assume that h<+. It follows from Lemma 3.1 that limt+I(,t)C([0,h(t)])=0. Moreover, limt+S(x,t)=Λδ uniformly in any bounded subset of [0,+).

    Since h0>h and R0>1, we have λ1<0, where λ1 is the principal eigenvalue of the eigenvalue problem (3.2). Choose ι>0 such that λ1+βι<0 and R0>1+βια+δ+γ (which implies that β(Λδι)δαγ>0). For this ι, there exists T>0 such that S(x,t)Λδι and I(x,t)<1 for x[0,h(t)] and t>T. Then I(x,t) satisfies

    {ItdIxx[β(Λδι)δαγ]I(1I),0<x<h0,t>T,Ix(0,t)=0,I(h0,t)0,t>T,I(x,T)>0,0x<h0.

    It is easy to see that I(x,t)I_(x,t), where I_(x,t) satisfies

    {I_tdI_xx=[β(Λδι)δαγ]I_(1I_),0<x<h0,t>T,I_x(0,t)=0,I_(h0,t)=0,t>T,I_(x,T)=I(x,T),0x<h0. (3.3)

    Consider the following eigenvalue problem

    {dϕxx+[β(Λδι)δαγ]ϕ+λϕ=0,0<x<h0,ϕx(0)=ϕ(h0)=0,

    whose principal eigenvalue is

    ˆλ1=dπ24h20[β(Λδι)δαγ]=λ1+βι<0.

    Employing Proposition 3.2 and Proposition 3.3 of [16], we obtain that limt+I_(t,x)=I_(x) uniformly in x[0,h0], where I_(x)>0 satisfies

    {dI_xx=[β(Λδι)δαγ]I_(1I_),0<x<h0,I_x(0)=0,I_(h0)=0.

    It follows that lim inft+I(x,t)limt+I_(x,t)=I_(x)>0 uniformly in x[0,h0], which contradicts with limt+I(,t)C([0,h(t)])=0. Therefore, we have proved h=+.

    Remark 3.1. Obviously, h0>h is equivalent to d<d4h20βΛ(11R0)δπ2. As a result, if R0>1 and 0<d<d, then h=+.

    In this case, the basic reproduction number ˜R0 of problem (1.3) is given by

    ˜R0=βΛ(δ+η)δ[γδ+(δ+η)(α+δ)].

    As in the case where η=0, we start with disease vanishing.

    Theorem 3.5. If ˜R01, then limt+I(,t)C([0,h(t)])=0. Moreover, limt+R(,t)C([0,h(t)])=0 and limt+S(x,t)=Λδ uniformly in any bounded subset of [0,+).

    Proof. Consider the following system of ordinary differential equations,

    {dS(t)dt=ΛβS(t)I(t)δS(t),dI(t)dt=βS(t)I(t)(α+γ+δ)I(t)+ηR(t),dR(t)dt=γI(t)(δ+η)R(t), (3.4)

    with (S(0),I(0),R(0))=(S0,I0,R0). As in the proof of Theorem 3.2, a result similar as [12, Lemma 4.1] implies that S(x,t)S(t) for (x,t)[0,+)×(0,+), and I(x,t)I(t) and R(x,t)R(t) for (x,t){(x,t):x[0,h(t)],t(0,+)}.

    Obviously, (3.4) has a disease-free equilibrium E0=(Λδ,0,0), which is globally asymptotically stable. Indeed, consider V:R3+R defined by

    V(S,I,R)=(δ+η)(SS0S0lnSS0)+(δ+η)I+ηR. (3.5)

    It is clear that V(S,I,R) reaches its global minimum in R3+ only at E0. Moreover, the derivative of (3.5) with respect to t along solutions of (3.4) is

    ddtV(S,I,R)=(δ+η)SS0SdSdt+(δ+η)dIdt+ηdRdt=(δ+η)SS0S(ΛβSIδS)+(δ+η)[βSI(α+γ+δ)I+ηR]+η[γI(δ+η)R]=(δ+η)SS0S(ΛβSIδS)+(δ+η)[βSI(α+γ+δ)I+ηR]+η[γI(δ+η)R].

    Using the expression

    βSI(SS0)S0=βI(SS0)2S0+βI(SS0),

    we obtain

    ddtV(S,I,R)=(δ+η)SS0S(ΛβSIδS)+(δ+η)[βSI(α+γ+δ)I+ηR]+η[γI(δ+η)R]=(δ+η)(SS0)2S+[γδ+(δ+η)(α+δ)]I[(δ+η)S0β(γδ+δ+η)(α+δ)1]=(η+δ)(SS0)2S[γδ+(δ+η)(α+δ)]I(1˜R0).

    Since ˜R01, we have ddtV(S,I,R)0 for S>0. Moreover, if ddtV(S,I,R)=0 holds then S=S0. It is easy to verify from this that the disease-free equilibrium E0 is the largest invariant set in the set where ddtV(S,I,R)=0. Therefore, by LaSalle's invariance principle [17], E0 is globally asymptotically stable. This, combined with the above estimates, gives us

    lim supt+S(x,t)limt+S(t)=Λδuniformly for x[0,+)lim suptI(x,t)limtI(t)=0uniformly in any bounded subset of [0,h)lim suptR(x,t)limtR(t)=0uniformly in any bounded subset of [0,h)

    which implies that

    limt+I(,t)C([0,h(t)])=limt+R(,t)C([0,h(t)])=0.

    Then it follows from Lemma 3.1 that limt+S(x,t)=Λδ uniformly in any bounded subset of [0,+) and this completes the proof.

    Now we provide a sufficient condition on disease spreading.

    Theorem 3.6. If ˜R0>R0>1 and h0>h:=dδπ24βΛ(11R0), then h=+.

    Proof. We know that (S(x,t),I(x,t),R(x,t);h(t)) satisfies

    {St(x,t)=dSxx(x,t)+ΛβS(x,t)I(x,t)δS(x,t),x>0,t>0,It(x,t)dIxx(x,t)+βS(x,t)I(x,t)(α+γ+δ)I(x,t),0<x<h(t),t>0,Rt(x,t)=dRxx(x,t)+γI(x,t)(δ+η)R(x,t),0<x<h(t),t>0,Sx(0,t)=Ix(0,t)=Rx(0,t)=0,t>0,I(h(t),t)=R(h(t),t)=0,xh(t),t>0,h(t)=μIx(h(t),t),t>0,h(0)=h0,S(x,0)=S0(x)0,I(x,0)=I0(x)0,R(x,0)=R0(x)0,x0.

    A result similar as [12, Lemma 4.1] for lower solutions gives S(x,t)S_(x,t) for 0<x<+ and t>0; I(x,t)I_(x,t) and R(x,t)R_(x,t) for 0<x<h_(t) and t>0; and h(t)h_(t) for t>0, where (S_(x,t),I_(x,t),R_(x,t);h_(t)) satisfies

    {S_t(x,t)=dS_xx+ΛβS_(x,t)I_(x,t)δS_(x,t),x>0,t>0,I_t(x,t)=dI_xx+βS_(x,t)I_(x,t)(α+γ+δ)I_(x,t),0<x<h_(t),t>0,R_t(x,t)=dR_xx+γI_(x,t)(δ+η)R_(x,t),0<x<h_(t),t>0,S_x(0,t)=I_x(0,t)=R_x(0,t)=0,t>0,I_(h(t),t)=R_(h(t),t)=0,xh_(t),t>0,h_(t)=μIx(h_(t),t),t>0,h_(0)=h0,S_(x,0)=S0(x)0,I_(x,0)=I0(x)0,R_(x,0)=R0(x)0,x0.

    It follows from Theorem 3.4 that if ˜R0>R0>1 and h0>h then h_=+, which implies h=+.

    In this paper, we proposed and analyzed a free boundary problem of a reaction-diffusion SIRI model with the bilinear incidence rate. We first obtained the existence and uniqueness of global solutions. Then we established several criteria on disease vanishing and spreading. Roughly speaking, for the case without relapse, the disease will vanish if one of the following three conditions holds. (a) The basic reproduction number R0<1; (b) R0>1 and the initial infected area h0 and the boundary moving rate μ are small enough; (c) R0>1 together with the initial values I0, R0, and h0 being small enough. The disease will spread to the whole area if R0>1 and either h0 is large enough or the diffusion rate d is small enough. For the case with relapse, the disease will die out if the basic reproduction number ˜R01 whereas the disease will spread to the whole area if ˜R0>R0>1 and h0 is large enough. Unfortunately, we have not considered the case where ˜R0>1>R0. In this case, the disease transmission is complex, which we are working on. Moreover, when the free boundaries can extend to the whole area, we also gave an estimate on the spreading speed.

    Compared with the ordinary differential equation model (1.1), the model we studied with free boundary allows more reasonable sufficient conditions on the disease spreading and vanishing. With the main results obtained, we can better understand the phenomenon of relapse. To illustrate this, we demonstrate how the basic reproduction numbers rely on the relapse rate η. For system (1.3), fix other parameters except η, we see that R0(η)=˜R0=βΛ(δ+η)δ(γδ+(δ+η)(α+δ)), which is a strictly increasing function of η. Thus there exists an η[0,+) such that R0(η)1 when ηη and R0(η)<1 when η<η. Then the relapse rate η plays an important role in R0(η). In other words, when η varies, disease spreading and vanishing will change. Since R0(η)>R0 always holds, with relapse the disease will be more easily spread to the whole area than without relapse.

    The authors would like to thank the two anonymous reviewers for their valuable suggestions and comments, which greatly improve the presentation of the paper. QD, YL, and ZG were supported by the National Natural Science Foundation of China (No. 11771104) and by the Program for Chang Jiang Scholars and Innovative Research Team in University (IRT-16R16). YC was supported partially by NSERC.

    All authors declare no conflicts of interest in this paper.



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