Loading [MathJax]/jax/output/SVG/jax.js
Research article

Traveling waves of di usive disease models with time delay and degeneracy

  • Received: 05 February 2018 Accepted: 12 December 2018 Published: 22 March 2019
  • In this paper, we propose a diffusive epidemic model with a standard incidence rate and distributed delays in disease transmission. We also consider the degenerate case when one of the diffusion coe cients vanishes. By establishing existence theory of traveling wave solutions and providing sharp lower bound for the wave speeds, we prove linear determinacy of the proposed model system. Sensitivity analysis suggests that disease propagation is slowed down by transmission delay but fastened by spatial diffusion. The existence proof is based on the construction of a suitable convex set which is invariant under the integral map of traveling wave equations. An innovative argument is formulated to study the boundary value problems of nonlinear elliptic equations satisfied by the traveling wave solutions, which enables us to prove that there does not exist a positive traveling wave connecting two nontrivial equilibria.

    Citation: Xiao-Min Huang, Xiang-ShengWang. Traveling waves of di usive disease models with time delay and degeneracy[J]. Mathematical Biosciences and Engineering, 2019, 16(4): 2391-2410. doi: 10.3934/mbe.2019120

    Related Papers:

    [1] Shiqiang Feng, Dapeng Gao . Existence of traveling wave solutions for a delayed nonlocal dispersal SIR epidemic model with the critical wave speed. Mathematical Biosciences and Engineering, 2021, 18(6): 9357-9380. doi: 10.3934/mbe.2021460
    [2] Wenhao Chen, Guo Lin, Shuxia Pan . Propagation dynamics in an SIRS model with general incidence functions. Mathematical Biosciences and Engineering, 2023, 20(4): 6751-6775. doi: 10.3934/mbe.2023291
    [3] Tong Li, Zhi-An Wang . Traveling wave solutions of a singular Keller-Segel system with logistic source. Mathematical Biosciences and Engineering, 2022, 19(8): 8107-8131. doi: 10.3934/mbe.2022379
    [4] Ran Zhang, Shengqiang Liu . Traveling waves for SVIR epidemic model with nonlocal dispersal. Mathematical Biosciences and Engineering, 2019, 16(3): 1654-1682. doi: 10.3934/mbe.2019079
    [5] Danfeng Pang, Yanni Xiao . The SIS model with diffusion of virus in the environment. Mathematical Biosciences and Engineering, 2019, 16(4): 2852-2874. doi: 10.3934/mbe.2019141
    [6] Xixia Ma, Rongsong Liu, Liming Cai . Stability of traveling wave solutions for a nonlocal Lotka-Volterra model. Mathematical Biosciences and Engineering, 2024, 21(1): 444-473. doi: 10.3934/mbe.2024020
    [7] M. B. A. Mansour . Computation of traveling wave fronts for a nonlinear diffusion-advection model. Mathematical Biosciences and Engineering, 2009, 6(1): 83-91. doi: 10.3934/mbe.2009.6.83
    [8] Felicia Maria G. Magpantay, Xingfu Zou . Wave fronts in neuronal fields with nonlocal post-synaptic axonal connections and delayed nonlocal feedback connections. Mathematical Biosciences and Engineering, 2010, 7(2): 421-442. doi: 10.3934/mbe.2010.7.421
    [9] Maryam Basiri, Frithjof Lutscher, Abbas Moameni . Traveling waves in a free boundary problem for the spread of ecosystem engineers. Mathematical Biosciences and Engineering, 2025, 22(1): 152-184. doi: 10.3934/mbe.2025008
    [10] Guo Lin, Shuxia Pan, Xiang-Ping Yan . Spreading speeds of epidemic models with nonlocal delays. Mathematical Biosciences and Engineering, 2019, 16(6): 7562-7588. doi: 10.3934/mbe.2019380
  • In this paper, we propose a diffusive epidemic model with a standard incidence rate and distributed delays in disease transmission. We also consider the degenerate case when one of the diffusion coe cients vanishes. By establishing existence theory of traveling wave solutions and providing sharp lower bound for the wave speeds, we prove linear determinacy of the proposed model system. Sensitivity analysis suggests that disease propagation is slowed down by transmission delay but fastened by spatial diffusion. The existence proof is based on the construction of a suitable convex set which is invariant under the integral map of traveling wave equations. An innovative argument is formulated to study the boundary value problems of nonlinear elliptic equations satisfied by the traveling wave solutions, which enables us to prove that there does not exist a positive traveling wave connecting two nontrivial equilibria.


    The study of deterministic epidemic model can be dated back in 1927 when Kermack and McKendrik [1,Eq. (29)] proposed a simple ordinary differential system with three compartments: susceptible individuals (S), infected individuals (I) and removed/recovered individuals (R). The corresponding three-dimensional system can be decoupled into two subsystems:

    S(t)=φ(S(t),I(t)), (1.1)
    I(t)=φ(S(t),I(t))γI(t), (1.2)

    and R(t)=γI(t). Here, the incidence rate is chosen as the mass-action function:

    φ(S(t),I(t))=βS(t)I(t),

    The two parameters β>0 and γ>0 in the model denote the disease transmission rate and the removal/recovery rate, respectively. Given initial values {S0,I0,R0} at t=0, one can integrate the R-equation to obtain

    R(t)=R0+t0γI(s)ds,

    where I(t) can be solved from the subsystem for S and I:

    I(t)=S0+I0S(t)+γβlnS(t)S0.

    Substituting this into (1.1) gives a separable equation for S(t), whose solution can be expressed as the inverse of an integral:

    S(t)S0[u(S0+I0u+γβlnuS0)]1du=βt.

    As noted by Kermack and McKendrik, there is no explicit formula, in terms of elementary functions, instead of the integral representation as given above, for the solution of their SIR model.

    In real application, however, it is more desirable to have a closed formula for the functions of interest so as to fit the epidemic wave data collected during a disease outbreak. The Richards empirical model [2], which is also called the theta-logistic model [3,4], has been used to estimate the key parameters in recent outbreaks of SARS [5], dengue fever [6] and pandemic influenza H1N1 [7]. This model has the advantage that it can be solved in terms of elementary functions. However, the Richards model was originally introduced in ecological studies, and it did not have any epidemiological justification. To make a connection between Richards empirical model with Kermack-Mckendrick compartmental model, it was proposed in [8] that the following standard incidence rate shall be used:

    φ(S(t),I(t))=βS(t)I(t)S(t)+I(t).

    Here, the removed individuals are assumed to be isolated from the community so that the denominator of the incidence function is S(t)+I(t), instead of S(t)+I(t)+R(t). This assumption ensures that the R-equation can be decoupled from the model system. The above system can be integrated with solutions explicitly given by elementary functions. Actually, if we add (1.1) to (1.2) and then divide both sides of the resulting equation by both sides of (1.1) respectively, we arrive at a separable equation

    d(S+I)dS=γ(S+I)βS.

    Solving the above equation gives S(t)+I(t)=cS(t)γ/β, where c>0 is a constant of integration which depends on initial conditions. Now, we use this algebraic equation to eliminate I(t) in (1.1) and obtain a Richards-type equation:

    S(t)=βS(t)[1S(t)1γ/β/c],

    whose solution has a closed form.

    Moreover, an intrinsic relation between the parameters in Kermack-Mckendrick model and Richards model was obtained in [8], which provided a satisfactory interpretation of Richards model in epidemiology.

    It is worth to remark that "standard incidence is not realistic for small population sizes" [9]. This is because "for small population, the contact rate increases with population size" and mass action is more appropriate [10]. However, when population increases, the contact rate cannot increase linearly but it should be saturated as population size goes to infinity [11]. So, our model is biologically relevant when the population size is large.

    If the transmission delay is taken into consideration [12], one should change the equation (1.2) to

    I(t)=0η(τ)q(τ)φ(S(tτ),I(tτ))dτγI(t), (1.3)

    where q(τ) is a probability density function which characterizes the possibility of a susceptible individual having a latency period of τ after being infected and η(τ)<1 is the survival probability. Note that the time delay does not appear in the S-equation because the susceptible individual should be removed from this group once being infected. We refer to the equations (1.1) and (1.3) as a type-Ⅰ delayed disease model. For convenience, we define θ=0η(τ)q(τ)dτ<1 and p(τ)=η(τ)q(τ)/θ. It is noted that p(τ) is still a probability density function and θ can be regarded as average survival rate during latency stage. We then adopt the convolution symbol as

    (pg)(t):=0p(τ)g(tτ)dτ,

    and rewrite (1.3) as I(t)=θ[pφ(S,I)](t)γI(t). When p(τ)=δr(τ) is a Dirac delta function, then (pg)(t)=g(tr) and the equation (1.3) has only one discrete delay:

    I(t)=θφ(S(tr),I(tr))γI(t).

    Let i(t)=I(t+r). One may further obtain a closed system for S and i:

    S(t)=φ(S(t),i(tr)),i(t)=θφ(S(t),i(tr))γi(t).

    In some literature, the term i(tr) in both of the above two equations is replaced by a general functional with distributed delays (pi)(t). We refer to such generalized delay system as a type-Ⅱ delayed disease model. Note that in the case of single discrete delay, type-Ⅰ and type-Ⅱ delayed disease models are equivalent with the relation i(t)=I(t+r); see [13]. However, if more general distributed delays are taken into consideration, these two types of delayed disease models are different.

    To study the spatial spread of infectious diseases, we assume random movement of each individual and propose the following delayed diffusive epidemic model:

    tS(x,t)=d1xxS(x,t)φ(S(x,t),I(x,t)), (1.4)
    tI(x,t)=d2xxI(x,t)+θ0p(τ)φ(S(x,tτ),I(x,tτ))dτγI(x,t), (1.5)

    where d10 and d20 are the diffusion rates of the susceptible and infective individuals, respectively. If both d1 and d2 vanish, the above system reduces to the delay differential system with spatial homogeneity. Throughout this paper, we assume that at least one diffusion rate is positive. However, it is biologically relevant that one of the diffusion rates may vanish. For instance, in the study of spatial spread of rabies among foxes [14], the uninfected foxes S(x,t) always stay in their territories, while the rabid foxes I(x,t) may loss their sense of direction and wonder randomly. Thus, we shall consider the degenerate case d1=0. Here, we also assume for simplicity that the individuals do not diffuse during the incubation period. This is reasonable because, in the rabies model, the fox moves rapidly and randomly only when it becomes infective. Note that for the special case when p(τ)=σeστ, the system (1.4)-(1.5) reduces to an SEIR diffusive model without delay but with degeneracy in the equation for the exposed class.

    Our objective is to provide existence theory for the traveling wave solutions of the proposed model system with time delay and degeneracy. Especially, we will calculate the minimal traveling speed and analyze the effect of disease transmission delay and spatial movement in the propagation of an infectious disease. We shall show that the delayed system with degeneracy is linearly determinant in the sense that the minimal traveling speed can be obtained by considering the linearized system at a disease-free equilibrium. The traveling wave solutions of (1.4)-(1.5) with speed c0 take the special forms S(x,t)=S(ξ) and I(x,t)=I(ξ), where ξ=x+ctR. The solution with zero wave speed is actually the steady state of the diffusive system. So, we always assume c>0. A simple application of chain rule gives the following nonlocal differential system

    cS(ξ)=d1S(ξ)φ(S(ξ),I(ξ)), (1.6)
    cI(ξ)=d2I(ξ)+θ0p(τ)φ(S(ξcτ),I(ξcτ))dτγI(ξ). (1.7)

    For simplicity, we denote pc(τ):=p(τ/c)/c. Then the integral on the right-hand side of (1.7) has the simple convolution expression: pcφ(S,I). We also note that pc is still a probability density function whose total integral on the positive real line equals to one.

    It follows from maximum principle that any non-constant and non-negative solutions of (1.6)-(1.7) should be positive at any finite point. In general, the traveling wave connects two equilibria at infinities. Note that the system (1.4)-(1.5) has infinitely many equilibrium points (S,0), where S is any nonnegative number. However, as we will demonstrate later, it is impossible to find a traveling wave solution connecting two different non-trivial equilibria. In other words, if a positive traveling wave solution of (1.6)-(1.7) satisfies the boundary conditions:

    S()=S0>0,I()=0,S()=I()=0 (1.8)

    then we should also have S()=I()=0.

    Unlike the rich theory and general tools developed for monotone systems [15,16,17,18,19], the study of non-monotone systems such as the diffusive disease models is far from complete and unified. Especially, the results on traveling wave solutions in epidemic models are obtained on a case-by-case basis. Most of the earlier studies only consider the non-delay case. When the incidence rate is a bilinear function in S and I, Källén [20] investigated the degenerate case d1=0, while Hosono and Ilyas [21] considered the degenerate case d2=0. Hosono and Ilyas [22] further studied the non-degenerate case by using a geometric shooting method introduced by Dunbar [23,24]. This technique was recently developed by Huang [25] in a work on non-monotone diffusive systems with general reaction functions. The applications of geometric shooting method are limited to the non-delay models. For the diffusive disease models with time delay, a commonly used method is the Schauder fixed point theorem; see [26] and [27] for the study of two delayed epidemic models with standard incidence rate and mass-action, respectively. In [28], a delayed diffusive SIR model with external supplies is considered. The disease models all have type-Ⅱ delays. A type-Ⅰ delayed disease model with mass-action and recruitment was proposed in [29], where a critical wave speed was calculated and numerical simulation was conducted to suggest that this critical value should be the minimal wave speed. This observation was recently proved in [30] with the aid of Schauder fixed point theorem. To the best of our knowledge, there is no results concerning the traveling wave solutions of type-Ⅰ delayed diffusive models with standard incidence rate. In this paper, we will fill in this gap and establish existence theory for the traveling wave equations (1.6)-(1.7). Moreover, we will extend our results to the degenerate cases when one of the diffusion coefficients vanishes.

    The rest of the paper is organized as follows. In Section 2, we obtain a critical wave speed by linearizing the traveling wave equations at a non-trivial equilibrium. Sensitivity analysis is conducted to study the dependence of this critical value on diffusion coefficient and time delay. In Section 3, we introduce some preliminary results about shifted Laplacian operators and their inverses. In Section 4, we provide an existence theorem for the positive traveling wave solutions. In Section 5, we investigate some properties of the positive traveling wave solutions. In Section 6, we conclude this paper with some discussions on the main results and an open problem.

    It is well known that the minimal wave speed for monotone systems can be determined from the corresponding linearized system at a certain equilibrium point [31,32,33]. However, for non-monotone systems, especially predator-prey type systems in disease models, it is still an open problem to find the conditions for linear determinacy [34,35]. In this section, we will calculate a critical wave speed c by linearization. As we shall see later, this critical value is exactly the minimal wave speed, which proves linear determinacy of our model system.

    By linearizing the I-equation (1.7) at the nontrivial equilibrium (S0,0), we find the characteristic function:

    f(λ,c):=d2λ2+cλθβ0p(τ)ecτλdτ+γ, (2.1)

    where λ0 and c0. Recall that p is a probability density function such that p(τ)0 for all τ0 and the total integral on the real line is one. Throughout this paper, we also assume that p(τ) decays exponentially at infinity; namely, there exists σ>0 such that p(τ)eστ is uniformly bounded for τ[0,). We will need to use this assumption to prove continuity of the integral map associated with traveling wave equations. As a simple consequence of the assumption, all moments of p exist. In this section, we are only interested in the case when θβ>γ; as we shall prove later, no positive traveling wave solution exists for the case θβγ. First, we note that if c=0, then f(λ,0)=d2λ2θβ+γ is always negative. Fix any c>0, we have

    λλf(λ,c)=2d2θβc20p(τ)τ2ecτλdτ<0,

    which implies that

    λf(λ,c)=2d2λ+c+θβc0p(τ)τecτλdτ

    is decreasing in λ.

    We have the following lemma about the critical wave speed.

    Lemma 1. If θβ>γ, then there exists c0 such that for any c>c, the equation f(λ,c)=0 has a unique positive solution, denoted by λ1=λ1(c), such that λf(λ1,c)>0, and for any 0<c<c, the equation f(λ,c)=0 has no positive solution. In the later case, f(λ,c) as λ. Especially, c=0 for the degenerate case d2=0.

    Proof. We consider the non-degenerate case d2>0 and degenerate case d2=0, respectively. If d2>0, then λf(0,c)>0 and λf(λ,c)<0 for sufficiently large λ>0. There exists a unique ˉλ(c)>0 such that λf(ˉλ(c),c)=0. Actually ˉλ(c) is the global maximum point of f(λ,c) for λ[0,). Moreover,

    2d2ˉλ=c+θβc0p(τ)τecτˉλdτ(c,c+θβcm1),

    where m1=0p(τ)τdτ is the first moment of the probability density function p. Biologically, the first moment m1 is interpreted as the average delay of disease transmission. Since f(0,c)=θβ+γ<0, the equation f(λ,c)=0 has at least one positive solution if and only if g(c):=f(ˉλ(c),c)0. By chain rule, we have

    g(c)=λf(ˉλ(c),c)ˉλ(c)+cf(ˉλ(c),c)=ˉλ+θβˉλ0p(τ)τecτˉλdτ>0,

    which implies that g(c) is increasing in c. As c0+, we obtain ˉλc0+ and g(c)f(0,0)=θβ+γ<0. As c, we have f(1,c). Thus, g(c)>0 for sufficiently large c. The equation g(c)=0 has a unique positive solution, denoted by c. For any c>c, we have g(c)=f(ˉλ(c),c)>0, f(0,c)<0, and λf(λ,c)>0 for λ(0,ˉλ(c)). There exists a unique λ1=λ1(c)(0,ˉλ(c)) such that of f(λ1(c),c)=0 and λf(λ1(c),c)>0. For any c<c, we have g(c)=f(ˉλ(c),c)<0 and the equation f(λ,c)=0 has no positive real solution. It is obvious that f(λ,c) as λ.

    If d2=0 and c>0, then λf(λ,c)>0 for all λ0. Moreover, f(0,c)<0 and f(λ,c)>cλθβ+γ>0 for sufficiently large λ>0. The existence of λ1 follows immediately.

    Remark 2. From the proof of Lemma 1, we observe that in the non-degenerate case d2>0, the equation f(λ,c)=0 with c>c has another positive solution λ2(c)>ˉλ(c). But, this solution is irrelevant because λf(λ2(c),c)<0. We only need to use λ1(c) to construct the upper and lower solutions. Especially, we observe from λf(λ1(c),c)>0 that f(λ1(c)+ε,c)>0 for any sufficiently small ε>0.

    To investigate the dependence of critical wave speed on the time delay, we assume that p(τ)=δr(τ) is a Dirac delta function such that (pg)(t)=g(tr). In this special case, we emphasize the dependence of f on r and write f(λ,c,r)=d2λ2+cλθβeλcr+γ. From the proof of Lemma 2.1, we note that c=c and λ:=ˉλ(c) are a pair of positive solutions to the equations f(λ,c,r)=0 and λf(λ,c,r)=0. We treat r as an independent variable, while λ and c are dependent variables. By chain rule, we have

    λf(λ,c,r)dλdr+cf(λ,c,r)dcdr+rf(λ,c,r)=0.

    On the other hand, since λf(λ,c,r)=0, we then have

    dcdr=rf(λ,c,r)cf(λ,c,r)<0

    because f is an increasing function in both c and r. This implies that the critical value c is a decreasing function of time delay r; namely, time delay in the transmission mechanism will reduce the spatial propagation of infectious diseases. Since f is decreasing in d2, a similar argument shows that the critical wave speed c is increasing in d2; that is, a larger spatial diffusion rate will lead to a faster traveling speed.

    Let α1>0 and α2>0 be two large constants to be determined later. We introduce the shifted Laplacian differential operators:

    Δih:=dih+ch+αih

    for any function h which is second-order differentiable on the whole real line except at finite many points. If di>0, then the inverse of Δi have the following integral representation:

    (Δ1ih)(ξ):=1di(μ+iμi)[ξeμi(ξy)h(y)dy+ξeμ+i(ξy)h(y)dy],

    where μ±i:=(c±c2+4diαi)/(2di) are the two characteristic roots for the differential operator Δi. The integrals on the right-hand side of the above formula are well-defined if the function h(ξ) belongs to the Banach space Bμ(R) which consists of all continuous functions h(ξ) whose weighted norm

    |h|μ:=supξReμ|ξ||h(ξ)|

    is finite. Here, μ is a positive number such that μi<μ<μ<μ+i. Now, we consider the degenerate case by taking limits. It is readily seen that μiαi/c, μ+i, and di(μ+iμi)c as di0+. The integral operator Δ1i becomes:

    (Δ1ih)(ξ)=1cξeαic(ξy)h(y)dy.

    For all hBμ(R), it follows from fundamental theorem of calculus that Δi(Δ1ih)=h. However, if hBμ(R) such that h and h may have finite number of points with jump discontinuity, it is now always true that Δ1i(Δih)=h. For convenience, we use ˇBμ(R) to denote the collection of all piecewise continuous functions with possible jump discontinuity at finite many points and finite norm ||μ. Obviously, the integral operator Δ1i is well defined on ˇBμ(R). It is noted that Bμ(R)=ˇBμ(R)C(R), but the normed space ˇBμ(R) is not complete. We introduce the jump function associated with hˇBμ(R) as

    [h](ξ):=h(ξ+)h(ξ)=limε0+h(ξ+ε)limε0h(ξ+ε).

    Note that [h] vanishes at all continuous points of h, and [h]=0 if hBμ(R). We have the following results.

    Lemma 3. If di>0, then Δ1i(Δih)h for hBμ(R) such that h,hˇBμ(R) and [h]0. If di=0, then Δ1i(Δih)h for hˇBμ(R) such that hˇBμ(R) and [h]0.

    Proof. We first consider the non-degenerate case di>0. For simplicity, we assume that h and h have only one jump point at ξ0. The argument can be easily extended to the case of multiple jump points. By shifting, we may further set ξ0=0. For ξ0, it follows from the definition that

    di(μ+iμi)[Δ1i(h)](ξ)=ξeμi(ξy)h(y)dy+0ξeμ+i(ξy)h(y)dy+0eμ+i(ξy)h(y)dy.

    Using integration by parts, we have

    di(μ+iμi)[Δ1i(h)](ξ)=(μiμ+i)h(ξ)[h](0)eμ+iξ+ξ(μi)2eμi(ξy)h(y)dy+ξ(μ+i)2eμ+i(ξy)h(y)dy.

    Similarly, we obtain

    di(μ+iμi)[Δ1i(h)](ξ)=ξμieμi(ξy)h(y)dy+ξμ+ieμ+i(ξy)h(y)dy.

    Since μ±i are characteristic roots for the differential operator Δi, we have

    [Δ1i(Δih)](ξ)=h(ξ)+[h](0)eμ+iξμ+iμih(ξ).

    In a similar manner, we can show that for ξ0,

    [Δ1i(Δih)](ξ)=h(ξ)+[h](0)eμiξμ+iμih(ξ).

    Next, we consider the degenerate case di=0. For ξ0, by definition and integration by parts, we have

    c[Δ1i(h)](ξ)=ξeαic(ξy)h(y)dy=h(ξ)αicξeαic(ξy)h(y)dy,

    which, together with Δih=ch+αih, implies that

    [Δ1i(Δih)](ξ)=h(ξ).

    For ξ0, we have

    c[Δ1i(h)](ξ)=0eαic(ξy)h(y)dy+ξ0eαic(ξy)h(y)dy=h(ξ)eαiξ/c[h](0)αicξeαic(ξy)h(y)dy,

    which implies that

    [Δ1i(Δih)](ξ)=h(ξ)eαiξ/c[h](0)h(ξ).

    This completes the proof.

    Remark 4. By replacing h with h, we also obtain Δ1i(Δih)h if [h]0 in the non-degenerate case or [h]0 in the degenerate case. Especially, if h is continuous in the non-degenerate case or h is continuous in the degenerate case, we have Δ1i(Δih)=h.

    Throughout this section, we assume that θβ>γ and c>c. Making use of the integral representation for the inverse of shifted Laplacian operator Δi, we define the integral map: F=(F1,F2)T, where

    F1(u,v):=Δ11(α1uφ(u,v)), (4.1)
    F2(u,v):=Δ12(α2v+θpcφ(u,v)γv), (4.2)

    for any u,vBμ(R). Recall that pc(τ)=p(τ/c)/c and

    (pcg)(ξ)=0p(τ)g(ξcτ)dτ.

    It is readily seen that a traveling wave solution of (1.6)-(1.7) is the same as a fixed point of the integral map F. We then use the following upper and lower solutions to construct an invariant convex set. Let f(λ)=f(λ,c) be given as in (2.1). By Lemma 2.1, there exists a positive λ1=λ1(c) such that f(λ1)=0 and f(λ1+ε)>0 for any sufficiently small ε>0. We define

    S+(ξ):=S0,S(ξ):=max{S0[1eε(ξξ1)],0},I+(ξ):=min{eλ1ξ,S0(θβ/γ1)},I(ξ):=max{eλ1ξ[1eε(ξξ2)],0},

    where ε>0,ξ1<0,ξ2<0 are constants to be determined later. It is readily seen that SBμ(R) and S,SˇBμ(R) with [S]0. By Lemma 3.1, Δ11(Δ1S)S. Similarly, we note that Δ11(Δ1S+)=S+, Δ12(Δ2I)I and Δ12(Δ2I+)I+.

    The convex set Γ is chosen as the collection of all function pairs (u,v)Bμ(R)×Bμ(R) such that SuS+ and IvI+. To show that Γ is invariant under the map F, we first choose α1β, α2γ, and 0<ε<min{λ1,c/d1} such that f(λ1+ϵ)>0. Then, we let ξ1<0 be negatively large such that

    βeλ1ξ1ε(cd1ε)S0.

    Finally, we choose ξ2<ξ1 such that

    f(λ1+ε)S0[1eε(ξ2ξ1)]θβeλ1ξ2.

    Lemma 5. Assume θβ>γ and c>c. Let α1,α2,ε,ξ1,ξ2 be chosen as above. For any (u,v)Γ, we have F(u,v)Γ.

    Proof. We need to prove four inequalities SF1(u,v)S+ and IF2(u,v)I+, respectively. First, we note that

    α1uφ(u,v)α1uα1S+=Δ1S+,

    which, together with (4.1), implies that

    F1(u,v)Δ11(Δ1S+)=S+.

    Next, we choose α1β such that α1uφ(u,v) is increasing in u and decreasing in v. Thus,

    α1uφ(u,v)α1Sφ(S,I+).

    On the other hand, we have

    (Δ1S)(ξ)=α1S(ξ)ε(cd1ε)S0eε(ξξ1)

    for ξξ1, and (Δ1S)(ξ)=0 for ξξ1. We need to show α1uφ(u,v)Δ1S. Note that φ(S,I+)βSα1S and φ(S,I+)βI+βeλ1ξ. It suffices to prove

    βeλ1ξε(cd1ε)S0eε(ξξ1)

    for all ξξ1. We choose 0<ε<min{λ1,c/d1}. The above inequality is a consequence of the inequality

    βeλ1ξ1ε(cd1ε)S0,

    which can be achieved by letting ξ1=ξ1(ε)<0 be negatively large. Therefore, in view of (4.1), we obtain

    F1(u,v)Δ11(Δ1S)S.

    Now, we observe from monotonicity of φ(u,v) in both u and v that

    φ(u(ξ),v(ξ))φ(S+(ξ),I+(ξ))min{βeλ1ξ,S0(βγ/θ)}.

    Consequently, if α2γ, then

    α2v(ξ)+θ[pcφ(u,v)](ξ)γv(ξ)(α2γ)I+(ξ)+min{θβeλ1ξ0p(τ)eλ1cτdτ,S0(θβγ)}.

    Recall that λ1 is the root of the characteristic equation f(λ1)=0. We have

    (Δ2I+)(ξ)=(α2γ)I+(ξ)+θβeλ1ξ0p(τ)eλ1cτdτ

    for ξln[S0(θβ/γ1)]/λ1, and

    (Δ2I+)(ξ)=(α2γ)I+(ξ)+S0(θβγ)

    for ξln[S0(θβ/γ1)]/λ1. In either case, we have

    (Δ2I+)(ξ)α2v(ξ)+θ[pcφ(u,v)](ξ)γv(ξ),

    which, together with (4.2), yields

    F2(u,v)Δ12(Δ2(I+))I+.

    Finally, we show that

    α2v(ξ)+θ[pcφ(u,v)](ξ)γv(ξ)(Δ2I)(ξ).

    Since α2γ and I(ξ)=0 for ξξ2, we only need to verify the above inequality for ξξ2. Note that

    α2v(ξ)+θ[pcφ(u,v)](ξ)γv(ξ)(α2γ)I(ξ)+θ[pcφ(S,I)](ξ),

    and

    (Δ2I)(ξ)=(α2γ)I(ξ)+θβ[pcI](ξ)f(λ1+ε)eλ1ξ+ε(ξξ2).

    It suffices to prove

    f(λ1+ε)eλ1ξ+ε(ξξ2)θ[pc(βIφ(S,I))](ξ)

    for all ξξ2. Since an increasing exponential function is always greater than its convolution with pc, the above inequality is satisfied if we can show that

    f(λ1+ε)eλ1ξ+ε(ξξ2)θ[βIφ(S,I)](ξ)=θβ[I(ξ)]2S(ξ)+I(ξ)

    for all ξξ2. Assuming ξ2<ξ1, we only need to prove

    f(λ1+ε)eλ1ξ+ε(ξξ2)S0[1eε(ξξ1)]θβe2λ1ξ

    for all ξξ2. This can be done by choosing ξ2=ξ2(ξ1,ε)<0 negatively large such that

    f(λ1+ε)S0(1eε(ξ2ξ1)]θβeλ1ξ2.

    Hence, on account of (4.2), we obtain

    F2(u,v)Δ12(Δ2(I))I.

    This completes the proof.

    We shall use Schauder fixed point theorem to establish existence of traveling wave solution. Recall that Γ is invariant under the integral map. We have to prove that F is continuous and compact on Γ with respect to the induced norm in Bμ(R)×Bμ(R) for some sufficiently small μ>0. Recall that the probability density function p(τ) decays exponentially as τ. Especially, there exists a σ>0 such that p(τ)eστ is bounded as τ. We choose μ(0,σ/c) such that p(τ)eμcτ is integrable on the positive real line. Recall that we also require μi<μ<μ<μ+i with i=1,2.

    Lemma 6. For any μ>0 such that μ<σ/c and μi<μ<μ<μ+i with i=1,2, the integral map F is continuous and compact on Γ with respect to the induced norm in Bμ(R)×Bμ(R).

    Proof. For any (u1,v1) and (u2,v2) in Γ, we then have

    |[φ(u1,v1)φ(u2,v2)](ycτ)|βeμ|y|+μcτ(|u1u2|μ+|v1v2|μ)

    for any yR and τ0. Consequently,

    |[pcφ(u1,v1)pcφ(u2,v2)](y)|βC0eμ|y|(|u1u2|μ+|v1v2|μ)

    for all yR, where

    C0:=0p(τ)eμcτdτ<.

    Let g(y):=eμ|y|. It is readily seen that

    |(Δ12g)(ξ)|C2eμ|ξ|,

    where

    C2:=1d2(μ+2μ2)(1μ+2μ1μ2+μ).

    For the degenerate case d2=0, the above formula is still valid by taking limit d20+. Note that d2(μ+2μ2)c, μ2α2/c and μ+2 as d20+. It follows that C2=1/(α2μc) if d2=0. We then have

    |[F2(u1,v1)F2(u2,v2)](ξ)|(θβC0+α2γ)C2eμ|ξ|(|u1u2|μ+|v1v2|μ)

    for all ξR. Similarly, we can prove that

    |[F1(u1,v1)F1(u2,v2)](ξ)|(α1+β)C1eμ|ξ|(|u1u2|μ+|v1v2|μ)

    for all ξR, where

    C1:=1d1(μ+1μ1)(1μ+1μ1μ1+μ)

    is the upper bound for the operator norm of Δ11 in Bμ(R) with respect to the norm ||μ. For the degenerate case d1=0, we write C1=1/(α1μc). Thus, we have prove continuity of the map F=(F1,F2) on Γ. To prove compactness, we shall make use of Arzela-Ascolli theorem. Note that Γ is bounded, it suffices to show that the image F(Γ) is pre-compact. First, we note that F(Γ)Γ is bounded by the upper and lower solutions, which are uniformly bounded functions on the whole real line. For any ε>0, there exists a M>0 such that

    |[F1(u1,v1)F1(u2,v2)](ξ)|eμ|ξ|+|[F2(u1,v1)F2(u2,v2)](ξ)|eμ|ξ|<ε, (4.3)

    for any (u1,v1)Γ, (u2,v2)Γ and |ξ|M. On the other hand, the functions in F(Γ) continuous and uniformly bounded on the compact interval [M,M]. Moreover, they are equi-continuous because

    |[F1(u,v)](ξ)|2(α1+β)S0d1(μ+1μ1),|[F2(u,v)](ξ)|2(α2+θβγ)S0(θβ/γ1)d2(μ+2μ2),

    for all (u,v)Γ and ξR. By Arzela-Ascolli theorem, there exists a finite ε-net of F(Γ) with respect to the supremum norm in C[M,M]×C[M,M]. In view of (4.3), this net is also a finite ε-net of F(Γ) with respect to the weighted norm in Bμ(R)×Bμ(R). Thus, F(Γ) is precompact, which implies that F is a compact map on Γ.

    By Schauder fixed point theorem, F possesses a fixed point, denoted by (S,I), in Γ. Since S±()=S0 and I±()=0, by squeeze theorem, we have S()=S0 and I()=0. Actually, I(ξ)eλ1ξ1 as ξ, which implies that I can not be a constant function. From the integral representation of the operator F and L'H\^opital's rule, we note that S and I are differentiable, and S()=I()=0. Thus, (S,I) satisfies the boundary conditions (1.8). To sum up, we have the following existence result.

    Theorem 7. Assume θβ>γ and c>c. For any S0>0, there exists a positive and uniformly bounded traveling wave solution of (1.6)-(1.7) with the boundary conditions (1.8).

    In this section, we shall derive some general properties for a positive and uniformly bounded traveling wave solution of (1.6)-(1.7) with the boundary conditions (1.8). Especially, we will show that θβ>γ and cc are necessary conditions for the existence of traveling wave solutions.

    We denote the traveling wave solution by (S,I). An integration of (1.6) gives

    d1S(ξ)=c[S(ξ)S0]+ξφ(S(y),I(y))dy,

    which, together with boundedness of S, implies that φ(S,I) is integrable, S is differentiable, and S is uniformly bounded on the real line. If d1>0, we integrate (1.6) to obtain

    S(ξ)=ξec(ξy)/d1d1φ(S(y),I(y))dy<0.

    If d1=0, then (1.6) gives S(ξ)=φ(S(ξ),I(ξ))/c<0. In either case, S is strictly decreasing on R. Moreover, it follows from above formulas that S()=0 and

    c[S0S()]=φ(S(y),I(y))dy.

    Now, we solve (1.7) by variation of parameters. For non-degenerate case d2>0, we let μ±=(c±c2+4d2γ)/(2d2) be the two roots of the characteristic equation d2μ2+cμ+γ=0. On account of uniform boundedness of I and φ(S,I), we have the following integral equation

    I(ξ)=θd2(μ+μ)[ξeμ(ξy)[pcφ(S,I)](y)dy+ξeμ+(ξy)[pcφ(S,I)](y)dy].

    By taking the limit d20+, we obtain an integral equation for I in the degenerate case:

    I(ξ)=θcξeγ(ξy)/c[pcφ(S,I)](y)dy.

    Since φ(S,I) is integrable on the real line, by Tonelli-Fubini theorem, I is also integrable on the real line, and

    I(ξ)dξ=θγφ(S(ξ),I(ξ))dξ.

    Note that φ(S(ξ),I(ξ))<βI(ξ) for all ξR. It follows from the above equation that θβ>γ. Since φ(S,I) is uniformly bounded, we obtain from the integral representation of I that I is also uniformly bounded, this together with integrability of I implies that I()=0.

    Recall that S(ξ)<0 and S(ξ)>0 for all ξR. So, the limit S() exists. We claim that S()=0. If not, by monotonicity of φ in the first component, it follows from (1.7) that

    d2I(ξ)+cI(ξ)θ0pc(τ)φ(S(),I(ξτ))dτγI(ξ), (5.1)

    where pc(τ)=p(τ/c)/c is the scaled probability density function. The above inequality contradicts to the fact I()=0 by the following result.

    Lemma 8. If θβ>γ and S()>0, then there does not exist a positive and uniformly bounded function I(ξ) satisfying the inequality (5.1) and boundary condition I()=0.

    Proof. We prove by contradiction. Assume such function I(ξ) exists. We choose a sufficiently large T>0 and a sufficiently small δ>0 such that

    θφ(S(),δ)δT0pc(τ)dτ>γ(1+δ). (5.2)

    This could be done because, as T and δ0+, the left-hand side tends to θβ and the right-hand side tends to γ. Since I()=0, there exists ξ0 such that I(ξ)<δ for all ξξ0. Define a function

    IT(ξ):=min0τTI(ξτ).

    We claim IT(ξ)=I(ξ) for sufficiently large ξ. If not, there exist an infinite sequence ξk with k=1,2,, such that ξk>ξk1+T, I(ξk)<I(ξk1) and IT(ξk)<I(ξk) for all k1. For each k1, let ηk be the minimum point of I(ξ) in the interval [ξk,ξk+1]. Since IT(ξk+1)<I(ξk+1)<I(ξk), ηk lies in the open interval (ξk,ξk+1). Moreover, I(ηk)=0 and I(ηk)0. It follows from (5.1) and (5.2) that

    0θT0pc(τ)φ(S(),IT(ηk))dτγI(ηk)>γ(1+δ)IT(ηk)γI(ηk)>γ[IT(ηk)I(ηk)].

    Thus, I(ηk)>IT(ηk) for all k1. For each k2, since I(ηk)I(ξ) for all ξ(ξk,ξk+1), IT(ηk)=min0τTI(ηkτ) should be achieved at some point in (ξk1,ξk). Especially, I(ηk)>IT(ηk)I(ηk1) for all k2. Thus, we have found a sequence ηk, such that I(ηk) is increasing, which contradicts to the fact I()=0. This prove our claim that IT(ξ)=I(ξ) for sufficiently large ξ. For simplicity, we shift ξ0 such that IT(ξ)=I(ξ) for all ξξ0, which is the same as I(ξ)0 for ξξ0. By (5.1) and (5.2), we have

    d2I(ξ)+cI(ξ)θT0pc(τ)φ(S(),I(ξ))dτγI(ξ)>γδI(ξ),

    for all ξξ0. Since I(ξ)>0, we may define w(ξ):=I(ξ)/I(ξ). It is readily seen that w(ξ)0 and w(ξ)=I(ξ)/I(ξ)w2(ξ). For the degenerate case d2=0, the above inequality reads w(ξ)>γδ/c>0, which contradicts to the non-positiveness of w(ξ). For the non-degenerate case, the above inequality can be written as

    d2w(ξ)>γδcw(ξ)+d2w2(ξ)>d2w2(ξ),ξξ0.

    The solution of above differential inequality with nonpositive initial value at ξ=ξ0 will blow up at finite value of ξ>ξ0. Actually, an integration of the above inequality gives

    1w(ξ)>1w(ξ0)+ξξ0,

    where we have assumed without loss of generality that w(ξ0)<0. The right-hand side of the above inequality becomes positive for large ξ, but the left-hand side is always negative, a contradiction. This completes the proof.

    Remark 9. A similar result was obtained in [26] for a type-Ⅱ delayed disease model. Their proof is based on comparison principle and asymptotic stability results of quisi-monotone diffusive equations on the real line. It is noted that our proof has the potential to be generalized for more complicated nonlinear elliptic differential equations when the corresponding reaction-diffusion system is difficult to handle.

    Recall that θβ>γ is a necessary condition for the existence of positive traveling wave solutions. In what follows, we will show that c can not be smaller than c. Note that c=0 for the degenerate case d2=0. We only need to consider the non-degenerate case d2>0. In view of the boundary conditions S()=S0>0 and I()=0, we can find a ξ1R such that

    φ(S(ξ),I(ξ))>(θβ+γ)I(ξ)/2

    for all ξξ1. It then follows from (1.7) that

    d2I(ξ)+cI(ξ)>θβ+γ2[(pcI)(ξ)I(ξ)]+θβγ2I(ξ).

    Let K(ξ)=ξI(y)dy. Integrating the above inequality twice gives

    d2I(ξ)+cK(ξ)>θβ+γ2ξ[(pcK)(y)K(y)]dy+θβγ2ξK(y)dy.

    Note that

    ξ[(pcK)(y)K(y)]dy=ξ0τ0pc(τ)I(ys)dsdτdy=0τ0pc(τ)K(ξs)dsdτ>cm1K(ξ),

    where m1=0τp(τ)dτ is the first moment (also called the average delay). We then have

    c+cm1(θβ+γ)/2(θβγ)/2K(ξ)>d2(θβγ)/2I(ξ)+ξK(y)dy>sK(ξs)

    for any s0 and ξξ1. Especially, by choosing s=4[c+cm1(θβ+γ)/2]/(θβγ), we have K(ξs)<K(ξ)/2. By iterating this inequality, we obtain from monotonicity and uniform boundedness of K that the function K(ξ)eμ0ξ with μ0=ln2/s is uniformly bounded on the real line. Note from above inequality that for all ξξ1, I(ξ) is bounded by a constant multiplication of K(ξ). Thus, I(ξ)eμ0ξ is also uniformly bounded on the real line. Similarly, we have uniform boundedness of I(ξ)eμ0ξ and I(ξ)eμ0ξ on the real line. Furthermore, φ(S(ξ),I(ξ))eμ0ξ is uniformly bounded on the real line. For each μ(0,μ0), we introduce two-sided Laplace transform on (1.7) to obtain

    f(μ,c)eμξI(ξ)dξ=eμξ˜βI2(ξ)S(ξ)+I(ξ)dξ,

    where f is the characteristic function defined in (2.1) and ˜β=θβ0eμcτp(τ)dτ. If c<c, then f(μ,c) has no zero on the positive real line. By analytic continuation [36,Theorem 5b,p.58], the two integrals on both sides of the above equation are well defined for all μ>0. However, by rewriting the above equation as

    0=eμξI(ξ)[f(μ,c)+˜βI(ξ)S(ξ)+I(ξ)]dξ,

    we note from f(μ,c) as μ that the integrand is always negative for large μ, a contradiction. Thus, we require f(μ,c) to have at least one positive zero such that the analytic continuation fails to extend beyond this zero. This means that cc is a necessary conditions for the existence of positive traveling wave solutions. We then state the properties of traveling wave solutions in the following theorem.

    Theorem 10. If (S,I) is a positive and uniformly bounded solution pair of (1.6)-(1.7) with boundary conditions (1.8), then we have θβ>γ, cc, and S()=I()=0. The following identities are satisfied:

    γθI(ξ)dξ=φ(S(ξ),I(ξ))dξ=cS0.

    Moreover, S(ξ)<0 for all ξR.

    We classify the delayed epidemic models into two types. The first type impose delays only on the I-equation, while the second type assumes delayed infective terms in both equations for S and I. Though these two types are mathematically equivalent in the case of one discrete delay, they are different when general distributed delay is taken into consideration. We note that the Susceptible-Infected-Recovered model with type-Ⅰ distributed delay can be reduced to the classical Susceptible-Exposed-Infected-Recovered model with no delay by assuming that the probability density function takes a special form p(τ)=σeστ. It is thus reasonable to study the model systems with type-Ⅰ delay. However, not much has been done even for the non-diffusive model. On the contrast, there is a rich literature of works on the type-Ⅱ delayed non-diffusive epidemic models; see for example, [37,13,38]. To establish global stability results for the model systems with type-Ⅱ delays, one should take advantage of the fact that the differential equation for S+I is simple and has no delayed terms. However, this phenomenon disappears in the model system with type-Ⅰ delay and thus poses a challenge in the global analysis of model dynamics [39].

    In this paper, we consider a diffusive epidemic model with type-Ⅰ distributed delay. It is noted that when the probability density function for the distributed delay takes the special form of exponential function, then a standard application of linear chain trick reduces our model system to a diffusive delay-free model with an exposed compartment which does not diffuse. We also consider the degenerate cases when the diffusion coefficient of either susceptible individuals or infected individuals vanishes. We prove linear determinacy of our model system by establishing existence theory of positive traveling wave solutions. To be more specific, we calculate a critical value from the linearized I-equation and verify that this value is the sharp lower bound for the speeds of traveling wave solutions. Sensitivity analysis indicates that the critical wave speed is increasing in the diffusion coefficient but decreasing in time delay. Biological interpretation of this result is that random movement of infected individuals will enforce disease propagation, while time delay during transmission mechanism will inhibit spatial spread of infectious diseases.

    In Lemma 8, we also provide a novel and elementary proof of a conjecture proposed in [40] that a positive traveling wave can only connect a nontrivial equilibrium with a trivial equilibrium. We should mention that this conjecture was first proved in [26] by using comparison principle and global stability result for quasi-monotone reaction-diffusion equations on an unbounded domain. In comparison, our proof is simpler and more natural, and it provides a new idea of understanding nonlocal elliptic differential equations/inequalities.

    An open problem for our model system, and for many other diffusive epidemic models, is the existence of a positive traveling wave solution for the critical wave speed c=c. Due to the lack of monotonicity, the traditional limiting argument for the monotone systems fails. It is exciting to see some recent achievements in [41,42] for existence proofs of weak traveling wave solutions with critical speed in non-cooperative diffusive systems, and in [43] for an existence result of traveling waves in a nonlocal dispersal epidemic model with critical speed. We conjecture that, for our proposed epidemic model with type-Ⅰ distributed delays, a positive traveling wave solution with critical speed should exist.

    We would like to thank the anonymous referee for careful reading and helpful suggestions which led to an improvement to our original manuscript. XMH is partially supported by International Program of Project 985 from Sun Yat-Sen University.

    The authors declare there is no conflict of interest.



    [1] W. O. Kermack and A. G. McKendrick, A contribution to the mathematical theory of epidemics, Proc. R. Soc. Lond. A, 115 (1927), 700–721.
    [2] F. J. Richards, A flexible growth function for empirical use, J. Experi. Botany, 10 (1959), 290–300.
    [3] M. E. Gilpin and F. J. Ayala, Global models of growth and competition, Proc. Natl. Acad. Sci., 70 (1973), 3590–3593.
    [4] J. V. Ross, A note on density dependence in population models, Ecol. Model., 220 (2009), 3472–3474.
    [5] Y. H. Hsieh and Y. S. Cheng, Real-time forecast of multiphase outbreak, Emerg. Infect. Dis., 12 (2006), 122–127.
    [6] Y.H. Hsieh and C.W.S. Chen, Turning points, reproduction number, and impact of climatological events for multi-wave dengue outbreaks, Trop. Med. Int. Health, 14 (2009), 628–638.
    [7] Y.H. Hsieh, Pandemic influenza A (H1N1) during winter influenza season in the southern hemisphere, Influenza Other Respir. Viruses, 4 (2010), 187–197.
    [8] X.S.Wang, J.Wu and Y. Yang, Richards model revisited: validation by and application to infection dynamics, J. Theoret. Biol., 313 (2012), 12–19.
    [9] F. Brauer, Some simple epidemic models, Math. Biosci. Eng., 3 (2006), 1–15.
    [10] C.M. Kribs-Zaleta, To switch or taper off: The dynamics of saturation, Math. Biosc., 192 (2004), 137–152.
    [11] O. Diekmann, M. C. M. de Jong and A. A. de Koeijer, et al., The force of infection in populations of varying size: a modelling problem, J. Biol. Systems, 3 (1995), 519–529.
    [12] Y. Kuang, Delay differential equations with applications in population dynamics, Math. Sci. Eng., 191. Academic Press, Inc., Boston, MA, 1993.
    [13] G. Huang and Y. Takeuchi, Global analysis on delay epidemiological dynamics models with nonlinear incidence, J. Math. Biol., 63 (2011), 125–139.
    [14] J. D. Murray, E. A. Stanley and D. L. Brown, On the spatial spread of rabies among foxes, Proc. Roy. Soc. London Ser. B, 229 (1986), 111–150.
    [15] J. Al-Omari and S. A. Gourley, Monotone travelling fronts in an age-structured reaction-diffusion model of a single species, J. Math. Biol., 45 (2002), 294–312.
    [16] O. Diekmann, Thresholds and traveling waves for the geographical spread of infection, J. Math. Biol., 6 (1978), 109–130.
    [17] T. Faria and S. Trofimchuk, Nonmonotone travelling waves in a single species reaction-diffusion equation with delay, J. Differ. Equations, 228 (2006), 357–376.
    [18] X. Liang and X.Q. Zhao, Asymptotic speeds of spread and traveling waves for monotone semiflows with applications, Comm. Pure Appl. Math., 60 (2007), 1–40.
    [19] H. Thieme and X. Q. Zhao, Asymptotic speeds of spread and traveling waves for integral equations and delayed reaction-diffusion models, J. Differ. Equations, 195 (2003), 430–470.
    [20] A. Källén, Thresholds and travelling waves in an epidemic model for rabies, Nonlinear Anal., 8 (1984), 851–856.
    [21] Y. Hosono and B. Ilyas, Existence of traveling waves with any positive speed for a diffusive epidemic model, Nonlinear World, 1 (1994), 277–290.
    [22] Y. Hosono and B. Ilyas, Traveling waves for a simple diffusive epidemic model, Math. Models Methods Appl. Sci., 5 (1995), 935–966.
    [23] S. R. Dunbar, Travelling wave solutions of diffusive Lotka-Volterra equations, J. Math. Biol., 17 (1983), 11–32.
    [24] S. R. Dunbar, Traveling wave solutions of diffusive Lotka-Volterra equations: A heteroclinic connection in R4, Trans. Amer. Math. Soc., 286 (1984), 557–594.
    [25] W. Huang, A geometric approach in the study of traveling waves for some classes of nonmonotone reaction-diffusion systems, J. Differ. Equations, 260 (2016), 2190–2224.
    [26] W. T. Li, G. Lin, C. Ma and F. Y. Yang, Traveling wave solutions of a nonlocal delayed SIR model without outbreak threshold, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 467–484.
    [27] Z. C. Wang and J. Wu, Travelling waves of a diffusive Kermack-McKendrick epidemic model with non-local delayed transmission, Proc. R. Soc. Lond. Ser. A, 466 (2010), 237–261.
    [28] K. Zhou, M. Han and Q. Wang, Traveling wave solutions for a delayed diffusive SIR epidemic model with nonlinear incidence rate and external supplies, Math. Methods Appl. Sci., 40 (2017), 2772–2783.
    [29] J. Li and X. Zou, Modeling spatial spread of infectious diseases with a fixed latent period in a spatially continuous domain, Bull. Math. Biol., 71 (2009), 2048–2079.
    [30] Z. Xu, Wave propagation in an infectious disease model, J. Math. Anal. Appl., 449 (2017), 853–871.
    [31] M. A. Lewis, B. Li and H. F.Weinberger, Spreading speed and linear determinacy for two-species competition models, J. Math. Biol., 45 (2002), 219–233.
    [32] H. F.Weinberger, M. A. Lewis and B. Li, Analysis of linear determinacy for spread in cooperative models, J. Math. Biol., 45 (2002), 183–218.
    [33] H. F. Weinberger, M. A. Lewis and B. Li, Anomalous spreading speeds of cooperative recursion systems, J. Math. Biol., 55 (2007), 207–222.
    [34] H. Wang, Spreading speeds and traveling waves for non-cooperative reaction-diffusion systems, J. Nonlinear Sci., 21 (2011), 747–783.
    [35] H. Wang and X.S. Wang, Traveling wave phenomena in a Kermack-McKendrick SIR model, J. Dynam. Differ. Equations, 28 (2016), 143–166.
    [36] D. V. Widder, The Laplace transform, Princeton University Press, Princeton, N. J., 1941.
    [37] Y. Enatsu, Y. Nakata and Y. Muroya, Lyapunov functional techniques for the global stability analysis of a delayed SIRS epidemic model, Nonlinear Anal. Real World Appl., 13 (2012), 2120– 2133.
    [38] C. C. McCluskey, Complete global stability for an SIR epidemic model with delay-distributed or discrete, Nonlinear Anal. Real World Appl., 11 (2010), 55–59.
    [39] C. Paulhus and X.S. Wang, Global stability analysis of a delayed susceptible-infected-susceptible epidemic model, J. Biol. Dyn., 9 (2015), S45–S50.
    [40] X. S. Wang, H. Wang and J. Wu, Traveling waves of diffusive predator-prey systems: disease outbreak propagation, Discrete Contin. Dyn. Syst., 32 (2012), 3303–3324.
    [41] T. Zhang, Minimal wave speed for a class of non-cooperative reaction-diffusion systems of three equations, J. Differ. Equations, 262 (2017), 4724–4770.
    [42] T. Zhang, W. Wang and K. Wang, Minimal wave speed for a class of non-cooperative diffusionreaction system, J. Differ. Equations, 260 (2016), 2763–2791.
    [43] F. Y. Yang and W. T. Li, Traveling waves in a nonlocal dispersal SIR model with critical wave speed, J. Math. Anal. Appl., 458 (2018), 1131–1146.
  • Reader Comments
  • © 2019 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(4307) PDF downloads(462) Cited by(0)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog