"Exploring the bifurcations in a COVID-19 epidemiological model – the failure of the quadratic equation analysis"
In this study, we aim to investigate the nature of bifurcations in an extended version of an SVEIR type compartmental model with differential morbidity. Since all existing COVID-19 vaccines are imperfect, we consider vaccine efficacy as a pivotal parameter in the study. The endemic equilibrium of the model was analyzed by explicitly constructing a quadratic equation which was then manipulated appropriately in order to derive R0 using an alternative approach. This also permitted a comprehensive categorization of the number of endemic equilibria based on the threshold condition R0 = 1, which also seemed to imply potential existence of the backward bifurcation phenomenon. However, numerical simulations and application of center manifold theory showed that the bifurcation at R0 = 1 is forward. Therefore, an analysis based on the existence of a quadratic equation at the endemic equilibria is not sufficient in establishing backward bifurcations. We then explored what causes the equation of endemic equilibria to become quadratic and the bifurcation diagram to have a non-linear shape. In this respect, it was shown that the underlying equation is not quadratic (but linear) when the vaccine is perfect which also yields a linear bifurcation diagram.
Keywords: COVID-19 vaccination, Compartmental models, Basic reproduction number, Quadratic equation of endemic equilibria, Bifurcations
Additional authors: Kevin E. M. Church (Centre de Recherches Mathématiques, Universit´e de Montr´eal, Montréal, Québec, Canada.); Elissa J. Schwartzc (Department of Mathematics & Statistics and School of Biological Sciences, Washington State University, Pullman, Washington, USA.)