Traveling waves of a delayed epidemic model with spatial diffusion

In this paper, we study the existence and non-existence of traveling waves for a delayed epidemic model with spatial diffusion. That is, by using Schauder’s fixedpoint theorem and the construction of Lyapunov functional, we prove that when the basic reproduction number R0 > 1, there exists a critical number c∗ > 0 such that for all c > c∗, the model admits a non-trivial and positive traveling wave solution with wave speed c. And for c < c∗, by the theory of asymptotic spreading, we further show that the model admits no non-trivial and non-negative traveling wave solution. And also, some numerical simulations are performed to illustrate our analytic results.


Introduction
In [3], the authors derived the following delayed epidemic model with the Beddington-DeAngelis incidence rate where S(t), I(t) represent the number of susceptible individuals and infective individuals at time t, respectively.A is the recruitment rate of the population, µ is the natural death of the population, α is the death rate due to disease, β is the transmission rate, α 1 and α 2 are the parameters that measure the inhibitory effect, γ is the recovery rate of the infectious individuals, and τ is the incubation period.
By constructing the suitable Lyapunov functional, the authors [3] determined the global asymptotic stability of model (1.1).Clearly, model (1.1) is one of ODE type, which could only reflect the epidemiological and demographic process as the time changes.We note that the spatial content of the environment has been ignored in model (1.1).To closely match the reality, considering a diffusive epidemic model of PDE type is natural and reasonable, therefore, it gives us the motivation to investigate the PDE type of model (1.1).Here, we propose the following delayed disease model with spatial diffusion = d 1 ∆S(t, x) + A − µS(t, x) − βS(t, x)I(t, x) 1 + α 1 S(t, x) + α 2 I(t, x) , in which S(t, x) and I(t, x) denote the number of susceptible individuals and infective individuals at time t and position x ∈ R n , respectively.d 1 , d 2 > 0 are the diffusion rates, ∆ is the Laplacian operator.The parameters A, µ, β, α 1 , α 2 , γ, τ are positive constants as in model (1.1).
In the biological context, to better understand the geographic spread of infectious diseases, epidemic waves play a key role in studying the spatial spread of infectious diseases.Biologically speaking, the existence of an epidemic wave implies that the disease can invade successfully and an epidemics arises.The traveling wave describes the epidemic wave moving out from an initial disease-free equilibrium to the endemic equilibrium with a constant speed.The wave speed c may explain the spatial spread speed of the disease, which may measure how fast the disease invades geographically.Recently, many authors have studied the existence of traveling wave solutions of various epidemic models, see, for example, [1,2,4,5,7,9,10,13,[15][16][17][18][19][21][22][23][24][25] and references therein.
In this paper, we will study the existence and non-existence of traveling waves for model (1.2).We employ Schauder's fixed point theorem combining with the upper-lower solutions to establish the existence theorem (Theorem 3.2).Namely, we will show that when the basic reproduction number R 0 > 1, there exists c * > 0 such that (1.2) has a positive traveling wave solution if c > c * .Further, we shall construct the appropriate Lyapunov functional to show that the traveling wave converges to the endemic steady state E * = (S * , I * ) as t → +∞.Moreover, by the theory of asymptotic spreading, we conclude the non-existence of traveling wave solutions for model (1.2) when R 0 > 1 and c ∈ (0, c * ) (Theorem 3.3).Some numerical simulations are carried out to validate the theoretical results.
This paper is organized as follows.In Section 2, we give some preliminaries, that is, we establish the well-posedness for model (1.2), construct a pair of upper-lower solutions, and verify the conditions of the Schauder fixed point theorem.In Section 3, we give and show the existence and non-existence of traveling waves of model (1.2).Some numerical simulations are given in Section 4.

The well-posedness
For simplicity, let and dropping the bars on S, I, α 1 , α 2 , β, we obtain the following model where under the initial conditions As in [3], we define the basic reproduction number R 0 as .
By a direct computation, we get the following conclusion.
(2) If R 0 > 1, then system (2.1) has a unique endemic equilibrium E * = (S * , I * ), where Next, we consider the positive invariance and uniform boundedness of solutions for the initial value problem of system (2.1)-(2.2).
Let X := BUC(R n , R 2 ) be the set of all bounded and uniformly continuous functions from R n to R 2 , and X + := BUC(R n , R 2 + ).Then X + is a closed cone of X and induces a partial ordering on X.With the usual supremum norm, it follows that (X, • X ) is a Banach space.Clearly, any vector in R 2 can be regarded as an element in X.
0], X) with the supremum norm and C + = C([−τ, 0], X + ).Then (C, C + ) is an ordered Banach space.As usual, we identify an element ϕ ∈ C as a function from [6,Theorem 1.5], it follows that X-realization D∆X of D∆ generates an analytic semigroup T (t) on X.

The wave equations and the upper and lower solutions
In this paper, we mainly deal with the existence of traveling waves of system (2.1) connecting the disease-free equilibrium E 0 (1, 0) and the endemic equilibrium E * (S * , I * ).Without loss generality, we consider n = 1.A traveling wave solution of (2.1) is a special type of solution of system (2.1) with the form (S(t, x), I(t, x)) = (S(x + ct), I(x + ct)), here c > 0 is the wave speed, and letting x + ct by t, which satisfies the following wave equation and the boundary conditions Linearizing of the second equation of (2.3) at E 0 (1, 0), we get the characteristic equation It is easy to show the following lemma, see, [13,Lemma 4.4] or [21,Lemma 3.1].
Then there exist two positive constants λ * > 0 and c * > 0 such that ( In this subsection, we assume that R 0 > 1.In addition, we fix a positive constant c > c * and always denote λ i (c) = λ i , i = 1, 2. Now, we define four continuous functions as following for t ∈ R, where σ, M, ε are three positive constants to be determined in the following lemmas.
Lemma 2.4.The functions S(t) and I(t) satisfy the inequality ) Proof.If t < t 1 , then I(t) = e λ 1 t .Note that S(t) = 1 and I(t) ≤ e λ 1 t for all t ∈ R, then In view of the fact that S(t) = 1 and This completes the proof.

The verification of the Schauder fixed point theorem
In this subsection, we will use the upper and lower solutions (S(t), I(t)) and (S(t), I(t)) constructed in Section 2.2 to verify that the conditions of Schauder fixed point theorem hold.Denote Choose two constants β 1 > µ + β α 2 and β 2 > r such that H 1 is nondecreasing with respect to the first variable S(t) ∈ [0, 1] and H 1 is nonincreasing with respect to the second variable ] for all t ∈ R. Clearly, (2.3) is equal to Define the set Then the set Γ is nonempty, closed and convex in [0, M] C .Furthermore, define an operator where and Lemma 2.7.The operator F maps Γ into Γ.
Proof.For (S, I) ∈ Γ, we only need to prove the following inequalities hold.
We only prove the first inequality since the proof of the second inequality is similar to that of the first.Indeed, according to the monotonicity of H 1 with respect to S and I, we have Thus it is sufficient to verify (2.9) In fact, for t = t 2 , by (2.6), we have For t > t 2 , since S (t 2 −) ≤ 0 and λ 12 > 0 > λ 11 , we have Similarly, we also have F 1 (S, I)(t) ≥ S(t) for t < t 2 .By the continuity of both S(t) and F 1 (S, I)(t), we obtain F 1 (S, I)(t) ≥ S(t) for all t ∈ R.
On the other hand, note that H 1 is nondecreasing with respect to S(t) ∈ [0, 1], we get This completes the proof of (2.9).

Existence and non-existence of the traveling wave solution
First, using the ideas in [7], we derive some boundedness property of the solution (S(t), I(t)) of system (2.1).That is, we give the following lemma.
For case (i), by the definition of Φ(t) and in view of Φ(t) ≤ 0, we get Together with 0 < S ≤ 1 and which implies that S (t) is decreasing on [t 1 , +∞).Hence This implies that S(t) is decreasing and convex, which contradicts the boundedness of S(t).
For case (ii), since Φ(t 2 ) = 0, Φ (t 2 ) ≥ 0, we get Hence, we obtain This is a contradiction.Similarly, we also can show that the other inequations of (3.1) hold for t ≥ 0. Now we are in a position to state and show our main results.
Theorem 3.2.Assume that R 0 > 1 holds.Then there exists a constant c * > 0 such that for every c > c * , system (2.1) admits a nontrivial positive traveling wave solution S(x + ct), I(x + ct) satisfying the asymptotic boundary condition (2.4), and Proof.In view of Lemmas 2.6-2.9, it follows from Schauder's fixed point theorem that there exists a pair of (S, I) ∈ Γ, which is a fixed point of the operator F. Further, (S, I) is a solution of (2.1).Consequently, the solution S(x + ct), I(x + ct) is a traveling wave solution of system (2.1).Moreover, (S, I) satisfies the following inequalities Note that (S, I) ∈ Γ is a fixed point of the operator of F. Applying L'Hospital's rule to the maps F 1 and F 2 , it is easy to see that S (−∞) = 0 and I (−∞) = 0. Integrating both sides of the second equation of (2.3) from −∞ to t gives Hence, by lim Next we claim that, for all t ∈ R, That is, the traveling wave solution of (2.1) is nontrivial positive.Indeed, Similarly, we can prove another inequality is also true.
To simplify the notation, let it is easy to see that g(x) ≥ 0, x > 0, and g(x) = 0 if and only if x = 1.
By (3.2), we see that (S, I) is a positive and bounded solution of (2.1).Define D = (S, I) : By Lemma 3.1, we see that D = ∅.For each (S, I) ∈ D, we consider the following Lyapunov functional V(S, I) : R + → R as follows where By a direct calculation, we have From the fact that Hence, we get Finally, we apply the ideas of [10] to establish the non-existence of traveling wave solutions of system (2.1).Theorem 3.3.Assume that R 0 > 1 holds.Then there exists a constant c * > 0 such that for c ∈ (0, c * ), system (2.1) does not admit a traveling wave solution (S(x + ct), I(x + ct)) satisfying (2.4).
Thus, for t < T( ), we have According to (3.2), there exists a constant h > 0 large enough, such that In fact, it is equivalent to the following inequality  Let −x = c 1 +c * 2 t, then t → ∞ implies that x + c 1 t → −∞.Consequently, lim t→∞ I(x, t) = 0, which contradicts (3.9).This completes the proof.

Numerical simulations
In this section, we carry out numerical simulations to illustrate the theoretical results obtained in Sections 3.For simplify, we use the following trivial functions as initial conditions