Master of Science
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 behaviors 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.
Summary for Lay Audience
Infectious diseases are a global problem that harms people’s health and well-being and severely threatens human survival. In epidemiology, SIR models are widely used to predict the spread of diseases, the number of infected individuals, epidemic duration, and other factors.
We have studied the stability of epidemic SIR models, the conditions for the existence of limit cycle bifurcations, as well as Hopf and B-T bifurcations as a result of the increasing interest in the dynamic behavior of epidemic anaysis. We therefore examined the dynamics of the systems in this paper and reconsider the two epidemic models. Our analysis focused on the equilibrium solutions, stability, and bifurcation of the two models and we obtained explicit conditions that permit us to classify each bifurcation. The center manifold theorem, normal form theory, bifurcation theory and limit cycle theory were used to demonstrate the complex dynamics of both models. Hopf, codimension 2 and codimension 3 B-T bifurcations were studied to demonstrate the complex dynamics of these models.
Ruan, Yuzhu, "Complete Hopf and Bogdanov-Takens Bifurcation Analysis on Two Epidemic Models" (2022). Electronic Thesis and Dissertation Repository. 8876.