Heteroclinic and Traveling Wave Solutions for a SIR Epidemic Model with Nonlocal Response

WANG Zongyi(王宗毅)

( College of Mathematics and Big Data,Huizhou University,Guangdong 516007,China)

Abstract: The existence of positive heteroclinic solutions is proved for a class of sir epidemic model with nonlocal interaction and non monotone property.Applying the theory of Fredholm operator decomposition and nonlinear perturbation developed by Faria and HUANG(2006),we study a connection between traveling wave solutions for the reaction-diffusion system and heteroclinic solutions of the associated differential equations.Existence and dynamics of wavefront profile are obtained as a consequence.

Key words: Delay ordinary differential equation; Reaction-diffusion equation; SIR epidemic model; Heteroclinic solution; Traveling wave solution

1.Introduction

As we all know,delay differential equations(DDEs) have been extensively used as models in biology and other sciences,with particular emphasis in population dynamics.Such equations serve as models for the growth of a single species population,in ecology problems or in disease modeling[1−3,9−13].Recently,Faria et al.proposed the theory to obtain traveling wave solutions for scalar delay reaction-diffusion equations[4],which can be viewed as perturbations of heteroclinic solutions connecting two hyperbolic equilibria of the associated equation without diffusion.Our study was motivated by the following diffusive population model with stage structure[5]

whereu1(t,x) andu2(t,x) denote respectively the densities of juvenile,mature individuals at timetand locationx.αu2is a birth function,andγu1,βu22represent respectively the death functions of juvenile and mature individuals.The adult recruitment term iswhere the delayτis the time taken from birth to maturity.Gourley and KUANG[5]study the minimal speed of equations (1.1) atτ=0.They also discuss the relations betweenτand the minimal speed,and the monotonicity of the traveling waves for such given model.

We note that the spread of disease for species is possibly related to the stage structure.Juveniles have more opportunities to contract some diseases such as measles and mumps,while some other diseases may spread in adults.Thus only the juveniles are assumed to be susceptible to the infection in many SIR epidemic models with stage structure and nonlinear incidence.Furthermore,a general incidence can beU(u)vwith susceptible populationuand infectious populationv.However,we are concerned on the infectious agentsU(u) instead of the nonlinear incidenceU(u)vfor simplicity,and study a SIR epidemic model with stage structure and nonlocal response with the form as follows

whereu1(t,x),u2(t,x) andu3(t,x) denote the densities of juvenile,susceptible mature and infectious mature individuals at timet ∈R respectively,locationx ∈R,γ,ρ >0 denote the death rate of juvenile and infectious adults respectively,andr >0 denote the recovery rate of infectious adults.HereB(u2)is the birth function,andε∫

Rfα(x −y)B(u2(t−τ,y))dyrepresents the recruitment term whereε ∈[0,1],andfα(·)is the kernel function.For instance,we can letandε=e−γτ,coinciding with those of maturation ageτand nonlocal response.We assume that only the mature individuals are susceptible,and the susceptible individuals,once infected by infectious adults,can carry germs and then transmit the infections.Since it act similarly as the infectious agent and thus letU(u2) be the force of infection on the mature population due to a concentration of the susceptible adultsu2.

The purpose of this paper is to study connections between the traveling wave solutions and heteroclinic solutions for such different type systems.Roughly speaking,by detailed discussion on the dynamics of the corresponding ordinary differential equations,we expect to find the long time behaviors of the epidemic models (1.2).This paper is organized as follows.In Section 2,we introduce some assumptions and establish the well-posedness of nondiffusive system.We show that there is a positive heteroclinic solution connecting two hyperbolic equilibria,provided that one of them is global attractive.In Section 3,we obtain our main results on the existence of the positive heteroclinic solutions for the ordinary differential equations.However,for sufficient largec,the set of all traveling wave solutions propagating at speedcforms aC1-smooth manifold in someC([−τ,0],R)-neighborhood of the heteroclinic solution.As a consequence,the existence of traveling wave solutions for the given SIR epidemic model follows immediately.

2.Well-Posedness and Positive Heteroclinic Solutions

LetC(R,R) be the space of continuous functions on R.Let=C([−τ,0],R) be a Banach space,=C([−τ,0],R+).For anyK >0,let [0,K]C={ϕ ∈C(R,R):0≤ϕ(t)≤K,t ∈R}.In the system (1.2),the first and the third equations can be solved independently onceu2(t,x) is determined.Thus,we first consider the equation foru2(t,x)

The corresponding nondiffusive system is

Foru ∈C(R,R),define

We always assume thatB(0)=0,U(0)=0 and

(H1) There is au∗2>0 such thatg(u∗2)=0,g(u)>0 foru ∈(0,u∗2) andg(u)<0 foru ∈(u∗2,∞).

According to the assumption (H1),(2.2) has a unique positive equilibriau∗2.Thus (1.2)has two spatially unform equilibriaE0(0,0,0) andE∗(u∗1,u∗2,u∗3),whereu∗1,u∗2,u∗3>0 and satisfy

In addition to (H1),we need more assumptions as follows

(H2)B(u) is differentiable and 0< |B′(u)| ≤k1foru ∈[0,u∗2] withk1>0;B′(u∗2)<0;B′(0)u −ρ1u1+ν1≤B(u)≤B′(0)uforu ∈[0,u∗2] with someν1∈(0,1],ρ1>0;

(H3)U(u) is differentiable and 0< U′(u)≤k2foru ∈[0,u∗2] withk2>0;U′(0)u ≤U(u)≤U′(0)u+ρ2u1+ν2foru ∈[0,u∗2] with someν2∈(0,1],ρ2>0;

(H4)εB′(0)−U′(0)>d.

Notice that the conditions (H2)-(H4) imposed on functionsB(u) andU(u) are natural,and they are not more restrictive conditions.For example,we take population growth rateB(u)=pue−auwith two positive constantsp,a>0,thenB′(u)=pe−au(1−au),B′(0)=pandB(u)≤B′(0)uforu ∈[0,1].If we takeU(u)=δu,thenU(u) satisfies the assumption(H3).However,it also follows thatprovidedthusB(u) is an nonmonotone function and satisfies the assumption (H2).

Sinceg(ϕ)=−dϕ(0)+εB(ϕ(−τ))−U(ϕ(0)) and the assumptions (H1)-(H3),for anyL ≥u∗2,gis global Lipschitz continuous and quasi-monotone on [0,K]Cin the sense that

for allϕ1,ϕ2∈[0,K]Cwithϕ1≥ϕ2.

In fact,it follows that

Hence,for anyh>0 with 1>h(d+k2),we have

from which (2.4) follows.

For (2.2),define the differential operatorL

For the linearized equation of (2.2) about zero,

The characteristic equation is

Lemma 2.1Letλ1be the unique real root of (2.6),forϵ ∈(0,λ1) sufficiently small,we have

(i) ForM=M(ϵ) sufficiently large,the function

witht1=−(logM)/ϵ,is a lower solution of(2.2),i.e.,ϕ∗: R→R is continuous,differentiable almost everywhere on R,and satisfiesLϕ∗≤0 a.e.t ∈R.

(ii) Moreover,0≤ϕ∗≤u∗2.

ProofLetM ≥1.Fort>t1,we haveLϕ∗(t)=−εB(ϕ∗(t −τ))≤0.Consider nowt ≤t1.It is easy to see that 0≤1−Meϵt<1 and 0<1−Meϵ(t−τ)<1 fort ≤t1.Since(H2),(H3) and (2.6) hold,we obtain

whereρ∗:=ρ1+ρ2,and we use the inequality eλ1(1+νi)s≤e(λ1+ϵ)sfor anys<0 andi=1,2 provided thatϵ>0 is sufficiently small.Since ∆(λ1+ϵ)>0,hence we obtain thatLϕ∗(t)≤0 ifM=M(ϵ) is chosen so thatM ≥1 andM ≥∆(λ1+ϵ)−1ρ∗.This proves (i).

Lett0be suchSinceϕ′∗(t0)=0,we haveλ1=M(λ1+ϵ)eϵt0,and therefore for 0<ϵ ≪1,

According to (2.3) and (2.4),gis quasi-monotone on [0,K]C.However,we can choose sufficiently largeh >0 such thathφ(s)+g(φ(·)) is non-decreasing function forφ ∈[0,K]C.Thus,we can define the operatorT:C(R,R)→C(R,R) by

Clearly,a positive functionφ(t)is a global bounded solution of(2.2)if and only ifφ=Tφ,t ∈R.Our goal in the remainder of the section is to show thatTis completely continuous on a suitable convex,closed set of a Banach space,in which we shall apply Schauder’s fixed point theorem to find a fixed point ofTsatisfying

withK=u∗2.

Definet0=(logK)/λ1.Using the assumptions (H2) and (H3),we have the following results.

Lemma 2.2For allφ ∈C(R,R),φnon-negative,thenTφis bounded and differentiable,with

Moreover,ifφ ∈C(R,R) with 0≤φ(t)≤eλ1t,t ≤t0,then for some positive constantk,0≤Tφ(t)≤keλ1t,t ≤t0,whereλ1is the unique positive real root of (2.6).

ProofRecall thathφ(s)+g(φ(·))is non-decreasing function.Consider any non-negativeφ ∈[0,K]C.Then,fort ∈R,

and (Tφ)′(t)=−hTφ(t)+hφ(t)+g(φ(·−τ)).It follows that|(Tφ)′(t)|≤hK.This proves(2.9).

From (H2),(H3) and the definition ofg,forφ ≥0,we have

withh′:=h+d+k2.Then,from (2.8),(2.9) and (2.10),we obtain

This completes the proof.

Letϕ∗be as in (2.7),withϵ>0 andM ≥1 chosen in Lemma 2.1.Then we have

Lemma 2.3The following statements hold.

(i)Tϕ∗(t)≥ϕ∗(t),for allt ∈R;

(ii) forφ ∈C(R,R) satisfyingϕ∗(t)≤φ(t)≤K,t ∈R,thenϕ∗(t)≤Tφ(t),t ∈R.

ProofDefineϕ1:=Tϕ∗.We have

Letw(t)=ϕ1(t)−ϕ∗(t).Since (2.11) andϕ∗is a lower solution of (2.2),it follows that

andr(t) is continuous and bounded from Lemma 2.2.We obtain

for some constantc ∈R.On the other hand,w(t) is bounded on R,implying thatc=0.Hencew(t)≥0 fort ∈R.This completes the proof of (i).

Notice thathφ+g(φ(·−τ)) is non-decreasing for anyφ ∈C(R,R).Forϕ∗(t)≤φ(t)≤K,t ∈R,by (i) we obtain

and (ii) follows immediately.

Define

We equipped the spaceC(R,R) with the norm whereρ ∈(0,min{λ1,h}).Thus (C(R,R),||·||) is a Banach space.

Lemma 2.4The setSis||·||ρ-closed,convex and non-empty.

ProofFrom Lemma 2.1,we haveϕ∗(t)≤eλ1tandϕ∗(t)≤u∗2(t),t ∈R,thusϕ∗(t)∈S.It is clear thatSis convex and||φ||ρ≤Kforφ ∈S.Since the||φ||ρconvergence implies the uniform convergence in any compact set of R,it follows thatSis||·||ρ-closed.

Lemma 2.5Consider the spaceC(R,R),equipped with the norm|| · ||ρ.Then,T:S →C(R,R) is lipschitz continuous.

ProofConsiderφ,ψ ∈S.Fort ≤τ,

With a simple computation,we have

Fort>τ,

Hence we have

Lemma 2.6ForSdefined in (2.12),the setT(S) is relatively compact in (C(R,R),||·||ρ).

ProofFor any compact intervalI ∈Sandφn∈I,letψn=Tφn,n ∈N.From Lemma 2.2,(ψn) is uniformly bounded on R and equicontinuous.By Ascoli-Arzel`a theorem,there is a subsequence of (ψn) which converges uniformly onIto someψI∈C(I,R).DenoteIk=[−k,k],k ∈N.We take a convergent subsequences (ψαk(n)) such that (ψαk(n)) is a subsequences of (ψα(k−1)(n)) andαk: N→N is increasing.It follows thatψαk(n)→ψkuniformly onIkandψk+1|Ik=ψkfork ≥1.Defineϕ ∈C(R,R) byϕ(t)=ψk(t) for|t|≤k,t ∈R.

Now we show that the“diagonal”subsequence(ψαn(n))convergence toϕ(t)with respect to the norm||·||ρ.Letϵ >0 be given.Choosen0∈N such that e−ρn0≤ϵ/K.By Lemma 2.2,0≤ψαn(n),ϕ(t)≤K,thus if|t|≥n0we have

On the other hand,ψαn(n)→ϕ(t) uniformly on [−n0,n0].Consequently,there existsn1≥n0such that

forn ≥n1and|t|≤n0.Hence|ψαn(n)−ϕ(t)|e−ρ|t|→0.This completes the proof.

Theorem 2.1Assume that conditions(H1)-(H4)are satisfied.Then,there is a positive solutionu(t) of (2.2),defined on R and satisfyingu(−∞)=0 andu(t)=O(eλ1t)ast →−∞,whereλ1is the positive root of (2.6).Furthermore,if there exists a globally attractive equilibriumu∗∈(0,u∗2],there is a positive heteroclinic solution of (2.1) connecting 0 tou∗.

ProofConsiderSas in (2.12).From Lemmas 2.1-2.3,T(S)⊂S.From Lemma 2.4 and Lemma 2.5,T:S →Sis||·||ρcompletely continuous.Lemma 2.6 allows us to use the Schauder’s fixed-point theorem to conclude that there isu ∈Ssuch thatTu=u.Thus,u(t)is a positive global solution of (2.2) satisfyingϕ∗(t)≤u(t)≤eλ1tfort ≤t0.Moreover,ifu∗(t) is globally attractive,it follows that limt→∞u(t)=u∗(t).This complete the proof.

We need another lemma which is cited from [6].

Lemma 2.7[6]Assume that (i) the functionalV:C([−τ,0],Rn)→R is continuous,V(0)=0;

(ii) there exist nonnegative and continuous functionsu(s) andv(s) such thatu(s)→∞(s →∞),v(0)=0;

(iii)u(|ϕ(0)|)≤ V(ϕ) forϕ ∈C;

(iv) ˙V(ϕ)≤ −v(|ϕ(0)|) forϕ ∈C,where

Then all solutions of (2.2) are bounded and the zero solution of (2.2) is stable.If in addition,v(s) is positive definite,then the all solutions of (2.2) tend to zero ast →+∞.

Using Lemma 2.7,we can prove the existence of positive heteroclinic solutions of the given model.

Theorem 2.2Assume that (H1)-(H3) hold.Furthermore,if there exists a positive constantksatisfying

then equation (2.2) has a heteroclinic solutionu∗such that

ProofConsider the initial problem

whereλ0is the positive real root ofΛ1(λ)=0.Express the solution of (2.17) and (2.18) asuT(t),t ∈R.For allT ∈(−∞,0],we obtain a set of functions{uT(t)}T∈(−∞,0].Define

Thenu∗(t) satisfies the following Properties.

(1o)u∗(t) is a solution of (2.2);

(2o)

Hence{uT(t)}T∈(−∞,0]is equi-continuous on R.For anyN >0,{uT(t)}T∈(−∞,0]has subsequence (without loss of generality,we may assume that it is{uT(t)}T∈(−∞,0]itself) which is uniformly convergent on [−N,N].Suppose that the limit function isu∗(t).SinceNis arbitrary,noting the definition of{uT(t)}T∈(−∞,0],we claim thatu∗(t) is defined on R,and is a solution of (2.2).

For anyϵ>0,choosingT <0,if|T|is large enough andt

Therefore,we obtain

Letx(t)=u(t)−u∗(t),t ∈R.Then the equation forxis

Define a functional

Then we have

Calculating the right derivative along the solutions of (2.21),we obtain

wher eζ(t) is betweenx(t)+u∗(t) andu∗(t) fort ∈R.From (2.15),we haveandk >12.Noting that 0

Defineu(s) :=s2,andv(s) :=[(2k −1)−k22]s2.Then

andv(s) is positive definite.On the other hand,we obtain from (2.23) that

Therefore,by Lemma 2.7,we know that any solutionu(t)=x(t)+u∗(t) of (2.2) tends tou∗(t) ast →∞.Thus (2o) holds.

We conclude from (1o) and (2o) thatu∗(t) is a solution of (2.2) satisfying (2.16).This completes the proof.

3.Existence of Traveling Wave Solution

Now we are in a position to study traveling wave solutions for the reaction-diffusive SIR model(2.1).To the end we shall use the method developed in[4].The results obtained in the paper tell us if the nondiffusive equation has a heteroclinic connection betweenE1andE2,then the diffusive system has a family of traveling wavefronts fromE1toE2with large speed.For convenience of discussion,we denote two positive equilibria byE1=0,E2=u∗2(t),respectively.We have the following results.

Lemma 3.1E1is hyperbolic.

ProofConsider the characteristic equationΛ1(λ)=0 of (2.1) atE1,where

Since

We know thatΛ1(λ) is an increasing function with respect toλ,andΛ1(λ)=0 has a positive real rootλ0>0.Therefore the unstable manifold associated withE1is at least one dimensional.Note that the equationΛ1(λ)=0 has only finite rootsλwith Reλ >0.ThusΛ1(λ)=0 has exactm(m ≥1) roots with positive real parts.SinceΛ1(iβ)=0 (β >0) is equivalent to

which leads to

We obtain from (3.1) thatβτis in the first quarter,and

wheren ∈N0:={0}∪N.LetIf 0≤τ <τ′,thenE1is hyperbolic.

LetΛ2(λ)=λ+d+U′(u∗2)−εB′(u∗2)e−λτ,andλ=α+iβ,then we have the following result.

Lemma 3.2All roots ofΛ2(λ)=0 have negative real parts.

ProofFromΛ2(λ)=0,we have

If|εB′(u∗2)|≤d+U′(u∗2),the the first equation in (3.2) can not have nonnegative solutionα.In fact,if there isα>0 such that (3.2) holds,then we have

which is a contradiction.Ifα=0,then we haved+U′(u∗2)=e−ατB′(u∗2)cosβτ,which can not hold either forβ >0 orβ=0.Thus all zeros ofΛ2(λ) have negative real parts.

IfεB′(u∗2)|>d+U′(u∗2),we can also show that Reλ<0 for all roots ofΛ2(λ)=0 whileτis sufficiently small.However,ifτ=0,we haveα+d+U′(u∗2)=εB′(u∗2),which leads toα<0.Letα=0,β >0,then (3.2) leads to

We obtain from (3.3) and (H1) thatβτis in the second quarter,and

wheren ∈N0.LetTherefore if 0≤ τ < τ′′,then all roots ofΛ2(λ)=0 have negative real parts.This completes the proof.

Summarizing the above discussion,we obtain the following results.

Theorem 3.1Assume that (H1),(H2) and (H3) hold.Then as either|εB′(u∗2)| ≤d+U′(u∗2),0≤τ < τ′,or|εB′(u∗2)| > d+U′(u∗2),0≤τ < τ′′,ifτ∗:=min{τ′,τ′′},(H2)and (H3) of Theorem 1.1 in [4] are satisfied for (2.2).

Theorem 3.2Assume the assumptions (H1)-(H4) hold.Letτ∗=min{τ′,τ′′},where

Then as either|εB′(u∗2)|≤d+U′(u∗2),0≤τ <τ∗,or|εB′(u∗2)| ≥d+U′(u∗2),0≤τ < τ∗,there exists a constantc∗>0,such that for everyc > c∗,the equation (2.1) has a traveling wave,which connects the trivial equilibriumE1to the positive equilibriumE2.

ProofNotice that if there is no diffusion,the equation (2.1) reduces to (2.2).From Lemma 3.1,Lemma 3.2,Theorem 3.1,we know that the equilibriaE1andE2are hyperbolic,and,in particular,all the eigenvalues toE2have negative real parts.From Theorem 3.2,the equation (2.2) has a heteroclinic connection.Thus if 0≤ τ < τ∗,the conditions (H1),(H2) and (H3) of Theorem 1.1 in [4] are satisfied.In fact,for our kernel functionfα(x),for instance,it is easy to see that

So all conditions of Theorem 1.1 in[4]are satisfied.Hence by Theorem 1.1 in[4],we conclude that there exists a constantc∗>0 so that for anyc > c∗,the equation (2.1) has a traveling wave solution which connectsE1toE2.This completes the proof.

Remark 3.1In fact,as a consequence of Theorem 1.1 in [4],for eachc>c∗,the set of all traveling wave solutions of (2.1) connecting zero toE2and propagating at speedcforms aC1-smoothM-dimensional manifold in someC([−τ,0],R)-neighborhood of the heteroclinic solution in Theorem 3.2.

Now we return to study the first and the third equations of the given SIR model (1.2),

Lets=x+ctandφ(s)=u2(x+ct) be the traveling wave solution of (2.1).Defineϕ(z) :=u1(x+ct) andψ(z) :=u3(x+ct).We have wave profile equations for (1.2)

where

It is easy to see that equations (3.5) are independent DDEs with boundary value such that

and

whereu∗2is the unique positive root ofg(u)=0.Thus,we have the following result.

Theorem 3.3Let the conditions (H1)-(H4) hold.Then there exists a constantc∗>0,such that for everyc>c∗,the equations(1.2)have traveling wave solution(φ(s),ϕ(s),ψ(s))connecting the trivial equilibriumE0(0,0,0) to the positive equilibriumE∗(u∗1,u∗2,u∗3) withs=x+ctandφ(·)∈[0,K]C.