# Fast Diffusion Inhibits Disease Outbreaks

## TL;DR

This paper demonstrates that increasing the diffusion rate of infected individuals in a multi-patch SIS epidemic model generally decreases the basic reproduction number, thereby reducing infection risk, and provides new bounds and results for such models.

## Contribution

It fully resolves and extends a previous conjecture by analyzing the impact of diffusion on the reproduction number in multi-patch epidemic models.

## Key findings

- Fast diffusion reduces the basic reproduction number.
- Provides improved bounds on the multipatch reproduction number.
- Generalizes results to models with single infected class and one transmission route.

## Abstract

We show that the basic reproduction number of an SIS patch model with standard incidence is either strictly decreasing and strictly convex with respect to the diffusion coefficient of infected subpopulation if the patch reproduction numbers of at least two patches in isolation are distinct or constant otherwise. Biologically, it means that fast diffusion of the infected people reduces the risk of infection. This completely solves and generalizes a conjecture by Allen et al. ({\it SIAM J Appl Math}, 67: 1283-1309, 2007). Furthermore, a substantially improved lower bound on the multipatch reproduction number, a generalized monotone result on the spectral bound the Jacobian matrix of the model system at the disease-free equilibrium, and the limiting endemic equilibrium are obtained. The approach and results can be applied to a class of epidemic patch models where only one class of infected compartments migrate between patches and one transmission route is involved.