# Multi-Patch and Multi-Group Epidemic Models: A New Framework

## TL;DR

This paper introduces a comprehensive multi-patch, multi-group epidemic model incorporating host mobility based on epidemiological status, deriving the basic reproduction number and analyzing how heterogeneity and mobility influence disease spread.

## Contribution

It presents a novel framework for modeling complex host interactions and mobility patterns, including explicit derivation of the basic reproduction number in this context.

## Key findings

- Reproduction number increases with the number of patches and mobility complexity.
- Explicit formula for $ ext{R}_0$ when mobility matrices are rank one.
- Mobility patterns significantly impact disease prevalence and $ ext{R}_0$.

## Abstract

We develop a multi-patch and multi-group model that captures the dynamics of an infectious disease when the host is structured into an arbitrary number of groups and interacts into an arbitrary number of patches where the infection takes place. In this framework, we model host mobility that depends on its epidemiological status, by a Lagrangian approach. This framework is applied to a general SEIRS model and the basic reproduction number $\mathcal{R_0}$ is derived. The effects of heterogeneity in groups, patches and mobility patterns on $\mathcal{R_0}$ and disease prevalence are explored. Our results show that for a fixed number of groups, the basic reproduction number increases with respect to the number of patches and the host mobility patterns. Moreover, when the mobility matrix of susceptible individuals is of rank one, the basic reproduction number is explicitly determined and was found to be independent of the latter. The cases where mobility matrices are of rank one capture important modeling scenarios. Additionally, we study the global analysis of equilibria for some special cases. Numerical simulations are carried out to showcase the ramifications of mobility pattern matrices on disease prevalence and basic reproduction number.