Exactly solvable SIR models, their extensions and their application to sensitive pandemic forecasting

Area of Science:

• Mathematical epidemiology
• Dynamical systems theory
• Statistical modeling

Background:

• The classic SIR (Susceptible-Infectious-Recovered) model is a cornerstone of epidemic dynamics.
• Generalizing epidemic models to time-dependent parameters and complex interactions is crucial for real-world applicability.
• Accurate fitting and forecasting of infectious disease data are essential for public health interventions.

Purpose of the Study:

• To provide complete analytical solutions for the SIR model and its generalizations using quadratures and integral transforms.
• To develop and apply a generalized logistic (Richards) model for fitting and forecasting COVID-19 data.
• To investigate the chaotic dynamics of discretized epidemic models and critique their robustness for pandemic analysis.

Main Methods:

• Solving the SIR model using quadratures and time integral transforms (incomplete gamma functions).
• Generalizing the SIR model for time-dependent infection rates and interacting regions.
• Fitting the Richards model to Mexico's COVID-19 data and analyzing bifurcation diagrams for chaotic behavior.

Main Results:

• Explicit solutions were obtained for generalized SIR models with time-dependent infection rates.
• The Richards model successfully fitted Mexico's COVID-19 data up to day 134, enabling forecasting scenarios.
• A discretized Richards model exhibited period-doubling bifurcations, leading to chaotic dynamics, and its robustness was critiqued.

Conclusions:

• The study offers advanced mathematical tools for analyzing and predicting epidemic dynamics.
• The Richards model provides a valuable framework for fitting and forecasting infectious disease outbreaks.
• Understanding the potential for chaotic behavior in epidemic models is critical for robust pandemic response.