Epidemic models with discrete state structures

Abstract

The state of an infectious disease can represent the degree of infectivity of infected individuals, or susceptibility of susceptible individuals, or immunity of recovered individuals, or a combination of these measures. When the disease progression is long such as for HIV, individuals often experience switches among different states. We derive an epidemic model in which infected individuals have a discrete set of states of infectivity and can switch among different states. The model also incorporates a general incidence form in which new infections are distributed among different disease states. We discuss the importance of the transmission–transfer network for infectious diseases. Under the assumption that the transmission–transfer network is strongly connected, we establish that the basic reproduction number R0 is a sharp threshold parameter: if R0≤1, the disease-free equilibrium is globally asymptotically stable and the disease always dies out; if R0>1, the disease-free equilibrium is unstable, the system is uniformly persistent and initial outbreaks lead to persistent disease infection. For a restricted class of incidence functions, we prove that there is a unique endemic equilibrium and it is globally asymptotically stable when R0>1. Furthermore, we discuss the impact of different state structures on R0, on the distribution of the disease states at the unique endemic equilibrium, and on disease control and preventions. Implications to the COVID-19 pandemic are also discussed.