Mathematical modeling of the COVID-19 epidemic with fear impact

Many studies have shown that faced with an epidemic, the effect of fear on human behavior can reduce the number of new cases. In this work, we consider an SIS-B compartmental model with fear and treatment effects considering that the disease is transmitted from an infected person to a susceptible person. After model formulation and proving some basic results as positiveness and boundedness, we compute the basic reproduction number ${\mathcal{R}}_0$ and compute the equilibrium points of the model. We prove the local stability of the disease-free equilibrium when ${\mathcal{R}}_0<1$. We study then the condition of occurrence of the backward bifurcation phenomenon when ${\mathcal{R}}_0\le 1$. After that, we prove that, if the saturation parameter which measures the effect of the delay in treatment for the infected individuals is equal to zero, then the backward bifurcation disappears and the disease-free equilibrium is globally asymptotically stable. We then prove, using the geometric approach, that the unique endemic equilibrium is globally asymptotically stable whenever the ${\mathcal{R}}_0>1$. We finally perform several numerical simulations to validate our analytical results.