# Threshold phenomenon and traveling waves for heterogeneous integral equations and epidemic models

## TL;DR

This paper investigates threshold phenomena and traveling wave solutions in spatially periodic heterogeneous integral equations relevant to epidemiology, providing conditions for wave existence and speeds, and applying findings to a spatial SIR model.

## Contribution

It introduces new results on threshold behavior and traveling wave existence in heterogeneous integral equations, with explicit formulas for wave speeds and applications to epidemic models.

## Key findings

- Identification of threshold phenomena in heterogeneous equations
- Existence and non-existence criteria for traveling waves
- Explicit formulas for admissible wave speeds

## Abstract

We study some anisotropic heterogeneous nonlinear integral equations arising in epidemiology. We focus on the case where the heterogeneities are spatially periodic. In the first part of the paper, we show that the equations we consider exhibit a "threshold phenomenon". In the second part, we study the existence and non-existence of "traveling waves", and we provide a formula for the admissible speeds. In a third part, we apply our results to a spatial heterogeneous SIR model.