Séverine Bernard

University of the French Antilles | Reunion · Research Group on Mathematics and Computer Science

Obesity and diabetes are diseases that are increasing every year in the world and their control is an important problem faced by health systems. In this work, we present an optimal control problem based on a model for overweight and obesity and its impact on the diagnosis of diabetes using fractional order derivatives in the Caputo sense. The contr...

Today, the use of renewable energy sources such as sun, wind and hot water to generate electricity is increasingly becoming a global priority. In this paper, we propose a deterministic dynamical system reporting the transition from oil to solar energy, in which we then add the randomness of this phenomenon. First, we study the positivity of the det...

Overweight and obesity are current problems humankind faces and have serious health consequences because they contribute to diseases such as heart diseases and diabetes. In this paper, we present a mathematical model for the study of overweight and obesity in a population and its impact on the growth of the number of diabetics. For the construction...

In this paper, we present a deterministic mathematical model for the study of overweight, and obesity in a population and its impact on the growth of the number of diabetics. For the construction of the model, we take into account social factors and the interactions between the elements of society. We find the basic reproduction number and prove th...

In this paper, we present a deterministic mathematical model for the study of overweight, and obesity in a population and its impact on the growth of the number of diabetics. For the construction of the model, we take into account social factors and the interactions between the elements of society. We find the basic reproduction number and prove th...

In this paper, we develop a deterministic compartmental model for the obesity dynamics. Contrary to other contributions on this subject we explore the impact of the media on the spreading of this phenomenon in a constant population. Stability analysis shows that the disease-free equilibrium point is globally asymptotically stable when the number of...

In this paper, we investigate the effects of a Lévy jump on the dynamic of propagation of a rumor on a social network. The random environment is characterized by white noises and Lévy jump and we establish suffcient conditions for extinction and persistence in the mean of an e-rumor. At the end, we compare our study with our previous one[7] to see...

La diabetes, debido a sus complicaciones, es una de las enfermedades que más problemas plantean en la salud pública actual mundial. En este trabajo se parte de una población de diabéticos con y sin complicaciones y se asocia un problema de control optimal no lineal que describe la dinámica de la población. Para este modelo se prueba la existencia d...

The emergence of new communication tools leads us to have public discussions on social networks. These public spaces of exchange are firmly established in our societies but are strong sensors of both human behaviors and collective feelings. The propagation of rumors, namely e-rumor, is very fast and is one of the most dangerous for a society becaus...

This paper deals with variational inclusions of the form \(0 \in K-f(x)\) where \(f : \mathbb{R}^{n} \rightarrow \mathbb{R} ^{m}\) is a semismooth function and \(K\) is a nonempty closed convex cone in \(\mathbb{R}^{m}\). We show that the previous problem can be solved by a Newton-type method using the Clarke generalized Jacobian of \(f\). The resu...

The understanding and the control of the phenomena of rumour on social networks, mainly e-rumour nowadays, are big challenges for communities, organizations and states. For example, rumour’s propagation can jeopardise their public opinion and affect their financial markets. In these last decades, many mathematical theories have been developed on th...

Social networks have a significant role in spreading rumors. Such phenomena of e-rumor are big challenges for communities, organizations and states, since the spread of rumors can rapidly jeopardise their public opinion and their economic and financial markets. In these last decades, many mathematical theories have been developed on this topic both...

In this paper, we consider a population of diabetics and divide it into two subcategories, one of diabetics with complications and another one of diabetics without complications. From a model examining the complications of individu- als diagnosed with diabetes, we associate a nonlinear optimal control problem. Considering this last one, we prove th...

In this paper, via the algebra of generalized functions, we investigate the generalized Riemann’s problem associated to conservation laws with analytical coefficients. This allows us to transform the problem into a system of ordinary differential equations. In some particular cases, such as Burgers’ and conservative Richard’s equation, approximated...

The aim of this paper is to prove that the framework of generalized functions of Sobolev type is more suitable to pose and solve some PDEs problems with very irregular data, than the one introduced by J.-F. Colombeau, when C∞ estimates are not available or out of reach. In such type of algebras, one shows the existence and some qualitative properti...

In the last two decades, many algebras of generalized functions have been constructed, particularly the so-called generalized Sobolev algebras. Our goal is to study the latter and some of their main properties. In this framework, we pose and solve a nonlinear degenerated Dirichlet problem with irregular data such as Dirac generalized functions.

We extend the theory of distributional kernel operators to a framework of generalized functions, in which they are replaced by integral kernel operators. Moreover, in contrast to the distributional case, we show that these generalized integral operators can be composed unrestrictedly. This leads to the definition of the exponential of a subclass of...

We extend the theory of distributional kernel operators to a framework of generalized functions, in which they are replaced by integral kernel operators. Moreover, in contrast to the distributional case, we show that these generalized integral operators can be composed unrestrictedly. This leads to the definition of the exponential, and more genera...

This paper deals with the existence and uniqueness of an entropy solution of the Cauchy problem for the quasilinear equation u t +a(f(u)) x =0 in one space dimension, where a is a non-smooth coefficient.

In order to model phenomena arising in matter flows in electromagnetic fields, engineers join systems of conservation laws with discontinuous coefficients. The simplest quasi-linear equation is ut+a(f(u))x=0, where a is a given discontinuous coefficient function and f is a smooth function. We begin our study by solving the Riemann problem, and then...