Susceptible-infectious-susceptible (SIS) model with virus mutation in a variable population size

The complex dynamics of a contagious disease in which populations experience horizontal and vertical transmissions, size variation, and virus mutations are of considerable practical and theoretical interest. We model such a system by dividing a population into three distinct groups: susceptibles ($S$), $C$-infected ($C$) and $F$-infected ($F$), based on the Susceptible-Infectious-Susceptible ($SIS$) model. Once the individuals in the $C$-infected group recover from the disease, they gain no permanent immunity. The virus can mutate in the group $C$. When it does, the individuals become members of the $F$-infected group. The mutated virus causes a lethal and incurable disease with a high mortality rate. We discuss the model for two cases. For the first case, all the newborns from infected mothers develop the disease shortly after their birth. For the second case, there exist equal transmission rates and the $C$-infected population is lifelong infectious. Our analysis shows that both systems have positive solutions, and the first model possesses four equilibrium points, the trivial one (extinction of the species), $C$-free equilibrium (extinction of the ancestor virus) and two endemic equilibria of different properties. We identify the net population growth rates of the susceptible and $C$-infected groups for the existence of the equilibria of the first model. We define the conditions of parameters for which species extinction and endemic equilibria are locally asymptotically stable. We observe that bifurcation occurs at the $C$-free equilibrium. For the second model, we find that there is only one endemic equilibrium and it is always locally asymptotically stable. We also determine the region for the net population growth rates of the susceptible and $F$-infected groups for the existence of the endemic equilibrium.