SIR Model with Dependent Infectivity and Death Rates

This work constructs, analyzes and simulates a new general 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 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 construct the SIR-ID model for a generic disease in which the rate coefficients are state variables as a part of the systems’s evolution in time. The model consists of a coupled system of five differential equations, where the equations for the infectivity and death rate have general source functions. The analysis shows the existence, positivity and boundedness of the solutions. A discussion of the Endemic (EE) and Disease-Free (DFE) equilibria and their stability is provided. A bifurcation analysis of the DFE and EE is conducted, as well as a sensitivity analysis. Then, computer simulations depict two typical cases of dynamic behavior, one when the DFE is stable and attracting, and one in which the EE is stable and attracting. These also show the way the system approaches the steady states.