SIARD model and effect of lockdown on the dynamics of COVID-19 disease with non total immunity

We propose a new compartmental mathematical model describing the transmission and the spreading of COVID-19 epidemic with a special focus on the non-total immunity. The model (called SIARD) is given by a system of differential equations which model the interactions between five populations “susceptible”, “reported infectious”, “unreported infectious”, “recovered with/without non total immunity” and “death”. Depending on the basic reproduction number, we prove that the total immunity induces local stability-instability of equilibria and the epidemic may disappear after a first epidemic wave and more epidemic waves may appear in the case of non-total immunity. Using the sensitivity analysis we identify the most sensitive parameters. Numerical simulations are carried out to illustrate our theoretical results. As an application, we found that our model fits well the Moroccan epidemic wave, and predicts more than one wave for French case.