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

This paper focuses on numerical approximation of the basic reproduction number R₀, which is the threshold defined by the spectral radius of the next-generation operator in epidemiology. Generally speaking, R₀ 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 R_0,n . Using the spectral approximation theory, we obtain that R_0,n → R₀ as n → +∞. Some numerical simulations are provided to certify the theoretical results.