Numerical threshold of linearly implicit Euler method for nonlinear infection-age SIR models

Abstract

<p style='text-indent:20px;'>In this paper, we consider a numerical threshold of a linearly implicit Euler method for a nonlinear infection-age SIR model. It is shown that the method shares the equilibria and basic reproduction number <inline-formula><tex-math id="M1">\begin{document}$ R_0 $\end{document}</tex-math></inline-formula> of age-independent SIR models for any stepsize. Namely, the disease-free equilibrium is globally stable for numerical processes when <inline-formula><tex-math id="M2">\begin{document}$ R_0&lt;1 $\end{document}</tex-math></inline-formula> and the underlying endemic equilibrium is globally stable for numerical processes when <inline-formula><tex-math id="M3">\begin{document}$ R_0&gt;1 $\end{document}</tex-math></inline-formula>. A natural extension to nonlinear infection-age models is presented with an initial mortality rate and the numerical thresholds, i.e., numerical basic reproduction numbers <inline-formula><tex-math id="M4">\begin{document}$ R^h $\end{document}</tex-math></inline-formula>, are presented according to the infinite Leslie matrix. Although the numerical basic reproduction numbers <inline-formula><tex-math id="M5">\begin{document}$ R^h $\end{document}</tex-math></inline-formula> are not quadrature approximations to the exact threshold <inline-formula><tex-math id="M6">\begin{document}$ R_0 $\end{document}</tex-math></inline-formula>, the disease-free equilibrium is locally stable for numerical processes whenever <inline-formula><tex-math id="M7">\begin{document}$ R^h&lt;1 $\end{document}</tex-math></inline-formula>. Moreover, a unique numerical endemic equilibrium exists for <inline-formula><tex-math id="M8">\begin{document}$ R^h&gt;1 $\end{document}</tex-math></inline-formula>, which is locally stable for numerical processes. It is much more important that both the numerical thresholds and numerical endemic equilibria converge to the exact ones with accuracy of order 1. Therefore, the local dynamical behaviors of nonlinear infection-age models are visually displayed by the numerical processes. Finally, numerical applications to the influenza models are shown to illustrate our results.</p>