Oscillation in the SIRS model.

Area of Science:

• Epidemiology
• Mathematical Biology
• Computational Science

Background:

• The SIRS (Susceptible-Infected-Recovered-Susceptible) model describes disease dynamics.
• Traditional models identify stable states: disease-free (I=0) and endemic (I>0).
• External factors like seasonality are often invoked to explain epidemic fluctuations.

Purpose of the Study:

• To investigate the intrinsic dynamics of the SIRS model.
• To explore the emergence of oscillatory behavior in epidemic models.
• To determine if waning immunity alone can drive regular epidemic cycles.

Main Methods:

• Analytical investigation of the SIRS model.
• Numerical simulations on a square lattice with noise.
• Analysis of model solutions to identify stable states and oscillations.

Main Results:

• The SIRS model exhibits two stable states: disease-free and endemic.
• Implementation with noise or on a lattice reveals a third state: regular oscillations.
• These oscillations represent intrinsic boom-and-bust epidemic cycles driven by waning immunity.

Conclusions:

• Oscillatory epidemic behavior is an intrinsic property of the SIRS model.
• Waning immunity, on a timescale of approximately ten weeks, can explain non-seasonal oscillations.
• This intrinsic oscillatory behavior offers an explanation for patterns observed in diseases like COVID-19 (e.g., Omicron variant).