Stochastic modeling of nonlinear epidemiology

Area of Science:

• Epidemiology
• Computational Biology
• Mathematical Modeling

Background:

• Infectious disease spread is often nonlinear, challenging traditional linear stochastic models.
• The classical Kermack-McKendrick (SIR) model categorizes populations into Susceptible, Infective, and Removed.
• Previous stochastic analyses frequently relied on simplifying linearity assumptions.

Purpose of the Study:

• To analyze, model, and simulate infectious disease spread using modern stochastic algorithms.
• To overcome the linearity limitations of previous epidemic process models.
• To derive governing equations for population dynamics and uncertainties in epidemic processes.

Main Methods:

• Formulation of master equations for the SIR process using probabilistic population balance.
• Application of system-size expansion to nonlinear master equations to derive equations for means, variances, and covariance.
• Numerical simulation using derived algorithms and an event-driven Monte Carlo algorithm.

Main Results:

• The methodology successfully derives equations for the means, variances, and covariance of population variables.
• Analysis provides insights into population means and inherent uncertainties during an epidemic.
• Simulations from the master equation-derived algorithm and Monte Carlo method yielded identical results.

Conclusions:

• The developed stochastic approach accurately models nonlinear epidemic processes.
• The method provides a robust framework for understanding disease dynamics and population fluctuations.
• Validation through Monte Carlo simulation confirms the reliability of the derived algorithms.