Hopf bifurcation in the shadow of extinction: Collaborating with epidemic dynamics through lethal mutations and declining ancestor infections

This study delves into the intricate realm of controlling Hopf and degenerate Hopf bifurcations within a Susceptible–Infectious–Susceptible model. Employing Braga's control theory as our cornerstone, we embark on an exploration of the model's dynamics, particularly focusing on an equilibrium point under the influence of control inputs. Our specific aim is to induce limit cycles associated with Hopf bifurcations of co-dimension 1 and 2. Through the integration of controllability principles, we endeavor to unravel the underlying mechanisms governing the manipulation of parameters to shape the occurrence and attributes of these periodic fluctuations. By examining how the behavior of infectious diseases changes in response to various control parameters, our study aims to provide a practical example of their application.