Numerical threshold of split-step θ methods for stochastic age-structured population models

This study delves into the numerical threshold properties of split-step θ methods applied to stochastic age-structured population models. We establish that these methods can maintain the invariance of the total population, a pivotal attribute, by employing appropriate boundary conditions. The convergence of these methods is affirmed in both the mean- and mean-square senses under suitable boundary conditions. To evaluate the stability of numerical solutions, the numerical threshold is introduced, paralleling the significance of the analysis threshold in stochastic age-structured population models. Numerical solutions are considered stable for and unstable for. Furthermore, the method is shown to maintain the basic reproduction number for any sufficiently large step size, allowing the asymptotic behavior of these models to be represented graphically through numerical processes. The theoretical findings are corroborated with illustrative examples.