A theta-scheme approximation of basic reproduction number for an age-structured epidemic system in a finite horizon

Abstract

This paper focuses on numerical approximation of the basic reproduction number $\mathcal{R}_0$, which is the threshold defined by the spectral radius of the next-generation operator in epidemiology. Generally speaking, $\mathcal{R}_0$ cannot be explicitly calculated for most age-structured epidemic systems. In this paper, for a deterministic age-structured epidemic system and its stochastic version, we discretize a linear operator produced by the infective population with a theta scheme in a finite horizon, which transforms the abstract problem into the problem of solving the positive dominant eigenvalue of the next-generation matrix. This leads to a corresponding threshold $\mathcal{R}_{0, n}$. Using the spectral approximation theory, we obtain that $\mathcal{R}_{0, n}\rightarrow\mathcal{R}_0$ as $n\rightarrow+\infty$. Some numerical simulations are provided to certify the theoretical results.

Keywords

• numerical approximation
• basic reproduction number
• age-structure epidemic system
• spectral radius
• theta scheme