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 → $\mathcal{R}_0$ as n → +∞. Some numerical simulations are provided to certify the theoretical results.

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