A non-standard numerical scheme for an age-of-infection epidemic model
Eleonora Messina, Mario Pezzella, Antonia Vecchio
TL;DR
The paper tackles numerical integration of the Kermack–McKendrick age-of-infection model, which involves a Volterra-type integral term, by introducing a non-standard finite difference discretization that preserves positivity and key dynamical features for any step length. The NSFD scheme yields first-order consistency, bounded and monotone discrete solutions, and a discrete reproduction number $R_0(h)$ that governs invasion and final size through a discrete analogue of the continuous relations, with convergence to the continuous dynamics as $h\to0$. Theoretical results are complemented by numerical experiments demonstrating robust convergence and improved qualitative behavior compared to standard direct quadrature methods. This approach offers a computationally efficient and reliable tool for real-time simulations of epidemic processes with memory effects.
Abstract
We propose a numerical method for approximating integro-differential equations arising in age-of-infection epidemic models. The method is based on a non-standard finite differences approximation of the integral term appearing in the equation. The study of convergence properties and the analysis of the qualitative behavior of the numerical solution show that it preserves all the basic properties of the continuous model with no restrictive conditions on the step-length $h$ of integration and that it recovers the continuous dynamic as $h$ tends to zero.