Computation of the domain of attraction for suboptimal immunity epidemic models using the maximal Lyapunov function method

Abstract

We are concerned with the estimation of the domain of attraction (DOA) for suboptimal immunity epidemic models. We establish a procedure to determine the maximal Lyapunov function in the form of rational functions. Based on the definition of DOA and the maximal Lyapunov function, a theorem and subsequently a numerical procedure are established to determine the maximal Lyapunov function and the DOA. Determination of the domain of attraction for epidemic models is very important for understanding the dynamic behaviour of the disease transmission as a function of the state of population distribution in different categories of disease states. We focus on suboptimal immunity epidemic models with saturated treatment rate and nonlinear incidence rate. Different from classical models, suboptimal immunity models are more realistic to explain the microparasite infection diseases such as Pertussis and Influenza A. We show that, for certain values of the parameter, larger k value (i.e., the model is more toward the SIR model) leads to a smaller DOA.