Bivariate collocation for computing $R_{0}$ in epidemic models with two structures

TL;DR

This paper introduces a numerical method using bivariate collocation to accurately compute the basic reproduction number $R_{0}$ in complex structured epidemic models with two traits, demonstrating high convergence for smooth eigenfunctions.

Contribution

The paper presents a novel bivariate collocation technique for approximating $R_{0}$ in epidemic models with two structures, enabling efficient eigenvalue computation for complex PDE-based models.

Findings

Convergence is infinite for smooth eigenfunctions.

Finite convergence observed for less regular eigenfunctions.

Method effectively applied to a demographic-age and immunity structured model.

Abstract

Structured epidemic models can be formulated as first-order hyperbolic PDEs, where the "spatial" variables represent individual traits, called structures. For models with two structures, we propose a numerical technique to approximate $R_{0}$, which measures the transmissibility of an infectious disease and, rigorously, is defined as the dominant eigenvalue of a next-generation operator. Via bivariate collocation and cubature on tensor grids, the latter is approximated with a finite-dimensional matrix, so that its dominant eigenvalue can easily be computed with standard techniques. We use test examples to investigate experimentally the behavior of the approximation: the convergence order appears to be infinite when the corresponding eigenfunction is smooth, and finite for less regular eigenfunctions. To demonstrate the effectiveness of the technique for more realistic applications, we present a new epidemic model structured by demographic age and immunity, and study the approximation of $R_{0}$ in some particular cases of interest.