Numerical analysis of linearly implicit Euler method for age-structured SIS model

In this paper, we consider a numerical threshold for a coupled age-structured SIS epidemic model. We deal with the existence, convergence and stability of a linearly implicit Euler method for an age-independent SIS model firstly. It is shown that the numerical processes replicate the global dynamical behaviors of SIS model for any time-step size. We continue analyzing an age-structured SIS model similarly by introducing the numerical total population \(p_{j}^{n}\) $p_j^n$ to overcome the coupling. With the help of Krein–Rutman’s theorem, the numerical basic reproduction numbers \(R_{\Delta t}^{1}\) $R_{\mathrm{\Delta }t}^1$ and \(R_{\Delta t}^{2}\) $R_{\mathrm{\Delta }t}^2$ are presented, which play a decisive role in the numerical dynamical behavior of disease-free equilibrium and endemic equilibrium. Moreover, instead of the convergence of numerical solutions, it is much more interesting that the numerical basic reproduction numbers converge to the exact ones with accuracy of order 1. Finally, numerical applications to SIS epidemic models illustrate the verification and the efficiency of our results.