Stability and bifurcation analysis of a SIRI model with Crowley–Martin-type incidence rate and Holling-type II treatment rate

We introduce and analyze an infectious disease model that combines the Crowley–Martin incidence rate with the Holling-type II treatment rate. This model aims to capture the transmission dynamics of infectious diseases and takes into account the potential for reinfections. We thoroughly examine the behavior and characteristics of this model to gain insights into the spread and control of infectious diseases. We obtained the case of transcritical and backward bifurcations conditionally. The global stability of a unique endemic steady state is obtained under appropriate parametric conditions, as demonstrated using a geometric approach. When the basic reproduction number exceeds one, the system exhibits the coexistence of numerous endemic steady states. These steady states give birth to complex and sophisticated dynamics because they display a variety of bifurcations, oscillations (via Hopf bifurcation), bistability, and hysteresis. Through the use of sensitivity analysis, the parameters that are most sensitive to the basic reproduction number are identified. The theoretical results are accompanied by their corresponding numerical simulations. This work highlights the importance of ensuring proper access to effective treatment and preventing reinfection to achieve the ultimate goal of disease eradication.