# Traveling Wave Solutions in a Reaction-Diffusion Epidemic Model

Authors: Sheng Wang; Wenbin Liu; Zhengguang Guo; Weiming Wang
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

We investigate the traveling wave solutions in a reaction-diffusion epidemic model. The existence of the wave solutions is derived through monotone iteration of a pair of classical upper and lower solutions. The traveling wave solutions are shown to be unique and strictly monotonic. Furthermore, we determine the critical minimal wave speed.


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## Body

## 1. Introduction

Recently, great attention has been paid to the study of the traveling wave solutions in reaction-diffusion models [1–17]. In the sense of epidemiology, the traveling wave solutions describe the transition from a disease-free equilibrium to an endemic equilibrium; the existence and nonexistence of nontrivial traveling wave solutions indicate whether or not the disease can spread [11]. The results contribute to predicting the developing tendency of infectious diseases, to determining the key factors of the spread of infectious disease, and to seeking the optimum strategies of preventing and controlling the spread of the infectious diseases [18–21].Some methods have been used to derive the existence of traveling wave solutions in reaction-diffusion models, and the monotone iteration method has been proved to be an effective one. Such a method reduces the existence of traveling wave solutions to that of an ordered pair of upper-lower solutions [6, 7, 9, 10, 14, 15].In [22], Berezovsky and coworkers introduced a simple epidemic model through the incorporation of variable population, disease-induced mortality, and emigration into the classical model of Kermack and McKendrick [23]. The total population (N) is divided into two groups of susceptible (S) and infectious (I); that is to say, N=S+I. The model describing the relations between the state variables is
(1)dSdt=rN(1-NK)-βSIN-(μ+m)S,dIdt=βSIN-(μ+d)I,
where the reproduction of susceptible follows a logistic equation with the intrinsic growth rate r and the carrying capacity K, β denotes the contact transmission rate (the infection rate constant), μ is the natural mortality; d denotes the disease-induced mortality, and m is the per-capita emigration rate of uninfected.For model (1), the epidemic threshold, the so-called basic reproduction number R0, is then computed as R0=β/(μ+d). The disease will successfully invade when R0>1 but will die out if R0<1. R0=1 is usually a threshold whether the disease goes to extinction or goes to an endemic. Large values of R0 may indicate the possibility of a major epidemic [19]. In addition, the basic demographic reproductive number Rd is given by Rd=r/(μ+m). It can be shown that if Rd>1 the population grows, while Rd≤1 implies that the population does not survive [22].For simplicity, rescaling model (1) by letting S→S/K, I→I/K, and t→t/(μ+d) leads to the following model:
(2)dSdt=νRd(S+I)[1-(S+I)]-R0SIS+I-νS,dIdt=R0SIS+I-I,
where ν=(μ+m)/(μ+d) is defined by the ratio of the average life span of susceptibles to that of infectious.For details, we refer the reader to [20, 22].In this paper, we are interested in the existence of traveling wave solutions in the following reaction-diffusion epidemic model [20]:
(3)∂S∂t=νRd(S+I)[1-(S+I)]-R0SIS+I-νS+d∂2S∂x2,∂I∂t=R0SIS+I-I+d∂2I∂x2,S(x,0)=S0(x),I(x,0)=I0(x),
where ν,R0,Rd are all positive constants, d is the diffusion coefficient, and (x,t)∈R×R+.We are looking for the traveling wave solutions of model (3) with the following form:
(4)S(x,t)=S(ξ),I(x,t)=I(ξ),ξ=x+ct,
satisfying the following boundary value conditions:
(5)(S(-∞),I(-∞))T=E1,(S(+∞),I(+∞))T=E2,
where E1,E2 are the equilibrium points of model (3).This paper is arranged as follows. In Section2, we construct a pair of ordered upper-lower solutions of model (3) and establish the uniqueness and strict monotonicity of the traveling wave solutions.


## 2. Existence and Asymptotic Decay Rates

In this section, we will establish the existence of traveling wave solutions of model (3) by constructing a pair of ordered upper-lower solutions. The definition of the upper solution and the lower solution is standard. We assume that the inequality between two vectors throughout this paper is componentwise.Setting(6)S^=Rd-1Rd-S,I^=I,
then model (3) can be written as
(7)∂S^∂t=-νRd(Rd-1Rd-S^+I^)[1-(Rd-1Rd-S^+I^)]+R0((Rd-1)/Rd-S^)I^(Rd-1)/Rd-S^+I^+ν(Rd-1Rd-S^)+d∂2S^∂x2,∂I^∂t=R0((Rd-1)/Rd-S^)I^(Rd-1)/Rd-S^+I^-I^+d∂2I^∂x2,(S^,I^)T(-∞)=(0,0)T,(S^,I^)T(+∞)=(S^*,I^*)T.For model (3), the equilibria are E1=((Rd-1)/Rd,0) and E2=(S*,I*), where
(8)S*=νR0Rd-R0-ν+1νR02Rd,I*=(R0-1)S*,
and for model (7), the equilibria are E1^=(0,0) and E2^=(S^*,I^*), where
(9)S^*=(R0-1)(νR0Rd-νR0-ν+1)νR02Rd,I^*=(R0-1)(νR0Rd-R0-ν+1)νR02Rd.
Obviously,
(10)I^*-S^*=(ν-1)(R0-1)νR0Rd,S^*=Rd-1Rd-S*,I^*=I*.
For simplicity, we define the following functions and constants:
(11)α0=Rd-1Rd,ϕ(I)=α0I^*+(I^*-S^*)I;β0=α0(I^*)2R0-1R0(R0-ν+1);γ0=2ν(Rd-1)(I^*-S^*)-[νI^*+(R0-1)S^*];ψ(I)=νRd(I^*-S^*)2I2+γ0I^*I+β0;η0=-γ0I^*2νRd(I^*-S^*)2;φ(I)=1,I>0;φ(I)=-1,I≤0;B=I^*2[1+φ(I^*2-η0)].
And we will always assume the following hypotheses throughout the rest of this paper: [H1]
(12)R0>1,1<Rd<2R02+2R0-23R02-2R0,max{27R0(Rd-1)2Rd3,R0-1R0Rd-1}<ν<-1R0Rd-R0-1.[H2]
(13)ν≥max{R02-Rd,R03-2R02+4R0-22R02+2R0-2-(3R02-2R0)Rd},ψ(B)≤0.Then we can obtain the following.Lemma 1.
If [H1] holds, thenE2 and E2^ are endemic points of model (3) and model (7), respectively.Lemma 2.
For model (7), if [H1] holds, then E1^ is unstable, and E2^ is stable.For the sake of convenience, letx=dx~. For simplicity, we still use the variables S, I, and x instead of S^, I^, and x~, respectively, then model (7) could be rewritten as
(14)∂S∂t=-νRd(α0-S+I)[1-(α0-S+I)]+R0(α0-S)Iα0-S+I+ν(α0-S)+∂2S∂x2,∂I∂t=R0(α0-S)Iα0-S+I-I+∂2I∂x2,(S,I)T(-∞)=(0,0)T,(S,I)T(+∞)=(S^*,I^*)T.Following the definition of quasi-monotonicity [17], we can obtain the following results.Lemma 3.
Model (14) is a quasi-monotone decreasing system in (S,I)∈[E1^,E2^].Proof.
Let(15)F1(S,I)=-νRd(α0-S+I)[1-(α0-S+I)]+R0(α0-S)Iα0-S+I+ν(α0-S),F2(S,I)=R0(α0-S)Iα0-S+I-I.
From [17], we can know that the functions F1(S,I) and F2(S,I) are said to possess a quasi-monotone nonincreasing system, if the sign of ∂F1(S,I)/∂I and ∂F2(S,I)/∂S are both nonpositive.
Since(16)∂F2(S,I)∂S=-R0(Iα0-S+I)2≤0,∂F1(S,I)∂I=R0(α0-Sα0-S+I)2+2νRd(α0-S+I)-νRd,∂∂S(∂F1(S,I)∂I)=-2νRd-2R0(α0-S)I(α0-S+I)3≤-2νRd<0.
Then,
(17)∂F1(S,I)∂I≤R0(α0α0+I)2+2νRd(α0+I)-νRd.
Let
(18)G(z)=α02R0z2+2νRdz-νRd,z∈[α0,α0+I*^],
then
(19)G′(z)=2νRd-2α02R0z3=0,
obviously, z*=α02R0/νRd3 is the unique real root of G′(z).
Sinceν>27R0(Rd-1)2/Rd3, consider α0=(Rd-1)/Rd, then we can get
(20)G(z*)=(α02R0)2/3[(27α02R0)1/3-(νRd)1/3](z*)2<0.
And
(21)limz→0+G(z)=limz→+∞G(z)=+∞;
hence, G(z) has two positive roots.
Sinceν≥R0/(2-Rd), thus G(α0)=R0+νRd-2ν≤0.
According to conditions[H1] and [H2], we can get
(22)G(α0+I*^)=νRd-2ν+2νRdI*^+R0(α0α0+I*^)2<νRd-2ν+2νRdI*^+R0=(3R02Rd-2R02-2R0Rd-2R0+2)νR02+(R03-2R02+4R0-2)R02≤0.
Then, G([α0,α0+I*^])≤0. Hence, ∂F1(S,I)/∂I≤0.
That is to say, model (14) is a quasi-monotone system in (S,I)∈[E1^,E2^].Since the traveling wave solution of model (14) has the following form
(23)S(ξ)=S(x+ct),I(ξ)=I(x+ct),ξ=x+ct,c>0;
substituting (23) into model (14), we can get the following model:
(24)S′′-cS′-νRd(α0-S+I)[1-(α0-S+I)]+R0(α0-S)Iα0-S+I+ν(α0-S)=0,I′′-cI′+R0(α0-S)Iα0-S+I-I=0,(S,I)T(-∞)=(0,0)T,(S,I)T(+∞)=(S^*,I^*)T.Obviously, we can know the following.Remark 4.
Model (24) is also a quasi-monotone system in (S,I)∈[E1^,E2^].Now we establish the existence of traveling wave solutions of model (24) through monotone iteration of a pair of smooth upper and lower solutions. Following [17], we give the definitions of the upper and lower solutions of model (24) as follows, respectively.Definition 5.
A smooth function(S¯(ξ),I¯(ξ))T  (ξ∈ℝ) is an upper solution of model (24) if its derivatives (S¯′,I¯′)T and (S¯′′,I¯′′) are continuous on ℝ, and (S¯,I¯)T satisfies
(25)S′′-cS′-νRd(α0-S+I)[1-(α0-S+I)]+R0(α0-S)Iα0-S+I+ν(α0-S)≤0,I′′-cI′+R0(α0-S)Iα0-S+I-I≤0,
with the following boundary value conditions
(26)(SI)(-∞)=(00),(SI)(+∞)≥(S^*I^*).Definition 6.
A smooth function(S_(ξ),I_(ξ))T  (ξ∈ℝ) is a lower solution of model (24) if its derivatives (S_′,I_′)T and (S_′′,I_′′) are continuous on ℝ, and (S_,I_)T satisfies
(27)S′′-cS′-νRd(α0-S+I)[1-(α0-S+I)]+R0(α0-S)Iα0-S+I+ν(α0-S)≥0,I′′-cI′+R0(α0-S)Iα0-S+I-I≥0,
with the following boundary value conditions
(28)(SI)(-∞)=(00),(SI)(+∞)≤(S^*I^*).The construction of the smooth upper-lower solution pair is based on the solution of the following KPP equation:(29)w′′-cw′+f(w)=0,w(-∞)=0,w(+∞)=b,
where f∈C2([0,b]) and f>0 in the open interval (0,b) with f(0)=f(b)=0, f′(0)=a1>0, and f′(b)=-b1<0 [15]. First, let us recall the following result.Lemma 7 (see [1, 15]).
Corresponding to everyc≥2a1, model (29) has a unique (up to a translation of the origin) monotonically increasing traveling wave solution w(ξ) for ξ∈R. The traveling wave solution w has the following asymptotic behaviors.(i)
For the wave solution with noncritical speedc>2a1, one has
(30)w(ξ)=aωe((c-c2-4a1)/2)ξ+o(e((c-c2-4a1)/2)ξ)asξ⟶-∞,w(ξ)=b-bωe((c-c2+4b1)/2)ξ+o(e((c-c2+4b1)/2)ξ)asξ⟶+∞,where aω and bω are positive constants.(ii)
For the wave with critical speedc=2a1, one has
(31)w(ξ)=(ac+dcξ)ea1ξ+o(ξea1ξ)asξ⟶-∞,w(ξ)=b-bce(a1-a1+b1)ξ+o(e(a1-a1+b1)ξ)asξ⟶+∞, where the constant dc is negative, bc is positive, and ac∈R.For constructing the upper solution of the model (24), we start with the following model:
(32)I′′-cI′+I(I^*-I)α0(R0-1)α0I^*+(I^*-S^*)I=0,I(-∞)=0,I(+∞)=I^*.Definef(I)=I(I^*-I)(α0(R0-1)/ϕ(I)), I∈[0,I^*], one can verify that all of the following conditions are satisfied:(i)
f(I)=I(I^*-I)(α0(R0-1)/ϕ(I))∈C2([0,I^*]);(ii)
f(I)>0,for all I∈(0,I^*) and f(0)=f(I^*)=0;(iii)
f′(0)=R0-1>0, f′(I^*)=-α0(R0-1)2/R0I^*<0.From Lemma7, we know that, for each c≥2R0-1, equation (32) has a unique traveling wave solution I¯(ξ) (up to a translation of the origin), satisfying the given boundary value conditions (26).Define(33)(S¯(ξ)I¯(ξ))=(S^*I^*I¯(ξ)I¯(ξ)),ξ∈R,
then we can get the following result.Lemma 8.
For eachc≥2R0-1, (33) is a smooth upper solution of model (24).Proof.
On the boundary,(34)(S¯I¯)(-∞)=(00),(S¯I¯)(+∞)≥(S^*I^*).
As for the I component, we have
(35)I¯′′-cI¯′+R0(α0-S¯)I¯α0-S¯+I¯-I¯=-I¯(I^*-I¯)α0(R0-1)ϕ(I¯)+R0(α0-S¯)I¯α0-S¯+I¯-I¯=-I¯(I^*-I¯)α0(R0-1)ϕ(I¯)+R0α0I^*-S^*I¯ϕ(I¯)I¯-I¯=-I¯(I^*-I¯)α0(R0-1)ϕ(I¯)+I¯α0(R0-1)I^*-(R0S^*-S^*+I^*)I¯ϕ(I¯)=-I¯(I^*-I¯)α0(R0-1)ϕ(I¯)+I¯α0(R0-1)I^*-α0(R0-1)I¯ϕ(I¯)=0.
As for the S component, since ν>1, then I^*-S^*=(ν-1)(R0-1)/νR0Rd>0. And(i)
ifη0<I^*/2, then maxξ∈[0,I^*]ψ(I¯)=ψ(I^*)=ψ(B);(ii)
ifη0≥I^*/2, then maxξ∈[0,I^*]ψ(I¯)=ψ(0)=ψ(B).
Thus we can get:(36)S¯′′-cS¯′-νRd(α0-S¯+I¯)[1-(α0-S¯+I¯)]+R0(α0-S¯)I¯α0-S¯+I¯+ν(α0-S¯)=S^*I^*(I¯′′-cI¯′)-νRd(α0-S¯+I¯)[1-(α0-S¯+I¯)]+R0(α0-S¯)I¯α0-S¯+I¯+ν(α0-S¯)=S^*I^*(-R0(α0-S¯)I¯α0-S¯+I¯+I¯)-νRd(α0-S¯+I¯)×[1-(α0-S¯+I¯)]+R0(α0-S¯)I¯α0-S¯+I¯+ν(α0-S¯)=(1-S^*I^*)R0(α0-S¯)I¯α0-S¯+I¯+S^*I^*I¯-νRd(α0-S¯+I¯)×[1-(α0-S¯+I¯)]+ν(α0-S¯)=I^*-S^*I^*R0α0I^*-S^*I¯ϕ(I¯)I¯+S^*I^*I¯-νRdϕ(I¯)I^*[1-ϕ(I¯)I^*]+να0I^*-S^*I¯I^*=(I^*-S^*)I¯ψ(I¯)(I^*)2ϕ(I¯)≤0.
Hence, (S¯,I¯) forms a smooth upper solution for model (24).For constructing the lower solution of the model (24), we start with the following model:
(37)I′′-cI′+I[I^*-(1+ɛ)I]α0(R0-1)α0I^*+(I^*-S^*)I=0,I(-∞)=0,I(+∞)=I^*1+ɛ.Defineg(I)=I[I^*-(1+ɛ)I](α0(R0-1)/(α0I^*+(I^*-S^*)I)), I∈[0,I^*/(1+ɛ)]. One can easily verify that all of the following conditions hold:(i)
g(I)=I[I^*-(1+ɛ)I](α0(R0-1)/(α0I^*+(I^*-S^*)I))∈C2([0,I^*/(1+ɛ)]);(ii)
g(I)>0, for all I∈(0,I^*/(1+ɛ)) and g(0)=g(I^*/(1+ɛ))=0;(iii)
g′(0)=R0-1>0, g′(I^*/(1+ɛ))=-(1+ɛ)α0(R0-1)/(ɛα0+(R0/(R0-1))I^*)<0.From Lemma7, we know that, for each fixed c≥2R0-1, model (37) has a unique traveling wave solution I_(ξ) (up to a translation of the origin), satisfying the given boundary value conditions (28).Define(38)(S_(ξ)I_(ξ))=(S^*I^*I_(ξ)I_(ξ)),ξ∈R,
then we have the following result:Lemma 9.
For each fixedc≥2R0-1, (38) is a lower solution of model (24).Proof.
On the boundary,(39)(S_I_)(-∞)=(00),(S_I_)(+∞)=(S^*1+ɛI^*1+ɛ)≤(S^*I^*).
As for the I component, we have
(40)I_′′-cI_′+R0(α0-S_)I_α0-S_+I_-I_=-I_[I^*-(1+ɛ)I_]α0(R0-1)ϕ(I_)+R0(α0-S_)I_α0-S_+I_-I_=-I_[I^*-(1+ɛ)I_]α0(R0-1)ϕ(I_)+I_(I^*-I_)α0(R0-1)ϕ(I_)=ɛ(I_)2α0(R0-1)ϕ(I_)≥0.
As for the S component, we have
(41)S_′′-cS_′-νRd(α0-S_+I_)[1-(α0-S_+I_)]+R0(α0-S_)I_α0-S_+I_+ν(α0-S_)=S^*I^*(I_′′-cI_′)-νRd(α0-S_+I_)[1-(α0-S_+I_)]+R0(α0-S_)I_α0-S_+I_+ν(α0-S_)=S^*I^*{[ɛ(I_)2α0(R0-1)ϕ(I_)]-[R0(α0-S_)I_α0-S_+I_-I_]}-νRd(α0-S_+I_)[1-(α0-S_+I_)]+R0(α0-S_)I_α0-S_+I_+ν(α0-S_)=ɛS^*I^*(I_)2α0(R0-1)ϕ(I_)+(I^*-S^*)I_ψ(I_)(I^*)2ϕ(I_)≥0.
Thus (S_,I_) forms a smooth lower solution for model (24).Next, we show that, by shifting the upper solution far enough to the left, then the upper-lower solution in Lemmas8 and 9 are ordered.Lemma 10.
Letc≥2R0-1, (S¯,I¯)T and (S_,I_)T be the upper solution and the lower solution defined in (33) and (38), then there exists a positive number r, such that (S¯,I¯)T(ξ+r)≥(S_,I_)T(ξ) for all ξ∈R.Proof.
Our proof is only forc>2R0-1, and the proof for the case of c=2R0-1 is similar to it.
First, we derive the asymptotic behaviors of the upper solution and the lower solution at infinities.
According to Lemma7, when ξ→-∞, we can obtain:
(42)(S¯I¯)(ξ)=(S^*I^*A1A1)e((c-c2-4(R0-1))/2)ξ+o(e((c-c2-4(R0-1))/2)ξ),(S_I_)(ξ)=(S^*I^*B1B1)e((c-c2-4(R0-1))/2)ξ+o(e((c-c2-4(R0-1))/2)ξ).
And letσ0=(1/2)(c-c2+4(α0(R0-1)2/R0I^*)<0, δ0=(1/2)(c-c2+4((1+ɛ)α0(R0-1)/(ɛα0+(R0/(R0-1))I^*))<0, when ξ→+∞, we can get
(43)(S¯I¯)(ξ)=(S^*I^*)-(S^*I^*A2A2)eσ0ξ+o(eσ0ξ),(S_I_)(ξ)=11+ɛ(S^*I^*)-(S^*I^*B2B2)eδ0ξ+o(eδ0ξ),
where, A1, A2, B1, B2 are all positive constants.
Since for anyr~>0, I¯r~(ξ)≡I¯(ξ+r~) is also a solution of model (32). Thus, (S¯r~,I¯r~)T(ξ) is an upper solution of model (24). So, according to Lemma 7, when ξ→-∞, we can get:
(44)(S¯r~I¯r~)(ξ)=(S^*I^*A1A1)e((c-c2-4(R0-1))/2)r~e((c-c2-4(R0-1))/2)ξ+o(e((c-c2-4(R0-1))/2)ξ).
Since(c-c2-4(R0-1))/2>0, we can choose a large enough number r~≫0, such that
(45)(S^*I^*A1A1)e((c-c2-4(R0-1))/2)r~>(S^*I^*B1B1),
hence, there exists a large number N1≫1, such that
(46)(S¯r~(ξ)I¯r~(ξ))>(S_(ξ)I_(ξ)),ξ∈(-∞,-N1].
By using a similar argument as above, there exists a large enough numberN2≫1, such that
(47)(S¯r~(ξ)I¯r~(ξ))>(S_(ξ)I_(ξ)),ξ∈[N2,+∞).
Second, we show that(48)(S¯r~(ξ)I¯r~(ξ))>(S_(ξ)I_(ξ)),ξ∈[-N1,N2].
We deal with such two possible cases:
Case  1. If
(49)(S¯r~(ξ)I¯r~(ξ))>(S_(ξ)I_(ξ)),ξ∈[-N1,N2],
then, the proof is completed.
Case  2. If there exists a point ξ0∈(-N1,N2), such that
(50)(S¯r~(ξ0)I¯r~(ξ0))≤(S_(ξ0)I_(ξ0))
satisfying  S¯r~(ξ0)<S_(ξ0) or I¯r~(ξ0)<I_(ξ0).
In this case, we use the Sliding Domain method [15].
Step  1. we shift (S¯r~,I¯r~)T to the left by increasing the number r~ until finding a new number r1>r~ such that (S¯r1,I¯r1)T>(S_,I_)T on the smaller interval [-N1,N2-(r1-r~)].
Step  2. we shift (S¯r1,I¯r1)T back to the right by decreasing r1 to a smaller number r~<r2<r1 such that one of the branches of the upper solution touches its counterpart of the lower solution at some point ξ1 in the interval (-N1+(r1-r2),N2-(r1-r~)). On the endpoints of the interval (-N1+(r1-r2),N2-(r1-r~)), we still have (S¯r2,I¯r2)T>(S_,I_)T.
LetW→(ξ)=(S¯r2,I¯r2)T-(S_,I_)T and F→=(F1,F2)T, where
(51)F1=-νRd(α0-S+I)[1-(α0-S+I)]+R0(α0-S)Iα0-S+I+ν(α0-S),F2=R0(α0-S)Iα0-S+I-I.
Forξ∈(-N1+(r1-r2),N2-(r1-r~)), we get that
(52)W→′′-cW→′+(∂F1∂S(S_+ζ1ω1,I¯r2)∂F1∂I(S¯r2,I_+ζ2ω2)∂F2∂S(S_+ζ3ω1,I¯r2)∂F2∂I(S¯r2,I_+ζ4ω2))W→=0,
where ζi∈[0,1], i=1,2,3,4. Since the above model is monotone and the cube [(0,0),(S^*,I^*)] is convex, thus we can deduce by Maximum Principle that W→>0 for ξ∈[-N1+(r1-r2),N2-(r1-r~)]. So ξ1 does not exist and we can decrease r2 further to r~. It is calculated that the point ξ0 does not exist either. The proof of this lemma is completed.To ease the burden of notations, we still use(S¯,I¯)T to denote the shifted upper solution as given in Lemma 8. Let
(53)D11=-R02+νR0Rd+νR0-4R0-2ν+3R0,D12=νR0Rd-2R0-2ν+3R0,D21=-(R0-1)2R0,D22=-R0-1R0,μ1=-(D11+D22)+(D11-D22)2+4D12D212,μ2=-(D11+D22)-(D11-D22)2+4D12D212.With such constructed ordered upper-lower solution pair, we can get the following.Theorem 11.
Forc≥2R0-1, model (24) has a unique (up to a translation of the origin) traveling wave solution. The traveling wave solution is strictly increasing and has the following asymptotic properties:(i)
ifc>2R0-1, when ξ→-∞,
(54)(SI)(ξ)=(A1A2)e((c-c2-4(R0-1))/2)ξ+o(e((c-c2-4(R0-1))/2)ξ).when ξ→+∞, and if μ1≠μ2, then
(55)(SI)(ξ)=(S^*I^*)-(A1¯A2¯)e((c-c2+4μ)/2)ξ+o(e((c-c2+4μ)/2)ξ),
while μ1=μ2:
(56)(SI)(ξ)=(S^*I^*)-(A11¯¯+A12¯¯ξA21¯¯+A22¯¯ξ)e((c-c2+4μ)/2)ξ+o(e((c-c2+4μ)/2)ξ),
where, μ=min{μ1,μ2}>0, A11¯¯,A21¯¯∈ℝ, A1, A2, A1¯, A2¯, A12¯¯ and A22¯¯ are all positive constants.(ii)
ifc=2R0-1, when ξ→-∞,
(57)(SI)(ξ)=(B11+B12ξB21+B22ξ)eR0-1ξ+o(ξeR0-1ξ),when ξ→+∞, and if μ1≠μ2, then
(58)(SI)(ξ)=(S^*I^*)-(B11¯B22¯)e(R0-1-R0-1+μ)ξ+o(e(R0-1-R0-1+μ)ξ),
while μ1=μ2,
(59)(SI)(ξ)=(S^*I^*)-(B11¯¯+B12¯¯ξB21¯¯+B22¯¯ξ)e(R0-1-R0-1+μ)ξ+o(e(R0-1-R0-1+μ)ξ),
where μ=min{μ1,μ2}>0, B12,B22<0, B11,B21,B11¯¯,B21¯¯∈ℝ, and B11¯, B22¯, B12¯¯, B22¯¯ are all positive constants.Proof.
From Lemma3 and Remark 4, we know that model (24) is a quasi-monotone nonincreasing system in (S,I)∈[E1^,E2^], and by using the monotone iteration scheme given in [3, 13], we can obtain the existence of the solution (S,I)T to the first two equations in model (24) for every c≥2R0-1, which satisfies
(60)(S_(ξ)I_(ξ))≤(S(ξ)I(ξ))≤(S¯(ξ)I¯(ξ)).
According to the above inequality, we can get that, on the boundary, the solution tends to(0,0)T as ξ→-∞ and (S^*,I^*)T as ξ→+∞.
To derive the asymptotic decay rate of the traveling wave solutions asξ→±∞, we just let c>2R0-1 and
(61)U(ξ)=(S(ξ),I(ξ))T,-∞<ξ<+∞
be the traveling wave solution of model (24) generated form the monotone iteration, since the case of (ii) c=2R0-1 is similar to it.
We differentiate model (24) with respect to ξ, and note that U′(ξ)=(χ1,χ2)T(ξ) satisfies
(62)χ1′′-cχ1′+C11(S,I)χ1+C12(S,I)χ2=0,χ2′′-cχ2′+C21(S,I)χ1+C22(S,I)χ2=0,
where
(63)C11(S,I)=νRd[1-(α0-S+I)]-νRd(α0-S+I)-R0Iα0-S+I+R0(α0-S)I(α0-S+I)2-ν,C12(S,I)=-νRd[1-(α0-S+I)]+νRd(α0-S+I)+R0(α0-S)α0-S+I-R0(α0-S)I(α0-S+I)2,C21(S,I)=-R0Iα0-S+I+R0(α0-S)I(α0-S+I)2,C22(S,I)=R0(α0-S)α0-S+I-R0(α0-S)I(α0-S+I)2-1.
Now, we study the exponential decay rate of the traveling wave solution asξ→-∞. The asymptotic model of model (62) as ξ→-∞ is
(64)λ′′-cλ′+E11λ+E12μ=0,μ′′-cμ′+E21λ+E22μ=0,
where
(65)E11=-ν(Rd-1),E12=νRd+R0-2ν,E21=0,E22=R0-1.
The second equation of model (64) has two independent solutions with the following form:
(66)μ(1)(ξ)=e((c-c2-4(R0-1))/2)ξ,μ(2)(ξ)=e((c+c2-4(R0-1))/2)ξ.
Relating the second equation of model (62) with the second equation of model (64), we can deduce that χ2(ξ) has the following property as ξ→-∞:
(67)χ2(ξ)=α[1+o(1)]e((c-c2-4(R0-1))/2)ξ+β[1+o(1)]e((c+c2-4(R0-1))/2)ξ
for some constants α and β. Thus, we can obtain that
(68)χ2(ξ)=αe((c-c2-4(R0-1))/2)ξ+βe((c+c2-4(R0-1))/2)ξ+Υ1(ξ)+Υ2(ξ),
where
(69)limξ→-∞Υ1(ξ)e((c-c2-4(R0-1))/2)ξ=0,limξ→-∞Υ2(ξ)e((c+c2-4(R0-1))/2)ξ=0.
So we obtain that
(70)limξ→-∞χ2(ξ)-αe((c-c2-4(R0-1))/2)ξe((c-c2-4(R0-1))/2)ξ=limξ→-∞Υ1(ξ)+Υ2(ξ)+βe((c+c2-4(R0-1))/2)ξe((c-c2-4(R0-1))/2)ξ=limξ→-∞Υ1(ξ)e((c-c2-4(R0-1))/2)ξ+βlimξ→-∞ec2-4(R0-1)ξ+limξ→-∞Υ2(ξ)e((c+c2-4(R0-1))/2)ξlimξ→-∞ec2-4(R0-1)ξ=0.
Thus, χ2(ξ)=α[1+o(1)]e((c-c2-4(R0-1))/2)ξ.
Now, we consider the first equation of model (64). We rewrite it as
(71)λ′′-cλ′-ν(Rd-1)λ=-(νRd+R0-2ν)μ.
One can verify that(c-c2-4(R0-1))/2 is not a characteristic of
(72)λ′′-cλ′-ν(Rd-1)λ=0.
The above equation has two independent solutions of the following form:(73)λ(1)(ξ)=e((c-c2+4ν(Rd-1))/2)ξ,λ(2)(ξ)=e((c+c2+4ν(Rd-1))/2)ξ.
Thus, when ξ→-∞, χ1(ξ) has the following property:
(74)χ1(ξ)=α¯[1+o(1)]e((c+c2+4ν(Rd-1))/2)ξ+β¯[1+o(1)]e((c-c2+4ν(Rd-1))/2)ξ+γ[1+o(1)]e((c-c2-4(R0-1))/2)ξ
for some constants α¯,β¯; γ≠0. Since χ1(-∞)=0, thus β¯=0. So, when ξ→-∞, we have the following formula:
(75)(χ1(ξ)χ2(ξ))=(γ[1+o(1)]e((c-c2-4(R0-1))/2)ξα[1+o(1)]e((c-c2-4(R0-1))/2)ξ).
Then, we study the exponential decay rate of the traveling wave solution asξ→+∞. The asymptotic model of model (62) as ξ→+∞ is
(76)ψ1′′-cψ1′+D11ψ1+D12ψ2=0,ψ2′′-cψ2′+D21ψ1+D22ψ2=0.
By setting(ψi)′(ξ)=ψ~i, i=1,2, we rewrite model (76) as a first order model of ordinary differential equation in the four components (ψ1,ψ~1,ψ2,ψ~2)T:
(77)ψ1′=ψ~1,ψ~1′=cψ~1-D11ψ1-D12ψ2,ψ2′=ψ~2,ψ~2′=cψ~2-D21ψ1-D22ψ2.
In the case of (i)μ1≠μ2, we can obtain that the solution of model (77) has the following form:
(78)(ψ1,ψ~1,ψ2,ψ~2)T=∑i=14‍cihieλiξ,
where
(79)λ1=c+c2+4μ12,λ2=c-c2+4μ12,λ3=c+c2+4μ22,λ4=c-c2+4μ22,
and hi(i=1,2,3,4) are the eigenvectors of the constant matrix with λi(i=1,2,3,4) as the corresponding eigenvalues, ci(i=1,2,3,4) are arbitrary constants. Since
(80)limξ→+∞(ψ1,ψ~1,ψ2,ψ~2)T=0,
thus (ψ1,ψ~1,ψ2,ψ~2)T=c2h2eλ2ξ+c4h4eλ4ξ, so when ξ→+∞, we can get that
(81)(χ1(ξ)χ2(ξ))=(κ1[Λ1+o(1)]e((c-c2+4μ1)/2)ξκ1[Γ1+o(1)]e((c-c2+4μ1)/2)ξ)+(κ2[Λ2+o(1)]e((c-c2+4μ2)/2)ξκ2[Γ2+o(1)]e((c-c2+4μ2)/2)ξ).
Furthermore, we can obtain that(82)χ1(ξ)=κ1Λ1e((c-c2+4μ1)/2)ξ+κ2Λ2e((c-c2+4μ2)/2)ξ+Ω11(ξ)+Ω12(ξ),χ2(ξ)=κ1Γ1e((c-c2+4μ1)/2)ξ+κ2Γ2e((c-c2+4μ2)/2)ξ+Ω21(ξ)+Ω22(ξ),
where
(83)limξ→+∞Ω11(ξ)e((c-c2+4μ1)/2)ξ=0,limξ→+∞Ω12(ξ)e((c-c2+4μ2)/2)ξ=0,limξ→+∞Ω21(ξ)e((c-c2+4μ1)/2)ξ=0,limξ→+∞Ω22(ξ)e((c-c2+4μ2)/2)ξ=0.κ1, κ2, Λ1, Λ2, Γ1, and Γ2 are all constants.
Letμ=min{μ1,μ2}, then
(84)limξ→+∞χ1(ξ)e((c-c2+4μ)/2)ξ=κ1Λ1limξ→+∞e((c2+4μ-c2+4μ1)/2)ξ+κ2Λ2limξ→+∞e((c2+4μ-c2+4μ2)/2)ξ+limξ→+∞Ω11(ξ)e((c-c2+4μ1)/2)ξlimξ→+∞e((c2+4μ-c2+4μ1)/2)ξ+limξ→+∞Ω12(ξ)e((c-c2+4μ2)/2)ξlimξ→+∞e((c2+4μ-c2+4μ2)/2)ξ=Δ1(κ1,κ2,Λ1,Λ2),limξ→+∞χ2(ξ)e((c-c2+4μ)/2)ξ=κ1Γ1limξ→+∞e((c2+4μ-c2+4μ1)/2)ξ+κ2Γ2limξ→+∞e((c2+4μ-c2+4μ2)/2)ξ+limξ→+∞Ω21(ξ)e((c-c2+4μ1)/2)ξlimξ→+∞e((c2+4μ-c2+4μ1)/2)ξ+limξ→+∞Ω22(ξ)e((c-c2+4μ2)/2)ξlimξ→+∞e((c2+4μ-c2+4μ2)/2)ξ=Δ2(κ1,κ2,Γ1,Γ2),
where
(85)Δ1(κ1,κ2,Λ1,Λ2)={κ1Λ1,μ1<μ2,κ2Λ2,μ1>μ2,Δ2(κ1,κ2,Γ1,Γ2)={κ1Γ1,μ1<μ2,κ2Γ2,μ1>μ2;
thus, when ξ→+∞, we can get that
(86)(χ1(ξ)χ2(ξ))=(Δ1(κ1,κ2,Λ1,Λ2)(1+o(1))e((c-c2+4μ)/2)ξΔ2(κ1,κ2,Γ1,Γ2)(1+o(1))e((c-c2+4μ)/2)ξ).
In the case of (ii)μ1=μ2, we can obtain that the solution of model (77) has the following form:
(87)(ψ1,ψ~1,ψ2,ψ~2)T=(G1+G2ξ)H1,2eλ1¯ξ+(G3+G4ξ)H3,4eλ3¯ξ,
where H1,2 is the eigenvector of the constant matrix with λ1¯=λ2¯ as the corresponding eigenvalues, H3,4 is the eigenvector of the constant matrix with λ3¯=λ4¯ as the corresponding eigenvalues, Gi(i=1,2,3,4) are arbitrary constants.
Sincelimξ→+∞(ψ1,ψ~1,ψ2,ψ~2)T=0, thus
(88)(ψ1,ψ~1,ψ2,ψ~2)T=(G3+G4ξ)H3,4eλ3¯ξ.
So, when ξ→+∞, we can get that
(89)(χ1(ξ)χ2(ξ))=((G1,3+G1,4ξ)eλ3¯ξ+o(eλ3¯ξ)(G2,3+G2,4ξ)eλ4¯ξ+o(eλ4ξ¯)).
By comparing the upper solution and roughness of the exponential dichotomy [24], we obtain the asymptotic decay rate of the traveling wave solutions at +∞ given in Theorem 11.
According to the monotone iteration process [3], the traveling wave solution U(ξ) is increasing; thus U′(ξ)≥0 and U′(ξ)=(χ1,χ2)T(ξ) hold
(90)χ1′′-cχ1′+∂F1∂Sχ1+∂F1∂Iχ2=0,χ2′′-cχ2′+∂F2∂Sχ1+∂F2∂Iχ2=0,
satisfying
(91)(χ1,χ2)T(ξ)≥0,(χ1,χ2)T(±∞)=0.
The strong Maximum Principle implies that(χ1,χ2)T(ξ)>0. So the strict monotonicity of the traveling wave solutions is concluded.
Now, we use the Sliding domain method to prove the uniqueness of the traveling wave solution. LetU1(ξ)=(S1,I1)T(ξ) and U2(ξ)=(S2,I2)T(ξ) be the traveling wave solution of model (24), with c>2R0-1. Thus, there are some positive numbers Ai,Bi(i=1,2,3,4), such that for a big enough number N≫1, when ξ<-N, we have
(92)(S1(ξ)I1(ξ))=(A1e((c-c2-4(R0-1))/2)ξA2e((c-c2-4(R0-1))/2)ξ)+o(e((c-c2-4(R0-1))/2)ξ),(S2(ξ)I2(ξ))=(A3e((c-c2-4(R0-1))/2)ξA4e((c-c2-4(R0-1))/2)ξ)+o(e((c-c2-4(R0-1))/2)ξ),
when ξ>N,
(93)(S1(ξ)I1(ξ))=(S^*-B1e((c-c2+4μ)/2)ξI^*-B2e((c-c2+4μ)/2)ξ)+o(e((c-c2+4μ)/2)ξ),(S2(ξ)I2(ξ))=(S^*-B3e((c-c2+4μ)/2)ξI^*-B4e((c-c2+4μ)/2)ξ)+o(e((c-c2+4μ)/2)ξ).
Since the traveling wave solutions of model (24) are translation-invariant, then for any θ∈R, U1θ(ξ)≡U1(ξ+θ)≡(S1(ξ+θ),I1(ξ+θ))T is also a traveling wave solution of model (24). Thus, by using the same method as above, when ξ<-N, we can get
(94)(S1(ξ+θ)I1(ξ+θ))=(A1e((c-c2-4(R0-1))/2)θe((c-c2-4(R0-1))/2)ξA2e((c-c2-4(R0-1))/2)θe((c-c2-4(R0-1))/2)ξ)+o(e((c-c2-4(R0-1))/2)ξ),
when ξ>N,
(95)(S1(ξ+θ)I1(ξ+θ))=(S^*-B1e((c-c2+4μ)/2)θe((c-c2+4μ)/2)ξI^*-B2e((c-c2+4μ)/2)θe((c-c2+4μ)/2)ξ)+o(e((c-c2+4μ)/2)ξ).
If θ is large enough, then we can obtain the following inequalities:
(96)A1e((c+c2-4(R0-1))/2)θ>A3,A2e((c+c2-4(R0-1))/2)θ>A4,B1e((c-c2+4μ)/2)θ<B3,B2e((c-c2+4μ)/2)θ<B4.
Thus, if θ is large enough, then U1θ(ξ)>U2(ξ), for all ξ∈R∖[-N,N].
Now, we consider model (24) on the interval [-N,N].
First, suppose that(97)W(ξ)≡U1θ(ξ)-U2(ξ)≥0,ξ∈[-N,N],
then
(98)W′′-cW′+(∂F1∂S(S2+ζ1ω1,I1)∂F1∂I(S1,I2+ζ2ω2)∂F2∂S(S2+ζ3ω1,I1)∂F2∂I(S1,I2+ζ4ω2))W=0,W(-N)>0,W(N)>0,
where, ζi∈(0,1)(i=1,2,3,4), ξ∈(-N,N). Since the above model is monotone, by the Maximum Principle, we can deduce that W(ξ)>0,ξ∈[-N,N]. Consequently, we get that U1θ(ξ)>U2(ξ),ξ∈R.
Second, we suppose that there exists a pointξ*∈(-N,N) such that
(99)S1θ(ξ*)<S2(ξ*)
or
(100)I1θ(ξ*)<I2(ξ*).
In this case, we increaseθ, that is shifting U1θ to the left, so that U1θ(-N)>U2(-N) and U1θ(N)>U2(N). According to the monotonicity of U1θ and U2, we can find a number θ¯∈(0,2N) such that U1θ(ξ+θ¯)>U2(ξ), ξ∈(-N,N). Shifting U1θ(ξ+θ¯) back until one component of U1θ(ξ+θ¯) touches its counterpart of U2(ξ) at some point ξ¯¯∈(-N,N). Since U1θ(ξ+θ¯) and U2(ξ) are strictly increasing, ξ¯¯∈(-N,N), thus, we get that U1θ(ξ+θ¯)>U2(ξ), ξ=±N. However, by the Maximum Principle for that component again, we find that components of U1θ and U2 are identically equal for all ξ∈[-N,N] for a larger number θ. This is a contradiction, thus U1θ(ξ)>U2(ξ), ξ∈R. Here, θ is a new number which is chosen by the above mean.
Now, decrease theθ until one of the following happens.
Case (a). There is a θ¯≥0, such that U1θ¯=U2(ξ), ξ∈R. In this case, we have finished the proof.
Case (b). There are a θ¯ and a point ξ1∈R, such that one of the components of U1θ¯ and U2 are equal. And U1θ¯≥U2, ξ∈R. On R for that component, according to the Maximum Principle, we find that U1θ¯ and U2 must be identical on that component. We can return to Case (a).
Consequently, in either situation, their exists a numberθ¯≥0 such that
(101)U1θ¯(ξ)=U2(ξ),ξ∈(-∞,+∞).
This ends of the proof.By Theorem11, we can get the following theorem:Theorem 12.
For eachc≥2R0-1, model (3) has a unique (up to a translation of the origin) traveling wave solution. The traveling wave solution is strictly increasing and has the following asymptotic properties:(i)
c>2R0-1: when ξ→-∞,
(102)S(ξ)=Rd-1Rd-A1e((c-c2-4(R0-1))/2)ξ+o(e((c-c2-4(R0-1))/2)ξ),I(ξ)=A2e((c-c2-4(R0-1))/2)ξ+o(e((c-c2-4(R0-1))/2)ξ).when ξ→+∞, and if μ1≠μ2, then
(103)S(ξ)=S*+A1¯e((c-c2+4μ)/2)ξ+o(e((c-c2+4μ)/2)ξ),I(ξ)=I*-A2¯e((c-c2+4μ)/2)ξ+o(e((c-c2+4μ)/2)ξ),
if μ1=μ2,
(104)S(ξ)=S*+(A11¯¯+A12¯¯ξ)e((c-c2+4μ)/2)ξ+o(e((c-c2+4μ)/2)ξ),I(ξ)=I*-(A21¯¯+A22¯¯ξ)e((c-c2+4μ)/2)ξ+o(e((c-c2+4μ)/2)ξ),
where μ=min{μ1,μ2}>0, A11¯¯,A21¯¯∈ℝ  A1, A2, A1¯, A2¯, A12¯¯ and A22¯¯ are all positive constants.(ii)
c=2R0-1: when ξ→-∞,
(105)S(ξ)=Rd-1Rd-(A11+A12ξ)eR0-1ξ+o(ξeR0-1ξ),I(ξ)=(A11+A12ξ)eR0-1ξ+o(ξeR0-1ξ). when ξ→+∞, and if μ1≠μ2, then
(106)S(ξ)=S*+B11¯e(R0-1-R0-1+μ)ξ+o(e(R0-1-R0-1+μ)ξ),I(ξ)=I*-B22¯e(R0-1-R0-1+μ)ξ+o(e(R0-1-R0-1+μ)ξ),
if μ1=μ2, then
(107)S(ξ)=S*+(B11¯¯+B12¯¯ξ)e(R0-1-R0-1+μ)ξ+o(e(R0-1-R0-1+μ)ξ),I(ξ)=I*-(B21¯¯+B22¯¯ξ)e(R0-1-R0-1+μ)ξ+o(e(R0-1-R0-1+μ)ξ),
where μ=min{μ1,μ2}>0, B12,B22<0, B11,B21,B11¯¯,B21¯¯∈ℝ, B11¯, B22¯, B12¯¯, B22¯¯ are all positive constants.Theorem 13.
There is no monotone traveling wave solution of model (24) for any 0<c<2R0-1. In other words, there is no monotone traveling wave solution of model (3) for any 0<c<2R0-1.Proof.
Suppose there is a monotone traveling wave solution𝕃(ξ)=(l1(ξ),l2(ξ))T of model (24) with the wave speed c0, where c0∈(0,2R0-1).
The asymptotic model of𝕃(ξ)=(l1(ξ),l2(ξ))T as ξ→-∞ is
(108)λ¯′′-c0λ¯′-ν(Rd-1)λ¯+(νRd+R0-2ν)μ¯=0,μ¯′′-c0μ¯′+(R0-1)μ¯=0.
The second function of (108) has two characteristics as the following ones: (c0+4(R0-1)-c02i)/2, (c0-4(R0-1)-c02i)/2. Thus it has two independent solutions of the following form:
(109)μ11¯=e(c0/2)ξcos(4(R0-1)-c022ξ),μ22¯=e(c0/2)ξsin(4(R0-1)-c022ξ).
Similar to the proof of Theorem11, we can get that, when ξ→-∞, χ2(ξ) can be described as the following equation:
(110)χ2(ξ)=K1e(c0/2)ξcos(4(R0-1)-c022ξ)+K2e(c0/2)ξsin(4(R0-1)-c022ξ)+h.o.t,=K12+K22e(c0/2)ξsin(4(R0-1)-c022ξ+τ(ξ))+h.o.t,
where tan(τ(ξ))=K1/K2, and h.o.t is the short notation for the higher order terms.
That is to say,l2(ξ) is oscillating. Thus, any solution of model (24) with 0<c<2R0-1 is not strictly monotone.Theorems12 and 13 indicate that c=2R0-1 is the critical minimal wave speed.



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*Source: 216913-2013-04-09.xml*