Numerical threshold stability analysis of a positivity-preserving IMEX numerical scheme for a nonlinear age-space structured SIR epidemic model

This paper focuses on the numerical threshold stability of a nonlinear age-space structured SIR epidemic model under Neumann boundary condition. The space domain is firstly discretized to derive a numerical semi-discrete system in accordance with the method of lines. It is shown that the semi-discrete system shares the same stability threshold value (basic reproduction number \(\mathcal {R}_0\) ${\mathcal{R}}_0$ ) of the model. A fully discrete positivity-preserving numerical scheme is established by applying an Implicit–Explicit (IMEX) numerical method to discretize the semi-discrete system. A numerical stability threshold value \(\mathcal {R}_0^{h}\) ${\mathcal{R}}_0^h$ is proposed and named as numerical basic reproduction number. Moreover, it is proved that \(\mathcal {R}_0^{h}\) ${\mathcal{R}}_0^h$ converges to the basic reproduction number \(\mathcal {R}_0\) ${\mathcal{R}}_0$ of the original model while the stepsize vanishing, which implies \(\mathcal {R}_0^{h}\) ${\mathcal{R}}_0^h$ provides a realistic approximation to \(\mathcal {R}_0\) ${\mathcal{R}}_0$ , besides the role plays in numerical stability analysis. Eventually, the numerical conclusions are testified through some numerical simulations.