The Equilibrium Analysis for Competitive Spreading Over Networks With Mutations

Epidemic models have been used to analyze various spreading phenomena in the population, and usually consider that the spreading object does not change in the spreading process. Yet generally, the virus may evolve due to the influence of environments and medical interventions, or the information may be modified by individuals in networks. In this letter, we investigate the spread of two competing viruses in a network, where one of the viruses can mutate into the other one with a certain probability. Based on the multi-group susceptible-infected-susceptible (SIS) model, a mathematical model is proposed to describe the spread of viruses with mutations. We provide a necessary and sufficient condition for the uniqueness of the zero equilibrium, and the conditions for the existence, uniqueness, and local exponential stability of the coexisting equilibrium. Our results demonstrate that the mutation can affect the spreading ability of the virus and the coexistence of viruses. Moreover, we show the effect of mutation on the proportion of infected individuals by comparing it with the model without mutation.