Citation: Ran Zhang, Shengqiang Liu. Traveling waves for SVIR epidemic model with nonlocal dispersal[J]. Mathematical Biosciences and Engineering, 2019, 16(3): 1654-1682. doi: 10.3934/mbe.2019079
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As one of the most basic models in modeling infectious diseases, the SIR epidemiological model was introduced by Kermack and McKendrick [1] in 1927. Since then, a lot of differential equations have been studied as models for the spread of infectious diseases. Considering a continuous vaccination strategy, let
{dS(t)dt=Λ−β1S(t)I(t)−αS(t)−μ1S(t),dV(t)dt=αS(t)−β2V(t)I(t)−(γ1+μ1)V(x,t),dI(t)dt=β1S(t)I(t)+β2V(t)I(t)−γI(x,t)−μ3I(x,t),dR(t)dt=γ1V(t)+γI(t)−μ1R(t), | (1.1) |
where
On the other hand, in order to understand the geographic spread of infectious disease, the spatial effect would give insights into disease spread and control. Due to this fact, many literatures have studied the spatial effects on epidemics by using reaction-diffusion equations (see, for instance, [8,9,10,11,12,13,14,15,16,17] and the references therein). In the study of reaction-diffusion models, the Laplacian operator describes the random diffusion of each individual, but it can not describe the long range diffusion. Therefore, a nonlocal dispersal term has been established, which is by a convolution operator:
J∗ϕ(x)−ϕ(x)=∫IRJ(x−y)ϕ(y)dy−ϕ(x), | (1.2) |
where
Recently, Li et al. [23] proposed a nonlocal dispersal SIR model with delay:
{∂S(x,t)∂t=d1(J∗S(x,t)−S(x,t))+Λ−β1S(x,t)I(x,t−τ)1+θI(x,t−τ)−μ1S(x,t),∂I(x,t)∂t=d2(J∗I(x,t)−I(x,t))+β1S(x,t)I(x,t−τ)1+θI(x,t−τ)−γI(x,t)−μ3I(x,t),∂R(x,t)∂t=d3(J∗R(x,t)−R(x,t))+γI(x,t)−μ1R(x,t), | (1.3) |
where
Motivated by [2] and [23], in this paper, we consider a nonlocal dispersal epidemic model with vaccination and delay. Precisely, we study the following model.
{∂S(x,t)∂t=d1(J∗S(x,t)−S(x,t))+Λ−β1S(x,t)I(x,t−τ)−αS(x,t)−μ1S(x,t),∂V(x,t)∂t=d2(J∗V(x,t)−V(x,t))−β2V(x,t)I(x,t−τ)+αS(x,t)−(γ1+μ1)V(x,t),∂I(x,t)∂t=d3(J∗I(x,t)−I(x,t))+β1S(x,t)I(x,t−τ)+β2V(x,t)I(x,t−τ)−γI(x,t)−μ3I(x,t),∂R(x,t)∂t=d4(J∗R(x,t)−R(x,t))+β1S(x,t)I(x,t−τ)+γ1V(x,t)+γI(x,t)−μ4R(x,t), | (1.4) |
where
Assumption 1.1. [23,24] The kernel function
(J1)
(J2) There exists a constant
∫IRJ(x)e−λxdx<+∞, for any λ∈[0,λM) |
and
limλ→λM−0∫IRJ(x)e−λxdx→+∞. |
The organization of this paper is as follows. In section 2, we proved the existence of traveling wave solutions of (1.4) for
In this section, we study the existence of traveling wave solutions of system (1.4). Since we have assumed that the recovered have gained permanent immunity and
{∂S(x,t)∂t=d1(J∗S(x,t)−S(x,t))+Λ−β1S(x,t)I(x,t−τ)−αS(x,t)−μ1S(x,t),∂V(x,t)∂t=d2(J∗V(x,t)−V(x,t))−β2V(x,t)I(x,t−τ)+αS(x,t)−μ2V(x,t),∂I(x,t)∂t=d3(J∗I(x,t)−I(x,t))+(β1S(x,t)+β2V(x,t))I(x,t−τ)−γI(x,t)−μ3I(x,t). | (2.1) |
where
ℜ0=β1S0+β2V0μ3+γ. | (2.2) |
Furthermore, there exists another equilibrium
{Λ−β1S∗I∗−αS∗−μ1S∗=0,β2V∗I∗+αS∗−μ2V∗=0,(β1S∗+β2V∗)I∗−γI∗−μ3I∗=0. | (2.3) |
From [2,Theorem 2.1], system (2.1) has a unique positive equilibrium
Let
{cS′(ξ)=d1(J∗S(ξ)−S(ξ))+Λ−β1S(ξ)I(ξ−cτ)−αS(ξ)−μ1S(ξ),cV′(ξ)=d2(J∗V(ξ)−V(ξ))+αS(ξ)−β2V(ξ)I(ξ−cτ)−μ2V(ξ),cI′(ξ)=d3(J∗I(ξ)−I(ξ))+β1S(ξ)I(ξ−cτ)+β2V(ξ)I(ξ−cτ)−γI(ξ)−μ3I(ξ). | (2.4) |
We want to find traveling wave solutions with the following asymptotic boundary conditions:
limξ→−∞(S(ξ),V(ξ),I(ξ))=(S0,V0,0) | (2.5) |
and
limξ→+∞(S(ξ),V(ξ),I(ξ))=(S∗,V∗,I∗). | (2.6) |
Consider the following linear system of system (2.4) at infection-free equilibrium
cI′(ξ)=d3(J∗I(ξ)−I(ξ))+β1S0I(ξ−cτ)+β2V0I(ξ−cτ)−(γ+μ3)I(ξ). | (2.7) |
Let
Δ(λ,c)≜d3∫IRJ(x)e−λxdx−(d3+γ+μ3)−cλ+β1S0e−cτλ+β2V0e−cτλ=0. | (2.8) |
By some calculations, we obtain
Δ(0,c)=β1S0+β2V0−γ−μ3, limc→+∞Δ(λ,c)=−∞ for λ>0,∂Δ(λ,c)∂λ|(0,c)=−c−cτ(β1S0+β2V0)<0 for c>0,∂Δ(λ,c)∂c=−λ−τλe−cτλ(β1S0+β2V0)<0 for λ>0,∂2Δ(λ,c)∂λ2=d3∫IRJ(x)x2e−λxdx+(cτ)2e−cτλ(β1S0+β2V0)>0. |
For any
Lemma 2.1. Let
(ⅰ) If
(ⅱ) If
(ⅲ) If
Denote
0<λc<λ∗<λ2(c)<λc,τ. |
For the followings in this section, we always fix
{¯S(ξ)=S0,¯V(ξ)=V0,¯I(ξ)=eλcξ, {S_(ξ)=max{S0−M1eε1ξ,0},V_(ξ)=max{V0−M2eε2ξ,0},I_(ξ)=max{eλcξ(1−M3eε3ξ),0}, |
where
Lemma 2.2. The function
cI′(ξ)≥d3(J∗I(ξ)−I(ξ))+β1S0I(ξ−cτ)+β2V0I(ξ−cτ)−γI(ξ)−μ3I(ξ). | (2.9) |
Lemma 2.3. The functions
{cS′(ξ)≥d1(J∗S(ξ)−S(ξ))+Λ−β1S(ξ)I_(ξ−cτ)−αS(ξ)−μ1S(ξ),cV′(ξ)≥d2(J∗V(ξ)−V(ξ))+αS(ξ)−β2V(ξ)I_(ξ−cτ)−μ2V(ξ). | (2.10) |
The proof is trivial, so we omitted the above two lemmas.
Lemma 2.4. For each
cS′(ξ)≤d1(J∗S(ξ)−S(ξ))+Λ−β1S(ξ)¯I(ξ−cτ)−(μ1+α)S(ξ), | (2.11) |
with
Proof. See Appendix A.
Lemma 2.5. For each
cV′(ξ)≤d2(J∗V(ξ)−V(ξ))+αS_(ξ)−β2V(ξ)¯I(ξ−cτ)−μ2V(ξ), | (2.12) |
with
The proof is similar with Lemma 2.4.
Lemma 2.6. Let
cI′(ξ)≤d3(J∗I(ξ)−I(ξ))+β1S_(ξ)I(ξ−cτ)+β2V_(ξ)I(ξ−cτ)−γI(ξ)−μ3I(ξ), | (2.13) |
with
Proof. See Appendix B.
Let
ΓX={(ϕφψ)∈C([−X,X],IR3)|S_(ξ)≤ϕ(ξ)≤S0, ϕ(−X)=S_(−X), for ξ∈[−X,X];V_(ξ)≤φ(ξ)≤V0, φ(−X)=V_(X), for ξ∈[−X,X];I_(ξ)≤ψ(ξ)≤¯I(ξ), ψ(−X)=I_(X), for ξ∈[−X,X].}. |
For given
ˆϕ(ξ)={ϕ(X), for ξ>X,ϕ(ξ), for ξ∈[−X−cτ,X],S_(ξ), for ξ≤−X−cτ, ˆφ(ξ)={φ(X), for ξ>X,φ(ξ), for ξ∈[−X−cτ,X],V_(ξ), for ξ≤−X−cτ, |
and
ˆψ(ξ)={ψ(X), for ξ>X,ψ(ξ), for ξ∈[−X−cτ,X],I_(ξ), for ξ≤−X−cτ. |
We have
{S_(ξ)≤ˆϕ(ξ)≤S0,V_(ξ)≤ˆφ(ξ)≤V0,I_(ξ)≤ˆψ(ξ)≤¯I(ξ). |
For any
{cS′(ξ)=d1∫IRJ(y)ˆϕ(ξ−y)dy+Λ−β1S(ξ)ψ(ξ−cτ)−(d1+μ1+α)S(ξ),cV′(ξ)=d2∫IRJ(y)ˆφ(ξ−y)dy+αϕ(ξ)−β2V(ξ)ψ(ξ−cτ)−(d2+μ2)V(ξ),cI′(ξ)=d3∫IRJ(y)ˆψ(ξ−y)dy+β1ϕ(ξ)ψ(ξ−cτ)+β2φ(ξ)ψ(ξ−cτ)−(d3+γ+μ3)I(ξ),S(−X)=S_(−X), V(−X)=V_(−X), I(−X)=I_(−X). | (2.14) |
From the standard theory of functional differential equations (see [32]), the initial value problem (2.14) admits a unique solution
(SX,VX,IX)∈C1([−X,X]), |
this defines an operator
SX=A1(ϕ,φ,ψ), VX=A2(ϕ,φ,ψ), IX=A3(ϕ,φ,ψ). |
Next we show the operator
Lemma 2.7. The operator
Proof. Firstly, we show that
cS_′(ξ)−d1∫IRJ(y)ˆϕ(ξ−y)dy−Λ+β1S_(ξ)ψ(ξ−cτ)−(d1+μ1+α)S_(ξ)≤cS_′(ξ)−d1∫IRJ(y)S_(ξ−y)dy−Λ+β1S_(ξ)¯I(ξ−cτ)−(d1+μ1+α)S_(ξ))≤0, |
which implies that
Secondly, we show that
d1∫IRJ(y)ˆϕ(ξ−y)dy+Λ−β1S0ψ(ξ−cτ)−(d1+μ1+α)S0≤d1∫IRJ(y)S0dy+Λ−β1S0I_(ξ−cτ)−(d1+μ1+α)S0≤0, |
thus
Similarly,
Lemma 2.8. The operator
Proof. Suppose
SX,i(ξ)=A1(ϕi(ξ),φi(ξ),ψi(ξ)),VX,i(ξ)=A2(ϕi(ξ),φi(ξ),ψi(ξ)),IX,i(ξ)=A3(ϕi(ξ),φi(ξ),ψi(ξ)), |
We show the operator
SX(ξ)=S_(−X)exp{−1c∫ξ−X(d1+μ1+α+β1ψ(s−cτ))ds}+1c∫ξ−Xexp{−1c∫ξη(d1+μ1+α+β1ψ(s−cτ))ds}fϕ(η)dη, |
VX(ξ)=V_(−X)exp{−1c∫ξ−X(d2+μ2+β2ψ(s−cτ))ds}+1c∫ξ−Xexp{−1c∫ξη(d2+μ2+β2ψ(s−cτ))ds}fφ(η)dη, |
and
IX(ξ)=I_(−X)exp{−(d3+γ+μ3)(ξ+X)c}+1c∫ξ−Xexp{−(d3+γ+μ3)(ξ−η)c}fψ(η)dη. |
where
fϕ(η)=d1∫IRJ(η−y)ˆϕ(y)dy+Λ,fφ(η)=d2∫IRJ(η−y)ˆφ(y)dy+αϕ(η),fψ(η)=d3∫IRJ(η−y)ˆψ(y)dy+(β1ϕ(η)+β2φ(η))ψ(η−cτ). |
For any
|fϕ1(η)−fϕ2(η)|=d1|∫IRJ(η−y)[ˆϕ1(y)−ˆϕ2(y)]dy|≤d1|∫X−XJ(ξ−y)(ϕ1(y)−ϕ2(y))dy|+d1|∫∞XJ(ξ−y)(ϕ1(X)−ϕ2(X))dy|≤2d1maxy∈[−X,X]|ϕ1(y)−ϕ2(y)|, |
|fφ1(η)−fφ2(η)|=d2|∫IRJ(η−y)[ˆφ1(y)−ˆφ2(y)]dy+α(ϕ1(η)−ϕ2(η))|≤2d2maxy∈[−X,X]|φ1(y)−φ2(y)|+αmaxy∈[−X,X]|ϕ1(y)−ϕ2(y)|, |
|fψ1(η)−fψ2(η)|≤(2d2+β1S0+β2V0)maxy∈[−X,X]|ψ1(y)−ψ2(y)|+β1eλcξmaxy∈[−X,X]|ϕ1(y)−ϕ2(y)|+β2eλcξmaxy∈[−X,X]|φ1(y)−φ2(y)|. |
Here we use
|β1ϕ2(ξ)ψ2(ξ−cτ)−β1ϕ1(ξ)ψ1(ξ−cτ)|≤|β1ϕ2(ξ)ψ2(ξ−cτ)−β1ϕ2(ξ)ψ1(ξ−cτ)|+|β1ϕ2(ξ)ψ1(ξ−cτ)−β1ϕ1(ξ)ψ1(ξ−cτ)|≤β1S0maxy∈[−X,X]|ψ1(y)−ψ2(y)|+β1eλcξmaxy∈[−X,X]|ϕ1(y)−ϕ2(y)|. |
and
|β2φ2(ξ)ψ2(ξ−cτ)−β2φ1(ξ)ψ1(ξ−cτ)|≤β2V0maxy∈[−X,X]|ψ1(y)−ψ2(y)|+β2eλcξmaxy∈[−X,X]|φ1(y)−φ2(y)|. |
Thus, we obtain that the operator
c(S′X,1(ξ)−S′X,2(ξ))+(d1+μ1+α)(SX,1(ξ)−SX,2(ξ))=d1∫IRJ(ξ−y)(ˆϕ1(y)−ˆϕ2(y))dy+β1ϕ2(ξ)ψ2(ξ−cτ)−β1ϕ1(ξ)ψ1(ξ−cτ)≤(2d1+β1eλcξ)maxy∈[−X,X]|ϕ1(y)−ϕ2(y)|+β1S0maxy∈[−X,X]|ψ1(y)−ψ2(y)|. |
Same arguments with
Obviously,
Theorem 2.1. There exists
(SX(ξ),VX(ξ),IX(ξ))=A(SX,VX,IX)(ξ) |
for
Now we are in position to show the existence of traveling wave solutions, before that we do some estimates for
Define
C1,1([−X,X])={u∈C1([−X,X])|u,u′are Lipschitz continuous} |
with norm
‖u‖C1,1([−X,X])=maxx∈[−X,X]|u|+maxx∈[−X,X]|u′|+supx,y∈[−X,X]x≠y|u′(x)−u′(y)||x−y|. |
Lemma 2.9. There exists a constant
‖SX‖C1,1([−Y,Y])≤C(Y), ‖VX‖C1,1([−Y,Y])≤C(Y), ‖IX‖C1,1([−Y,Y])≤C(Y) |
for
Proof. Recall that
cS′X(ξ)=d1∫+∞−∞J(y)ˆSX(ξ−y)dy+Λ−β1SX(ξ)IX(ξ−cτ)−(d1+μ1+α)SX(ξ), | (2.15) |
cV′X(ξ)=d2∫+∞−∞J(y)ˆVX(ξ−y)dy+αSX(ξ)−β2VX(ξ)IX(ξ−cτ)−(d2+μ2)VX(ξ), | (2.16) |
cI′X(ξ)=d3∫+∞−∞J(y)ˆIX(ξ−y)dy+β1SX(ξ)IX(ξ−cτ)+β2VX(ξ)IX(ξ−cτ)−(d3+μ3)IX(ξ), | (2.17) |
where
(ˆSX(ξ),ˆVX(ξ),ˆIX(ξ))={(SX(X),VX(X),IX(X)), for ξ>X,(SX(ξ),VX(ξ),IX(ξ)), for ξ∈[−X−cτ,X],(S_(ξ),V_(ξ),I_(ξ)), for ξ≤−X−cτ, |
following that
|S′X(ξ)|≤d1c|∫+∞−∞J(y)ˆSX(ξ−y)dy|+Λc+d1+μ1+αc|SX(ξ)|+β1c|SX(ξ)||IX(ξ−cτ)|≤2d1+μ1+αcS0+Λc+β1S0ceλcY,|V′X(ξ)|≤2d2+μ2cV0+αS0c+β2V0ceλcY,|I′X(ξ)|≤(d3+μ3c+β1S0c+β2V0c)eλcY. |
Thus, there exists some constant
‖SX‖C1([−Y,Y])≤C1(Y), ‖VX‖C1([−Y,Y])≤C1(Y), ‖IX‖C1([−Y,Y])≤C1(Y). |
Then for any
|SX(ξ1)−SX(ξ2)|≤C1(Y)|ξ1−ξ2|, |VX(ξ1)−VX(ξ2)|≤C1(Y)|ξ1−ξ2|, |IX(x1)−IX(x2)|≤C1(Y)|ξ1−ξ2|. |
From (2.15), we have
c|S′X(ξ1)−S′X(ξ2)|≤d1|∫+∞−∞J(y)(ˆSX(ξ1−y)−ˆSX(ξ2−y))dy|+(d1+μ1+α)|SX(ξ1)−SX(ξ2)|+S0|IX(ξ1)−IX(ξ2)|. |
Recall (J1) of Assumption 1.1, we know
d1|∫+∞−∞J(y)(ˆSX(ξ1−y)−ˆSX(ξ2−y))dy|=d1|∫R−RJ(y)ˆSX(ξ1−y)dy−∫R−RJ(y)ˆSX(ξ2−y)dy|=d1|∫ξ1+Rξ1−RJ(ξ1−y)ˆSX(y)dy−∫ξ2+Rξ2−RJ(y)ˆSX(y)dy|=d1|(∫ξ2−Rξ1−R+∫ξ2+Rξ2−R+∫ξ1+Rξ2+R)J(ξ1−y)ˆSX(y)dy−∫ξ2+Rξ2−RJ(y)ˆSX(y)dy|≤d1|∫ξ1+Rξ2+RJ(ξ1−y)ˆSX(y)dy|+d1|∫ξ2−Rξ1−RJ(ξ1−y)ˆSX(y)dy|+d1|∫ξ2+Rξ2−R(J(ξ1−y)−J(ξ2−y))ˆSX(y)dy|≤d1(2S0‖J‖L∞+2RLS0)|ξ1−ξ2|. |
Thus there exists some constant
|S′X(ξ1)−S′X(ξ2)|≤C2(Y)|ξ1−ξ2|. |
Similarly
|V′X(ξ1)−V′X(ξ2)|≤C2(Y)|ξ1−ξ2|, |I′X(ξ1)−I′X(ξ2)|≤C2(Y)|ξ1−ξ2|. |
From the above discussion, there exists some constant
‖SX‖C1,1([−Y,Y])≤C(Y), ‖VX‖C1,1([−Y,Y])≤C(Y), ‖IX‖C1,1([−Y,Y])≤C(Y). |
Now let
SXnk→S, VXnk→V and IXnk→I in C1loc(IR) as k→+∞. |
Since
limk→+∞∫IRJ(y)ˆSXnk(ξ−y)dy=∫IRJ(y)S(ξ−y)dy=J∗S(ξ), |
limk→+∞∫IRJ(y)ˆVXnk(ξ−y)dy=∫IRJ(y)V(ξ−y)dy=J∗V(ξ) |
and
limk→+∞∫IRJ(y)ˆIXnk(ξ−y)dy=∫IRJ(y)I(ξ−y)dy=J∗I(ξ). |
Moreover,
S_(ξ)≤S(ξ)≤S0, V_(ξ)≤V(ξ)≤V0, I_(ξ)≤I(ξ)≤eλcξ. |
Next, we show that
Lemma 2.10. There exists some positive constant
∫IRJ(y)I(ξ−y)I(ξ)dy<C, I(ξ−cτ)I(ξ)<C and |I′(ξ)I(ξ)|<C. |
Proof. Let
θ(ξ)≥d3c(∫IRJ(y)I(ξ−y)I(ξ)dy−1)−γ+μ3c=d3c∫IRJ(y)e∫ξ−yξθ(s)dsdy−(d3+γ+μ3c). |
Set
W′(ξ)=(ϖ+θ(ξ))W(ξ)≥d3c∫IRJ(y)e∫ξ−yξθ(s)dsdyW(ξ), |
that is
W(ξ)≥d3cR0∫IRJ(y)eϖyW(ξ−R0−y)dy |
and
W(ξ+R0)≤σ0W(ξ) for all ξ∈IR, |
where
σ0≜d3cR0∫−2R0−∞J(y)eϖydy. |
Thus
∫IRJ(y)I(ξ−y)I(ξ)dy=∫0−∞J(y)I(ξ−y)I(ξ)dy+∫+∞0J(y)I(ξ−y)I(ξ)dy=∫0−∞J(y)eϖyW(ξ−y)W(ξ)dy+∫+∞0J(y)eϖyW(ξ−y)W(ξ)dy≤σ0∫0−∞J(y)eϖyW(ξ−y−R0)W(ξ)dy+∫+∞0J(y)eϖydy≤cσ0d3R0+∫+∞0J(y)eϖydy. |
Again with the third equation of (2.4), we have
I′(ξ)+ϖI(ξ)=d3J∗I(ξ)+β1S(ξ)I(ξ−cτ)+β2V(ξ)I(ξ−cτ)>0 for all ξ∈IR. |
Let
I(ξ−cτ)I(ξ)≤eϖcτ. |
Furthermore,
|I′(ξ)I(ξ)|≤d3c∫IRJ(y)I(ξ−y)I(ξ)dy+(β1S0+β2V0)I(ξ−cτ)I(ξ)+ϖ. |
This completes the proof.
Lemma 2.11. Choose
Proof. Assume that there is a subsequence of
S′k(ξ)≤2d1S0+Λc∗≜delta0 in IR. |
It follows that
Sk(ξ)≥ε2, ∀ξ∈[ξk−delta,ξk], |
for all
Ik(ξk)Ik(ξ−cτ)=exp{∫ξkξ−cτI′k(s)Ik(s)ds}≤eC0(cτ+delta), ∀ξ∈[ξk−delta,ξk] |
for all
minξ∈[ξk−delta,ξk]Ik(ξ−cτ)≥e−C0(cτ+delta)Ik(ξk), |
which give us
minξ∈[ξk−delta,ξk]Ik(ξ−cτ)→+∞ as k→+∞ |
since
maxξ∈[ξk−delta,ξk]S′k(ξ)≤delta0−β1ε2minξ∈[ξk−delta,ξk]Ik(ξ−cτ)→−∞ as k→+∞. |
Moreover, there exists some
S′k(ξ)≤−2S0delta, ∀k≥K and ξ∈[ξk−delta,ξk]. |
Note that
Lemma 2.12. If
The proof is similar to that of [34,Lemma 2.4], so we omit the details. With the previous lemmas, we can show that
Theorem 2.2.
Proof. Assume that
cθ(ξ)=d3∫IRJ(y)e∫ξ−yξθ(s)ds−(d3+γ+μ3)+B(ξ), |
where
B(ξ)=[β1S(ξ)+β2V(ξ)]I(ξ−cτ)I(ξ). |
Since
f(λ,c)≜d3(∫IRJ(y)e−λy−1)−cλ−(γ+μ3). |
By some calculations, we obtain
f(0,c)<0, ∂f(λ,c)∂λ|λ=0<0, ∂2f(λ,c)∂λ2>0 and limλ→+∞f(λ,c)=−∞. |
Thus,
Since
Proposition 2.1.
Λμ1+α+β1ρ≤S(ξ)≤S0, αΛ(μ1+α+β1ρ)(μ2+β2ρ)≤V(ξ)≤V0, I_(ξ)≤I(ξ)≤ρ |
for
The following lemma is to show that
Lemma 2.13. Assume that
lim infξ→∞I(ξ)>0. |
Proof. We only need to show that if
Sk(ξ)≜S(ξk+ξ), Vk(ξ)≜V(ξk+ξ) and Ik(ξ)≜I(ξk+ξ). |
Thus we have
Let
ψ′k(ξ)=I′k(ξ)Ik(0)=I′k(ξ)Ik(ξ)ψk(ξ), |
we have
cψ′∞(ξ)=d3∫IRJ(y)ψ∞(ξ−y)dy+(β1S0+β2V0)ψ∞(ξ−cτ)−(d3+γ+μ3)ψ∞(ξ). |
One can have
0=d3∫IRJ(y)ψ∞(ξ0−y)dy+(β1S0+β2V0)ψ∞(ξ0−cτ)>0, |
which is a contradiction.
Denote
cZ(ξ)=d3∫IRJ(y)e∫ξ−yξZ(s)dsdy+(β1S0+β2V0)e∫ξ−cτξZ(s)ds−(d3+γ+μ3). | (2.18) |
Then by similar discussion in [25,Theorem 2.9], for
0<ψ′∞(0)=limk→+∞ψ′n(0)=limk→+∞I′n(0)In(0). |
Thus,
Remark 2.1. In the proof of Lemma 2.13, we need to show that
Now, we can give the main result in this section.
Theorem 2.3. Suppose
Proof. First, it is easy to verify that
Next, we will show
Let
α+(y)≡0 and α−(y)≡0 for |y|≥R. | (2.19) |
Define the following Lyapunov functional
L(S,V,I)(ξ)=cS∗L1(ξ)+cV∗L2(ξ)+cI∗L3(ξ)+d1S∗U1(ξ)+d2V∗U2(ξ)+d3I∗U3(ξ) |
where
L1(ξ)=g(S(ξ)S∗); L2(ξ)=g(V(ξ)V∗);L3(ξ)=g(I(ξ)I∗)+(μ3+γ)I∗∫cτ0g(I(ξ−θ)I∗)dθ;U1(ξ)=∫+∞0α+(y)g(S(ξ−y)S∗)dy−∫0−∞α−(y)g(S(ξ−y)S∗)dy;U2(ξ)=∫+∞0α+(y)g(V(ξ−y)V∗)dy−∫0−∞α−(y)g(V(ξ−y)V∗)dy;U3(ξ)=∫+∞0α+(y)g(I(ξ−y)I∗)dy−∫0−∞α−(y)g(I(ξ−y)I∗)dy. |
Thanks to [38,Theorem 1] and
dU1(ξ)dξ=ddξ∫+∞0α+(y)g(S(ξ−y)S∗)dy−ddξ∫0−∞α−(y)g(S(ξ−y)S∗)dy=∫+∞0α+(y)ddξg(S(ξ−y)S∗)dy−∫0−∞α−(y)ddξg(S(ξ−y)S∗)dy=−∫+∞0α+(y)ddyg(S(ξ−y)S∗)dy+∫0−∞α−(y)ddyg(S(ξ−y)S∗)dy=g(S(ξ)S∗)−∫+∞−∞J(y)g(S(ξ−y)S∗)dy. |
Similarly,
dU2(ξ)dξ=g(V(ξ)V∗)−∫+∞−∞J(y)g(V(ξ−y)V∗)dy;dU3(ξ)dξ=g(I(ξ)I∗)−∫+∞−∞J(y)g(I(ξ−y)I∗)dy. |
By some calculations, it can be shown that
ddξ∫cτ0g(I(ξ−θ)I∗)dθ=∫cτ0ddξg(I(ξ−θ)I∗)dθ=−∫cτ0ddθg(I(ξ−θ)I∗)dθ=I(ξ)I∗−I(ξ−cτ)I∗+lnI(ξ−cτ)I(ξ). |
Thus
dL(ξ)dξ=(1−S∗S(ξ))(d1(J∗S(ξ)−S(ξ))+Λ−β1S(ξ)I(ξ−cτ)−(α+μ1)S(ξ))+(1−V∗V(ξ))(d2(J∗V(ξ)−V(ξ))+αS(ξ)−β2V(ξ)I(ξ−cτ)−μ2E(ξ))+(1−I∗I(ξ))(d3(J∗I(ξ)−I(ξ))+β1S(ξ)I(ξ−cτ)+β2V(ξ)I(ξ−cτ)−(γ+μ3)I(ξ))+(μ3+γ)I∗(I(ξ)I∗−I(ξ−cτ)I∗+lnI(ξ−cτ)I(ξ))+d1S∗g(S(ξ)S∗)−d1S∗∫+∞−∞J(y)g(S(ξ−y)S∗)dy+d2V∗g(V(ξ)V∗)−d2V∗∫+∞−∞J(y)g(V(ξ−y)V∗)dy+d3I∗g(I(ξ)I∗)−d3I∗∫+∞−∞J(y)g(I(ξ−y)I∗)dy≜B1+B2, |
where
B1=(1−S∗S(ξ))d1(J∗S(ξ)−S(ξ))+d1S∗g(S(ξ)S∗)−d1S∗∫+∞−∞J(y)g(S(ξ−y)S∗)dy+(1−V∗V(ξ))d2(J∗V(ξ)−V(ξ))+d2V∗g(V(ξ)V∗)−d2V∗∫+∞−∞J(y)g(V(ξ−y)V∗)dy+(1−I∗I(ξ))d3(J∗I(ξ)−I(ξ))+d3I∗g(I(ξ)I∗)−d3I∗∫+∞−∞J(y)g(I(ξ−y)I∗)dy, |
and
B2=(1−S∗S(ξ))(Λ−β1S(ξ)I(ξ−cτ)−(α+μ1)S(ξ))+(1−V∗V(ξ))(αS(ξ)−β2V(ξ)I(ξ−cτ)−μ2E(ξ))+(1−I∗I(ξ))(β1S(ξ)I(ξ−cτ)+β2V(ξ)I(ξ−cτ)−(γ+μ3)I(ξ))+(μ3+γ)I∗(I(ξ)I∗−I(ξ−cτ)I∗+lnI(ξ−cτ)I(ξ)). |
For
(1−S∗S(ξ))d1(J∗S(ξ)−S(ξ))+d1S∗g(S(ξ)S∗)−d1S∗∫+∞−∞J(y)g(S(ξ−y)S∗)dy=d1S∗∫+∞−∞J(y)[S(ξ−y)S∗−S(ξ−y)S(ξ)−lnS(ξ)S∗]−d1S∗∫+∞−∞J(y)g(S(ξ−y)S∗)dy=d1S∗∫+∞−∞J(y)[g(S(ξ−y)S∗)−g(S(ξ−y)S(ξ))]−d1S∗∫+∞−∞J(y)g(S(ξ−y)S∗)dy=−d1S∗∫+∞−∞J(y)g(S(ξ−y)S(ξ))dy. |
Then
B1=−d1S∗∫+∞−∞J(y)g(S(ξ−y)S(ξ))dy−d2V∗∫+∞−∞J(y)g(V(ξ−y)V(ξ))dy−d3I∗∫+∞−∞J(y)g(I(ξ−y)I(ξ))dy. | (2.20) |
For
B2=μ1S∗(2−S(ξ)S∗−S∗S(ξ))−β1S∗I∗g(S(ξ)I(ξ−cτ)S∗I(ξ))−β2V∗I∗[g(V(ξ)I(ξ−cτ)V∗I(ξ))+g(S(ξ)V∗S∗V(ξ))]−μ2V∗[g(V(ξ)V∗)+g(S(ξ)V∗S∗V(ξ))]−(αS∗+β1S∗I∗)g(S∗S(ξ)), |
here we use
Consider an increasing sequence
{Sn(ξ)=S(ξ+ξn)}n≥0, {Vn(ξ)=V(ξ+ξn)}n≥0, and {In(ξ)=I(ξ+ξn)}n≥0. |
We can assume that
ˆC≤L(Sn,Vn,In)(ξ)=L(S,V,I)(ξ+ξn)≤L(S,V,I)(ξ). |
Therefore there exists some
limn→+∞L(Sn,Vn,In)(ξ)=L(S∞,V∞,I∞)(ξ), ξ∈IR. |
Thus
L(S∞,V∞,I∞)(ξ)=~delta. |
Note that
(S∞,V∞,I∞)≡(S∗,V∗,I∗). |
This completes the proof.
In this section, we investigate the existence of traveling wave solutions for the case
Theorem 3.1. Suppose
limξ→+∞(S(ξ),V(ξ),I(ξ))=(S∗,V∗,I∗). |
Furthermore, if we assume that
limξ→−∞(S(ξ),V(ξ),I(ξ))=(S0,V0,0). |
Proof. Let
In(0)=delta∗, In(ξ)≤delta∗, ξ<0 |
with
Similar to [23,39], we can find a subsequence of
\lim\limits_{n\rightarrow\infty}J*S_n = J*S, \ \ \lim\limits_{n\rightarrow\infty}J*V_n = J*V, \ \ \textrm{and}\ \ \lim\limits_{n\rightarrow\infty}J*I_n = J*I |
on every bounded interval. Then we get that
\lim\limits_{\xi\rightarrow+\infty}S(\xi) = S^*, \ \ \lim\limits_{\xi\rightarrow+\infty}V(\xi) = V^*, \ \ \lim\limits_{\xi\rightarrow+\infty}I(\xi) = I^*. |
Moreover, we have
I(0) = \rm{d}elta^*, \ \ I(\xi)\leq\rm{d}elta^*, \ \ \xi \lt 0. |
Let
S_{sup} = \limsup\limits_{\xi\rightarrow-\infty}S(\xi), \ \ V_{sup} = \limsup\limits_{\xi\rightarrow-\infty}V(\xi), \ \ I_{sup} = \limsup\limits_{\xi\rightarrow-\infty}I(\xi) |
and
S_{inf} = \liminf\limits_{\xi\rightarrow-\infty}S(\xi), \ \ V_{inf} = \liminf\limits_{\xi\rightarrow-\infty}V(\xi), \ \ I_{inf} = \liminf\limits_{\xi\rightarrow-\infty}I(\xi). |
Next, we show that
\lim\limits_{n\rightarrow+\infty}I(x_n) = I_{inf}\ \ \textrm{}\ \ \lim\limits_{n\rightarrow+\infty}I(y_n) = I_{sup}. |
Since we assumed that
S(-\infty)\leq\liminf\limits_{n\rightarrow\infty}J*S(\xi_n)\leq\limsup\limits_{n\rightarrow\infty}J*S(\xi_n)\leq S(-\infty). |
and
V(-\infty)\leq\liminf\limits_{n\rightarrow\infty}J*V(\xi_n)\leq\limsup\limits_{n\rightarrow\infty}J*V(\xi_n)\leq V(-\infty). |
Thus, we have
\lim\limits_{n\rightarrow\infty}[J*S(\xi_n)-S(\xi_n)] = 0\ \ \textrm{and}\ \ \lim\limits_{n\rightarrow\infty}[J*V(\xi_n)-V(\xi_n)] = 0 |
Taking
\begin{equation} \left\{ \begin{array}{l} \displaystyle \Lambda - \beta_1 S(-\infty)I(-\infty) - \alpha S(-\infty) - \mu_1 S(-\infty = 0), \\ \displaystyle \alpha S(-\infty)- \beta_2 V(-\infty)I(-\infty) - \mu_2 V(-\infty) = 0, \\ \displaystyle \beta_1 S(-\infty)I(-\infty) + \beta_2 V(-\infty)I(-\infty)) - \gamma I(-\infty) - \mu_3 I(-\infty) = 0. \end{array}\right. \end{equation} | (3.1) |
In the view of
\lim\limits_{\xi\rightarrow-\infty}S(\xi) = S_0, \ \ \lim\limits_{\xi\rightarrow-\infty}V(\xi) = V_0, \ \ \lim\limits_{\xi\rightarrow-\infty}I(\xi) = 0. |
This completes the proof.
Remark 3.1. For the case
In this section, we show the nonexistence of traveling waves when
Theorem 4.1. If
Proof. Since
\begin{equation}\label{Equ1} c I'(\xi) \geq d_3(J*I(\xi) - I(\xi)) +\frac{\beta_1S_0 + \beta_2V_0 - (\gamma + \mu_3)}{2}I(\xi-c\tau) +(\gamma+\mu_3)(I(\xi-c\tau) - I(\xi)) \end{equation} | (4.1) |
holds. Let
\begin{align}\label{Equ2} d_3\int_{-\infty}^{\xi}J * I(s)\rm{d} s = &d_3\int_{-\infty}^{\xi} \int_{{\rm IR}} J(y) I(s-y)\rm{d} y\rm{d} s\\ \nonumber = &d_3\int_{{\rm IR}} \int_{-\infty}^{\xi} J(y) I(s-y)\rm{d} s\rm{d} y\\ \nonumber = &d_3\int_{{\rm IR}} J(y) \int_{-\infty}^{\xi} I(s-y)\rm{d} s\rm{d} y\\ \nonumber = & d_3 J * K(\xi).\nonumber \end{align} | (4.2) |
Integrating the both sides of (4.1) from
\begin{align}\label{Equ3} \nonumber cI(\xi) \geq &d_3(J*K(\xi) - K(\xi)) + (\gamma + \mu_3)[K(\xi-c\tau) - K(\xi)]\\ & + \frac{\beta_1S_0 + \beta_2V_0 - (\gamma + \mu_3)}{2} K(\xi-c\tau). \end{align} | (4.3) |
Furthermore, the following two equations hold.
\begin{align}\label{Equ4} \nonumber\int_{-\infty}^\xi[K(\eta-c\tau) - K(\eta)]\rm{d} \eta = &\int_{-\infty}^\xi (-c\tau) \int_0^1\frac{\partial K(\eta-c\tau s)}{\partial s}\rm{d} s\rm{d} \eta\\ = &- c\tau \int_0^1 K(\xi-c\tau s)\rm{d} s \end{align} | (4.4) |
and
\begin{align}\label{Equ5} d_3\nonumber\int_{-\infty}^\xi[J*K(\eta) - K(\eta)]\rm{d} \eta = &d_3\int_{-\infty}^\xi \int_{-\infty}^{+\infty}(-x)J(x)\int_0^1\frac{\partial K(\eta-x s)}{\partial s}\rm{d} s\rm{d} x\rm{d} \eta\\ = &d_3\int_{-\infty}^{+\infty} (-x) J(x)\int_0^1 K(\xi-x s)\rm{d} s \rm{d} x. \end{align} | (4.5) |
Integrating both sides of inequality (4.3) from
\begin{align}\label{Equ6} \nonumber&\frac{\beta_1S_0 + \beta_2V_0 - (\gamma + \mu_3)}{2} \int_{-\infty}^\xi K(\eta-c\tau)\rm{d} \eta\\ \nonumber\leq&c K(\xi) + (\gamma+\mu_3)c\tau \int_0^1 K(\xi-c\tau s)\rm{d} s\\ \nonumber& + d_3\int_{-\infty}^{+\infty} x J(x)\int_0^1 K(\xi-x s)\rm{d} s \rm{d} x\\ \leq&\left(c+d_3\int_{{\rm IR}}xJ(x)\rm{d} x+(\gamma+\mu_3)c\tau\right)K(\xi), \end{align} | (4.6) |
Here we use
\begin{align}\label{Equ7} \nonumber&\frac{\beta_1S_0 + \beta_2V_0 - (\gamma + \mu_3)}{2} \int_0^{+\infty} K(\xi - \eta - c\tau)\rm{d} \eta\\ \leq&(c+(\gamma+\mu_3)c\tau)K(\xi), \end{align} | (4.7) |
For the non-decreasing function
\begin{align}\label{Equ8} \nonumber&\frac{\beta_1S_0 + \beta_2V_0 - (\gamma + \mu_3)}{2} (\tilde{\eta} + c\tau) K(\xi - \tilde{\eta} - c\tau)\\ \leq&(c+(\gamma+\mu_3)c\tau)K(\xi), \end{align} | (4.8) |
Thus there exists a sufficiently large constant
K(\xi-\theta -c\tau)\leq\varepsilon K(\xi), \ \ \xi\leq-M. |
Let
p(\xi) = K(\xi)e^{-\nu\xi}, |
where
0 \lt \nu\triangleq\frac{1}{\theta+c\tau}\ln \frac{1}{\varepsilon} \lt \lambda_c, |
By some simple calculation, we have
p(\xi-\theta-c\tau)\leq p(\xi). |
Using L'Hospital's rule yields
\lim\limits_{\xi\rightarrow+\infty}p(\xi) = \lim\limits_{\xi\rightarrow+\infty}\frac{K(\xi)}{e^{\nu\xi}} = \lim\limits_{\xi\rightarrow+\infty}\frac{I(\xi)}{\nu e^{\nu\xi}} = 0, |
Note that
\begin{align}\label{Equ9} p(\xi) = K(\xi)e^{-\nu\xi}\leq p_0, \ \ \xi\in {\rm IR}. \end{align} | (4.9) |
On the other hand, since
\begin{align}\label{Equ91} \nonumber cI'(\xi) = &d_3(J*I(\xi)-I(\xi)) + \beta_1 S(\xi)I(\xi-c\tau) + \beta_2 V(\xi)I(\xi-c\tau) - \gamma I(\xi) - \mu_3 I(\xi)\\ \leq&d_3(J*I(\xi)-I(\xi)) + \beta_1 S_0I(\xi-c\tau) + \beta_2 V_0I(\xi-c\tau) - \gamma I(\xi) - \mu_3 I(\xi). \end{align} | (4.10) |
Integrating the both sides of (4.10) from
\begin{align}\label{Equ10} cI(\xi)\leq d_3J*K(\xi)-(\gamma+\mu_3+d_3)K(\xi) + (\beta_1S_0+\beta_2V_0)K(\xi-c\tau). \end{align} | (4.11) |
From (4.9), using
\begin{align}\label{Equ11} \nonumber (d_3J*K(\xi))e^{-\nu \xi} = &d_3\int_{{\rm IR}}J(y)e^{-\nu \xi}K(\xi-y)\rm{d} y\\ = & d_3\int_{{\rm IR}}J(y)e^{-\nu y}K(\xi-y) e^{-\nu (\xi-y)}\rm{d} y\\ \nonumber\leq& d_3p_0\int_{{\rm IR}} J(y)e^{-\nu y} \rm{d} y\\ \nonumber\leq&M_1. \end{align} | (4.12) |
Thus there exists a constant
\begin{align} I(\xi) e^{-\nu \xi}\leq M_2, \ \ \xi\in{\rm IR}, \end{align} | (4.13) |
since (4.9), (4.11) and (4.12) hold. Then
\begin{align} \sup\limits_{\xi\in{\rm IR}}\{I(\xi) e^{-\nu \xi}\} \lt +\infty. \end{align} | (4.14) |
By the same procedure in (4.12), there exists a positive constant
\begin{align}\label{Equ12} (d_3J*I(\xi))e^{-\nu \xi}\leq&M_2. \end{align} | (4.15) |
Hence
\begin{align} \sup\limits_{\xi\in{\rm IR}}\{I'(\xi) e^{-\nu \xi}\} \lt +\infty. \end{align} | (4.16) |
For
\begin{align} \nonumber \mathcal{L}_I(\lambda):& = \int_{{\rm IR}}I(\xi)e^{-\lambda \xi}\rm{d} \xi. \end{align} |
From (2.4), we have
\begin{align} \nonumber &d_3(J*I(\xi)-I(\xi)) - cI'(\xi) + (\beta_1 S_0 + \beta_2V_0) I(\xi-c\tau) - (\gamma+\mu_3) I(\xi)\\ = &\beta_1 (S_0 - S(\xi))I(\xi-c\tau) + \beta_2 (V_0 - V(\xi))I(\xi-c\tau). \end{align} | (4.17) |
Take the two-side Laplace transform to the above equation, thus
\begin{align}\label{Equ13} \Delta(\lambda, c)\mathcal{L}_I(\lambda) = \int_{\rm IR}e^{-\lambda \xi}[\beta_1 (S_0 - S(\xi))I(\xi-c\tau) + \beta_2 (V_0 - V(\xi))I(\xi-c\tau)]\rm{d} \xi \end{align} | (4.18) |
for
c L'(\xi) = d_1 (J*L(\xi)-L(\xi)) + \beta_1 S_(\xi)I(\xi-c\tau) + (\alpha+\mu_1)S(\xi). |
Let
c \int_{{\rm IR}}L'(\xi)e^{-\nu_0 \xi}\eta_N\rm{d} \xi = d_1\int_{{\rm IR}}(J*L(\xi)-L(\xi))e^{-\nu_0 \xi}\eta_N\rm{d} \xi + \int_{{\rm IR}}S(\xi)[\beta_1I(\xi-c\tau) + \alpha + \mu_1] e^{-\nu_0 \xi}\eta_N\rm{d} \xi. |
By the argument in [22,Theorem 3.1], there exists a constant
\int_{{\rm IR}}L(\xi)e^{-\nu_0 \xi}\rm{d} \xi \leq \Xi. |
Thus,
\int_{{\rm IR}}\beta_1(S_0 - S(\xi))I(\xi-c\tau)e^{-(\nu+\nu_0)\xi}\rm{d} \xi\leq \beta_1\sup\limits_{\xi\in{\rm IR}}\{I(\xi) e^{-\nu \xi}\}\int_{{\rm IR}}L(\xi)e^{-\nu_0 \xi}\rm{d} \xi \lt \infty. |
Similarly,
\int_{{\rm IR}}\beta_2(V_0 - V(\xi))I(\xi-c\tau)e^{-(\nu+\nu_0)\xi}\rm{d} \xi \lt \infty. |
From the property of Laplace transform [41],
\begin{align}\label{Equ16} \int_{\rm IR}e^{-\lambda \xi}\left[\Delta(\lambda, c)I(\xi)+\beta_1 (S_0 - S(\xi))I(\xi-c\tau) + \beta_2 (V_0 - V(\xi))I(\xi-c\tau)\right]\rm{d} \xi = 0. \end{align} | (4.19) |
Recall (J2) of Assumption 1.1, then
As traveling wave solutions describe the transition from disease-free equilibrium to endemic equilibrium when the wave speed is larger than the minimal wave speed. Now, we focus on how the parameters in system (2.1) can affect the wave speed. Suppose
\Delta(\hat{\lambda}, \hat{c}) = d_3\int_{{\rm IR}}J(x)e^{-\hat{\lambda} x}\rm{d} x-(d_3+\gamma+\mu_3)-\hat{c}\hat{\lambda}+\beta_1S_0e^{-\hat{c}\tau\hat{\lambda}}+\frac{\beta_2\Lambda\alpha}{(\mu_1 +\gamma_1)(\mu_1+\alpha)}e^{-\hat{c}\tau\hat{\lambda}} = 0. |
By some calculations, we obtain
\frac{\rm{d} \hat{c}}{\rm{d} d_3} = \frac{\int_{{\rm IR}}J(x)[e^{-\hat\lambda x}-1]\rm{d} x}{\hat\lambda(1+ [\beta_1 S_0 + \beta_2 V_0] \tau e^{-\hat{c}\tau\hat{\lambda}})} \gt 0, \ \ \frac{\rm{d} \hat{c}}{\rm{d} \tau} = -\frac{\beta_1S_0 + \beta_2V_0}{e^{\hat{c}\tau\hat{\lambda}} + \beta_1S_0\tau + \beta_2V_0\tau} \lt 0, |
\frac{\rm{d} \hat{c}}{\rm{d} \beta_1} = \frac{S_0e^{-\hat{c}\tau\hat{\lambda}}}{\hat\lambda(1+ [\beta_1 S_0 + \beta_2 V_0] \tau e^{-\hat{c}\tau\hat{\lambda}})} \gt 0, \ \ \frac{\rm{d} \hat{c}}{\rm{d} \beta_2} = \frac{V_0e^{-\hat{c}\tau\hat{\lambda}}}{\hat\lambda(1+ [\beta_1 S_0 + \beta_2 V_0] \tau e^{-\hat{c}\tau\hat{\lambda}})} \gt 0, |
and
\frac{\rm{d} \hat{c}}{\rm{d} \gamma_1} = -\frac{\beta_2V_0 e^{-\hat{c}\tau\hat{\lambda}}}{(\mu_1+\gamma_1)\hat\lambda(1+ [\beta_1 S_0 + \beta_2 V_0] \tau e^{-\hat{c}\tau\hat{\lambda}})} \lt 0, |
that is,
Ⅰ. The more successful the vaccination, the slower the disease spreads;
Ⅱ. The longer the latent period, the slower the disease spreads;
Ⅲ. The faster infected individuals move, the faster the disease spreads;
Ⅳ. The more effective the infections are, the faster the disease spreads.
Now, we are in a position to make the following summary:
Mathematically, we investigated a nonlocal dispersal epidemic model with vaccination and delay; The existence of traveling wave solutions is studied by applying Schauder fixed point theorem with upper-lower solutions, that is there exists traveling wave solutions when
Biologically, our results imply that the nonlocal dispersal and infection ability of infected individuals can accelerate the spreading of infectious disease, while the latent period and successful rate of vaccination can slow down the disease spreads.
The authors are very grateful to the editors and three reviewers for their valuable comments and suggestions that have helped us improving the presentation of this paper. We would also very grateful to Prof.Shigui Ruan, Dr. Sanhong Liu and Dr.Wen-Bing Xu for their valuable comments and helpful advice. This work is supported by Natural Science Foundation of China (No.11871179; No.11771374), and the first author was also partially supported by China Scholarship Council (No.201706120216). R. Zhang acknowledges the kind hospitality received from the Department of Mathematics at the University of Miami, where part of the work was completed.
All authors declare no conflicts of interest in this paper.
Proof. If
\begin{align*} &c{\underline{S}}'(\xi)- d_1(J*\underline{S}(\xi)-\underline{S}(\xi)) - \Lambda + \beta_1\underline{S}(\xi)\overline{I}(\xi-c\tau) +(\mu_1+\alpha) \underline{S}(\xi)\\ = &-c\varepsilon_1M_1e^{\varepsilon_1\xi}+d_1M_1e^{\varepsilon_1\xi}\int_{{\rm IR}}J(x)e^{-\varepsilon_1x}\rm{d} x-d_1M_1e^{\varepsilon_1\xi}-\Lambda\\ &+\beta_1(S_0-M_1e^{\varepsilon_1\xi})e^{\lambda_c(\xi-c\tau)}+(\mu_1+\alpha)(S_0-M_1 e^{\varepsilon_1 \xi})\\ \leq&e^{\varepsilon_1\xi}\left[-c\varepsilon_1M_1e^{\varepsilon_1\xi}+d_1M_1e^{\varepsilon_1\xi}\int_{{\rm IR}}J(x)e^{-\varepsilon_1x}\rm{d} x-d_1M_1e^{\varepsilon_1\xi}+\beta_1S_0\left(\frac{S_0}{M_1}\right)^{\frac{\lambda-\varepsilon_1}{\varepsilon_1}}\right]. \end{align*} |
Here we use
e^{(\lambda_c-\varepsilon_1)\xi} \lt \left(\frac{S_0}{M_1}\right)^{\frac{\lambda_c-\varepsilon_1}{\varepsilon_1}}\ \ \ \textrm{for}\ \ \ \xi \lt \mathfrak{X}_1. |
Keeping
\begin{equation*} c{\underline{S}}'(\xi)- d_1(J*\underline{S}(\xi)-\underline{S}(\xi)) - \Lambda + \beta_1\underline{S}(\xi)\overline{I}(\xi-c\tau) +(\mu_1+\alpha) \underline{S}(\xi)\leq0. \end{equation*} |
This completes the proof.
Proof. If
Case Ⅰ:
In this case,
c{\underline{I}}'(\xi) \leq d_3(J*\underline{I}(\xi)-\underline{I}(\xi)) - \gamma \underline{I}(\xi) - \mu_3 \underline{I}(\xi), |
that is
\begin{align*} &c\lambda_c-d_3\int_{{\rm IR}}J(y)e^{-\lambda_cy}\rm{d} y+d_3+\gamma+\mu_3\\ \leq& M_3e^{\varepsilon_3\xi}\left[c(\lambda+\varepsilon_3)-d_3\int_{{\rm IR}}J(y)e^{-(\lambda_c+\varepsilon_3)y}\rm{d} y+d_3+\gamma+\mu_3\right]. \end{align*} |
From
\beta_1S_0e^{-c\tau\lambda_c}+\beta_2V_0e^{-c\tau\lambda_c}\\ \leq M_3e^{\varepsilon_3\xi}\left[-\Delta(\lambda_c+\varepsilon_3, c)+\beta_1S_0e^{-c\tau(\lambda_c+\varepsilon_3)}+\beta_2V_0e^{-c\tau(\lambda_c+\varepsilon_3)}\right], |
Because
\begin{align*} \beta_1S_0+\beta_2V_0\leq M_3e^{\varepsilon_3\xi}\left[-\Delta(\lambda_c+\varepsilon_3, c)+\beta_1S_0e^{-c\tau(\lambda_c+\varepsilon_3)}+\beta_2V_0e^{-c\tau(\lambda_c+\varepsilon_3)}\right]. \end{align*} |
Since
\beta_1S_0+\beta_2V_0\leq-\Delta(\lambda_c+\varepsilon_3, c)M_3\left(\frac{S_0}{M_1}\right)^{\frac{1}{2}}\left(\frac{V_0}{M_2}\right)^{\frac{1}{2}}. |
Thus, Equation (2.13) holds for sufficiently large
M_3\geq \frac{\beta_1S_0+\beta_2V_0}{-\Delta(\lambda_c+\varepsilon_3, c)}\sqrt{\frac{S_0}{M_1}}\sqrt{\frac{V_0}{M_2}}\triangleq\Pi_1. |
Case Ⅱ:
In this case,
c{\underline{I}}'(\xi) \leq d_3(J*\underline{I}(\xi)-\underline{I}(\xi)) - \gamma \underline{I}(\xi) - \mu_3 \underline{I}(\xi) + \beta_1\underline{S}(\xi)\underline{I}(\xi-c\tau), |
that is
\begin{align*} &c\lambda_c-d_3\int_{{\rm IR}}J(y)e^{-\lambda_cy}\rm{d} y+d_3+\gamma+\mu_3-\beta_1S_0e^{-\lambda_cc\tau}+\beta_1M_1e^{\varepsilon_1\xi-\lambda_cc\tau}\\ \leq& M_3e^{\varepsilon_3\xi}\left[c(\lambda+\varepsilon_3)-d_3\int_{{\rm IR}}J(y)e^{-(\lambda_c+\varepsilon_3)y}\rm{d} y+d_3+\gamma+\mu_3-\beta_1S_0e^{-(\varepsilon_3+\lambda_c)c\tau}+\beta_1M_1e^{\varepsilon_1\xi-(\varepsilon_3+\lambda_c)c\tau}\right], \end{align*} |
we need to prove
\begin{equation*} \beta V_0\leq -\Delta(\lambda_c+\varepsilon_3, c)M_3e^{\varepsilon_3\xi}. \end{equation*} |
Choose
M_3\geq \frac{\beta_2 \sqrt{V_0M_2}}{-\Delta(\lambda_c+\varepsilon_3, c)}\triangleq\Pi_2. |
Case Ⅲ:
In this case,
M_3\geq \frac{\beta_1 \sqrt{S_0M_1}}{-\Delta(\lambda_c+\varepsilon_3, c)}\triangleq\Pi_3 |
large enough.
Case Ⅵ:
In this case,
c{\underline{I}}'(\xi) \leq d_3(J*\underline{I}(\xi)-\underline{I}(\xi)) - \gamma \underline{I}(\xi) - \mu_3 \underline{I}(\xi) + \beta_1\underline{S}(\xi)\underline{I}(\xi-c\tau) + \beta_2\underline{V}(\xi)\underline{I}(\xi-c\tau), |
that is
\begin{align*} c\lambda_c-d_3&\int_{{\rm IR}}J(y)e^{-\lambda_cy}\rm{d} y+d_3+\gamma+\mu_3-\beta_1S_0e^{-\lambda_cc\tau}-\beta_2V_0e^{-\lambda_cc\tau}+\beta_1M_1e^{\varepsilon_1\xi-\lambda_cc\tau}+\beta_2M_2e^{\varepsilon_2\xi-\lambda_cc\tau}\\ \leq M_3e^{\varepsilon_3\xi}&\left(c(\lambda+\varepsilon_3)-d_3\int_{{\rm IR}}J(y)e^{-(\lambda_c+\varepsilon_3)y}\rm{d} y+d_3+\gamma+\mu_3-\beta_1S_0e^{-(\varepsilon_3+\lambda_c)c\tau}\right.\\ &+\left.\beta_1M_1e^{\varepsilon_1\xi-(\varepsilon_3+\lambda_c)c\tau}-\beta_2V_0e^{-(\varepsilon_3+\lambda_c)c\tau}+\beta_2M_2e^{\varepsilon_1\xi-(\varepsilon_3+\lambda_c)c\tau}\right) \end{align*} |
we only need to ensure
M_3\geq\frac{\beta_1M_1e^{(\varepsilon_1-\varepsilon_3)\xi-\lambda_cc\tau}+\beta_2M_2e^{(\varepsilon_2-\varepsilon_3)\xi-\lambda_cc\tau}}{-\Delta(\lambda_c+\varepsilon_3, c)+\beta_1M_1e^{\varepsilon_1\xi-(\varepsilon_3+\lambda_c)c\tau}+\beta_2M_2e^{\varepsilon_2\xi-(\varepsilon_3+\lambda_c)c\tau}} |
Since
\frac{\beta_1M_1e^{(\varepsilon_1-\varepsilon_3)\xi-\lambda_cc\tau}+\beta_2M_2e^{(\varepsilon_2-\varepsilon_3)\xi-\lambda_cc\tau}}{-\Delta(\lambda_c+\varepsilon_3, c)+\beta_1M_1e^{\varepsilon_1\xi-(\varepsilon_3+\lambda_c)c\tau}+\beta_2M_2e^{\varepsilon_2\xi-(\varepsilon_3+\lambda_c)c\tau}} \lt \frac{\beta_1\sqrt{S_0M_1}+\beta_2\sqrt{V_0M_2}}{-\Delta(\lambda_c+\varepsilon_3, c)}. |
Then Equation (2.13) holds if we choose
M_3\geq\frac{\beta_1\sqrt{S_0M_1}+\beta_2\sqrt{V_0M_2}}{-\Delta(\lambda_c+\varepsilon_3, c)}\triangleq\Pi_4. |
Through the above discussion, Equation (2.13) holds if we choose
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