Analytically and numerically, the study examines the stability and local bifurcations of a discrete-time $SIR$ epidemic model. For this model, a number of bifurcations are studied, including the transcritical, flip bifurcations, Neimark–Sacker bifurcations, and strong resonances. These bifurcations are checked, and their non-degeneracy conditions are determined by using the normal form technique (computing of critical normal form coefficients). We use the MATLAB toolbox MatcontM, which is based on the numerical continuation method, to confirm the obtained analytical results and specify more complex behaviors of the model. Numerical simulation is employed to present a closed invariant curve emerging from a Neimark–Sacker point and its breaking down to several closed invariant curves and eventually giving rise to a chaotic strange attractor by increasing the bifurcation parameter.

To describe the main propagation of the COVID-19 and has to find the control for the rapid spread of this viral disease in real life, in current manuscript a discrete form of the SEIR model is discussed. The main aim of this is to describe the viral disease in simplest way and the basic properties that are related with the nature of curves for susceptible and infected individuals are discussed here. The elementary numerical examples are given by using the real data of India and Algeria.