Propagation dynamics in an SIRS model with general incidence functions

Abstract

This paper studies the initial value problems and traveling wave solutions in an SIRS model with general incidence functions. Linearizing the infected equation at the disease free steady state, we can define a threshold if the corresponding basic reproduction ratio in kinetic system is larger than the unit. When the initial condition for the infected is compactly supported, we prove that the threshold is the spreading speed for three unknown functions. At the same time, this threshold is the minimal wave speed for traveling wave solutions modeling the disease spreading process. If the corresponding basic reproduction ratio in kinetic system is smaller than the unit, then we confirm the extinction of the infected and the nonexistence of nonconstant traveling waves.

Keywords

• comparison principle
• upper and lower solutions
• asymptotic spreading
• minimal wave speed
• spreading speed