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 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.

We propose a non-standard numerical method for the solution of a system of integro-differential equations describing an epidemic of an infectious disease with behavioral changes in contact patterns. The method is constructed in order to preserve the key characteristics of the model, like the positivity of solutions, the existence of equilibria, and asymptotic behavior. We prove that the numerical solution converges to the exact solution as the step size h of the discretization tends to zero. Furthermore, the method is first-order accurate, meaning that the error in the discretization is O(h), it is linearly implicit, and it preserves all the properties of the continuous problem, unconditionally with respect to h. Numerical simulations show all these properties and confirm, also by means of a case-study, that the method provides correct qualitative information at a low computational cost

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

Epidemic models structured by the age of infection can be formulated in terms of a system of renewal equations and represent a very general mathematical framework for the analysis of infectious diseases ([1, 2]). Here, we propose a formulation of renawal equations that takes into account of the behavioral response of individuals to infection. We use the so called "information index", which is a distributed delay that summarizes the information available on current and past disease trend, and extend some results regarding compartmental behavioral models [3, 4, 5]. For the numerical solution of the equations we propose a non-standard approach [6] based on a non local discretization of the integral term characterizing the mathematical equations. We discuss classical problems related to the behaviour of this scheme and we prove the positivity invariance and the unconditional preservation of the stability nature of equilibria, with respect to the discretization parameter. These properties, together with the fact that the method can be put into an explicit form, actually make it a computationally attractive tool and, at the same time, a stand-alone discrete model describing the evolution of an epidemic. This is a joint work with Bruno Buonomo and Claudia Panico from University of Naples "Federico II", and Antonia Vecchio from IAC-CNR, Naples.

We propose a numerical method for a general integro-differential system of equations which includes a number of age-of-infection epidemic models in the literature [1, 2]. The numerical solution is obtained by a non-standard discretization of the nonlinear terms in the system, and agrees with the analytical solution in many important qualitative aspects. Both the behaviour at finite time and the asymptotic properties of the solution are preserved for any value of the discretization parameter. These properties, together with the fact that the method is linearly implicit, actually make it a computationally attractive tool and, at the same time, a stand-alone discrete model describing the evolution of an epidemic [3, 4]. References [1] F. Brauer. Age of infection in epidemiology models, Electronic Journal of Differential Equations, 2005. [2] D. Breda, O. Diekmann, W. F. de Graaf, A. Pugliese and R. Vermiglio, On the formulation of epidemic models (an appraisal of Kermack and McKendrick), J. of Biological Dynamics, 6:sup2, 103-117, 2012. [3] E. Messina, M. Pezzella and A. Vecchio, A non-standard numerical scheme for an age-of-infection epidemic model, J. Comput. Dyn., 9 (2), 239-252, 2022. [4] E. Messina, C. Panico and A. Vecchio, Global stability properties of nonstandard discretization for renewal epidemic models, in preparation.

Mathematical models based on non-linear integral and integro-differential equations are gaining increasing attention in mathematical epidemiology due to their ability to incorporate the past infection dynamic into its current development. This property is particularly suitable to represent the evolution of diseases where the dependence of infectivity on the time since becoming infected plays a crucial role. These renewal equation models contain an integral term describing the contribution of the force of infection to the total infectivity and need, in general, numerical simulations for a complete understanding and quantitative description. For a general model which includes demographic effects [1, 2], we propose a non-standard approach [3] based on a non local discretization of the integral term characterizing the mathematical equations. We discuss classical problems related to the behaviour of this scheme and we prove the positivity invariance and the unconditional preservation of the stability nature of equilibria, with respect to the discretization parameter. These properties, together with the fact that the method can be put into an explicit form, actually make it a computationally attractive tool and, at the same time, a stand-alone discrete model describing the evolution of an epidemic. This is a joint work with Bruno Buonomo and Claudia Panico from University of Naples Federico II, and Antonia Vecchio from IAC-CNR, Naples.

We present a novel approach to the system inversion problem for linear, scalar (i.e. single-input, single-output, or SISO) plants. The problem is formulated as a constrained optimization program, whose objective function is the transition time between the initial and the final values of the system's output, and the constraints are (i) a threshold on the input intensity and (ii) the requirement that the system's output interpolates a given set of points. The system's input is assumed to be a piecewise constant signal. It is formally proved that, in this frame, the input intensity is a decreasing function of the transition time. This result lets us to propose an algorithm that, by a bisection search, finds the optimal transition time for the given constraints. The algorithm is purely algebraic, and it does not require the system to be minimum phase or nonhyperbolic. It can deal with time-varying systems too, although in this case it has to be viewed as a heuristic technique, and it can be used as well in a model-free approach. Numerical simulations are reported that illustrate its performance. Finally, an application to a mobile robotics problem is presented, where, using a linearizing pre-controller, we show that the proposed approach can be applied also to nonlinear problems.

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 employ Lyapunov functions to study boundedness and stability of dynamic equationson time scales. Most of our Lyapunov functions involve the term |x| and its ?-derivative.In particular, we prove general theorems regarding qualitative analysis of solutions of delaydynamical systems and then use Lyapunov functionals that partially include |x| to provide examples.

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.

In this paper, the asymptotic behaviour of the numerical solution to the Volterra integralequations is studied. In particular, a technique based on an appropriate splitting of the kernel isintroduced, which allows one to obtain vanishing asymptotic (transient) behaviour in the numericalsolution, consistently with the properties of the analytical solution, without having to operaterestrictions on the integration steplength

Asymptotically convolution Volterra equations are characterized by kernel functions which exponentiallydecay to convolution ones. Their importance in the applications motivates a numerical analysis of theasymptotic behavior of the solution. Here the quasi-convolution nature of the kernel is exploited in orderto investigate the stability of .; / methods for general systems and in some particular cases.

This paper describes the effect of perturbation of the kernel on the solutions of linear Volterra integral equations on time scales and proposes a new perspective for the stability analysis of numerical methods.