Traveling waves for a nonlocal dispersal SIR epidemic model with the
mass action infection mechanism
Abstract
This paper is concerned with a nonlocal dispersal
susceptible–infected–recovered (SIR) epidemic model adopted with the
mass action infection mechanism. We mainly study the existence and
non-existence of traveling waves connecting the infection-free
equilibrium state and the endemic equilibrium state. The main
difficulties lie in the fact that the semiflow generated here does not
admit the order-preserving property. Meanwhile, this new model brings
some new challenges due to the unboundedness of the nonlinear term. We
overcome these difficulties to obtain the boundedness of traveling waves
with the speed $c>c_{\min}$ by some
analysis techniques firstly and then prove the existence of traveling
waves by employing Lyapunov–LaSalle theorem and Lebesgue dominated
convergence theorem. By utilizing a approximating method, we study the
existence of traveling waves with the critical wave speed
$c_{\min}$. Our results on this new model may
provide some implications on disease modelling and controls.