Traveling waves for a nonlocal dispersal SIR model with standard incidence

Abstract

This paper is concerned with traveling wave solutions of a nonlocal dispersal SIR epidemic model with standard incidence. We show that our results on existence and nonexistence of traveling wave solutions are determined by the basic reproduction number of the corresponding ordinary differential model and the minimal wave speed. These threshold dynamics are proved by constructing an invariant cone and applying Schauder's fixed point theorem on this cone and the Laplace transform. The main difficulties are the lack of an occurrence of a regularizing effect and the loss of the order-preserving property of this model. Introduction.The current paper is concerned with the existence and nonexistence of traveling wave solutions of the following nonlocal dispersal SIR model with standard incidence:where S, I and R denote the sizes of the susceptible, infected and removal individuals, respectively. The infection rate β and the removal rate γ are positive numbers. d i > 0 (i = 1, 2, 3) are dispersal rates for the susceptible, infected and removal individuals, respectively.

Traveling wave solutions for a delayed diffusive SIR epidemic model with nonlinear incidence rate and external supplies

“…They also obtained the existence and non-existence results of traveling wave solutions, which is related to the basic reproduction number. Li et al [12][13][14] further studied the traveling wave solutions for nonlocal delayed and nonlocal dispersal SIR model with this incidence rate. We mention that the incidence rateˇS I SCI here is an inseparable form, which is different from the other types of incidence.…”

“…considered the diffusive SIR epidemic model with the standard incidence rate They also obtained the existence and non‐existence results of traveling wave solutions, which is related to the basic reproduction number. Li et al . further studied the traveling wave solutions for nonlocal delayed and nonlocal dispersal SIR model with this incidence rate.…”

Traveling waves for a nonlocal dispersal SIR model with delay and external supplies

“…As we know, since Bartlett [1] in 1956 predicted a wave of infection moving out from the initial source of infection with a constant speed, epidemic waves have been extensively studied, see for example [10,29,22,45,15,38,40,42,20,12,19,11,23,46]. See also the monograph [30], the survey papers [32,33] and the references therein for more details.…”

“…[13,2,39,18]). Recently, traveling waves of nonlocal dispersal problems have attracted much attention, see for instance [29,22,45,2,6,5,[7][8][9]21,35,47] and references therein.…”

“…The much significant one is Medlock and Kot [29], where they computed the minimal wave speeds under linear conjecture assumption and gave some approximations to the shape of the traveling waves. Recently, Li and Yang [22] and Yang et al [45] considered traveling waves of nonlocal dispersal SIR models without the vital dynamics, which have rapid outbreak patterns and can only be used to model the fast diseases. However, many diseases in reality occur in more than one year and there is a renewal of the susceptible individuals (see Hethcote [14]).…”