Stability Analysis of a Multistage Compartmental Model of Infection

Mathematical epidemiology has gained significant interest in recent years, especially following the coronavirus outbreak. It helps us understand how diseases spread and supports the development of control measures to stop transmission. Compartmental models form the backbone of mathematical epidemiology. A key example is the SIRS model, which divides the population into categories such as susceptible, infected, and recovered. In this work, we study the dynamics of a multistage SIRS model. Using the Lyapunov method, we first revisit the stability of both the disease-free and endemic states of a single-stage SIRS model. We then extend these ideas to multistage infection models by introducing a systematic method for constructing Lyapunov functions through symbolic computation in MATLAB. Interestingly, the structure of these Lyapunov functions reflects the pathways through which the disease progresses. Finally, we present sufficient conditions for stability in two-stage and three-stage SIRS models, and we show how some of these conditions can be interpreted in meaningful epidemiological terms.