# Travelling Waves of a Diffusive Epidemic Model with Latency and Relapse

Authors: Zhiping Wang; Rui Xu
Journal: Discrete Dynamics in Nature and Society
 (2013)

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## Abstract

An SEIR epidemic model with relapse and spatial diffusion is studied. By analyzing the corresponding characteristic equations, the local stability of each of the feasible steady states to this model is discussed. The existence of a travelling wave solution is established by using the technique of upper and lower solutions and Schauder's fixed point theorem. Numerical simulations are carried out to illustrate the main results.


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## Body

## 1. Introduction

In most epidemic models, individuals are often divided into several classes such as susceptible, infective, and recovered classes. Some diseases such as human or bovine tuberculosis have a latency (see, e.g., [1, 2] and references therein): infected individuals do not infect other susceptible individuals until sometime later. Hence, it is reasonable to include a latent class in such disease models. Also, for some diseases, recovered individuals may relapse with reactivation of latent infection and revert back to the infective class. This recurrence of disease is an important feature of some animal and human diseases. For human tuberculosis, incomplete treatment can lead to relapse, but relapse can also occur in patients who took a full course of treatment and were declared cured [3–5]. Most tuberculosis in human adults (caused by Mycobacterium tuberculosis) in the USA results from reactivation of latent infection [3].In order for a model to be more realistic, taking into consideration a general exposed distribution and the possibility of relapse, van den Driessche et al. [6] formulated and studied the following model:
(1)dS(t)dt=μN-βS(t)I(t)N-μS(t),E(t)=∫0t‍βS(ξ)I(ξ)Ne-μ(t-ξ)P(t-ξ)dξ,dR(t)dt=γI(t)-(μ+δ)R(t),I(t)=N-S(t)-E(t)-R(t),
where N is the size of the population; S,E,I, and R are the population sizes of susceptible, latent, infective, and recovered classes, respectively; δ>0 is a constant rate at which an individual in the recovered class reverts to the infective class; μ>0 is the recruitment rate and the death rate (this guarantees that N can be assumed as a constant); β>0 is the transmission coefficient; γ⩾0 is the recovery rate; P(t) denotes the probability that a latent individual still remains in the latent class t time units after entering the latent class.As in [6], when P(t) is a step function, that is,
(2)P(t)={1,0<t<τ,0,t>τ,
by rescaling
(3)S(t)N⟶S(t),E(t)N⟶E(t),I(t)N⟶I(t),R(t)N⟶R(t),
model (1) reduces to the following delay differential equation system:
(4)S′(t)=μ-μS(t)-βS(t)I(t),I′(t)=βe-μτS(t-τ)I(t-τ)+δR(t)-(μ+γ)I(t),R′(t)=γI(t)-(μ+δ)R(t),
where τ>0 can be explained as the average latent period of the disease. The global stability of the steady states for system (1) and (4) was analyzed in [6, 7].Note that it is implicitly assumed that the population is well mixed, and the spatial mobility of individuals has been ignored in model (1) as well as most other epidemic models. In reality, the environment in which an individual lives is often heterogeneous making it necessary to distinguish the locations, and due to the large mobility of individuals within an area or even worldwide, spatially uniform models are not sufficient to give a realistic picture of a disease's transmission. For this reason, the spatial effects cannot be neglected in studying the spread of epidemics. If the environment is spatially continuous, random diffusion is often used to describe the mobility of the population, leading to models in the form of reaction-diffusion equations (see, e.g., [8–10]). Noble [9] applied reaction-diffusion theory to describe the spread of plague through Europe in the mid-14th century. Maidana and Yang [10] proposed a spatial model to analyze the West Nile Virus propagation across the USA, from east to west.Motivated by the work mentioned above, the main purpose of this paper is to study the effect of the spatial diffusion of the individuals on the dynamics of an epidemic model with latency and relapse. If we further consider the spatial diffusion, then the epidemic model corresponding to (4) should be described by the following delayed reaction diffusion system:
(5)∂S(x,t)∂t=d1∂2S(x,t)∂x2+μ-μS(x,t)-βS(x,t)I(x,t),∂I(x,t)∂t=d2∂2I(x,t)∂x2+βe-μτS(x,t-τ)I(x,t-τ)∂I(x,t)∂t=+δR(x,t)-(μ+γ)I(x,t),∂R(x,t)∂t=d3∂2R(x,t)∂x2+γI(x,t)-(μ+δ)R(x,t),
with initial conditions
(6)S(x,t)=ρ1(x,t),I(x,t)=ρ2(x,t),R(x,t)=ρ3(x,t),t∈[-τ,0],x∈Ω¯.
In problem (5) and (6), d1,d2, and d3 are the spatial diffusion rates of susceptible, infective, and recovered individuals, respectively. The functions ρi(x,t)∈𝒞=C([-τ,0],X)(i=1,2,3) and X is defined by
(7)X={S,I,R∈W2,2(Ω):∂S(x,t)∂n=∂I(x,t)∂n=∂R(x,t)∂n=0,x∈∂Ω},
with the inner product 〈·,·〉, where ∂/∂n denotes the outward normal derivative on ∂Ω.The organization of this paper is as follows. In the next section, by analyzing the corresponding characteristic equations, the local stability of an endemic steady state and a disease-free steady state of system (5) is discussed. In Section 3, by constructing a pair of upper-lower solutions, we use the cross-iteration method and the Schauder's fixed point theorem to prove the existence of travelling wave solutions to system (5). In Section 4, numerical simulations are carried out to illustrate the main theoretical results, and by combining with numeric simulations, we discuss the effects of some parameters on the dynamics of system (5). A brief conclusion is given in the last section.


## 2. Local Stability

In this section, by analyzing the corresponding characteristic equations, we investigate the local stability of steady states to system (5) with the initial conditions (6) and the homogeneous Neumann boundary conditions
(8)∂S(x,t)∂n=∂I(x,t)∂n=∂R(x,t)∂n=0,t⩾0,x∈∂Ω,
respectively. The homogeneous Neumann boundary conditions imply that the populations do not move across the boundary ∂Ω.Let(9)ℛ0=βe-μτ(μ+δ)μ(μ+δ+γ).ℛ0 is called the basic reproductive number of system (5), which describes the average number of newly infected individuals at the beginning of the infectious process. It is easy to show that system (5) always has a disease-free uniform steady state E0(1,0,0). Further, if ℛ0>1, system (5) has a unique endemic steady state E*(S*,I*,R*), where
(10)S*=μ(μ+δ+γ)eμτβ(μ+δ),I*=βe-μτ(μ+δ)-μ(μ+δ+γ)β(μ+δ+γ),R*=γμ+δI*.LetE^(S0,I0,R0) represent any feasible uniform steady state of system (5). If we set S¯=S-S0, I¯=I-I0, R¯=R-R0, and drop the bars for simplicity of notations, then (5) can be transformed into the following equivalent system:
(11)∂S(x,t)∂t=d1∂2S(x,t)∂x2-μS(x,t)-βS(x,t)(I(x,t)+I0)-βS0I(x,t),∂I(x,t)∂t=d2∂2I(x,t)∂x2+βe-μτ[S(x,t-τ)(I(x,t-τ)+I0)+S0I(x,t-τ)]+δR(x,t)-(μ+γ)I(x,t),∂R(x,t)∂t=d3∂2R(x,t)∂x2+γI(x,t)-(μ+δ)R(x,t).
Denote U=(S(·,t),I(·,t),R(·,t))T. Then the linearization of (11) can be rewritten as an abstract differential equation in the phase space 𝒞=C([-τ,0],X),
(12)U˙(t)=DΔU(t)+L(Ut),
where D=diag(d1,d2,d3), and L is defined by
(13)L(ϕ)=(-μ-βI0-βS000-(μ+γ)δ0γ-(μ+δ))(ϕ1(0)ϕ2(0)ϕ3(0))+(00βe-μτI0βe-μτS00000)(ϕ1(-τ)ϕ2(-τ)ϕ3(-τ))
for ϕ=(ϕ1,ϕ2,ϕ3)T∈𝒞, and its characteristic equation is
(14)λy-DΔy-L(eλy)=0,
where y∈dom(Δ) and y≠0, dom(Δ)⊂X.From the properties of the Laplacian operator defined on the bounded domain, the operatorΔ on X has the eigenvalues -k2 with the relative eigenfunctions
(15)βk1=(γk,0,0)T,βk2=(0,γk,0)T,βk3=(0,0,γk)T,
where γk=cos(kx), k∈N0={0,1,2,…}. Clearly, (βk1,βk2,βk3)k=0∞ construct a basis of the phase space X, and therefore, any element y in X can be expanded as Fourier series in the following form:
(16)y=∑k=0∞‍YKT(βk1,βk2,βk3)T,YKT=(〈y,βk1〉,〈y,βk2〉,〈y,βk3〉).
Some simple computations show that
(17)L(ϕT(βk1,βk2,βk3)T)=L(ϕT)(βk1,βk2,βk3)T,k∈N0.
From (16) and (17), (14) is equivalent to
(18)∑k=0∞‍YKT(λI2+Dk2-M)(βk1,βk2,βk3)T=0,
where
(19)M=(-μ-βI0-βS00βe-(μ+λ)τI0βe-(μ+λ)τS0-(μ+γ)δ0γ-(μ+δ)).
If E0=E0, (18) is equivalent to
(20)(λ+k2d1+μ)×[(λ+k2d2+μ+γ)(λ+k2d3+μ+δ)-δγ-(λ+k2d3+μ+δ)βe-μτe-λτ]=0.
Clearly, for any i⩾1, (20) has a negative root -k2d1-μ. Its other roots satisfy
(21)g1(λ)≜(λ+k2d2+μ+γ)(λ+k2d3+μ+δ)-δγ-(λ+k2d3+μ+δ)βe-μτe-λτ=0.Ifℛ0>1, then for i=1, we get that, for λ real,
(22)g1(0)=μ(μ+δ+γ)-(μ+δ)βe-μτ<0,limλ→+∞g1(λ)=∞.
Hence, when i=1, (21) has a positive real root, and accordingly, E0 is unstable.Ifℛ0<1 and τ=0, we have
(23)g1(λ)=λ2+(k2d2+k2d3+2μ+γ+δ-β)λ+(k2d2+μ+γ-β)(k2d3+μ+δ)-δγ>0,
which means that the roots of (21) are negative constants, then E0 is locally asymptotically stable. For τ>0, if iσ(σ>0) is a solution of (21), separating the real and imaginary parts, we derive that
(24)-σ2+A0=A1cosστ+A2σsinστ,A3σ=A2σcosστ-A1sinστ,
where
(25)A0=(k2d2+μ+γ)(k2d3+μ+δ)-δγ,A1=(k2d3+μ+δ)βe-μτ,A2=βe-μτ,A3=k2(d2+d3)+2μ+γ+δ.
Squaring and adding the two equations of (24), it follows that
(26)σ4+(A32-2A0-A22)σ2+A02-A12=0.Note that ifR0<1, we obtain
(27)A32-2A0-A22=(k2d2+μ+γ)2+(k2d3+μ+γ)2+2δγ-β2e-2μτ⩾(μ+γ)2+(μ+δ)2+2δγ-β2e-2μτ⩾μ2>0,A0-A1⩾(μ+δ)(μ+γ-βe-μτ)-δγ=μ(μ+δ+γ)-β(μ+δ)e-μτ>0.
Hence, (26) has no positive roots for all i⩾1. Therefore, if ℛ0<1, E0 is locally asymptotically stable for all τ>0.IfE0=E*, (18) is equivalent to
(28)λ3+p2λ2+p1λ+p0+(q2λ2+q1λ+q0)e-λτ=0,
where
(29)p0=(k2d1+μ+βI*)[(k2d2+μ+γ)(k2d3+μ+δ)-δγ],p1=(k2d2+μ+γ)(k2d3+μ+δ)-δγp1=+(k2d1+μ+βI*)(k2d2+k2d3+2μ+δ+γ),p2=(k2d1+μ+βI*)+(k2d2+μ+γ)+(k2d3+μ+δ),q0=-βS*e-μτ(k2d1+μ)(k2d3+μ+δ),q1=-βS*e-μτ(k2d1+k2d3+2μ+δ),q2=-βS*e-μτ.
When τ=0, (28) becomes
(30)λ3+(p2+q2)λ2+(p1+q1)λ+p0+q0=0.
It is not difficult to verify that
(31)p2+q2>μ+γ-βS*=δγμ+δ>0,p0+q0>(k2d1+μ)[(μ+γ)(k2d3+μ+δ)-δγ]p2+q2>-βS*(k2d1+μ)(k2d3+μ+δ),p0+q0>(μ+δ)(k2d1+μ)δγμ+δ-δγ(k2d1+μ)=0,(p1+q1)(p2+q2)-(p0+q0)p0+q0>[(k2d2+μ+γ)(k2d3+μ+δ)-δγp2+q2>+(k2d1+μ+βI*)(k2d2+k2d3+2μ+δ+γ)p2+q2>-βS*(k2d1+k2d3+2μ+δ)]p2+q2>×(k2d2+k2d3+2μ+δ+γ-βS*)>0,
which indicates that all roots λi,1,λi,2, and λi,3 of (30) have negative real parts.Next we can conclude that there exists a positive constantξ such that
(32)Re{λi,1},Re{λi,2},Re{λi,3}⩽-ξ,i⩾1.In fact, letλ=μiv, then
(33)λ3+(p2+q2)λ2+(p1+q1)λ+p0+q0=μi3v3+(p2+q2)μi2v2+(p1+q1)μiv+p0+q0≜hi(v).
Since μi→∞ as i→∞, it yields that
(34)limi→∞hi(v)μi3=v3+(d1+d2+d3)v2+(d1d2+d1d3+d2d3)v+d1d2d3≜h(v).
It is easy to see that these three roots v1,v2, and v3 of h(v) have negative real parts from the Routh-Hurwitz criterion again, and thus, there exists a positive constant ξ¯ such that
(35)Re{v1},Re{v2},Re{v3}⩽-ξ¯.By continuity, we see that there existsi0 such that the three roots vi,1,vi,2, and vi,3 of hi(v) satisfy
(36)Re{vi,1},Re{vi,2},Re{vi,3}⩽-ξ¯2,i⩾i0,
and thus,
(37)Re{λi,1},Re{λi,2},Re{λi,3}⩽-μiξ¯2⩽-μi0ξ¯2,i⩾i0.
Let
(38)-ξ~=max1⩽i⩽i0{Re{λi,1},Re{λi,2},Re{λi,3}},
then ξ~, and (32) holds for ξ∈(ξ~,μi0ξ¯/2).Consequently, the spectrum ofℒ, consisting only of eigenvalues, lies in Reλ⩽-ξ. Hence, the infected steady state is locally asymptotically stable when τ=0.Forτ>0, if iω(ω>0) is a solution of (28), separating the real and imaginary parts, we derive that
(39)ω3-p1ω=q1ωcosωτ+(q2ω2-q0)sinωτ,p2ω2-p0=q1ωsinωτ-(q2ω2-q0)cosωτ.
Squaring and adding the two equations of (39), it follows that
(40)ω6+(p22-2p1-q22)ω4+(p12-2p0p2-q12+2q0q2)ω2+p02-q02=0.
By calculation, it is not difficult to verify that, for all i⩾1,
(41)p22-2p1-q22=(k2d1+μ+βI*)2+(k2d2+μ+γ)2+(k2d3+μ+δ)2+2δγ-(βS*e-μτ)2>(μ+γ)2-β2S*2>0,p12-2p0p2-q12+2q0q2>[(k2d2+μ+γ)(k2d3+μ+δ)-δγ]2-(βS*e-μτ)2(k2d3+μ+δ)2>μ2(μ+δ+γ)2-[βS*e-μτ(μ+δ)]2=0,p02-q02=(p0-q0)(p0+q0)>0.Hence, (40) has no positive roots. Therefore, if ℛ0>1, E* is locally asymptotically stable for all τ>0.From the discussion above, we can obtain the following results.Theorem 1.
For system (5) with initial conditions (6) and homogeneous Neumann conditions (8), if ℛ0<1, E0 is locally asymptotically stable; if ℛ0>1, E0 becomes unstable, and E* is locally asymptotically stable.


## 3. Existence of Travelling Waves

In this section, we investigate the existence of travelling wave solutions to system (5). The technique of the proofs is to use the Schauder's fixed point theorem, the method of upper-lower solutions, and its associated cross iteration scheme.LetS^=1-S, then system (5) is transformed into (omitting the hat on S for simplicity)
(42)∂S(x,t)∂t=d1∂2S(x,t)∂x2-μS(x,t)+βI(x,t)(1-S(x,t)),∂I(x,t)∂t=d2∂2I(x,t)∂x2+βe-μτI(x,t-τ)(1-S(x,t-τ))+δR(x,t)-(μ+γ)I(x,t),∂R(x,t)∂t=d3∂2R(x,t)∂x2+γI(x,t)-(μ+δ)R(x,t).A travelling wave solution of (42) is a solution (S(x,t),I(x,t),R(x,t)) of the form S(x,t)=ϕ(x+ct), I(x,t)=φ(x+ct), R(x,t)=ψ(x+ct) where ϕ,φ,ψ∈C2(ℝ,ℝ) and c>0 is a constant corresponding to the wave speed. On substituting S(x,t)=ϕ(x+ct), I(x,t)=φ(x+ct), and R(x,t)=ψ(x+ct) denoting the travelling wave coordinate x+ct still by t, we derive from (42) that
(43)d1ϕ′′(t)-cϕ′(t)+fc1(ϕt,φt,ψt)=0,d2φ′′(t)-cφ′(t)+fc2(ϕt,φt,ψt)=0,d3ψ′′(t)-cψ′(t)+fc3(ϕt,φt,ψt)=0,
where
(44)fc1(ϕt,φt,ψt)=-μϕ(t)+βφ(t)(1-ϕ(t)),fc2(ϕt,φt,ψt)=βe-μτφ(t-cτ)(1-ϕ(t-cτ))+δψ(t)-(μ+γ)φ(t),fc3(ϕt,φt,ψt)=γφ(t)-(μ+δ)ψ(t).
Equation (43) will be solved subject to the following conditions:
(45)Limt→-∞(ϕ(t),φ(t),ψ(t))=(0,0,0),limt→+∞(ϕ(t),φ(t),ψ(t))=(k1,k2,k3)≜(1-S*,I*,R*).
We define the upper and lower solutions of system (43) as follows.Definition 2.
A pair of continuous functionsΦ¯=(ϕ¯,φ¯,ψ¯) and Φ_=(ϕ_,φ_,ψ_) are called a pair of upper-lower solutions of system (43) if Φ¯ and Φ_ are twice differential almost everywhere in ℝ and they are essentially bounded on ℝ, and there hold
(46)d1ϕ¯′′(t)-cϕ¯′(t)+fc1(ϕ¯t,φ¯t,ψ¯t)⩽0,a.e.inℝ,d2φ¯′′(t)-cφ¯′(t)+fc2(ϕ_t,φ¯t,ψ¯t)⩽0,a.e.inℝ,d3ψ¯′′(t)-cψ¯′(t)+fc3(ϕ¯t,φ¯t,ψ¯t)⩽0,a.e.inℝ,d1ϕ_′′(t)-cϕ_′(t)+fc1(ϕ_t,φ_t,ψ_t)⩾0,a.e.inℝ,d2φ_′′(t)-cφ_′(t)+fc2(ϕ¯t,φ_t,ψ_t)⩾0,a.e.inℝ,d3ψ_′′(t)-cψ_′(t)+fc3(ϕ_t,φ_t,ψ_t)⩾0,a.e.inℝ.As tofc1,fc2, and fc3, we have the following.Lemma 3.
f
c
1
,
f
c
2, and fc3 satisfy the following partial quasimonotonicity conditions.(C1)
There exist three positive constantsβ1,β2,β3 such that
(47)fc1(ϕ1,φ1,ψ1)-fc1(ϕ2,φ2,ψ2)+β1[ϕ1(0)-ϕ2(0)]⩾0,fc2(ϕ1,φ1,ψ1)-fc2(ϕ1,φ2,ψ2)+β2[φ1(0)-φ2(0)]⩾0,fc2(ϕ1,φ1,ψ1)-fc2(ϕ2,φ1,ψ1)⩽0,fc3(ϕ1,φ1,ψ1)-fc3(ϕ2,φ2,ψ2)+β3[ψ1(0)-ψ2(0)]⩾0,whereϕi,φi,ψi∈C([-τ,0],R), i=1,2 with (0,0,0)⩽(ϕ2,φ2,ψ2)⩽ (ϕ1,φ1,ψ1)⩽ (M1, M2, M3), Mj>kj(j=1,2,3) are positive constants.Proof.
It is not difficult to verify that(48)fc1(ϕ1,φ1,ψ1)-fc1(ϕ2,φ2,ψ2)=β(1-ϕ1(0))φ1(0)-μϕ1(0)+μϕ2(0)+β(1-ϕ2(0))φ2(0)⩾-(μ+βϕ2(0))(ϕ1(0)-ϕ2(0)),fc2(ϕ1,φ1,ψ1)-fc2(ϕ1,φ2,ψ2)=βe-μτ(1-ϕ1(-τ))(φ1(-τ)-φ2(-τ))+δ(ψ1(0)-ψ2(0))-(μ+γ)(φ1(0)-φ2(0))⩾-(μ+γ)(φ1(0)-φ2(0)),fc2(ϕ1,φ1,ψ1)-fc2(ϕ2,φ1,ψ1)=βe-μτφ1(-τ)(ϕ2(-τ)-ϕ1(-τ))⩽0,fc3(ϕ1,φ1,ψ1)-fc3(ϕ2,φ2,ψ2)⩾-(μ+δ)[ψ1(0)-ψ2(0)]⩾0.
Let β1=μ+βϕ2(0),β2=μ+γ, and β3=μ+δ, then the proof is completed.For the constantsβ1,β2,β3 above, define
(49)H1(ϕ,φ,ψ)(t)=fc1(ϕt,φt,ψt)+β1ϕ(t),H2(ϕ,φ,ψ)(t)=fc2(ϕt,φt,ψt)+β2φ(t),H3(ϕ,φ,ψ)(t)=fc3(ϕt,φt,ψt)+β3ψ(t),F1(ϕ,φ,ψ)(t)=1d1(λ2-λ1)×[∫-∞t‍eλ1(t-s)H1(ϕ,φ,ψ)(s)ds+∫t∞‍eλ2(t-s)H1(ϕ,φ,ψ)(s)ds],F2(ϕ,φ,ψ)(t)=1d2(λ4-λ3)×[∫-∞t‍eλ3(t-s)H2(ϕ,φ,ψ)(s)ds+∫t∞‍eλ4(t-s)H2(ϕ,φ,ψ)(s)ds],F3(ϕ,φ,ψ)(t)=1d3(λ6-λ5)×[∫-∞t‍eλ5(t-s)H3(ϕ,φ,ψ)(s)ds+∫t∞‍eλ6(t-s)H3(ϕ,φ,ψ)(s)ds],
where (0,0,0)⩽(ϕ,φ,ψ)⩽(k1,k2,k3) and
(50)λ1=c-c2+4β1d12d1,λ2=c+c2+4β1d12d1,λ3=c-c2+4β2d22d2,λ4=c+c2+4β2d22d2,λ5=c-c2+4β3d32d3,λ6=c+c2+4β3d32d3.
Define the following profile set:
(51)Γ((ϕ_,φ_,ψ_),(ϕ¯,φ¯,ψ¯))={(ϕ,φ,ψ)∈C(ℝ,ℝ3)∣(ϕ_,φ_,ψ_)⩽(ϕ,φ,ψ)⩽(ϕ¯,φ¯,ψ¯)}.
From Lemmas 3.1–3.6 of [11], we have the following results.Lemma 4.
F
(
Γ
(
(
ϕ
_
,
φ
_
,
ψ
_
)
,
(
ϕ
¯
,
φ
¯
,
ψ
¯
)
)
)
⊂
Γ
(
(
ϕ
_
,
φ
_
,
ψ
_
)
,
(
ϕ
¯
,
φ
¯
,
ψ
¯
)
), where F=(F1,F2).Lemma 5.
F
:
Γ
(
(
ϕ
_
,
φ
_
,
ψ
_
)
,
(
ϕ
¯
,
φ
¯
,
ψ
¯
)
)
→
Γ
(
(
ϕ
_
,
φ
_
,
ψ
_
)
,
(
ϕ
¯
,
φ
¯
,
ψ
¯
)
) is compact.From Lemmas4 and 5, we see that the existence of travelling wave solutions for system (5) follows from the existence of a pair of upper and lower solutions (ϕ¯,φ¯,ψ¯) and (ϕ_,φ_,ψ_) of (43) satisfying the following conditions.(C2)
(0,0,0)⩽(ϕ_(t),φ_(t),ψ_(t))⩽(ϕ¯(t),φ¯(t),ψ¯(t))⩽(M1,M2,M3),t∈ℝ.(C3)
limt→-∞(ϕ_(t),φ_(t),ψ_(t))=(0,0,0),limt→+∞(ϕ¯(t),φ¯(t),ψ¯(t))=(k1,k2,k3).Letd1=d2=d3 and c>c*≜max{2d1βk2,2d2βk1k2e-μτ}, then there exists λ0 such that
(52)d1λ02-cλ0+βk2<0,d2λ02-cλ0+βk1k2e-μτ<0.
We can choose ɛi>0(i=1,2,…,6) such that
(53)βɛ2(1-k1)-βɛ1k2-μɛ1<0,βe-μτ[ɛ2(1-k1+ɛ4)+k2ɛ4]+δɛ3-(μ+γ)ɛ2<0,γɛ2-(μ+δ)ɛ3<0,βɛ5(1-k1+ɛ4)-(βk2+μ)ɛ4<0,δɛ6-(μ+γ)ɛ5<0,γɛ5-(μ+δ)ɛ6<0.For the above constants and suitableti(i=1,2,…,6) satisfying t4>0 and t5<t6, we define the continuous functions Φ(t)=(ϕ1(t),φ1(t),ψ1(t)) and Ψ(t)=(ϕ2(t),φ2(t),ψ2(t)) as follows:
(54)ϕ1(t)={k1eλ0t,t⩽t1,k1+ɛ1e-λt,t>t1,φ1(t)={k2eλ0t,t⩽t2,k2+ɛ2e-λt,t>t2,ψ1(t)={k3eλ0t,t⩽t3,k3+ɛ3e-λt,t>t3,ϕ2(t)={0,t⩽t4,k1-ɛ4e-λt,t>t4,φ2(t)={0,t⩽t5,k2-ɛ5e-λt,t>t5,ψ2(t)={0,t⩽t6,k3-ɛ6e-λt,t>t6.Lemma 6.
Φ
(
t
)
=
(
ϕ
1
(
t
)
,
φ
1
(
t
)
,
ψ
1
(
t
)
) is an upper solution of system (43).Proof.
Denote(55)p1(t)=d1ϕ1′′(t)-cϕ1′(t)+βφ1(t)(1-ϕ1(t))-μϕ1(t),p2(t)=d2φ1′′(t)-cφ1′(t)+βe-μτφ1(t-cτ)(1-ϕ2(t-cτ))p2(t)=+δψ1(t)-(μ+γ)φ1(t),p3(t)=d3ψ1′′(t)-cψ1′(t)+γφ1(t)-(μ+δ)ψ1(t).
Ift⩽t1,  ϕ1(t)=k1eλ0t,φ1(t)⩽k2eλ0t, then
(56)p1(t)⩽d1k1λ02eλ0t-ck1λ0eλ0t+β(1-ϕ1(t))k2eλ0t-μk1eλ0t⩽k1eλ0t(d1λ02-cλ0+βk2)<0.
If t>t1,  ϕ1(t)=k1+ɛ1e-λt,φ1(t)⩽k2+ɛ2e-λt. It follows that
(57)p1(t)⩽e-λt[d1ɛ1λ2+cɛ1λ+βɛ2(1-k1)-βɛ1k2-μɛ1].
Since β(1-k1)ɛ2-βɛ1k2-μɛ1<0, then there exists a λ1*>0 such that p1(t)<0 for all λ∈(0,λ1*).
Ift⩽t2,  φ1(t)=k2eλ0t,φ1(t-cτ)=k2eλ0(t-cτ),ψ1(t)⩽k3eλ0t, then
(58)p2(t)⩽k2eλ0t[d2λ02-cλ0+βe-(μ+cλ0)τ+δk3k2-(μ+γ)]⩽k2eλ0t(d2λ02-cλ0+βk1k2e-μτ)<0.
If t>t2, we obtain that
(59)p2(t)⩽e-λt{(1-k1+ɛ4e-λ(t-cτ))d2ɛ2λ2+cɛ2λe-λt+βe-(μ-cλ)τe-λt×[ɛ2(1-k1+ɛ4e-λ(t-cτ))+k2ɛ4]e-λt+δɛ3-(μ+γ)ɛ2(1-k1+ɛ4e-λ(t-cτ))}⩽e-λt[d2ɛ2λ2+cɛ2λ+I2(λ)],
where I1(λ)=βe-(μ-cλ)τ[ɛ2(1-k1+ɛ4)+k2ɛ4]+δɛ3-(μ+γ)ɛ2, then I1(0)<0 indicates that there exists a λ2*>0 such that p2(t)<0 for all λ∈(0,λ2*).
Ift⩽t3,  φ1(t)⩽k2eλ0t,  ψ1(t)=k3eλ0t, then
(60)p3(t)⩽eλ0t[k3d3λ02-ck3λ0+γk2-(μ+δ)k3]=k3λ0eλ0t(d3λ0-c)<0.
If t>t3,  φ1(t)⩽k2+ɛ2e-λt,ψ1(t)=k3+ɛ3e-λt, then it follows
(61)p3(t)⩽e-λt[d3ɛ3λ2+cɛ3λ+γɛ2-(μ+δ)ɛ3].
Note that γɛ2-(μ+δ)ɛ3<0, then there exists a λ3*>0 such that p3(t)<0 for all λ∈(0,λ3*).
Clearly, for allλ∈(0,min{λi*,i=1,2,3}),  pi(t)<0(i=1,2,3). This completes the proof.Lemma 7.
Ψ
(
t
)
=
(
ϕ
2
(
t
)
,
φ
2
(
t
)
,
ψ
2
(
t
)
) is a lower solution of system (43).Proof.
Denote(62)q1(t)=d1ϕ2′′(t)-cϕ2′(t)+βφ2(t)(1-ϕ2(t))-μϕ2(t),q2(t)=d2φ2′′(t)-cφ2′(t)+βe-μτφ2(t-cτ)(1-ϕ1(t-cτ))q2(t)=+δψ2(t)-(μ+γ)φ2(t),q3(t)=d3ψ2′′(t)-cψ2′(t)+γφ2(t)-(μ+δ)ψ2(t).
If t⩽t4,  ϕ2(t)=0,q1(t)=βφ2(t)⩾0.
Ift>t4,  ϕ2(t)=k1-ɛ4e-λt,φ2(t)⩾k2-ɛ5e-λt, then
(63)q1(t)⩾-ɛ4e-λt[d1λ2+cλ+βɛ5(1-k1+ɛ4)ɛ4-βk2-μ].
Note that βɛ5(1-k1+ɛ4)/ɛ4-βk2-μ<0, then there exists a λ4*>0 such that q1(t)>0 for all λ∈(0,λ4*).
Ift⩽t5, q2(t)=0. If t>t5, φ2(t)=k2-ɛ5e-λt,  φ2(t-cτ)⩾k2-ɛ5e-λ(t-cτ),  ϕ1(t-cτ)⩽k1+ɛ1e-λ(t-cτ),   ψ2(t)⩾k3-ɛ6e-λt. It follows that
(64)q2(t)⩾-e-λt{d2ɛ5λ2+cɛ5λ+δɛ6-(μ+γ)ɛ5}.
Since δɛ6-(μ+γ)ɛ5<0, then there exists a λ5*>0 such that q2(t)>0 for all λ∈(0,λ5*).
Ift⩽t6, ψ2(t)=0,q3(t)=γφ2(t)⩾0. If t>t6, φ2(t)=k2-ɛ5e-λt,ψ2(t)=k3-ɛ6e-λt, then
(65)q3(t)⩾-ɛ6e-λt[d3λ2+cλ+γɛ5ɛ6-(μ+δ)].
Note that γɛ5/ɛ6-(μ+δ)<0, then there exists a λ6*>0 such that q3(t)>0 for all λ∈(0,λ6*).
Clearly, for allλ∈(0,min{λi*,i=4,5,6}), qi(t)>0(i=1,2,3). This completes the proof.Combining Lemmas3–7 with the Schauder's fixed point theorem, we know that there exists a fixed point (ϕ*(t),φ*(t),ψ*(t)) of F in Γ((ϕ_,φ_,ψ_),(ϕ¯,φ¯)), which gives a solution of (43). Furthermore, from (C3), we can obtain that
(66)Limt→-∞(ϕ*(t),φ*(t),ψ*(t))=(0,0,0),limt→+∞(ϕ*(t),φ*(t),ψ*(t))=(k1,k2,k3),
which indicates that the fixed point satisfies the asymptotic boundary conditions (45). Therefore, there exists a travelling wave solution for system (42) connecting the steady state (0,0,0) and (k1,k2,k3), and accordingly, we have the following conclusion.Theorem 8.
Supposeℛ0>1 and d1=d2=d3. For every c>c*, system (5) always has a travelling wave solution with speed c connecting the disease-free steady state E0(1,0,0) and the endemic steady state E*(S*,I*,R*).


## 4. Numerical Simulations

In this section, focusing on the travelling wave solutions of system (5), we perform some numerical simulations.Firstly, we illustrate the results stated in Theorems1 and 8. As in [6], we take parameter values motivated by bovine tuberculosis in a cattle herd, with time unit of one year. It is assumed that the exposed period is 9 months (i.e., τ=0.75). Taking μ=0.1, δ=γ=0.5 (equal periods of infection and recovery before relapse), β=0.6, and d1=d2=d3=0.01, system (5) with above coefficients has a unique endemic equilibrium E*(0.3294,0.3394,0.2828). For convenience, we truncate the spatial domain ℝ by [-10,10]. It is easy to show that ℛ0=3.0363>1. If we use the following trivial functions as initial conditions:
(67)(S(x,t),I(x,t),R(x,t))=(0.9,0.1,0),-10⩽x⩽10,-τ⩽t⩽0,
then the numerical simulations shown in Figures 1(a)–1(c) indicate that the endemic steady state E* is stable. If we change the initial conditions, E* keeps stable (see Figure 1(d); for simplicity, we show only the solutions of I(x,t)), which indicates that E* is in fact globally asymptotically stable, although we fail to give a proof theoretically at present.The temporal solution found by numerical integration of system (5) with parameters μ=0.1, δ=γ=0.5, β=0.6, τ=0.75,  d1=d2=d3=0.01, and the initial conditions are (0.9, 0.1, 0) in (a)–(c), (0.6, 0.3, 0.1) in (A) of (d), and (0.99, 0.01, 0) in (B) of (d).
(a)
(b)
(c)
(d)Now, to illustrate the existence of travelling wave solutions, we use the following piecewise functions as initial conditions:(68)(S(x,t),I(x,t),R(x,t))={(1,0,0),-10⩽x<0,-τ⩽t⩽0,E*,0⩽x⩽10,-τ⩽t⩽0.
The numerical simulations shown in Figure 2 indicate that system (5) has a travelling wave solution with speed c≈0.07 connecting E0 and E*.The traveling wave observed in system (5) with parameters: μ=0.1, δ=γ=0.5, β=0.6, d1=d2=d3=0.01, and τ=0.75.
(a)
(b)
(c)
(d)Secondly, we address the effects of parameters on the dynamics of system (5). For this part, the following parameter values will be fixed:
(69)μ=0.1,γ=0.5,
and the initial conditions are the same as in (68).We first investigate the impact of the diffusion rates. To this end, we fixδ=0.5, β=0.6,  τ=0.75, and let the diffusivity d1,d2, and d3 vary. In this case, the values of ℛ0 and E* are the same as those in Figure 2 (Note that ℛ0 and E* are independent of the diffusion rates). Figure 3 is the simulation results of (5). In Figure 3(a), d1=d2=d3=0.01, but in Figures 3(b)–3(d), we increase the values of d1,d2, and d3 to 0.04, respectively. By contrasting these figures, we can see that the travelling waves in Figures 3(c) and 3(d) are much fast than those in Figure 3(a), but the wave speed in Figure 3(b) is just the same as that in Figure 3(a), which indicates that both the diffusion rates of infective and recovered individuals can advance the time to arrive at the infection steady state, but the diffusion rate of susceptible individuals cannot. Even so, there is an obvious difference between the wave profiles in Figure 3(b) and the other three ones: the former has a hump in wave profile for I, but the latter does not seem to show that any more; that is, the diffusion rate of susceptible individuals could lead to nonmonotone travelling waves. In biological meaning, large diffusion of susceptible individuals may enhance the accumulative effect of infection.The traveling wave observed in system (5) with parameters: μ=0.1, δ=γ=0.5, β=0.6, τ=0.75, and d1=d2=d3=0.01 in (a); d1=0.04, d2=d3=0.01 in (b); d2=0.04, d1=d3=0.01 in (c); d3=0.04, d1=d3=0.01 in (d).
(a)
(b)
(c)
(d)Next, we study the impact ofτ, that is, the average latent period of the disease. To this end, we fix δ=0.5, β=0.6, d1=0.02, and  d2=d3=0.01, and let τ vary. In this case, the values of ℛ0 and E* are different. (note that both ℛ0 and E* are decreasing in τ). Figure 4 is the simulation results of (5). In Figure 4(a), τ=0.75, but in Figure 4(b), we increase τ to 7.5. By contrasting the two figures, we can see that the travelling wave speed in Figure 4(b) is less than that in Figure 4(a), and the humps in Figure 4(a) disappear in Figure 4(b), which indicates that the average latent period of the disease can impact not only the wave speed but also the monotonicity of the wave profile, as well as the level of the positive steady state. In biological meaning, large latent period of the disease could lead to low and slow infection, and may weaken the accumulative effect of infection.The traveling wave observed in system (5) with parameters: μ=0.1, δ=γ=0.5,β=0.6, d1=0.02, d2=d3=0.01, and τ=0.75 in (a); τ=7.5 in (b).
(a)
(b)Now we investigate the impact of transmission coefficientβ. To this end, we fix δ=0.5, τ=0.75, d1=d2=d3=0.01, and let β vary. In this case, the values of ℛ0 and E* are different (note that both ℛ0 and E* are increasing in β). Figure 5 is the simulation results of (5). In Figure 5(a), β=0.3, but in Figure 5(b), we increase β to 1.2. By contrasting the two figures, we can see that the travelling wave in Figure 5(b) is rather fast than that in Figure 5(a), but the travelling waves keep monotone as τ varies, which indicates that the transmission coefficient can obviously impact the wave speed, but it cannot change the monotonicity of the wave profile. In biological meaning, large transmission coefficient could lead to high and fast infection.The traveling wave observed in system (5) with parameters: μ=0.1, δ=γ=0.5, τ=0.75, d1=d2=d3=0.01, and β=0.3 in (a); β=1.2 in (b).
(a)
(b)At last, we study the impact of an important parameterδ, that is, the rate at which an individual in the recovered class reverts to the infective class. In the following, for convenience, we call δ the relapse rate. Now we fix β=0.6, τ=0.75, and d1=d2=d3=0.01, and let δ vary. In this case, the values of ℛ0 and E* are also different (note that both ℛ0 and E* are increasing in δ). Figure 6 is the simulation results of (5). In Figure 6(a), δ=0.1, but in Figure 6(b), we increase δ to 2. By contrasting the two figures, we can see that the travelling wave in Figure 6(b) is rather fast than that in Figure 6(a), but the travelling waves keep monotone as δ varies, which indicates that the relapse rate can obviously impact the wave speed, but it cannot change the monotonicity of the wave profile. In biological meaning, large transmission coefficient could lead to high and fast infection, which is similar to the case of transmission coefficient.The traveling wave observed in system (5) with parameters: μ=0.1, γ=0.5, β=0.6, τ=0.75, d1=d2=d3=0.01, and δ=0.1 in (a); δ=2 in (b).
(a)
(b)To end this section, we point out that for the parameters given above, the following properties of the spread speedc can be obtained from the simulations and discussions above: c is increasing in d2,d3,β, and δ but decreasing in τ. In addition, small τ and large d1 may lead to nonmonotone travelling waves.


## 5. Conclusions

In this paper, we formulated an SEIR model with latent period, relapse and spatial diffusion. The dynamics of problem (5) and (6) was addressed. It was shown that the basic reproductive number of system (5) is given by ℛ0=βe-μτ(μ+δ)/(μ(μ+δ+γ)), which describes the average number of newly infected individuals at the beginning of the infectious process. We have shown when the average number of newly infected individuals is less than unity, that is, ℛ0<1; system (5) has a unique disease-free steady state E0(1,0,0), which is asymptotically stable; when the average number of newly infected individuals is greater than unity, that is, ℛ0>1, system (5) admits two steady states, E0(1,0,0) and E*(S*,I*,R*), where S*,I*, and R* are defined in (10). In this case, the endemic steady state E* is always stable, while the disease-free steady state E0 is unstable. Clearly, the spatial diffusion cannot destabilize the spatially homogenous steady state. Then by using the technique of upper and lower solutions and Schauder's fixed point theorem, we derived the existence of a travelling wave solution connecting the disease-free steady state and the endemic steady state. Furthermore, we gave some numeric simulations to illustrate the main results, and combining with numeric simulations, we discussed the effects of some parameters on the dynamics of system (5).From the discussion in Section4, we could conclude that the travelling wave speed is increasing in the diffusion rates of infective and recovered individuals, the transmission coefficient, and the relapse rate, but decreasing in the average latent period of the disease. In addition, small latent period or large diffusion rate of susceptible individuals may lead to nonmonotone travelling waves. In other words, to slow down or weaken the infection, we can reduce the diffusion rates, the transmission coefficient, or the relapse rate.



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*Source: 869603-2013-04-03.xml*