A delayed hand–foot–mouth disease model with pulse vaccination strategy

Abstract

In this paper, we have considered a dynamical model of hand–foot–mouth disease (HFMD) with varying total population size, saturation incidence rate and discrete time delay to become infectious. It is assumed that there is a latent period of the disease and an immune period of the recovered individuals. It is reported that the first vaccine to protect children against enterovirus 71 or EV71 has been invented (Zhang and Teng, J Biol Dyn 2(1):64-84, 2008). Pulse vaccination is an effective and important strategy for the elimination of infectious diseases and so we have analyzed this model with pulse vaccination. It is also assumed that the time lag due to lose of immunity of recovered individuals is equal to the interval between two pulses. We have defined two positive numbers $$R_{1}$$ and $$R_{2}$$ . It is proved that there exists an infection-free periodic solution which is globally attractive if $$R_{1}<1$$ and the disease is permanent if $$R_{2}>1$$ . It is derived that the infectious population extinct if the pulse vaccination rate is larger than a critical vaccination proportion. Our analysis indicates that a small pulse vaccination rate will rise the permanence of the disease. The important mathematical findings for the dynamical behaviour of the HFMD model are also numerically verified using MATLAB. Finally epidemiological implications of our analytical findings are addressed critically.