We propose a numerical method for approximating integro-differential equations arising in age-of-infection epidemic models. The method is based on a non-standard finite differences approximation of the integral term appearing in the equation. The study of convergence properties and the analysis of the qualitative behavior of the numerical solution show that it preserves all the basic properties of the continuous model with no restrictive conditions on the step-length \begin{document}$ h $\end{document} of integration and that it recovers the continuous dynamic as \begin{document}$ h $\end{document} tends to zero.

In this paper we consider a non-standard discretization to a Volterra integro-differential system which includes a number of age-of-infection models in the literature. The aim is to provide a general framework to analyze the proposed scheme for the numerical solution of a class of problems whose continuous dynamic is well known in the literature and allow a deeper analysis in cases where the theory lacks.

In this paper, we study a dynamically consistent numerical method for the approximation of a nonlinear integro-differential equation modeling an epidemic with age of infection. The discrete scheme is based on direct quadrature methods with Gregory convolution weights and preserves, with no restrictive conditions on the step-length of integration h, some of the essential properties of the continuous system. In particular, the numerical solution is positive and bounded and, in cases of interest in applications, it is monotone. We prove an order of convergence theorem and show by numerical experiments that the discrete final size tends to its continuous equivalent as h tends to zero.

We show that the condition is not necessary, though sufficient, for the asymptotic stability of . We prove the existence of a class of Volterra difference equations (VDEs) that violate this condition but whose zero solutions are asymptotically stable.

In this paper we propose and study a hybrid discrete–continuous mathematical model of collective motion under alignment and chemotaxis effect. Starting from paper [23], in which the Cucker-Smale model [22] was coupled with other cell mechanisms, to describe the cell migration and self-organization in the zebrafish lateral line primordium, we introduce a simplified model in which the coupling between an alignment and chemotaxis mechanism acts on a system of interacting particles. In particular we rely on a hybrid description in which the agents are discrete entities, while the chemoattractant is considered as a continuous signal. The proposed model is then studied both from an analytical and a numerical point of view. From the analytic point of view we prove, globally in time, existence and uniqueness of the solution. Then, the asymptotic behaviour of a linearised version of the system is investigated. Through a suitable Lyapunov functional we show that for t → +∞, the migrating aggregate exponentially converges to a state in which all the particles have a same position with zero velocity. Finally, we present a comparison between the analytical findings and some numerical results, concerning the behaviour of the full nonlinear system.