Time-continuous and time-discrete SIR models revisited: theory and applications

Area of Science:

• Mathematical Epidemiology
• Computational Biology
• Infectious Disease Modeling

Background:

• The SIR (susceptible-infectious-recovered) model, introduced in 1927, is foundational in mathematical epidemiology.
• Rising global epidemics like COVID-19 highlight the need for advanced epidemiological modeling.
• Existing discrete-time SIR models often use explicit schemes, which can have limitations.

Purpose of the Study:

• To develop and analyze an implicit time-discrete SIR model with time-varying transmission and recovery rates.
• To demonstrate the unique solvability and desirable properties of the implicit discrete model.
• To validate the model's applicability using real-world COVID-19 data.

Main Methods:

• Formulation of a continuous-time SIR model with time-varying parameters.
• Derivation of an implicit time-discrete SIR model, contrasting with explicit schemes.
• Mathematical analysis to prove unique solvability and error bounds for the discrete model.

Main Results:

• The implicit time-discrete SIR model exhibits unique solvability.
• Key properties of the continuous SIR model are preserved in the derived discrete model.
• An upper error bound for the numerical scheme was mathematically proven.

Conclusions:

• The implicit time-discrete SIR model provides a robust and accurate numerical method for epidemiological studies.
• The model successfully captures disease dynamics, as demonstrated by its application to COVID-19 data.
• This approach offers a valuable tool for understanding and predicting infectious disease outbreaks.