The spline collocation method in piecewise polynomial spaces is applied to a class of third-kind Volterra integral equations (VIEs). Under certain conditions the operator associated with the equivalent second-kind VIE is compact, and this guarantees that the resulting algebraical system is uniquely solvable for all sufficiently small mesh diameters. However, the solvability of this system is not ensured, both on uniform and classical graded meshes, when the operator is noncompact (which typically is the case). Hence, we introduce a modified graded mesh to overcome the solvability problem. For such meshes we establish results on the optimal order of global convergence of the collocation solutions for third-kind VIEs. Numerical tests confirm the validity of these results.

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 \begin{document}$ R_0 $\end{document} of age-independent SIR models for any stepsize. Namely, the disease-free equilibrium is globally stable for numerical processes when \begin{document}$ R_0<1 $\end{document} and the underlying endemic equilibrium is globally stable for numerical processes when \begin{document}$ R_0>1 $\end{document}. A natural extension to nonlinear infection-age models is presented with an initial mortality rate and the numerical thresholds, i.e., numerical basic reproduction numbers \begin{document}$ R^h $\end{document}, are presented according to the infinite Leslie matrix. Although the numerical basic reproduction numbers \begin{document}$ R^h $\end{document} are not quadrature approximations to the exact threshold \begin{document}$ R_0 $\end{document}, the disease-free equilibrium is locally stable for numerical processes whenever \begin{document}$ R^h<1 $\end{document}. Moreover, a unique numerical endemic equilibrium exists for \begin{document}$ R^h>1 $\end{document}, 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.

In this article, we consider second-kind linear Volterra integral equations (VIEs) with noncompact operators, that is, μu = f + V _ϕ u, where with the core ϕ ∈L ¹(0, 1). We present some different properties of noncompact operators V _ϕ from compact operators, such as eigenvalues, eigenfunctions, null spaces, and ranges. In many applications, the cores belong to L ^p (0, 1) for some p > 1. In this case, we completely describe the eigenvalues of V _ϕ and the null space and the range of μI − V _ϕ. In addition, a necessary and sufficient condition is given such that μI − V _ϕ is a Fredholm operator. In the end, we discuss the regularity of solutions.