A Note on Quantifying the Contributions of Incidence Functions in Spatio-Temporal Epidemic Models

Reaction-diffusion epidemic models play a central role in understanding how infectious diseases propagate through space and time, offering valuable insight for public health analysis. A key element of such models is the incidence function, which governs the nonlinear interaction between susceptible and infected populations. Despite extensive studies on various incidence formulations, the systematic identification of a suitable one for a given setting remains an open question. This work introduces a theoretical framework that interprets the selection of an incidence function as quantifying the contribution of several plausible formulations to the overall transmission dynamics, inferred from observational data. The resulting problem takes the form of a PDE-constrained optimization, where the objective is to determine the optimal weights in a convex combination of incidence functions that best fit the observed epidemic patterns. The analysis establishes the Fréchet differentiability of the parameter-to-state operator and derives first-order optimality conditions via an adjoint system. A numerical illustration, based on the Landweber iteration method, highlights the framework's potential as a mathematical tool to enhance modeling accuracy and support strategies aimed at disease prevention.