Backward bifurcations, turning points and rich dynamics in simple disease models

Area of Science:

• Mathematical Biology
• Epidemiology
• Dynamical Systems Theory

Background:

• Disease models are crucial for understanding epidemiological dynamics.
• Previous models have shown complex behaviors like bistability and oscillations.
• Autoimmunity and in-host disease dynamics are areas of interest.

Purpose of the Study:

• To investigate complex dynamical behaviors in simple disease models using bifurcation theory.
• To elucidate the role of backward bifurcations in generating these behaviors.
• To explore the clinical and public health implications of disease model dynamics.

Main Methods:

• Application of dynamical systems theory and bifurcation theory.
• Analysis of 2- and 3-dimensional disease models.
• Utilizing a Maple program to determine limit cycle stability from Hopf bifurcations.
• Numerical simulations to illustrate observed phenomena.

Main Results:

• Backward bifurcations are identified as critical for disease model dynamics.
• The position and properties of backward and Hopf bifurcations determine model behaviors.
• Bistability, recurrence, and oscillations were observed and analyzed.
• Stability of limit cycles was assessed using computational methods.

Conclusions:

• Backward bifurcations play a pivotal role in the emergence of complex dynamics in disease models.
• Hopf bifurcations, influenced by backward bifurcations, dictate the specific behaviors observed.
• The findings have potential implications for public health strategies and clinical interventions.