Travelling wave solutions for an infection-age structured model with diffusion

Abstract

International audienc

Traveling waves in a nonlocal dispersal Kermack-McKendrick epidemic model

“…We find that a uniform bound of solutions is more hard to be obtained due to the nonlocal dispersal effect. Moreover, it is necessary to point out that we don't know the exact boundary behavior of susceptible S(ξ) at +∞, which is different from the work in [10,11,23,27,39]. However, we can give a sufficient condition to guarantee that S(+∞) exists.…”

“…Moreover, since the solutions are of class C 1 ([−X, X]), S X and I X are bounded according to equations (9) and (10). Thus, by the definition of Γ X , we have F is compact.…”

“…We refer to [9,10,11,26,28,39] and references therein for more results on the development of the study of epidemic problems.…”

“…In this paper, inspired by [39], we establish the existence of nontrivial traveling wave solutions with wave speed c > c * by constructing a suitable invariant set and then using the Schauder's fixed theorem (also see e.g., [11,23,27]). Here, we mainly apply the method in [10,40] to obtain the existence of nontrivial traveling wave solutions. That is, we construct an invariant cone in a large bounded domain with initial functions being defined on, then pass to the unbounded domain by a limiting argument.…”

Mathematical analysis for an epidemic model with spatial and age structure

“…[4,7,15,17,18]. If diffusion is taken into consideration epidemic models with age structure are studied in [5], where existence of periodic solutions is investigated in the case V = R n and with a time periodic forcing, in [12] and [13], where existence of traveling wave solutions is investigated in the scalar case and in [19], where approximate solutions using Galerkin methods are considered. More related to our model are the papers [10] and [16], where similar SIR models, with spatial and age structure, are studied in a L 2 setting or a C setting and under different assumptions.…”