SIR Model with Dependent Infectivity and Death Rates

This work constructs, analyzes and simulates a modified SIR epidemiological model for the spread of a generic long-time disease, in which the coefficients of infectivity and death rate are system variables. Diseases, such as COVID-19, have demonstrated very clearly that infectivity and death rates can change over time, even for the same variant of the virus, due to vaccination, improved treatments, better analysis, better medications, etc. This motivates us to model a generic disease where the infectivity and death rates are state variables as a part of the systems's evolving in time. The model consists of a coupled system of five differential equations. The analysis shows the existence, positivity and boundedness of the solutions. A short discussion of the Endemic (EE) and Disease-Free (DFE) equilibria and their stability is provided. Then, computer simulations depict two typical cases of dynamic behaviors, one when the DFE is stable and attracting, and one in which the EE is stable and attracting. These also show how the system approaches these steady states.