SIRC epidemic model with cross-immunity and multiple time delays.

Abstract

Multi-strain diseases lead to the development of some degree of cross-immunity among people. In the present paper, we propose a multi-delayed SIRC epidemic model with incubation and immunity time delays. Here we aim to examine and investigate the effects of incubation delay [Formula: see text] and the impact of vaccine which provides partial/cross-immunity with immunity delay parameter ([Formula: see text]) on the disease dynamics. Also, we study the impact of the strength of cross-immunity [Formula: see text] on the disease prevalence. The positivity and boundedness of the solutions of the epidemic model have been established. Two different types of equilibrium points (disease-free and endemic) have been deduced. Expression for basic reproduction number has been derived. The stability conditions and Hopf-bifurcation about both the equilibrium points in the absence and presence of both delays have been discussed. The Lyapunov stability conditions about the endemic equilibrium point have been established. Numerical simulations have been performed to support our analytical results. We quantitatively demonstrate how oscillations and Hopf-bifurcation allow time delays to alter the dynamics of the system. The combined impacts of both the delays on disease prevalence has been studied. Through parameter sensitivity analysis, we observe that the infected population decreases with an increase in vaccination rate and the system starts to stabilize early with the increase in cross-immunity rate. Global sensitivity analysis for the basic reproduction number has been performed using Latin hypercube sampling and partial rank correlation coefficients techniques. The combined effect of vaccination rate with transmission rate and vaccination rate with re-infection probability (i.e. strength of cross-immunity) on [Formula: see text] have been discussed. Our research underlines the need to take cross-immunity and time delays into account in the epidemic model in order to better understand disease dynamics.