Mouhcine Tilioua

Reaction–diffusion epidemic models play a central role in understanding how infectious diseases propagate through space and time, offering valuable insight for public health analysis. A key element of such models is the incidence function, which governs the nonlinear interaction between susceptible and infected populations. Despite extensive studies on various incidence formulations, the systematic identification of a suitable one for a given setting remains an open question. This work introduces a theoretical framework that interprets the selection of an incidence function as quantifying the contribution of several plausible formulations to the overall transmission dynamics, inferred from observational data. The resulting problem takes the form of a PDE-constrained optimization, where the objective is to determine the optimal weights in a convex combination of incidence functions that best fit the observed epidemic patterns. The analysis establishes the Fréchet differentiability of the parameter-to-state operator and derives first-order optimality conditions via an adjoint system. A numerical illustration, based on the Landweber iteration method, highlights the framework’s potential as a mathematical tool to enhance modeling accuracy and support strategies aimed at disease prevention.

The purpose of this paper is to establish the well-posedness of martingale (probabilistic weak) solutions to stochastic degenerate aggregation--diffusion equations arising in biological and public health contexts. The studied equation is of a stochastic degenerate parabolic type, featuring a nonlinear two-sidedly degenerate diffusion term accounting for repulsion, a locally Lipschitz reaction term representing competitive interactions, and a stochastic perturbation term capturing environmental noise and uncertainty in biological systems. The existence of martingale solutions is proved via an auxiliary nondegenerate stochastic system combined with the Faedo--Galerkin method. Convergence of approximate solutions is established through Prokhorov's compactness and Skorokhod's representation theorems, and uniqueness is obtained using a duality approach. Finally, numerical simulations are given to illustrate the impact of environmental noise on aggregation dynamics and the long-term behavior of the system, offering insights that may inspire medical innovation and predictive modeling in public health.

A crucial feature of reaction-diffusion epidemic models is the incidence function, which characterizes disease transmission dynamics. Over the past few decades, many studies have investigated the behavior of such models under various incidence functions. However, the question of how to appropriately select a suitable incidence function remains largely unexplored. This paper addresses this issue by proposing an intuitive theoretical framework that recasts the original problem as determining the contributions of different incidence functions to the dynamics based on given observations. Specifically, the choice of an incidence function is linked to a weight assigned to it. Mathematically, this leads to a PDE-constrained optimization problem, where the objective is to identify the weights of a convex combination of multiple incidence functions that best approximate the experimental observations generated by the model. We first establish the Fréchet differentiability of the parameter-to-state operator and then derive the optimality conditions for the weights using a suitable adjoint problem. Finally, we illustrate our approach with a numerical example based on the Landweber iteration algorithm.

This paper explores the exponential stability of two nonlinear wave equations coupled through their velocities. The analysis is divided into two main cases. First, we consider a system where one equation is damped, while the other experiences a discrete time delay. By reformulating the problem in an abstract framework, we use semigroup theory and energy methods to establish well-posedness and derive conditions that guarantee exponential energy decay. In the second case, we examine a scenario where a frictional damping term appears in the first equation, while the second equation contains an indefinite damping term, namely with a sign-changing coefficient. Although this setup can be viewed as a special case of the first, we analyze it separately and show that exponential stability still holds under a weaker condition.

This paper deals with an optimal control problem of a mathematical model for current-induced magnetization dynamics in ferromagnets at elevated temperatures described by the classical Landau-Lifshitz-Bloch (LLB) equation involving adiabatic torque where the control is given by the applied current. We prove the existence of an optimal control and then we derive the first-order necessary optimality system for the optimal solution by using the Lagrange multiplier method. Some regularity properties of the adjoint and the optimal control are also discussed.

This paper considers a generalized version of the Landau–Lifshitz–Bloch equation. We prove existence of measure-valued solutions for the model in a bounded domain of \mathbb{R}^{d} (d\geq1) . The main difficulties in this study are due to the loss of compactness in the equation and the presence of a nonlinear term of type \boldsymbol{u}\wedge\mathrm{div}(a(\nabla\boldsymbol{u})) which does not satisfy the monotonicity assumption in the sense of Leray–Lions. We use a compactness result proved in Landes et al. [Ark. Mat. 25 (1987), 29–40] and the concept of measure-valued solutions which are appropriate to solve this problem.