Pathogen evolution in switching environments: a hybrid dynamical system approach

TL;DR

This paper models pathogen evolution in fluctuating environments using a hybrid dynamical system, revealing conditions for stability of genotypes and maintenance of polymorphism under stochastic environmental switching.

Contribution

It introduces a hybrid dynamical system framework combining Fisher-Haldane-Wright equations with Markov switching to analyze pathogen evolution.

Findings

Genotype with highest mean fitness is stable in constantly fluctuating environments.

Genotype-dependent switching probabilities can maintain polymorphism.

Stability in probability depends on environmental fluctuation patterns.

Abstract

We propose a hybrid dynamical system approach to model the evolution of a pathogen that experiences different selective pressures according to a stochastic process. In every environment, the evolution of the pathogen is described by a version of the Fisher-Haldane-Wright equation while the switching between environments follows a Markov jump process. We investigate how the qualitative behavior of a simple single-host deterministic system changes when the stochastic switching process is added. In particular, we study the stability in probability of monomorphic equilibria. We prove that in a "constantly" fluctuating environment, the genotype with the highest mean fitness is asymptotically stable in probability. However, if the probability of host switching depends on the genotype composition of the population, polymorphism can be stably maintained. This is a corrected version of the paper that appeared in Mathematical Biosciences 240 (2012), p. 70-75. A corrigendum has appeared in the same journal.