# Traveling Wave Solutions for a Delayed SIRS Infectious Disease Model with Nonlocal Diffusion and Nonlinear Incidence

Authors: Xiaohong Tian; Rui Xu
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

A delayed SIRS infectious disease model with nonlocal diffusion and nonlinear incidence is investigated. By constructing a pair of upper-lower solutions and using Schauder's fixed point theorem, we derive the existence of a traveling wave solution connecting the disease-free steady state and the endemic steady state.


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## Body

## 1. Introduction

Mathematical modeling has been proven to be valuable in studying the transmission dynamics of infectious diseases in a host population. We note that in disease progression, the spatial content of the environment plays a crucial role; the spread of germs, bacteria, and pathogen in the area is the main reason which leads to the spread of infectious disease. Thus, due to the large mobility of people within a country or even worldwide, spatially uniform models are not sufficient to give a realistic picture of a disease’s diffusion. Considering the spatial effects, Gan et al. [1] considered the following SIRS epidemic model with spatial diffusion and time delay:
(1)∂S∂t=DS∂2S∂x2+A-dS(x,t)-βS(x,t)I(x,t-τ)+δR(x,t),∂I∂t=DI∂2I∂x2+βS(x,t)I(x,t-τ)-(d+γ+a)I(x,t),∂R∂t=DR∂2R∂x2+γI(x,t)-(d+δ)R(x,t),
where S(t) represents the number of individuals who are susceptible to the disease, I(t) represents the number of infected individuals who are infectious and are able to spread the disease by contact with susceptible individuals, and R(t) represents the number of individuals who have been removed from the possibility of infection through full immunity. The parameters A, a, d, β, γ, δ are positive constants in which A is the recruitment rate of the population, a is the death rate due to disease, d is the natural death rate of the population, β is the transmission rate, γ is the recovery rate of the infective individuals, and δ is the rate at which recovered individuals lose immunity and return to the susceptible class. τ>0 is a fixed time during which the infectious agents develop in the vector and it is only after that time that the infected vector can infect a susceptible human. DS, DI, and DR denote the corresponding diffusion rates for the susceptible, infected, and removed populations, respectively. In [1], by constructing a pair of upper-lower solutions, the existence of a traveling wave solution connecting the disease-free steady state and the endemic steady state was given. In recent years, there has been a fair amount of work on epidemiological models with spatial diffusion (see, e.g., [2–6]).In system (1), the Laplacian operator ∂2/∂x2 has been used to model the diffusion of the species, which suggests that the population at the location x can only be influenced by the variation of the population near the location x. However, in dynamics of infectious diseases, dispersal is better described as a long range process rather than as a local one. At the same time, studies of disease infections have also shown that reaction-diffusion equation does not accurately describe the spatial and temporal behavior of some diseases, for example, in the incubation period of SARS patients, who can move freely and the movement may transmit the disease to other people. Since the long range effect is taken into account, nonlocal diffusion equations have received great interest and have been recently intensively studied to analyze the long range effects of the dispersal (see, e.g., [7–12]). A basic nonlocal diffusion equation is of the form [13]
(2)∂∂tu(x,t)=J*u-u(x,t)+f(u),
where the kernel J of the convolution (J*u)(x,t)=∫ℝ‍J(x-y)u(y,t)dy is a nonnegative function of mass one and f a given nonlinearity. As stated in [9], if u(x,t) represents the density of a species at the point x and time t and J(x-y) is regarded as the probability distribution of jumping from location y to location x, then ∫ℝ‍J(x-y)u(y,t)dy is the rate at which individuals arrive at position x from all other places and -u(x,t)=-∫ℝ‍J(x-y)u(y,t)dy is the rate at which they leave location x to travel to all other sites. The diffusion is modeled by a convolution operator which looks to be biologically reasonable.We note that in system (1), Gan et al. used a bilinear incidence rate βSI based on the law of mass action. If the number of susceptible individuals is very large, it is unreasonable to consider the bilinear incidence within a certain limited time, because the number of effective contacts between infective individuals and susceptible individuals may saturate at high infective levels due to crowding of infective individuals or due to the protection measures by the susceptible individuals. After a study of the cholera epidemic spread in Bari in 1973, Capasso and Serio [14] introduced a saturated incidence rate g(I)S into epidemic models, where g(I) tends to a saturation level when I gets large; that is, g(I)=βI/(1+αI); here βI measures the force of infection of the disease, and 1/(1+αI) measures the inhibition effect from the behavioral change of the susceptible individuals when their number increases or from the crowding effect of the susceptible individuals. This incidence rate seems more reasonable than the bilinear incidence rate, because it includes the behavioral change and crowding effect of the infective individuals and prevents the unboundedness of the contact rate by choosing suitable parameters [15].Motivated by the works of Capasso and Serio [14], Gan et al. [1], and Li et al. [13], in this paper, we study the following delayed SIRS infectious disease model with nonlocal diffusion:
(3)∂S∂t=D[(J*S)(x,t)-S(x,t)]+A-dS(x,t)-βS(x,t)I(x,t-τ)1+αI(x,t-τ)+δR(x,t),∂I∂t=D[(J*I)(x,t)-I(x,t)]+βS(x,t)I(x,t-τ)1+αI(x,t-τ)-(d+γ+a)I(x,t),∂R∂t=D[(J*R)(x,t)-R(x,t)]+γI(x,t)-(d+δ)R(x,t),
where the parameter D denotes the corresponding diffusion rates for the three populations, respectively. Here, for simplicity, we assume DS=DI=DR=D. J(z) is a kernel function which is continuous satisfying(A1)
∫ℝ‍J(x)dx=1, J(x)≥0 and J(x)=J(-x), for x∈ℝ. For any fixed μ>0, Jμ:=∫-∞+∞‍J(x)eμ|x|dx<∞ and
(4)lim⁡μ→∞∫-∞+∞‍J(x)eμ|x|dx=∞.The initial conditions for system (3) take the form
(5)S(x,t)=ρ1(x,t),I(x,t)=ρ2(x,t),R(x,t)=ρ3(x,t),lllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllt∈[-τ,0].In the biological context, it is important to analyse the epidemic waves which are described by traveling wave solutions propagating with a certain speed. In this paper, our focus is on the existence of traveling wave solutions to the SIRS infectious disease model (3).The rest of this paper is organized as follows. In Section2, by constructing a pair of upper-lower solutions and using Schauder’s fixed point theorem, the existence of traveling wave solutions connecting the disease-free steady state and the endemic steady state of system (3) is established. In Section 3, a brief concluding remark is given to end this work.


## 2. Existence of Traveling Waves

In this section, we apply Schauder’s fixed point theorem, the method of cross-iteration scheme associated with upper-lower solutions, to study the existence of traveling wave solutions of system (3) connecting the disease-free steady state and the endemic steady state.Denote(6)ℛ0=Aβd(d+γ+a).ℛ0 is called the basic reproduction ratio of system (3), which describes the average number of newly infected cells generated from one infected cell at the beginning of the infectious process. This quantity determines the thresholds for disease transmissions. It is easy to show that system (3) always has a disease-free steady state E0(A/d,0,0). Further, if ℛ0>1, system (3) has a unique endemic steady state E*(S*,I*,R*), where
(7)S*=(d+γ+a)(1+αI*)β,I*=d(d+δ)(d+γ+a)(ℛ0-1)(αd+β)(d+δ)(d+γ+a)-βδγ,R*=γd+δI*.DenotingN=S+I+R, then system (3) is equivalent to the following system:
(8)∂N∂t=D[(J*N)(x,t)-N(x,t)]+A-dN(x,t)-aI(x,t),∂I∂t=D[(J*I)(x,t)-I(x,t)]+β[N(x,t)-I(x,t)-R(x,t)]I(x,t-τ)1+αI(x,t-τ)-(d+γ+a)I(x,t),∂R∂t=D[(J*R)(x,t)-R(x,t)]+γI(x,t)-(d+δ)R(x,t).By making a change of variablesN~=A/d-N, I~=I, R~=R and dropping the tildes, system (8) becomes
(9)∂N∂t=D[(J*N)(x,t)-N(x,t)]-dN(x,t)+aI(x,t),∂I∂t=D[(J*I)(x,t)-I(x,t)]+β[A/d-N(x,t)-I(x,t)-R(x,t)]I(x,t-τ)1+αI(x,t-τ)-(d+γ+a)I(x,t),∂R∂t=D[(J*R)(x,t)-R(x,t)]+γI(x,t)-(d+δ)R(x,t).
It is easy to show that if ℛ0>1, system (9) has two steady states (0,0,0) and (k1,k2,k3), where k1=A/d-S*-I*-R* and k2=I*, k3=R*.A traveling wave solution of (9) is a special translation invariant solution of the form (N(x,t),I(x,t),R(x,t))=(ϕ(x+ct),φ(x+ct),ψ(x+ct)), where (ϕ,φ,ψ)∈C(ℝ,ℝ3) is the profile of the wave that propagates through one-dimensional spatial domain at a constant speed c>0. On substituting N(x,t)=ϕ(x+ct), I(x,t)=φ(x+ct), R(x,t)=ψ(x+ct) into (9) and denoting the traveling wave coordinate x+ct still by t, we derive from (9) that
(10)cϕ′(t)=D∫R‍J(t-y)ϕ(y)dy-Dϕ(t)+fc1(ϕt,φt,ψt),cφ′(t)=D∫R‍J(t-y)φ(y)dy-Dϕ(t)+fc2(ϕt,φt,ψt),cψ′(t)=D∫R‍J(t-y)ψ(y)dy-Dψ(t)+fc3(ϕt,φt,ψt),
where
(11)fc1(ϕt,φt,ψt)=-dϕ(t)+aφ(t),fc2(ϕt,φt,ψt)=β[A/d-ϕ(t)-φ(t)-ψ(t)]φ(t-cτ)1+αφ(t-cτ)-(d+γ+a)φ(t),fc3(ϕt,φt,ψt)=γφ(t)-(d+δ)ψ(t).
Equation (10) will be solved subject to the following boundary value conditions:
(12)lim⁡t→-∞(ϕ(t),φ(t),ψ(t))=(0,0,0),lim⁡t→+∞(ϕ(t),φ(t),ψ(t))=(k1,k2,k3).Now, we give the definition of upper and lower solutions of system (10) as follows.Definition 1.
A pair of continuous functionsΦ¯=(ϕ¯,φ¯,ψ¯) and Φ_=(ϕ_,φ_,ψ_) are called a pair of upper-lower solutions of system (10), if there exist constants Ti(i=1,2,…,m) such that Φ¯ and Φ_ are twice differential on ℝ∖{Ti:i=1,2,…,m} and satisfy
(13)D∫R‍J(t-y)ϕ¯(y)dy-Dϕ¯(t)-cϕ¯′(t)+fc1(ϕ¯t,φ¯t,ψ¯t)≤0,D∫R‍J(t-y)φ¯(y)dy-Dφ¯(t)-cφ¯′(t)+fc2(ϕ_t,φ¯t,ψ_t)≤0,D∫R‍J(t-y)ψ¯(y)dy-Dψ¯(t)-cψ¯′(t)+fc3(ϕ¯t,φ¯t,ψ¯t)≤0,D∫R‍J(t-y)ϕ_(y)dy-Dϕ_(t)-cϕ_′(t)+fc1(ϕ_t,φ_t,ψ_t)≥0,D∫R‍J(t-y)φ_(y)dy-Dφ_(t)-cφ_′(t)+fc2(ϕ¯t,φ_t,ψ¯t)≥0,D∫R‍J(t-y)ψ_(y)dy-Dψ_(t)-cψ_′(t)+fc3(ϕ_t,φ_t,ψ_t)≥0,fort∈ℝ∖{Ti:i=1,2,…,m}.In what follows, we assume that there exist an upper solutionΦ¯(t)=(ϕ¯,φ¯,ψ¯)(t) and a lower solution Φ_(t)=(ϕ_,φ_,ψ_)(t) of system (10) satisfying (P1)-(P2):(P1)
0≤Φ_≤Φ¯≤M=(M1,M2,M3);(P2)
lim⁡t→-∞Φ¯(t)=0, lim⁡t→+∞Φ_(t)=lim⁡t→+∞Φ¯(t)=K=(k1,k2,k3).Let(14)C[0,M](ℝ,ℝ3)={(ℝ,ℝ3)Φ(t)=(ϕ,φ,ψ)(t)∈C(ℝ,ℝ3):0≤Φ(t)≤M,t∈ℝ(ℝ,ℝ3)(ℝ,ℝ3)},
where Mi>ki(i=1,2,3) satisfy
(15)Aβd-(γ+a)>aM2M1>d,Aβd-(γ+a)>γM2M3-δ>d,Ad≥M1+M2+M3.
We look for traveling wave solutions to system (10) in the following profile set:
(16)Γ={(ϕ,φ,ψ)(t)∈C[0,M](ℝ,ℝ3):(ϕ_,φ_,ψ_)(t)≤(ϕ,φ,ψ)(t)≤(ϕ¯,φ¯,ψ¯)(t)}.
Obviously, Γ is nonempty, convex, closed, and bounded.Furthermore, corresponding to (10), we make the following hypotheses.(A2)
There exist three positive constantsLi>0(i=1,2,3) such that
(17)|fci(ϕ1,φ1,ψ1)-fci(ϕ2,φ2,ψ2)|≤Li∥Φ(t)-Ψ(t)∥,forΦ(t)=(ϕ1,φ1,ψ1)(t) and Ψ(t)=(ϕ2,φ2,ψ2)(t)∈C([-τ,0],ℝ3) with (0,0,0)≤(ϕj(t),φj(t),ψj(t))≤(M1,M2,M3), j=1,2, t∈[-τ,0], Mi≥ki, are positive constants.ForΦ=(ϕ,φ,ψ)∈C[0,M](ℝ,ℝ3), we define two operators H=(H1,H2,H3) and F=(F1,F2,F3) from C[0,M](ℝ,ℝ3) to C(ℝ,ℝ3) by
(18)H1(ϕ,φ,ψ)(t)=D∫R‍J(t-y)ϕ(y)dy+aφ(t),H2(ϕ,φ,ψ)(t)=D∫R‍J(t-y)φ(y)dy+βM2φ(t)+β[A/d-ϕ(t)-φ(t)-ψ(t)]φ(t-cτ)1+αφ(t-cτ),H3(ϕ,φ,ψ)(t)=D∫R‍J(t-y)ψ(y)dy+γφ(t),Fi(ϕ,φ,ψ)(t)=1ce-(βi/c)t∫-∞t‍e(βi/c)sHi(ϕ,φ,ψ)(s)dsllllllllllllllllllllllllllllllllllllllllllllll(i=1,2,3).
Letting β1=D+d,β2=D+d+γ+a, and β3=D+d+δ, then system (10) can be rewritten as
(19)cϕ′(t)=-β1ϕ(t)+H1(ϕ,φ,ψ)(t),cφ′(t)=-β2φ(t)+H2(ϕ,φ,ψ)(t),cψ′(t)=-β3ψ(t)+H3(ϕ,φ,ψ)(t),
and then F is well defined such that
(20)cFi′(ϕ,φ,ψ)(t)=-βiFi(ϕ,φ,ψ)(t)+Hi(ϕ,φ,ψ)(t)llllllllllllllllllllllllllllllllllll(i=1,2,3).
Hence, a fixed point of F is a solution of (10), which is a traveling wave solution of (9) connecting 0=(0,0,0) with K=(k1,k2,k3) if it satisfies (P2).In the following, we introduce some lemmas to support our main results.Forμ>0, define
(21)Bμ(ℝ,ℝ3)={sup⁡t∈ℝ(ϕ,φ,ψ)∈C(ℝ,ℝ3):sup⁡t∈ℝ|(ϕ,φ,ψ)(t)|e-μ|t|<∞},|(ϕ,φ,ψ)(t)|μ=sup⁡t∈ℝ|(ϕ,φ,ψ)(t)|e-μ|t|.
Then it is easy to check that (Bμ(ℝ,ℝ3),|·|) is a Banach space.In view of the definition ofH and F, we can easily see that they admit the following properties.Lemma 2.
Let(A1) hold. One has
(i)(22)H1(ϕ,φ,ψ)(t)≥0,H3(ϕ,φ,ψ)(t)≥0,H1(ϕ2,φ2,ψ2)(t)≤H1(ϕ1,φ1,ψ1)(t),H2(ϕ1,φ1,ψ1)(t)≤H2(ϕ2,φ1,ψ1)(t),H2(ϕ1,φ2,ψ1)(t)≤H2(ϕ1,φ1,ψ1)(t),H2(ϕ1,φ1,ψ1)(t)≤H2(ϕ1,φ1,ψ2)(t),H3(ϕ2,φ2,ψ2)(t)≤H3(ϕ1,φ1,ψ1)(t),
(ii)(23)F1(ϕ2,φ2,ψ2)(t)≤F1(ϕ1,φ1,ψ1)(t),F2(ϕ1,φ1,ψ1)(t)≤F2(ϕ2,φ1,ψ1)(t),F2(ϕ1,φ2,ψ1)(t)≤F2(ϕ1,φ1,ψ1)(t),F2(ϕ1,φ1,ψ1)(t)≤F2(ϕ1,φ1,ψ2)(t),F3(ϕ2,φ2,ψ2)(t)≤F3(ϕ1,φ1,ψ1)(t),
for t∈ℝ with 0≤ϕ2(t)≤ϕ1(t)≤M1, 0≤φ2(t)≤φ1(t)≤M2, 0≤ψ2(t)≤ψ1(t)≤M3.By using a similar argument as in the proof of Lemmas 3.3–3.6 in [16], one can show the following lemmas.Lemma 3.
Assume that (A2) holds.F=(F1,F2,F3) is continuous with respect to the norm |·|μ in Bμ(ℝ,ℝ3).Lemma 4.
F
(
Γ
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ϕ
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ψ
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,
φ
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ψ
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)
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)
⊂
Γ
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ϕ
_
,
φ
_
,
ψ
_
)
,
(
ϕ
¯
,
φ
¯
,
ψ
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)
), where F=(F1,F2,F3).Lemma 5.
F
:
Γ
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ϕ
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φ
_
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ψ
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φ
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ψ
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→
Γ
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ϕ
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,
φ
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ψ
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)
,
(
ϕ
¯
,
φ
¯
,
ψ
¯
)
) is compact.We now consider the following equations:(24)Δ1(λ,c):=D∫ℝ‍J(y)e-λydy-D-cλ-d+aM2M1,Δ2(λ,c):=D∫ℝ‍J(y)e-λydy-D-cλ+Aβd-(d+γ+a),Δ3(λ,c):=D∫ℝ‍J(y)e-λydy-D-cλ-(d+δ)+γM2M3.
Since (A1) and (15) hold, direct calculations show that
(25)Δ1(0,c)=-d+aM2M1>0,Δ1(λ,+∞)=-∞,lllllllllllllllllllllllllllllllllllllllllllllllll∀λ>0;∂Δ1(λ,c)∂c=-λ<0,∀λ>0;∂2Δ1(λ,c)∂λ2=D∫ℝ‍y2J(y)e-λydy>0.
Therefore, we obtain that there exist c1*>0 and λ1*>0 such that ∂Δ1(λ,c)/∂λ|(λ1*,c1*)=0 and Δ1(λ1*,c1*)=0. Further, if c>c1*, there exist λ1(c)>0 and λ2(c)>0 satisfying
(26)0<λ1(c)<λ1*<λ2(c).
Similarly, we can show that there exist ci*,λi*>0 such that Δi(λi*,ci*)=0. If c>ci*, there exist λj(c)>0 satisfying 0<λ3(c)<λ2*<λ4(c) and 0<λ5(c)<λ3*<λ6(c)(i=2,3;j=3,4,5,6).Lemma 6.
Letc*=max⁡{c1*,c2*,c3*}. Assume that ℛ0>1; then one has λ1(c)<λ3(c) and λ5(c)<λ3(c).Proof.
Define(27)h(λ)=D∫ℝ‍J(y)e-λydy,g1(λ)=cλ+D+d-aM2M1,g2(λ)=cλ+D+d+γ+a-Aβd,g3(λ)=cλ+D+d+δ-γM2M3.
It is easy to show that
(28)h(λ1)=g1(λ1),h(λ3)=g2(λ3),h(λ5)=g3(λ5).
If ℛ0>1, then by (9), we see that g2(0)<g1(0) and g2(0)<g3(0). Note that h′′(λ)>0 for all λ; hence, we have λ1(c)<λ3(c) and λ5(c)<λ3(c).
Suppose thatℛ0>1 and a/d+γ/(d+δ)<1; we can choose εi>0(i=1,2,…,6), and ε1,ε2∈(0,k1), ε3,ε4∈(0,k2), ε5,ε6∈(0,k3) satisfying
(29)a(k2+ε3)-d(k1+ε1)<0,β(Ad-k1+ε2-k2-ε3-k3+ε6)<(d+γ+a)(1+α(k2+ε3)),γ(k2+ε3)-(d+δ)(k3+ε5)<0,-d(k1-ε2)+a(k2-ε4)>0,β(Ad-k1-ε1-k2+ε4-k3-ε5)>(d+γ+a)(1+α(k2-ε4)),γ(k2-ε4)-(d+δ)(k3-ε6)>0.
In fact, noting thatdk1=ak2, for ɛ3,ɛ4∈(0,k2), there exist ɛ1,ɛ2∈(0,k1) and ɛ5,ɛ6∈(0,k3) such that
(30)k1>ε1>adε3=adε3+adk2-k1,k1>ɛ2>adɛ4=adɛ4+k1-adk2,
which yield
(31)a(k2+ε3)-d(k1+ε1)<0,-d(k1-ε2)+a(k2-ε4)>0.
Sinceγk2=(d+δ)k3, for ɛ3,ɛ4∈(0,k2), we can find ɛ1,ɛ2∈(0,k1) and ɛ5,ɛ6∈(0,k3) such that
(32)k3>ε5>γd+δε3=γd+δε3+γd+δk2-k3⟹γ(k2+ε3)-(d+δ)(k3+ε5)<0,k3>ɛ6>γd+δɛ4=γd+δɛ4+k3-γd+δk2⟹γ(k2-ε4)-(d+δ)(k3-ε6)>0.
Ifa/d+γ/(d+δ)<1, then we can choose suitable values of ɛ3,ɛ4∈(0,k2) such that
(33)(ad+γd+δ)ε3<ε4,(ad+γd+δ)ε4<ε3.
Furthermore, we can choose ɛ1,ɛ2∈(0,k1), ɛ3,ɛ4∈(0,k2), ɛ5,ɛ6∈(0,k3) satisfying
(34)ɛ2>adɛ4,ɛ6>γd+δɛ4,adɛ4+γd+δɛ4<ɛ3,ɛ1>adɛ3,ɛ5>γd+δɛ3,adɛ3+γd+δɛ3<ɛ4.
Accordingly, there exist suitable constants ɛi>0(i=1,…,6) such that
(35)ɛ2+ɛ6-ɛ3<0,ɛ1+ɛ5-ɛ4<0.
By the second equation of system (10), we have β(A/d-k1-k2-k3)-(d+γ+a)(1+αk2)=0. It then follows from (35) that
(36)β(Ad-k1+ε2-k2-ε3-k3+ε6)<(d+γ+a)(1+α(k2+ε3)),β(Ad-k1-ε1-k2+ε4-k3-ε5)>(d+γ+a)(1+α(k2-ε4)).
Now, we define the continuous functionsΦ¯(t)=(ϕ¯(t),φ¯(t),ψ¯(t)) and Φ_(t)=(ϕ_(t),φ_(t),ψ_(t)) as follows:
(37)ϕ¯(t)={k1eλ1t,t≤t1,k1+ε1e-λt,t>t1,ϕ_(t)={0,t≤t2,k1-ε2e-λt,t>t2,φ¯(t)={lk2eλ3t,t≤t3,k2+ε3e-λt,t>t3,φ_(t)={0,t≤t4,k2-ε4e-λt,t>t4,ψ¯(t)={k3eλ5t,t≤t5,k3+ε5e-λt,t>t5,ψ_(t)={0,t≤t6,k3-ε6e-λt,t>t6,
where t1,t3,t5>0, t2,t4,t6<0 and λ>0 is a constant sufficiently small to be chosen later. Then we can choose λ>0 to be sufficiently small such that t1>0, t3>0, et5>0 satisfying
(38)k1+ε1>M1=sup⁡t∈ℝϕ¯(t)=k1eλ1t1>k1,k2+ε3>M2=sup⁡t∈ℝφ¯(t)=k2eλ3t3>k2,k3+ε5>M3=sup⁡t∈ℝψ¯(t)=k3eλ5t5>k3,
where M1,M2,M3 are defined in (15). Furthermore, we can choose l∈(0,1) such that t3≥max⁡{t1,t5}. If a/d+γ/(δ+d)<1, it is easy to show that t4≤min⁡{t2,t6}. Clearly, Φ¯(t) and Φ_(t) satisfy (P1) and (P2).Lemma 7.
Φ
¯
(
t
)
=
(
ϕ
¯
(
t
)
,
φ
¯
(
t
)
,
ψ
¯
(
t
)
) is an upper solution of system (10).Proof.
Denote(39)P1(t)=D∫ℝ‍J(t-y)ϕ¯(y)dy-Dϕ¯(t)-cϕ¯′(t)-dϕ¯(t)+aφ¯(t),P2(t)=D∫ℝ‍J(t-y)φ¯(y)dy-Dφ¯(t)-cφ¯′(t)+β[A/d-ϕ_(t)-φ¯(t)-ψ_(t)]φ¯(t-cτ)1+αφ¯(t-cτ)-(d+γ+a)φ¯(t),P3(t)=D∫ℝ‍J(t-y)ψ¯(y)dy-Dψ¯(t)-cψ¯′(t)+γφ¯(t)-(d+δ)ψ¯(t).
Ift≤t1, ϕ¯(t)=k1eλ1t and φ¯(t)=lk2eλ3t. By Lemma 6, it follows that
(40)P1(t)=k1eλ1t[D∫ℝ‍J(y)e-λ1ydy-D-cλ1-d+alk2k1e(λ3-λ1)t∫]≤k1eλ1t[D∫ℝ‍J(y)e-λ1ydy-D-cλ1-d+aM2M1∫]≤k1eλ1tΔ1(λ1,c)=0.
Ift1<t≤t3, ϕ¯(t)=k1+ε1e-λt and φ¯(t)=lk2eλ3t. Then, we have
(41)P1(t)=[Dε1∫ℝ‍J(y)eλydy-Dε1+cλε1]e-λt-d(k1+ε1e-λt)+alk2eλ3t.
Note that l∈(0,1) and ak2/d=k1. Hence, for λ sufficiently small, there exists λ1*>0 such that P1(t)<0 for all λ∈(0,λ1*).
Ift>t3, ϕ¯(t)=k1+ε1e-λt and φ¯(t)=k2+ε3e-λt. We obtain that
(42)P1(t)≤[Dε1∫ℝ‍J(y)eλydy-Dε1+cλε1]e-λt-d(k1+ε1e-λt)+a(k2+ε3e-λt):=I1(λ).
For λ sufficiently small, a(k2+ε3)<d(k1+ε1) implies that I1(0)<0, and there exists λ2*>0 such that P1(t)≤I1(λ)<0 for all λ∈(0,λ2*).
Ift≤t3, φ¯(t)=lk2eλ3t and φ¯(t-cτ)=lk2eλ3(t-cτ). It follows that
(43)P2(t)≤lk2eλ3t[D∫ℝ‍J(y)e-λ3ydy-D-cλ3+Aβd-(d+γ+a)∫ℝ‍Aβd]=lk2eλ3tΔ2(λ3,c)=0.
Ift3<t≤t3+cτ, φ¯(t)=k2+ε3e-λt, φ¯(t-cτ)=lk2eλ3(t-cτ), ϕ_(t)=k1-ε2e-λt, and ψ_(t)=k3-ε6e-λt. We obtain that
(44)P2(t)=ε3e-λt[D∫ℝ‍J(y)eλydy-D+cλ]+I2(λ),
where
(45)I2(λ)=(β[Ad-(k1-ε2e-λt)-(k2+ε3e-λt)-(k3-ε6e-λt)Ad]lk2eλ3(t-cτ))×(1+αlk2eλ3(t-cτ))-1-(d+γ+a)(k2+ε3e-λτ).
Then by (29), we have
(46)I2(0)=β(A/d-k1+ε2-k2-ε3-k3+ε6)lk21+αlk2-(d+γ+a)(k2+ε3)≤d+γ+a1+αlk2((l-1)k2-ε3).
Sincel∈(0,1), for λ sufficiently small, it is easy to show that I2(0)<0 and there exists λ3*>0 such that P2(t)<0 for all λ∈(0,λ3*).
Ift>t3+cτ, φ¯(t)=k2+ε3e-λt, φ¯(t-cτ)=k2+ε3e-λ(t-cτ), ϕ_(t)=k1-ε2e-λt, and ψ_(t)=k3-ε6e-λt. It follows that
(47)P2(t)=ε3e-λt[D∫ℝ‍J(y)eλydy-D+cλ]+I3(λ),
where
(48)I3(λ)=(β[Ad-(k1-ε2e-λt)-(k2+ε3e-λt)-(k3-ε6e-λt)Ad](k2+ε3e-λ(t-cτ)))×(1+α(k2+ε3e-λ(t-cτ)))-1-(d+γ+a)(k2+ε3e-λt).
For λ sufficiently small, by (29), we see that I3(0)<0 and there exists λ4*>0 such that P2(t)≤I3(λ)<0 for all λ∈(0,λ4*).
Ift≤t5, ψ¯(t)=k3eλ5t and φ¯(t)=lk2eλ3(t). Then, by Lemma 6, we have
(49)P3(t)=k3eλ5t[γlk2eλ3tk3eλ5tD∫ℝ‍J(y)e-λ5tdy-D-cλ5-(d+δ)+γlk2eλ3tk3eλ5t]≤k3eλ5t[D∫ℝ‍J(y)e-λ5tdy-D-cλ5-(d+δ)+γM2M3∫ℝ‍J(y)]=k3eλ5tΔ3(λ5,c)=0.
Ift5<t≤t3, ψ¯(t)=k3+ε5e-λt and φ¯(t)=lk2eλ3t. We derive that
(50)P3(t)=ε3e-λt[D∫ℝ‍J(y)eλydy-D+cλ]+γlk2eλ3t-(d+δ)(k3+ε5e-λt):=I4(λ).
Note that l∈(0,1); then we have lk2eλ3t≤k2eλ3t3<k2+ε3. By (29), for λ sufficiently small, it is easy to show that I4(0)<0 and there exists λ5*>0 such that P3(t)<0 for all λ∈(0,λ5*).
Ift>t3, ψ¯(t)=k3+ε5e-λt and φ¯(t)=k2+ε3e-λt. We obtain that
(51)P3(t)≤ε5e-λt[D∫ℝ‍J(y)eλydy-D+cλ]+γ(k2+ε3e-λt)-(d+δ)(k3+ε5e-λt):=I5(λ).
For λ sufficiently small, by (29), we see that γ(k2+ε3)-(d+δ)(k3+ε5)<0 implies that I5(0)<0 and there exists λ6*>0 such that P3(t)≤I5(λ)<0 for all λ∈(0,λ6*).
Clearly, for allλ∈(0,min⁡{λi*,i=1,…,6}), Pi(t)≤0(i=1,2,3). This completes the proof.Lemma 8.
Φ
_
(
t
)
=
(
ϕ
_
(
t
)
,
φ
_
(
t
)
,
ψ
_
(
t
)
) is a lower solution of system (10).Proof.
Denote(52)Q1(t)=D∫ℝ‍J(t-y)ϕ_(y)dy-Dϕ_(t)-cϕ_′(t)-dϕ_(t)+aφ_(t),Q2(t)=D∫ℝ‍J(t-y)φ_(y)dy-Dφ_(t)-cφ_′(t)+β[A/d-ϕ¯(t)-φ_(t)-ψ¯(t)]φ_(t-cτ)1+αφ_(t-cτ)-(d+γ+a)φ_(t),Q3(t)=D∫ℝ‍J(t-y)ψ_(y)dy-Dψ_(t)-cψ_′(t)+γφ_(t)-(d+δ)ψ_(t).
Ift≤t2, ϕ_(t)=0. It is easy to see that Q1(t)=aφ_(t)≥0.
Ift>t2, ϕ_(t)=k1-ɛ2e-λt and φ_(t)=k2-ɛ4e-λt. Then, we have
(53)Q1(t)=ɛ2e-λt[-D∫ℝ‍J(y)eλydy+D-cλ]-d(k1-ɛ2e-λt)+a(k2-ɛ4e-λt).
For λ sufficiently small, -d(k1-ε2)+a(k2-ε4)>0 implies that Q1(0)>0 and there exists λ7*>0 such that Q1(t)>0 for all λ∈(0,λ7*).
Ift≤t4, φ_(t)=0 and φ_(t-cτ)=0. Hence, Q2(t)=0.
Ift>t4, φ_(t)=k2-ɛ4e-λτ. Noting that φ_(t-cτ)≥k2-ɛ4e-λt, ϕ¯(t)≤k1+ɛ1e-λτ and ψ¯(t)≤k3+ɛ5e-λt. We obtain that
(54)Q2(t)≥ɛ4e-λt[-D∫ℝ‍J(y)eλydy+D-cλ]+I6(λ),
where
(55)I6(λ)=(β[Ad-k1-ɛ1e-λt-k2+ɛ4e-λt-k3-ɛ5e-λtAd](k2-ɛ4e-λt))×(1+α(k2-ɛ4e-λt))-1-(d+γ+a)(k2-ɛ4e-λt).
By (29), we have I6(0)>0. Accordingly, for λ sufficiently small, there exists λ8*>0 such that Q2(t)≥I6(λ)>0 for all λ∈(0,λ8*).
Ift≤t6, ψ_(t)=0. Then, we have Q3(t)=γφ_(t)≥0.
Ift>t6, ψ_(t)=k3-ɛ6e-λt and φ_(t)=k2-ɛ4e-λt. We obtain that
(56)Q3(t)=ɛ6e-λt[-D∫ℝ‍J(y)eλydy+D-cλ]+γ(k2-ɛ4e-λt)-(d+δ)(k3-ɛe-λt).
For λ sufficiently small, then, by (29), it is readily seen that Q3(0)>0 and there exists λ9*>0 such that Q3(t)>0 for all λ∈(0,λ9*).
Obviously, for allλ∈(0,min⁡{λ7*,λ8*,λ9*}),Qi(t)≥0(i=1,2,3). This completes the proof.Applying Lemmas2–8 and Schauder’s fixed point theorem, we know that if ℛ0>1 and a/d+γ/(d+δ)<1, system (9) has a traveling wave solution with speed c>c* connecting the steady states (0,0,0) and (k1,k2,k3). Accordingly, we have the following conclusion.Theorem 9.
Letℛ0>1. Assume that (A1) and a/d+γ/(d+δ)<1 hold. For every c>c*, system (3) always has a traveling wave solution with speed c connecting the disease-free steady state E0(A/d,0,0) and the endemic steady state E*(S*,I*,R*).


## 3. Concluding Remark

In this paper, we have discussed a delayed SIRS infectious disease model with nonlocal diffusion and nonlinear incidence. By constructing a pair of upper-lower solutions and using Schauder’s fixed point theorem, we investigated the existence of a traveling wave solution connecting the disease-free steady stateE0 and the endemic steady state E*. We now study the influence of the nonlocal diffusion terms and time delay describing the incubation period on the spreading speed c*. From the second equation of system (3), we have a linearized equation at E0 that takes the form
(57)cI′(ξ)=D[J*I(ξ)-I(ξ)]+AβdI(ξ-cτ)-(d+γ+a)I(ξ).
Letting I(ξ)=eλξ yields the following characteristic equation:
(58)Δ(λ,c)=D∫ℝ‍J(y)[e-λy-1]dy-cλ+Aβde-λcτ-(d+γ+a).
By direct calculations we have
(59)dc*dD=∫ℝ‍J(y)[e-λ*y-1]dyλ*(1+(Aβτ/d)e-λ*c*τ)>0,dc*dτ=-Aβc*de-c*λ*τ+Aβτ<0.
It is easy to show that the spreading speed c* is monotonically increasing for the nonlocal diffusion rate D and is monotonically decreasing for the time delay τ.



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*Source: 795320-2014-04-10.xml*