EXISTENCE OF TRAVELING WAVES BY MEANS OF FIXED POINT THEORY FOR AN EPIDEMIC MODEL WITH HATTAF-YOUSFI INCIDENCE RATE AND TEMPORARY IMMUNITY ACQUIRED BY VACCINATION

. The main aim of this work is to investigate the existence of traveling waves of an epidemic model with temporary immunity acquired by vaccination. The incidence rate of the disease used in the epidemic model is of the form Hattaf-Yousﬁ that includes many types existing in the literature. By means of Schauder ﬁxed point theorem and construction of a pair of upper and lower solutions, the existence of traveling wave solution that connects the disease-free equilibrium and the endemic equilibrium is obtained and characterized by two parameters that are the basic reproduction number and the minimal wave speed


INTRODUCTION
In epidemiology, the existence of traveling wave which describes the transition of diseasefree equilibrium to endemic equilibrium has been investigated by many authors. For instance, in 1995, Hosono and Ilyas [1] investigated the existence of traveling wave solution of diffusive epidemic model. In 2011, Yang et al. [2] interested to the existence of traveling waves to a SIR epidemic model with nonlinear incidence rate, spatial diffusion and time delay. In 2014, Xu [3] studied the traveling waves in a Kermack-Mckendrick epidemic model with diffusion and latent period. In [4], the authors studied the existence of solution of traveling waves of a delayed diffusive epidemic model with specific nonlinear incidence rate. In this study, we propose the following system with general incidence rate and temporary immunity acquired by vaccination: S(x,t − u)g(u)e −µu du, ∂ I(x,t) ∂t = D I ∆I + h(S(x,t), I(x,t − τ))I(x,t − τ) − (µ + d + r)I(x,t), where S(x,t), I(x,t) and R(x,t) represent respectively the densities of susceptible, infected and removed individuals at position x and time t. The constants D S , D I and D R denote the corresponding diffusion coefficients for the susceptible, infected and removed populations. A is the recruitment rate of susceptible population, µ is the natural death rate of the population, d is the death rate due to disease, τ is the latent period, and r is the recovery rate of the infected population. ρ is the rate of vaccination, and g(u)du is the probability of losing immunity between u and u + du, where g(u) is the density of probability satisfying +∞ 0 g(u)du = 1. The incidence rate of infection is modeled by Hattaf-Yousfi function h(S, I) = β S α 0 +α 1 S+α 2 I+α 3 SI [5], with α 0 , α 1 , α 2 , α 3 ≥ 0 are constants and β > 0 is the infection process. This general incidence rate generalizes many type of incidence rate used in [6,7,8,9].
Since the variable of removed individuals R does not appears in the first two equations of (1), then system (1) can be reduced to the following model: with homogeneous Neumann boundary conditions and initial conditions where Ω is a bounded domain in R n with smooth boundary ∂ Ω and ∂ ∂ ν denotes the outward normal derivative on ∂ Ω.
The rest of this paper is outlined as follows. Section 2 is devoted to the reproduction number and steady states. Section 3 deals with the existence of traveling waves by means of Schauder fixed point theorem. Section 4 treats the nonexistence of traveling waves. Finally, the paper ends with a conclusion.

REPRODUCTION NUMBER AND STEADY STATES
In this section, we study the existence of equilibria of system (2).
A µ+ρ−ρη , 0 is the unique disease-free steady state. Hence, the basic reproduction number is as follows Theorem 2.1.

EXISTENCE OF TRAVELING WAVES
In this section, we study the existence of traveling waves by means of Schauder fixed point theorem.
Using the following transformations in system (2) In this case, system (8) has always a disease-free equilibrium E 0 (1, 0) and an endemic equilibrium E * S * ς , I * ς . The traveling wave solution of our new system connecting the disease-free equilibrium and the endemic equilibrium is a special solution of the form where ζ , ξ ∈ C 2 (R, R) and c > 0 is a constant representing the wave speed. By substituting ζ (x + ct) and ξ (x + ct) into (8) and denoting x + ct by y, we obtain two equations with the boundary conditions, On the other hand, the characteristic equation of equation (10) at E 0 satisfies Hence, it is easy to prove the following result.
To establish the existence of traveling wave solution of equations (9)-(10), we construct a pair of upper and lower solutions. For the rest of this section, we assume that R 0 > 1 and c > c * .
Proof. Let be the following function We have Then, we conclude It remains to prove that the functionξ satisfies for all y = y 1 := 1 If y < y 1 , thenξ (y) = e λ 1 y . Let the following function We have By Lemma 3.1, we have ∆ (λ 1 , c) = 0. Then we deduce (16).
On the other hand if y < y 2 , then ζ (y) = 1 − 1 σ e σ y . Let the following function We have By the choice of σ we get Then we conclude the inequality (17).

Thenh
(ζ (y), ξ (y − cτ))ξ (y − cτ) ≥ β ξ (y − cτ) and we verify immediately that Let Γ the set defined as follows We can verify that Γ is nonempty, closed and convex in C R, R 2 . Consider the operator We verify easily that any fixed point of F is a solution of (19). Hence, the existence of solution of (9)-(10) is reduced to verify that the operator F satisfies the conditions of Schauder fixed point theorem. Here, we divide the proof into three lemmas.
Proof. We can verify easily by the choice of γ, that H 1 is monotone increasing in ζ and monotone decreasing in ξ , then we get for all y ∈ R (20) By (17), we get for all y = y 2 .
Let v > 0 be a constant such that v < min −λ 11 , −λ 21 , µ c , and we can verify that B v R, R 2 is a Banach space with the norm |.| v , defined by Lemma 3.6. The operator F is continuous with respect to the norm | · | v in B v R, R 2 . Then Therefore, We have y −∞ e λ 11 (y−x) + +∞ y e λ 12 (y−x) e −v|y| e v|x| dx ≤ 1 Then where η 2 = +∞ 0 g(u)e −(µ−cv)u du. Hence , F 1 : Γ −→ Γ is continuous with respect to the norm | · | v . similarly, we show that F 2 : Γ → Γ is continuous with respect to the norm | · | v . Therefore we conclude that, F is continuous with respect to the norm | · | v in B v R, R 2 .
Lemma 3.7. The operator F is compact with respect to the norm | · | v in B v R, R 2 .
Proof. Let (ζ , ξ ) ∈ Γ, we have On the other hand, For every n ∈ N, let F n an operator defined by we can verify that F n is uniformly bounded and equicontinuous for (ζ , ξ ) ∈ Γ. By applying Arzela-Ascoli theorem, we get that F n : Γ → Γ is compact with respect to the super norm in Hence, {F n } +∞ 0 is a compact series, In addition, we have Thus, {F n } +∞ 0 converges to F in Γ with respect to the norm | · | v . From Proposition 2.1 in [10], we deduce that F : Γ → Γ is also compact with respect to the norm | · |v.
Theorem 3.8. For R 0 > 1 and c > c * , system (8) admits a traveling wave solution (ζ (x + ct), ξ (x + ct)) connecting the disease-free equilibrium E 0 (1, 0) and the endemic equilibrium Proof. From Lemma 3.6 and 3.7, we conclude that the operator F satisfies the conditions of Schauder fixed point theorem. Then F admits a fixed point (ζ , ξ ) ∈ Γ. We verify immediately that this fixed point is a solution of (9) and (10).
On the other hand, we verify that the fixed point (ζ , ξ ) satisfies the boundary conditions (11).

NONEXISTENCE OF TRAVELING WAVES
In this section, we study the nonexistence of nontrivial traveling wave solutions of system (1), connecting the disease-free equilibrium and the endemic equilibrium. Proof. First, for R 0 > 1 and c ∈ (0, c * ), we suppose that problem (9)-(10) admits a positive solution (ζ , ξ ). By the result of Lemma 3.1, there exists a ε > 0 sufficiently small such that the equation has no real solution forc ∈ 0, c+2c * 3 .
According to the above theorem, we deduce immediately the following result.

CONCLUSION
In this work, we have proposed an epidemic model with diffusion, Hattaf-Yousfi incidence rate and temporary immunity acquired by vaccination. The proposed model contains two delays one is discrete representing the latent period and the other is infinite distributed delay modeling immunity period. We first determined the basic reproduction number and steady states of the proposed model. The existence of traveling waves describing the transition of disease-free equilibrium to endemic equilibrium has been established by means of Schauder fixed point theory. In addition, we have studied the nonexistence of nontrivial traveling wave solution of the proposed model.