An optimal network that promotes the spread of an advantageous variant in an SIR epidemic

In the course of epidemics, the pathogen may mutate to acquire a higher fitness. At the same time, such a mutant is automatically in an unfavorable position because the resident virus has a head start in accessing the pool of susceptible individuals. We considered a class of tunable small-world networks, where a parameter, p (the rewiring probability), characterizes the prevalence of non-local connections, and we asked, whether the underlying network can influence the fate of a mutant virus. Under an SIR model, we considered two measures of mutant success: the expected height of the peak of mutant infected individuals, and the total number of recovered from mutant individuals at the end of the epidemic. Using these measures, we have found the existence of an optimal (for an advantageous mutant virus) rewiring probability that promotes a larger infected maximum and a larger total recovered population corresponding to the advantageous pathogen strain. This optimal rewiring probability decreases as mean degree and the infectivity of the wild type are increased, and it increases with the mutant advantage. The non-monotonic behavior of the advantageous mutant as a function of rewiring probability may shed light into some of the complex patterns in the size of mutant peaks experienced by different countries during the COVID-19 pandemic.