Spreading dynamics of an SVIRS model

The geographic spread of a disease epidemic has long been a key focus of public attention. This article investigates the spreading properties of a reaction-diffusion system, which models the Susceptible-Vaccinated-Infected-Recovered-Susceptible (SVIRS for short) process. Assume that the habitat of the entire population expands or contracts in a wave front pattern. Then we study the corresponding initial value problems and traveling wave solutions, which model the spatial expanding ability of the disease. A constant associated with the disease's transmission capacity is given, enabling the exploration of practical factors that influence disease spreading. For example, both the vaccination rate and vaccines' effective protection rate can reduce the spatial transmission capacity of diseases. Moreover, we numerically find that the proportion of recovered individuals who lose their immunity does not affect the spreading ability but changes the prevalence scale.