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This paper investigates the minimal dynamic complexity required for simple SIR models to reproduce multiple epidemic waves, demonstrating that time-varying transmission rates are sufficient for peak-preserving representability. The authors introduce Minimal Peak-Preserving Complexity (MPPC) to quantify this parsimony and establish that while low-dimensional time-varying SIR models can capture epidemic wave topology, incidence-only data leads to non-identifiability of transmission and removal rates, limiting reliable long-term counterfactual prediction. The framework disentangles representability, complexity, identifiability, and projection validity for parsimonious epidemic modeling.
Simple SIR models can reproduce complex multi-wave epidemic patterns with surprisingly low dynamic complexity, but incidence data alone cannot reliably recover underlying transmission dynamics.
Multiple epidemic waves are often taken as evidence that simple compartmental models are structurally inadequate. Yet the key question is not whether a simple model can reproduce multiple peaks, but how much time-varying structure is minimally required to preserve observed epidemic-wave patterns and what can legitimately be inferred from such a representation. Using the susceptible-infectious-removed (SIR) model, we develop a framework for peak-preserving dynamic parsimony linking dynamical systems, inverse problems, statistical inference, and infectious-disease modeling. We establish exact multi-wave representability under time-varying transmission, characterize conditions for epidemic peaks and resurgence, and show the incidence-only non-identifiability of time-varying transmission and removal rates. Our main methodological contribution is minimal peak-preserving complexity (MPPC): the smallest dimension of a positive time-varying transmission function whose posterior-predictive trajectories preserve salient features, including peak count, timing, height, trough depth, and whole-curve agreement. We estimate MPPC using peak-aware approximate Bayesian computation and evaluate it through six reproducible stress tests, a Philippine COVID-19 reported-case proof of concept, and short-term future-knot projection. Across diverse data-generating mechanisms, low-dimensional time-varying SIR models can preserve aggregate epidemic-wave topology with modest dynamic complexity. However, accurate trajectory compression does not imply recovery of the underlying mechanism or reliable long-term counterfactual prediction. The framework therefore separates four often-conflated questions: representability, minimal dynamic complexity, mechanistic identifiability, and projection validity, providing a basis for parsimonious modeling of complex epidemic trajectories.