In this work, we introduce a quadratically convergent and dynamically consistent integrator specifically designed for the replicator dynamics. The proposed scheme combines a two-stage rational approximation with a normalization step to ensure confinement to the probability simplex and unconditional preservation of non-negativity, invariant sets and equilibria. A rigorous convergence analysis is provided to establish the scheme’s second-order accuracy, and an embedded auxiliary method is devised for adaptive time-stepping based on local error estimation. Furthermore, a discrete analogue of the quotient rule, which governs the evolution of component ratios, is shown to hold. Numerical experiments validate the theoretical results, illustrating the method’s ability to reproduce complex dynamics and to outperform well-established solvers in particularly challenging scenarios.

In this manuscript, we present a comprehensive theoretical and numerical framework for the control of production-destruction differential systems. The general finite horizon optimal control problem is formulated and addressed through the dynamic programming approach. We develop a parallel in space semi-Lagrangian scheme for the corresponding backward-in-time Hamilton-Jacobi-Bellman equation. Furthermore, we provide a suitable conservative reconstruction algorithm for optimal controls and trajectories. The application to two case studies, specifically enzyme catalyzed biochemical reactions and infectious diseases, highlights the advantages of the proposed methodology over classical semi-Lagrangian discretizations.

Modified Patankar schemes are linearly implicit time integration methods designed to be unconditionally positive and conservative. In the present work we extend the Patankar-type approach to linear multistep methods and prove that the resulting discretizations retain, with no restrictions on the step size, the positivity of the solution and the linear invariant of the continuous-time system. Moreover, we provide results on arbitrarily high order of convergence and we introduce an embedding technique for the Patankar weights denominators to achieve it.

In this paper we design high-order positivity-preserving approximation schemes for an integro-differential model describing photochemical reactions. Specifically, we introduce and analyze three classes of dynamically consistent methods, encompassing non-standard finite difference schemes, direct quadrature techniques and predictor- corrector approaches. The proposed discretizations guarantee the positivity, monotonicity and boundedness of the solution regardless of the temporal, spatial and frequency stepsizes. Comprehensive numerical experiments confirm the theoretical findings and demonstrate the efficacy of the proposed methods in simulating realistic photochemical phenomena.

In this work we present three classes of unconditionally positive numerical methods for a photochemical model governed by non-local integro-differential equations. Specifically, we design and compare dynamically-consistent approximation schemes based on non-standard finite differences discretizations, predictor-corrector approaches and direct quadrature integrators. A rigorous analysis is performed to establish the preservation of key physical properties, i.e. positivity, monotonicity and boundedness, regardless of the temporal, spatial and frequency stepsizes. Furthermore, theoretical results are provided to establish the high-order consistency and convergence of the methods. Comprehensive numerical experiments confirm the theoretical findings and allow for a detailed comparison of the performance and computational efficiency of the proposed discretizations. Applications to two case studies of interest, photoactivation of serotonin in left-right brain patterning and photodegradation of cadmium pigments in historical paintings, demonstrate the practical relevance of the proposed model and simulation techniques in addressing complex phenomena in photochemistry.

Many paintings from the 19th century have exhibited signs of fading and discoloration, often linked to cadmium yellow, a pigment widely used by artists during that time. In this work, we develop a mathematical model of the cadmium sulfide photo catalytic reaction responsible for these damages. By employing nonlo cal integral operators, we capture the interplay between chemical processes and environmental factors, offering a detailed representation of the degradation mechanisms. Furthermore, we present a second order positivity-preserving numerical method designed to accurately simulate the phenomenon and ensure reliable predictions across different scenarios, along with a comprehensive sensitivity analysis of the model.

In this manuscript we propose a numerical method for non-linear integro-differential systems arising in age-of-infection models in a heterogeneously mixed population. The discrete scheme is based on direct quadrature methods and provides an unconditionally positive and bounded solution. Furthermore, we prove the existence of the numerical final size of the epidemic and show that it tends to its continuous equivalent as the discretization steplength vanishes.

The mathematical modeling of various real-life phenomena often leads to the formulation of positive and conservative Production-Destruction differential Systems (PDS). Here we address a general finite horizon Optimal Control Problem (OCP) for PDS and delve into the properties of its continuous-time solution. Leveraging the dynamic programming approach, we recast the OCP as a backward-in-time Hamilton-Jacobi-Bellman (HJB) equation, whose unique viscosity solution corresponds to the value function [1]. We then propose a parallel-in-space semi-Lagrangian approximation scheme for the HJB equation [3] and derive the optimal control in feedback form. Finally, to reconstruct the optimal trajectories of the controlled PDS, we employ unconditionally positive and conservative modified Patankar linear multistep methods [2]. [1] CRANDALL, M. G.; ISHII, H.; LIONS, P.-L. User’s guide to viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc., 1992, 27.1: 1-67. [2] IZZO, G.; MESSINA, E.; PEZZELLA, M.; VECCHIO, A. TITOLO DA ACCERTARE LUNEDì. In preparation. [3] FALCONE, M.; FERRETTI, R. Semi-Lagrangian approximation schemes for linear and Hamilton—Jacobi equations. SIAM, 2013.

Numerous real-world phenomena involve the interplay between processes of production and decay or consumption and can be therefore modeled by positive and conservative Production-Destruction differential Systems (PDS). Patankar-type schemes are linearly implicit integrators specifically designed for PDS with the aim of retaining, with no restrictions on the stepsize, the positivity of the solution and the linear invariant of the system. In this work we extend the Patankar technique, already established for Runge-Kutta and deferred correction methods, to multistep schemes. As a result, we introduce the class of Modified Patankar Linear Multistep (MPLM) methods, for which a thorough investigation of the convergence is carried out. Furthermore, we design an embedding procedure for the computation of the Patankar weights and prove the high order of convergence of the resulting MPLM scheme. A comparative study on the simulation of selected test cases highlights the competitive performance of the MPLM methods with respect to other Patankar-type discretizations.

In this paper we study non-linear implicit Volterra discrete equations of convolutiontype and give sufficient conditions for their solutions to converge to a finite limit. Theseresults apply to the stability analysis of linear methods for implicit Volterra integralequations. An application is given to the numerical study of the final size of an epidemicmodelled by renewal equations

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

Age of infection epidemic models [1, 3], based on non-linear integro-dierential equations, naturally describe the evolution of diseases whose infectivity depends on the time since becoming infected. Here we consider a multi-group age of infection model [2] and we extend the investigations in [4], [5] and [6] to provide numerical solutions that retain the main properties of the continuous system. In particular, we use Direct Quadrature methods and prove that the numerical solution is positive and bounded. Furthermore, in order to study the asymptotic behavior of the numerical solution, we formulate discrete equivalents of the nal size relation and of the basic reproduction number and we prove that they converge to the continuous ones, as the step-size of the discretization goes to zero.

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 h of integration and that it recovers the continuous dynamic as h tends to zero.

In this paper, we study a dynamically consistent numerical method for the approximationof a nonlinear integro-differential equation modeling an epidemic with age of infection. The discretescheme 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 ofthe continuous system. In particular, the numerical solution is positive and bounded and, in casesof interest in applications, it is monotone. We prove an order of convergence theorem and show bynumerical experiments that the discrete final size tends to its continuous equivalent as h tends to zero.