Fast diffusion inhibits disease outbreaks

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 infected people reduces the risk of infection. This completely solves and generalizes a conjecture by Allen et al. [SIAM J. Appl. Math., 67 (2007) pp. 1283–1309]. Furthermore, a substantially improved and reachable lower bound on the multipatch reproduction number, a generalized monotone result on the spectral bound of the Jacobian matrix of the model system at the disease-free equilibrium, and the limit of the endemic equilibrium as the diffusion coefficient goes to infinity 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.