The effect of the force of infection and treatment on the disease dynamics of an SIS epidemic model with immigrants

Abstract

A basic simple SIS epidemic model is proposed and analyzed. The population is infected by a transmissible disease. The uniform boundedness of solutions is established. Local and global stability dynamics of the systems are discussed. The basic model is modified by incorporating a restricted supply of treatment and immigrants considering that a fraction of the immigrants arrive with the same infection into the system. We have calculated the basic reproduction numbers for the proposed model systems which play an important role in the existence of endemic equilibria. Through theoretical analysis, we have shown that the disease-free equilibrium point of the modified system possesses a transcritical bifurcation. An endemic equilibrium state exhibits a Hopf bifurcation. Sotomayor’s theorem on local bifurcation theory yields the explicit formula determining the characteristic of the bifurcating periodic solutions. A significant conclusion is that the force of infection and treatment have an opposite effect on the stability dynamics of the proposed SIS model system. The study reveals that the difference between the maximum treatment capacity and the infected immigrants plays a significant role on the disease endemicity of the epidemic system. Numerical simulations are presented to validate the analytical findings.

Keywords

• SIS model
• Force of infection
• Treatment
• Immigrants
• Bifurcations
• Limit cycle