Bogdanov-Takens Bifurcation in SIRI Model with Multiple Reinfection of COVID-19

In the presence of cases of COVID-19 reinfection, we propose a SIRI (Susceptible-Infected-Recovery-Infected) spread model of two COVID-19 variants. This model considers the possibility of individuals becoming reinfected with the same or different variants, although the risk of reinfection with the same variant remains lower due to natural immunity from previous infections. Besides analyzing the stability of equilibrium points, we focus on codimension-one bifurcations. Our initial numerical simulations used parameters obtained from real data collected through a British government survey. Our analysis revealed unstable disease-free equilibria and stable endemic equilibria. By varying the Case Fatality Rate parameter, we identified all codimension-one bifurcations. To further investigate the model's dynamics, we introduced a new parameter, the reinfection rate, and utilized AUTO software. Our research led to the discovery of codimension-two bifurcations, specifically the Bogdanov-Takens bifurcation. We identified the parameter domain where a stable limit cycle and homoclinic orbit occur in the presence of the Bogdanov-Takens bifurcation. We also simulated parameter variations that could trigger a pandemic resurgence. This highlights the possibility of emerging variants causing a pandemic return.