Subharmonic bifurcation in an S-I-R epidemic model

Area of Science:

• Epidemiology
• Mathematical Biology
• Dynamical Systems

Background:

• Analysis of SIR (Susceptible-Infected-Recovered) epidemic models is crucial for understanding disease dynamics.
• Oscillations in contact rates, such as annual seasonal variations, can significantly impact epidemic behavior.
• Previous work suggested complex dynamics, including multi-year cycles, in oscillating epidemic models.

Purpose of the Study:

• To investigate the existence of subharmonic solutions with a two-year period in an SIR model.
• To mathematically prove the conditions under which a two-year epidemic cycle arises.
• To provide rigorous validation for prior theoretical arguments on epidemic model oscillations.

Main Methods:

• Analysis of a Susceptible-Infected-Recovered (SIR) epidemic model.
• Incorporation of annual oscillations in the contact rate parameter.
• Mathematical bifurcation analysis to identify transitions in solution stability and periodicity.

Main Results:

• Demonstration of the existence of subharmonic solutions with a two-year period.
• Proof that a stable two-year periodic solution bifurcates from a stable one-year periodic solution.
• Identification of a threshold for the amplitude of contact rate oscillation triggering this bifurcation.

Conclusions:

• The study rigorously confirms that annual oscillations in contact rates can lead to biennial epidemic cycles in SIR models.
• A specific threshold in oscillation amplitude is identified as the critical factor for the emergence of two-year cycles.
• These findings validate and formalize earlier theoretical insights into complex epidemic dynamics.