A Study of a Class Continuous SIR Epidemic Model with History

The SIR model, an epidemiological model, divides a population into three compartments: Susceptible (S), Infected (I), and Recovered (R). It is widely used to understand the spread of infectious diseases and predict epidemic outcomes based on factors such as transmission rates and population dynamics. A deterministic epidemic mathematical model to describe the transmission dynamics of an infectious disease was constructed and analyzed by incorporating memory term which provides information on the current and past disease states. The model revealed two key equilibria: a disease-free equilibrium and an endemic equilibrium. The calculated basic reproductive number R0, was employed to establish that when R0<1, the disease-free equilibrium is locally asymptotically stable, while the endemic equilibrium is locally asymptotically stable when R0>1. Additionally, we explored the global stability of these equilibria using Lyapunov functions and Dulac's method, respectively. To validate our analytical findings, we conducted numerical simulations of the model which show the importance of history in the dynamic spread and elimination of disease.