<p>This study delves into the numerical threshold properties of split-step <i>θ</i> 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 <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12190_2025_2438_Article_IEq3.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="24" /> </InlineMediaObject> <EquationSource Format="TEX">\({R^h}\)</EquationSource> </InlineEquation> is introduced, paralleling the significance of the analysis threshold in stochastic age-structured population models. Numerical solutions are considered stable for <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12190_2025_2438_Article_IEq4.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="53" /> </InlineMediaObject> <EquationSource Format="TEX">\({R^h} < 1\)</EquationSource> </InlineEquation> and unstable for <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12190_2025_2438_Article_IEq5.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="53" /> </InlineMediaObject> <EquationSource Format="TEX">\({R^h} > 1\)</EquationSource> </InlineEquation>. Furthermore, the method is shown to maintain the basic reproduction number <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12190_2025_2438_Article_IEq6.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="TEX">\({R_0}\)</EquationSource> </InlineEquation> 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.</p>

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

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发表日期：

2025年3月13日

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Yongqi Wang,
Huizi Yang,
Zhanwen Yang

摘要

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 \({R^h}\) is introduced, paralleling the significance of the analysis threshold in stochastic age-structured population models. Numerical solutions are considered stable for \({R^h} < 1\) and unstable for \({R^h} > 1\) . Furthermore, the method is shown to maintain the basic reproduction number \({R_0}\) 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.