Analysis of SIRVI model with time dependent coefficients and the effect of vaccination on the transmission rate and COVID-19 epidemic waves

COVID-19 epidemic models with constant transmission rate cannot capture the patterns of the infection data in the presence of pharmaceutical and non-pharmaceutical interventions during a pandemic. Because of this, a new modification of SIR model that contain the vaccination compartment with time dependent coefficients and weak/loss-immunity is explored. Literature review confirms that the effect of vaccination on the time dependent transmission rate is still an open problem. This study answers this open problem. In this study, we first prove the well-posedness and investigate the model dynamics to show their continuous dependence on the model parameters. We then provide an algorithm to derive the time-dependent transmission function for the epidemiologic model and the data of the infected cases. The derived coupled nonlinear differential equations show the effect of vaccination on the transmission rate. Unlike previous studies, we first filter the published data and solve the nonlinear coupled differential equations using the finite difference technique, where the coefficient of the coupled nonlinear differential equations is a function of given data. We then show that time-dependent transmission function can be represented by linear combinations of Gaussian radial base function. We then validate the prediction of our models using numerical simulations, where we used the published data of COVID-19 confirmed cases by the Ministries of Health in Saudi Arabia and Poland. Finally, the numerical solutions of a SIRVI model with time dependent transmission rate show that the waves for currently active cases are in good agreement with the data of Saudi Arabia and Poland.