Learning Epidemic Threshold Equations from Simulation: REEBLRA, an Interpretable Bayesian Classification Framework for Reproduction Numbers

The basic reproduction number, $R_0$, is central to infectious-disease modelling, yet analytic expressions can be difficult to derive for high-dimensional, nonlinear, or structurally complex models. We introduce REEBLRA ($R_0$ Equation Estimation using Bayesian Logistic Regression Algorithm), an interpretable framework that learns explicit surrogate equations for epidemic thresholds from parameter--response data. Rather than directly regressing $R_0$, REEBLRA estimates $\Pr\{R_0(\boldsymbolθ)>c\}$ for parameter vector $\boldsymbolθ$ and threshold $c$. It provides a binary equation for classifying epidemic invasion ($R_0>1$) and successive threshold equations defining ordered brackets that approximate $R_0$, the expected number of secondary infections caused by a typical case in a fully susceptible population. The resulting equations support parameter screening, sensitivity interpretation, threshold classification, and exploratory analysis when analytic derivation is cumbersome. Bayesian logistic regression using standardized polynomial-like features and proper weakly informative priors yields regularized estimates, posterior uncertainty, and finite coefficients under complete or quasi-complete separation. We establish a mechanistic--surrogate concordance principle linking the full fitted-score gradient to the local gradient of $\log R_0$ under explicit regularity and tangency assumptions, allowing conditional interpretation of threshold directions. Across seven analytic-label benchmarks associated with ODE models, including a 15-compartment five-group SIR system, five-fold binary AUC was 0.999--1.000 and ordered-bracket accuracy was 0.951--0.998. REEBLRA complements rather than replaces mechanistic derivations, bridging simulation, machine learning, and reproduction-number analysis. An accompanying Shiny app supports implementation.