Bi-Virus SIS Epidemic Propagation under Mutation and Game-theoretic Protection Adoption

We study a bi-virus susceptible-infected-susceptible (SIS) epidemic model in which individuals are either susceptible or infected with one of two virus strains, and consider mutation-driven transitions between strains. The general case of bi-directional mutation is first analyzed, where we characterize the disease-free equilibrium and establish its global asymptotic stability, as well as the existence, uniqueness, and stability of an endemic equilibrium. We then present a game-theoretic framework where susceptible individuals strategically choose whether to adopt protection or remain unprotected, to maximize their instantaneous payoffs. We derive Nash strategies under bi-directional mutation, and subsequently consider the special case of unidirectional mutation. In the latter case, we show that coexistence of both strains is impossible when mutation occurs from the strain with lower reproduction number and transmission rate to the other strain. Furthermore, we fully characterize the stationary Nash equilibrium (SNE) in the setting permitting coexistence, and examine how mutation rates influence protection adoption and infection prevalence at the SNE. Numerical simulations corroborate the analytical results, demonstrating that infection levels decrease monotonically with higher protection adoption, and highlight the impact of mutation rates and protection cost on infection state trajectories.