A Universal Convolution-Based Pre-processor to Correct the Prevalence-Incidence Gap in SIR, SEIR, and SIRS Modeling

TL;DR

This paper identifies a fundamental flaw in calibrating prevalence-based compartmental models (SIR, SEIR, SIRS) with incidence data, leading to biased peak timing and underestimation of the infectious stock. It introduces a universal pre-processor based on an exponentially weighted convolution that reconstructs prevalence from incidence: $I(t) \approx \frac{1}{p} \int_{0}^{t} NDC(\tau) e^{-\gamma(t-\tau)} d\tau$, incorporating recovery rate $\gamma$ and ascertainment $p$. The approach yields a more accurate peak position and amplitude, and it remains essential when extending to SEIR and SIRS, as misalignment propagates through more complex models. The practical impact is a standardized preprocessing step that aligns clinical reporting with mechanistic models, improving predictive performance across compartmental epidemic models.

Abstract

Traditional compartmental models, including SIR, SEIR, and SIRS frameworks, remain the analytical standard for epidemic forecasting. However, real-world data validation consistently reveals significant predictive failures, such as peak underestimations of up to 50%. This research identifies a persistent fundamental methodological error: the calibration of prevalence-based (stock) models using raw daily incidence (flow) data without proper transformation. We propose an integrated protocol utilizing an exponentially weighted convolution to reconstruct active cases from reported incidence: $I(t) \approx \frac{1}{p} \int_{0}^{t} NDC(τ) e^{-γ(t-τ)} dτ$. This transformation accounts for the recovery rate $γ$ and the ascertainment rate $p$. We demonstrate that increasing structural complexity, such as adding latency (SEIR) or waning immunity (SIRS), fails to resolve the incidence-prevalence gap. Simulation results show that without the proposed universal pre-processor, these advanced models inherit the systematic biases of misaligned data types, leading to significant errors in estimating latent periods and the "heavy tail" of endemicity. The proposed convolution transformation must serve as a universal prerequisite for any compartmental framework, bridging the gap between clinical reporting and mechanistic modeling.

Figures (4)

• Figure 1: This figure illustrates the temporal "gap" between disease prevalence ($I(t)$) and incidence within a standard SIR (Susceptible-Infectious-Recovered) model. The solid red line represents the true prevalence—the total number of infectious individuals at any given time—which determines the rate of new infections in the differential equations. The dashed blue line represents the scaled daily incidence, derived from the rate of change of the susceptible population ($-\frac{dS}{dt}$). The visualization highlights that incidence typically peaks earlier and declines faster than prevalence, a distinction that is vital for researchers when fitting models to daily case reports versus active clinical data.
• Figure 2: The solid red line represents the Prevalence ($I(t)$). The dashed blue line shows Scaled Daily Incidence (New Case Data), highlighting the characteristic "peak offset" where incidence leads prevalence in time. The dotted green line shows the Estimated Prevalence, calculated by integrating new cases over a sliding window ($1/\gamma$) representing the average infectious period. This transformation successfully reconstructs the timing of the infectious pool.
• Figure 3: This figure shows the resolution of the "Incidence-Prevalence Gap" through an exponentially weighted transformation. The solid red line represents the True Prevalence ($I(t)$), the structural variable required for SIR model calibration. The dashed blue line depicts the Daily Incidence, which peaks prematurely and at a different magnitude compared to active cases. The dotted green line shows the Estimated Prevalence, reconstructed by weighting past incidence with an exponential survival function ($e^{-\gamma \tau}$)
• Figure 4: Workflow for practitioners to implement corrected SIR modeling. The process emphasizes the transition from New Daily Cases (Incidence) to Active Cases (Prevalence) through convolution.