Analysis of a delayed hand–foot–mouth disease epidemic model with pulse vaccination

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 time lag (τ) to account for the fact that an individual infected with virus is not infectious until after some time after exposure. The probability that an individual remains in the latency period (exposed class) at least t time units before becoming infectious is given by a step function with value 1 for 0 ≤ t τ. The probability that an individual in the latency period has survived is given by e−μτ, where μ denotes the natural mortality rate in all epidemiological classes. It is reported that the first vaccine to protect children against enterovirus 71, or EV71 has been discovered [Zhu, F. C., Meng, F. Y., Li, J. X., Li, X. L., Mao, A. Y., Tao, H., …, Shen, X. L. (2013, May 29). Efficacy, safety, and immunology of an inactivated alum-adjuvant enterovirus 71 vaccine in children in China: A multicentre, randomised, double-blind, placebo-controlled, phase 3 trial. The Lancet, 381, 2024–2032. doi:10.1016/S0140-6736(13)61049-1]. Pulse vaccination is an effective and important strategy for the elimination of infectious diseases and so we have analyzed this model with pulse vaccination. We have defined two positive numbers R1 and R2. It is proved that there exists an infection-free periodic solution which is globally attractive if R1 < 1 and the disease is permanent if R2 > 1. The important mathematical findings for the dynamical behavior of the HFMD model are also numerically verified using MATLAB. Finally epidemiological implications of our analytical findings are addressed critically.