Global dynamics of two-strain epidemic model with single-strain vaccination in complex networks

Contagious pathogens, such as influenza and COVID-19, are known to be represented by multiple genetic strains. Different genetic strains may have different characteristics, such as spreading more easily, causing more severe diseases, or even evading the immune response of the host. These facts complicate our ability to combat these diseases. There are many ways to prevent the spread of infectious diseases, and vaccination is the most effective. Thus, studying the impact of vaccines on the dynamics of a multi-strain model is crucial. Moreover, the notion of complex networks is commonly used to describe the social contacts that should be of particular concern in epidemic dynamics. In this paper, we investigate a two-strain epidemic model using a single-strain vaccine in complex networks. We first derive two threshold quantities, $$ and $$ , for each strain. Then, by using the basic tools for stability analysis in dynamical systems (i.e., Lyapunov function method and LaSalle's invariance principle), we prove that if $$ and $$ , then the disease-free equilibrium is globally asymptotically stable in the two-strain model. This means that the disease will die out. Furthermore, the global stability of each strain dominance equilibrium is established by introducing further critical values. Under these stability conditions, we can determine which strain will survive. Particularly, we find that the two strains can coexist under certain condition; thus, a coexistence equilibrium exists. Moreover, as long as the equilibrium exists, it is globally stable. Numerical simulations are conducted to validate the theoretical results.