Exploration of Bogdanov–Takens and Hopf bifurcation through coupling of nonlinear recovery with multiple reinfections of COVID-19

This study introduces a five-compartment model to account for the impacts of vaccination-induced recovery and nonlinear treatment rates in settings with limited hospital capacity. To reflect real-world scenarios, the model incorporates multiple reinfections in both vaccinated and recovered groups. It reveals a range of dynamics, including a disease-free equilibrium and up to six endemic equilibria. Notably, the model demonstrates that COVID-19 can persist even when the basic reproduction number is less than one, due to backward bifurcation, which conditions the global stability of the disease-free equilibrium. Various bifurcations are analyzed, including saddle-node, Bogdanov-Takens of codimension-2, and Hopf bifurcation of codimension-1. As transmission rates increase, unstable oscillations stabilize, with the Hopf bifurcation becoming supercritical. The model also highlights forward hysteresis, driven by the multistability of endemic equilibria. Key factors influencing the disease's local endemic behavior, such as effective transmission rates and reinfection rates among vaccinated and recovered individuals, are emphasized. Numerical simulations validate the model and underscore its practical relevance.