Regime dependent infection propagation fronts in an SIS model

TL;DR

This paper investigates the existence and properties of traveling front solutions in various regimes of a diffusive SIS epidemic model, revealing how diffusion rates influence infection spread dynamics and identifying connections to known equations like Burgers-FKPP.

Contribution

It provides a comprehensive analysis of regime-dependent traveling fronts in SIS models, including existence, structure, and speed bounds, and links to classical equations.

Findings

Traveling front solutions exist for each fixed positive speed in certain regimes.

The structure and speed dependence of solutions vary with diffusion rates.

A connection to the Burgers-FKPP equation is established in a specific case.

Abstract

We show the existence of traveling front solutions in a diffusive classical SIS epidemic model and the SIS model with a saturating incidence in the size of the susceptible population. We investigate the situation where both susceptible and infected populations move around at a comparable rates, but small compared to the spatial scale. In this case, we show that traveling front solutions exist for each fixed positive speed. In the regime where the infected population diffuses slower than the susceptible population, we show the existence of traveling wave solutions for each fixed positive speed and describe their structure and dependence on the wave speed which as it is varied from 0 to infinity. In the regime where the infected population diffuses faster than the susceptible population, we derive a bound for the speeds of the fronts in this regime in which the infection propagates as a front. Moreover, for the classical SIS model we show that there is a case when the spread of the disease is governed by the Burgers-FKPP equation.