Modeling the Seasonal Transmission Dynamics of Measles

Measles, a highly contagious infection caused by the measles virus, is a major public-health problem worldwide. Since the monthly data of measles cases exhibit a seasonally fluctuating pattern, based on the measles model in Earn et al. (Science 287(2000), 667-670) we propose a susceptible, exposed, infectious, and recovered (SEIR) model with periodic transmission rate to investigate the seasonal measles epidemics and the effect of vaccination. We calculate the basic reproduction number R_0, analyze the dynamical behavior of the model, and use the model to simulate the monthly data of measles cases reported in China. Then we consider an age-structured version of the periodic measles model. After establishing the well-posedness of the initial-boundary value problem, we study the existence of time periodic solutions of the model by using the fixed point theorem. We also carry out some sensitivity analysis of R_0 in the terms of model parameters showing that measles can be controlled and eventually eradicated by increasing the immunization rate, improving the effective vaccine management, and enhancing the awareness of people about measles. Finally we will briefly discuss optimal vaccination programs in age-structured measles models.