Optimal Virulence, Diffusion and Tradeoffs

In this work we propose a variant of a classical SIR epidemiological model where pathogens are characterized by a (phenotypic) mutant trait x. Imposing that the trait x mutates according to a random walk process and that it directly influences the epidemiological components of the pathogen, we studied its evolutionary development by interpreting the tenet of maximizing the basic reproductive number of the pathogen as an optimal control problem. Pontryagin's maximum principle was used to identify the possible optimal evolutionary strategies of the pathogen. Qualitatively, three types of optimal evolutionary routes were identified and interpreted in the context of virulence evolution. Each optimal solution imposes a different tradeoff relation among the epidemiological parameters. The results predict (mostly) two kinds of infections: short-lasting mild infections and long-lasting acute infections.