The mathematical models of infections are essential tools in understanding the dynamical behavior of disease transmission. In this paper, we establish a model of differential equations with piecewise constant arguments that explores the outbreak of Covid-19 including the control mechanisms such as health organizations and police supplements for the sake of controlling the pandemic spread and protecting the susceptible population. The local asymptotic stability of the equilibrium points, the disease-free equilibrium point, the apocalypse equilibrium point and the co-existing equilibrium point are analyzed by the aide of Schur-Cohn criteria.
Furthermore and by incorporating the Allee function at time t, we consider the extinction case of the outbreak to analyze the conditions for a strong Allee Effect. Our study has demonstarted that the awareness of the police personal and the management of professional health organizations play a vital role to protect the susceptible class and to prevent the spreading. Numerical simulations are presented to support our theoretical findings. We end the paper by a describtive conclusion.