# A two-patch epidemic model with nonlinear reinfection

## TL;DR

This paper develops a two-patch epidemic model with nonlinear reinfection to analyze how movement between urban and rural populations influences disease spread and stability, highlighting potential benefits of population movement in controlling outbreaks.

## Contribution

It introduces a novel two-patch SI extasciitilde S model with nonlinear reinfection and analyzes the effects of population movement on disease dynamics and stability.

## Key findings

- Population movement can expand the stability region of disease-free states.
- Nonlinear reinfection significantly affects disease persistence.
- Movement between patches can be beneficial for disease control.

## Abstract

The propagation of infectious diseases and its impact on individuals play a major role in disease dynamics, and it is important to incorporate population heterogeneity into efforts to study diseases. As a simplistic but illustrative example, we examine interactions between urban and rural populations in the dynamics of disease spreading. Using a compartmental framework of susceptible--infected--susceptible ($\mathrm{SI\widetilde{S}}$) dynamics with some level of immunity, we formulate a model that allows nonlinear reinfection. We investigate the effects of population movement in the simplest scenario: a case with two patches, which allows us to model movement between urban and rural areas. To study the dynamics of the system, we compute a basic reproduction number for each population (urban and rural). We also compute steady states, determine the local stability of the disease-free steady state, and identify conditions for the existence of endemic steady states. From our analysis and computational experiments, we illustrate that population movement plays an important role in disease dynamics. In some cases, it can be rather beneficial, as it can enlarge the region of stability of a disease-free steady state.