arXiv subjects

Fatima Mroue

Publications and source records attributed to Fatima Mroue.

4 recordsLinked to original sources

arXiv2026

Derivation, Analysis and Simulation of a Spatio-Temporal Epidemiology Model with Memory

In this paper, we propose an integro-differential model for the spatio-temporal evolution of infectious diseases with asymptomatic transmission. The model consists of a reaction-diffusion system with an integral memory term accounting for the distribution of the incubation period. We first analyze the asymptotic behavior and the properties of the integro-differential model. Then, we prove the local existence of a weak solution of the system by means of the Faedo-Galerkin method and a compactness argument. The model is applied to simulate the geographical evolution of a disease in Lebanon.

math.AP↗

arXiv2026

A Spline-based Physics-Informed Numerical Scheme: Accurate Smooth Solutions for Differential Equations

The rise of Physics-Informed Neural Networks (PINNs) has popularized the concept of solving differential equations via residual minimization. However, neural networks are often viewed as ``black boxes" requiring significant computational overhead and stochastic optimization. Moreover, PINNs typically treat boundary conditions (BCs) as ``soft constraints" within the loss function and this makes the optimization process struggling to enforce the BCs properly. This paper introduces the \textbf{Spline-based Physics-Informed Numerical Scheme (SPINS)}, a numerical framework designed to solve both initial and boundary value problems of ordinary differential equations (ODEs). By replacing the neural network architecture of traditional PINNs with a structured spline basis, SPINS achieves high accuracy and interpretability with a minimal parameter set. In addition, the BCs are automatically satisfied from the choice of the splines architecture. Therefore, SPINS provides smooth numerical solutions for ODEs allowing analytical differentiation. Moreover, SPINS benefits from the automatic differentiation where computing the gradient of the physics-informed loss function is an easy task making the optimization process very fast using gradient-based optimizers such as the L-BFGS-B algorithm. We demonstrate the efficacy of SPINS on nonlinear second order ODEs with several choices of BCs using cubic and quintic interpolating splines and present its natural extension to high order ODEs.

math.NA↗

arXiv2023

Stochastic electromechanical bidomain model

We analyze a system of nonlinear stochastic partial differential equations (SPDEs) of mixed elliptic-parabolic type that models the propagation of electric signals and their effect on the deformation of cardiac tissue. The system governs the dynamics of ionic quantities, intra and extra-cellular potentials, and linearized elasticity equations. We introduce a framework called the active strain decomposition, which factors the material gradient of deformation into an active (electrophysiology-dependent) part and an elastic (passive) part, to capture the coupling between muscle contraction, biochemical reactions, and electric activity. Under the assumption of linearized elastic behavior and a truncation of the nonlinear diffusivities, we propose a stochastic electromechanical bidomain model, and establish the existence of weak solutions for this model. To prove existence through the convergence of approximate solutions, we employ a stochastic compactness method in tandem with an auxiliary non-degenerate system and the Faedo--Galerkin method. We utilize a stochastic adaptation of de Rham's theorem to deduce the weak convergence of the pressure approximations.

math.AP↗

arXiv2020

Modeling and Simulation of the spread of coronavirus disease (COVID-19) in Lebanon

In this paper, we develop a probabilistic mathematical model for the spread of coronavirus disease (COVID-19). It takes into account the known special characteristics of this disease such as the existence of infectious undetected cases and the different social and infectiousness conditions of infected people. In particular, it considers the social structure and governmental measures in a country, the fraction of detected cases over the real total infected cases, and the influx of undetected infected people from outside the borders. Although the model is simple and allows a reasonable identification of its parameters, using the data provided by local authorities on this pandemic, it is also complex enough to capture the most important effects. We study the particular case of Lebanon and use its reported data to estimate the model parameters, which can be of interest for estimating the spread of COVID-19 in other countries. We show a good agreement between the reported data and the estimations given by our model. We also simulate several scenarios that help policy makers in deciding how to loosen different measures without risking a severe wave of COVID-19. We are also able to identify the main factors that lead to specific scenarios which helps in a better understanding of the spread of the virus.

q-bio.PE↗