Modelling Covid-19 Dynamics With Periodic Contact Rates, Waning Vaccine Immunity and Temporary Cross Immunity

Abstract This paper present a novel delay differential equation model with periodic contact rate to characterize the transmission dynamics of the Covid-19. The model account for an imperfect vaccine whose protective efficacy wanes over time. The disease free equilibrium is determined, and Floquet theory is employed to analyze the stability of the periodic system by evaluating the eigenvalue of time-dependent Jacobian matrix. From this, the basic reproduction number is derived and shown to fall below unity under specific parameter regimes, indicating the potential for decease eradication. Numerical simulation are conducted to corroborate the analytical result highlighting the influence of vaccination coverage, waning immunity and periodic variation in contact rates on the long-term dynamics of COVID-19.