Analysis of a delayed epidemic model with pulse vaccination

In this paper, we have considered a dynamical model of infectious disease that spread by asymptomatic carriers and symptomatically infectious individuals with varying total population size, saturation incidence rate and discrete time delay to become infectious. It is assumed that there is a time lag (tau) to account for the fact that an individual infected with bacteria or 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 <= tau and value zero for t > tau. The probability that an individual in the latency period has survived is given by e(-mu tau), where mu denotes the natural mortality rate in all epidemiological classes. 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 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. The important mathematical findings for the dynamical behaviour of the infectious disease model are also numerically verified using MATLAB. Finally epidemiological implications of our analytical findings are addressed critically. (C) 2014 Elsevier Ltd. All rights reserved.