The Study Of The Traveling Wave Solutions For Two Diffusive Epidemic Models

Because the individuals can move freely,the process and law of epidemic spread are usually described by diffusive partial differential equations.The existence and non-existence of the traveling wave solutions for these epidemic models determine whether the epidemic can spread and how much the speed of transmission is.Therefore,the study of existence and non-existence of traveling wave solutions for these epidemic models is beneficial to prevent and control epidemics.The first part is concerned with a diffusive epidemic model with standard incidence rate.The existence of a critical traveling wave solution to this model is obtained by the method of upper and lower solutions together with Schauder's fixed point theorem.The positiveness of the critical traveling wave solution is derived by the strong maximum principle.The asymptotic boundary of this solution at minus infinity is found with the aid of squeeze theorem.At last,some properties of the critical traveling wave solution are presented.The second part studies a discrete diffusive epidemic model with saturated incidence and time delay.The existence of a super-critical traveling wave solution on a large interval to this model is obtained by combining the method of upper and lower solutions and Schauder's fixed point theorem.Then the existence is extended to the whole real line by a limiting argument.The positiveness and asymptotic boundary of the super-critical traveling wave solution at infinity are found by reduction to absurdity.Subsequently,the existence of a non-trivial,positive and bounded critical traveling wave solution is established in a similar way.At last,the non-existence of non-trivial,positive and bounded traveling wave solutions is derived by two-sided Laplace transform.