A mathematical study of a model for childhood diseases with non-permanent immunity

Protecting children from diseases that can be prevented by vaccination is a primary goal of health administrators. Since vaccination is considered to be the most effective strategy against childhood diseases, the development of a framework that would predict the optimal vaccine coverage level needed to prevent the spread of these diseases is crucial. This paper provides this framework via qualitative and quantitative analysis of a deterministic mathematical model for the transmission dynamics of a childhood disease in the presence of a preventive vaccine that may wane over time. Using global stability analysis of the model, based on constructing a Lyapunov function, it is shown that the disease can be eradicated from the population if the vaccination coverage level exceeds a certain threshold value. It is also shown that the disease will persist within the population if the coverage level is below this threshold. These results are verified numerically by constructing, and then simulating, a robust semi-explicit second-order finite-difference method.