PhD Public Lecture (Math) - Yuzhu Ruan

Infectious diseases are a global problem that harms people’s health and well-being and severely threatens human survival. As an epidemiological model, the SIR model is commonly referred to as forecasting how illnesses will spread, how many people will become sick, and how long an epidemic will last. It is also possible to estimate other epidemiological parameters.

Bifurcation theory and limit cycle theory have played an important role in the study of nonlinear dynamical systems, especially for the infectious disease models. In particular, Hopf and Bogdanov-Takens (B-T) bifurcations are the two most prevalent bifurcations in real-world systems and should be considered in practical problems which require analysis of stability and bifurcation.

In this paper, we reconsider two SIR models and focus on the dynamical behaviours of the systems, which are not explored in the previous studies. Our main attention focuses on the stability and bifurcation of equilibrium solutions. Explicit conditions are obtained to classify different bifurcations, including forward bifurcation, backward bifurcation, Hopf bifurcation, and B-T bifurcation. The method of normal forms is applied to study Hopf, codimension-2 and codimension-3 B-T bifurcations, showing complex dynamics in these two models.