Modular arithmetic as an alternative to model seasonal time-varying transmission rates: Influenza as a case study

Seasonal infectious diseases like influenza pose a recurrent challenge to public health. While compartmental models, such as the susceptible-infectious-recovered-susceptible (SIRS) framework, are standard tools, representing the time-varying transmission rate, $ \beta(t) $, in an interpretable yet effective manner remains a key challenge. Existing established alternative methods for modeling seasonality use sinusoidal forcing functions, flexible splines, etc. In this paper, we propose and apply a modular approach where $ \beta(t) $ is defined using distinct, epidemiologically intuitive seasonal rates, with smooth transitions between them. To begin, we develop this framework as a theoretical tool, demonstrating its capacity to generate realistic, recurring seasonal outbreaks under plausible parameter assumptions. We then calibrate and assess this model against real-world, monthly laboratory-confirmed influenza surveillance data from Ontario, Canada, for the pre-pandemic period of 2014-2019. A systematic optimization using a coarse grid search followed by stochastic refinement calibrates the model to the observed data. The calibrated model, featuring a mean immunity duration of approximately 235 days, achieves a strong fit with the historical case data (Pearson correlation $ r = 0.80 $). Our results demonstrate that this modular arithmetic-based framework is a practical and effective tool for modeling real-world influenza dynamics, successfully bridging the gap between theory and empirical surveillance.