Stability Analysis of SVIRC Epidemic model with the Impact of Vaccination and Cross-Immunity

Abstract

Multi-strain diseases have been always a great threat to society for decades. Such diseases display labyrinthine cycles of disease spread due to the ever evolving nature of the virus’s structure. The
drift mechanism in the viral structure is responsible for the development of cross-immunity among the people. Vaccination is another factor that plays an important role in the disease dynamics, which is a cost-effective measure that helps individuals stay protected from the disease to some extent. In this paper, we propose a SVIRC epidemic model to study the intricate dynamics of multi-strain diseases. The positivity and boundedness of the model have been discussed. The disease-free and endemic equilibrium points have been obtained. The basic reproduction number Rvac has been evaluated. The stability of the equilibrium points of the proposed model has been discussed. The global stability of the disease-free equilibrium point has also been investigated. Condition for the existence of backward and forward bifurcations has been obtained. Finally, we perform numerical simulations to validate the analytical finding. We carry out local sensitivity analysis of reproduction number Rvac using Normalised Forward Sensitivity Analysis. Also, global sensitivity analysis for the basic reproduction number and the infective class has been conducted using the Latin Hypercube Sampling (LHS), and Partial Rank Correlation Coefficients (PRCCs). We have also studied the combined effect of vaccine efficacy with interaction rate and vaccine efficacy with vaccination rate on Rvac . The effect of different parameters such as infection rate, vaccine efficacy, recovery rate, and re-infection probability on the infective population have also been discussed. Based on the study, it is suggested that implementations of preventive measures such as washing hands, maintaining social distance, using face masks, sanitizing frequently, lockdown during the
peak infection period and the administration of timely vaccines are the key strategies that the health authorities should employ to manage the course of the infection.