Hopf-Hopf bifurcation and hysteresis in a COVID-19 transmission model implementing vaccination induced recovery and a modified Holling type-III treatment response

In times where treatment methods are overwhelmed and have reached a saturation state, it is necessary to examine the propagation patterns of COVID-19 to assist in the decision-making process. In light of its practical significance, this paper proposes a dynamical model while implementing vaccination of susceptibles and a modified Holling type - III treatment response in presence of waning immunity. The susceptible population is assumed to be vaccinated and are transferred to the recovered class. The model also accounts for the cases of imperfect vaccination resulting in the relapse of those individuals. To have a better comprehension of the new model, the non-negativity and boundedness of its solutions are studied. The model shows the presence of a maximum of three endemic equilibria along with a disease-free equilibrium. Transcritical bifurcation is evident for basic reproduction number greater than unity and there is the occurrence of Hopf bifurcation in the system via periodic oscillations. The direction of the Hopf bifurcation is supercritical and the unstable oscillations stabilize when the transmission rate increases. Formation of endemic bubbles in the system suggests the presence of Hopf-Hopf bifurcation. The model exhibits the phenomenon of forward hysteresis owing to the multistability of the endemic equilibria. Sensitivity analysis and data fitting illustrate the practical validity of the model along with numerical simulations. Based on these findings, the modified saturated treatment response is deemed valuable over the traditional response due to its practical relevance in the context of modern healthcare. With significant advancements in infrastructure, the limitations on medical resources are less pronounced, offering clearer insights into the evolving dynamics of COVID-19.