Global dynamic analyzes of the discrete SIS models with application to daily reported cases

Emerging infectious diseases, such as COVID-19, manifest in outbreaks of varying magnitudes. For large-scale epidemics, continuous models are often employed for forecasting, while discrete models are preferred for smaller outbreaks. We propose a discrete susceptible-infected-susceptible model that integrates interaction between parasitism and hosts, as well as saturation recovery mechanisms, and undertake a thorough theoretical and numerical exploration of this model. Theoretically, the model incorporating nonlinear recovery demonstrates complex behavior, including backward bifurcations and the coexistence of dual equilibria. And the sufficient conditions that guarantee the global asymptotic stability of a disease-free equilibrium have been obtained. Considering the challenges posed by saturation recovery in theoretical analysis, we then consider the case of linear recovery. Bifurcation analysis for of the linear recovery model displays a variety of bifurcations at the endemic equilibrium, such as transcritical, flip, and Neimark–Sacker bifurcations. Numerical simulations reveal complex dynamic behavior, including backward and fold bifurcations, periodic windows, period-doubling cascades, and multistability. Moreover, the proposed model could be used to fit the daily COVID-19 reported cases for various regions, not only revealing the significant advantages of discrete models in fitting, evaluating, and predicting small-scale epidemics, but also playing an important role in evaluating the effectiveness of prevention and control strategies. Furthermore, sensitivity analyses for the key parameters underscore their significant impact on the effective reproduction number during the initial months of an outbreak, advocating for better medical resource allocation and the enforcement of social distancing measures to curb disease transmission.