Revisiting the modeling of the 1918 influenza pandemic's three mortality waves in England and Wales: Issues with the integration method and model formulation

Mathematical modeling serves as an indispensable and effective tool in epidemiology to quantify the characteristics of an epidemic, predict transmission trends, and evaluate interventions. However, within the plug-and-play framework, model fitting results may be simultaneously affected by numerical integration algorithms and mechanistic model structure. Using data from the 1918–1919 influenza pandemic across 334 administrative units in England and Wales, this study investigated whether numerical integration algorithms or mechanistic model formulation plays a more important role in the mathematical modeling. We compared three numerical schemes (multinomial Euler, deterministic Euler, and fourth-order Runge-Kutta) and further tested whether different functional forms of the temperature-response term (bounded versus exponential form) and the behavioral-response term affect model performance. Our results showed that numerical algorithms and temperature formulation have only limited effects on the overall inference. By comparison, the behavioral-response formulation has a much stronger impact on model performance. These findings underscore that while changing numerical algorithms affects likelihood values, the structural specification of key mechanistic terms plays a far more pivotal role in determining model performance.