Eleonora Messina

EPITIME, a computational framework for the simulation of two classes of integral epidemic models, namely an age of infection model and an information-dependent behavioural model, is presented. The main contributions of this work are the design and implementation of a modular MATLAB/Python software environment built upon previously developed structure-preserving non-standard finite difference discretizations. The solvers are complemented by input parsing and validation routines, performance indicators, reproducibility tools and user-oriented graphical interfaces. The preserved structures are specific to the underlying model and include positivity, monotonicity, final-size behaviour and extinction of infectivity for the age of infection model, and positivity, boundedness, positively invariant regions and equilibrium/threshold structure for the behavioural model. The numerical schemes for both model classes and their main analytical properties, including first-order convergence, are outlined. The software architecture is then described and its use is illustrated through numerical experiments on asymptotic behaviour, inverse reconstruction of an infectivity kernel from COVID-19 incidence data, and behavioural dynamics under different memory kernels. Performance, scalability and computational complexity analyses, together with comparisons against quadrature-based approaches, are also presented. Overall, EPITIME provides a reliable and accessible computational environment for the numerical study of renewal epidemic models.

Recent advances in epidemiological modelling have increasingly emphasised the role of sociological factors in describing epidemic dynamics. In this paper, we first provide an overview of the main results regarding the formulation of minimal ordinary differential equations (ODEs) models with information-dependent contact patterns. We then discuss how the minimal ODE models may be extended to integral models through the formulation of specific constitutive equations. We recall the results concerning the integral model that generalises the ODE models with prevalence-dependent contact patterns. Furthermore, we provide a follow-up by also considering the integral model with incidence-dependent contact patterns. For this new model, we study the asymptotic properties of the solutions and obtain sufficient conditions for the stability of the steady states, expressed in terms of the memory kernel and the infectivity function. We numerically show how the memory kernel affects the dynamical outcomes of the model when specific infectivity functions are considered.

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.

We propose an integral model describing an epidemic of an infectious disease. The model is behavioural in the sense that the force of infection includes the information index that describes the opinion-driven human behavioural changes. The information index contains a memory kernel to mimic how the individuals maintain memory of the past values of the infection. We obtain sufficient conditions for the endemic equilibrium to be locally stable. In particular, we show that when the infectivity function is represented by an exponential distribution, stability is guaranteed by the weak Erlang memory kernel. However, through numerical simulations, we show that oscillations, possibly self-sustained, may arise when the memory is more focused in the disease’s past history, as exemplified by the strong Erlang kernel. We also show the model solutions in cases of different infectivity functions taken from studies where specific diseases like Influenza and SARS are considered.

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 weight denominators to achieve it.

Electronic Transactions on Numerical Analysis (ETNA) is an electronic journal for the publication of significant new developments in numerical analysis and scientific computing. Papers of the highest quality that deal with the analysis of algorithms for the solution of continuous models and numerical linear algebra are appropriate for ETNA, as are papers of similar quality that discuss implementation and performance of such algorithms. New algorithms for current or new computer architectures are appropriate provided that they are numerically sound. However, the focus of the publication should be on the algorithm rather than on the architecture. The journal is published by the Kent State University Library in conjunction with the Institute of Computational Mathematics at Kent State University, and in cooperation with the Johann Radon Institute for Computational and Applied Mathematics of the Austrian Academy of Sciences (RICAM). Reviews of all ETNA papers appear in Mathematical Reviews and Zentralblatt für Mathematik. Reference information for ETNA papers also appears in the expanded Science Citation Index. ETNA is registered with the Library of Congress and has ISSN 1068-9613.

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.

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