Carleman-linearized epidemic models as a route to a data-driven algebraic estimate of the effective reproduction number

Nonlinear models are fundamental in systems biology, yet their complexity often limits analytical solutions and real-time prediction. We introduce Carleman linearization as a practical method to approximate nonlinear differential systems by transforming them into infinite-dimensional linear systems and truncating for manageable approximate solutions. This approach, rarely applied in biology, enables interpretable formulas that retain essential dynamics without relying solely on numerical simulations. To demonstrate its utility, we apply Carleman linearization to the classical Susceptible-Infected-Removed epidemic model. The resulting approximations provide explicit expressions for infection dynamics and an algebraic formula for the effective reproduction number, requiring only simple averages of prevalence data. Strikingly, the widely used Gompertz growth law emerges naturally from the structure of Carleman approximants, offering a theoretical basis for its empirical success in epidemic modeling. These findings position Carleman linearization as a versatile tool for Systems Biology, delivering controlled approximations that support rapid and reliable estimates in complex dynamical processes.