Résumés des conférences

Veronica Anaya

Numerical approximation for SIR models with cross diffusion.

We are concerned with a model of the indirect transmission of an epidemic disease between two spatially distributed host populations, the epidemic disease transmission occurring through a contaminated environment. We address the questions of existence of weak solutions by using a regularization method. Moreover, we propose a finite volume scheme and proved the well posedness, nonnegativity and convergence of the discrete solution. The convergence proof is based on deriving a series of a priori estimates and by using a general Lp compactness criterion. Finally, we give some examples.

David Mora

Virtual Element Approximation for the Transmission Eigenvalue Problem

In this talk, we analyze a virtual element method (VEM) for solving a non-selfadjoint fourth-order eigenvalue problem derived from the transmission eigenvalue problem. We write a variational formulation and propose a $C^1$-conforming discretization by means of the VEM. We use the classical approximation theory for compact non-selfadjoint operators to obtain optimal order error estimates for the eigenfunctions and a double order for the eigenvalues. Finally, we present some numerical experiments illustrating the behavior of the virtual scheme on different families of meshes.

Ricardo Ruiz Baier

Pattern formation within poroelastic media and applications to developmental models

We propose a new mathematical model for the interaction of skin cell populations with fibroblast growth factor and bone morphogenetic protein, occurring within deformable porous media. The equations for feather primordia pattering are based on the recent work by K.J. Painter et al. [J. Theoret. Biol., 437 (2018) 225--238]. We perform a linear stability analysis to identify relevant parameters in the coupling mechanisms, focusing in the regime of infinitesimal strains. We also discuss the well-posedness of the problem, and then extend the model to the case of nonlinear poroelasticity including mechanisms of solid growth by means of Lee decompositions of the deformation gradient. We present a few illustrative computational examples in 2D and 3D. This is a joint work with Jennifer Dingwall (University of Oxford), Luis M. De Oliveira (University of Geneva), and Michel C. Milinkovitch (University of Geneva).

Fernando Luiz Pio dos Santos

Mathematical Modeling of Dengue Disease in Infants

Dengue virus (DENV) has four distinct serotypes (DENV 1-4), and any of these can cause distinct severities: dengue fever (DF), in the classical form and DHF in the most severe case. In a first infection by a serotype, the individual acquires antibodies specific for this serotype and in the second heterologous infection, the DHF can be developed. In particular, DHF may occur in the infant in the primary infection by either serotype, due to the vertical transfer of specific antibodies coming from its immune mother to the DENV [1]. These specific antibodies play an important role in the infant’s life, providing protection during the first months of life, but then, as their serum levels decrease, they may increase the chance of infection through the Antibody-Dependent Enhancement (ADE) [2]. A mathematical model was developed to describe the dynamics of the primary infection of the dengue virus in infants, that was born of a mother immune to some serotype of the dengue virus and who, therefore, acquired maternal antibodies during pregnancy. The mathematical model is described by a system of nonlinear ordinary differential equations, where the time-dependent variables are: the number of infant antibodies, transferred from their immune mother to some DENV, uninfected and infected monocytes, and dengue virus, over the time, and it was analyzed mathematically, establishing the conditions for the existence of the free equilibrium points and the persistence of the disease, as well as the basic reproductive number, R0.The numerical simulations obtained using the fourth order Runge-Kutta method were performed considering the scenarios R0 < 1 and R0 > 1, which illustrate the convergence of the numerical results for the free equilibrium and for the persistence of the disease, and corroborate with the local stability study of the model. As the main result of this work, it is possible to visualize the formation of peaks in the dynamics of the infected monocytes and of the dengue virus for the scenario R0 > 1, that in the situation where there is persistence of the virus, it characterizes the occurrence DHF in the infant. References [1] A. Jain and U. C. Chaturvedi, Dengue in infants: an overview, FEMS Immunol. Med. Microbiol., 59: 119–130, 2010. [2] R. Nikin-Beers and S. M. Ciupe, The role of antibody in enhancing dengue virus infection. Math. Biosci., 263: 83–92, 2015.

Mazen Saad

Asymptotic behavior of solutions of a nonlinear degenerate chemotaxis model.

Pattern formation in various biological systems has been attributed to Turing instabilities in systems of reaction-diffusion equations. In this paper, a rigorous mathematical description for the pattern dynamics of aggregating regions of biological individuals possessing the property of chemotaxis is presented. We identify a generalized nonlinear degenerate chemotaxis model where a destabilization mechanism may lead to spatially non homogeneous solutions. Given any general perturbation of the solution nearby an homogenous steady state, we prove that its nonlinear evolution is dominated by the corresponding linear dynamics along the finite number of fastest growing modes. The theoretical results are tested against two different numerical results in two dimensions showing an excellent qualitative agreement.

Noureddine Alaa

GENERIC REACTION-DIFFUSION MODEL WITH APPLICATION TO IMAGE RESTORATION AND ENHANCEMENT.

This conference deals with the mathematical analysis of a generic Anisotropic Reaction-Diffusion system. We analyse the global existence of weak solution when the nonlinearities satisfy only quasi-positivity and triangular structure condition. An example of application of our model is demonstrated on a novel bio-inspired image restoration model.

Mauricio Sepúlveda Cortés

A Virtual Element Method for a Nonlocal FitzHugh-Nagumo Model of Cardiac Electrophysiology.

We present a Virtual Element Method (VEM) for a nonlocal reaction-diffusion system of the cardiac electric field. For this system, we analyze an $H^1$-conforming discretization by means of VEM which can make use of general polygonal meshes. Under standard assumptions on the computational domain, we establish the convergence of the discrete solution by considering a series of a priori estimates and by using a general $L^p$ compactness criterion. Moreover, we obtain optimal order space-time error estimates in the $L^2$ norm. Finally, we report some numerical tests supporting the theoretical results.

Noureddine Igbida

On the Eikonal equation and some related problems

The aim of the talk is to give an overview concerning some connections between the Eikonal equation, the sandpile and the optimal mass transport problem. We’ll discuss the continuous as well as the discrete and non local approaches for these problems. Some numerical results will be given at the end of the presentation.

Said Hadd

New results on maximal L^p-regularity for evolutions equations in Banach spaces

In this talk, we first present existing results on maximal regularity mainly due to Lions using the variational approach in Hilbert spaces and Lutz Weis using the operator approach in UMD spaces. In the second part, we present some new results on maximal regularity for perturbed evolution equations in Banach spaces. As an application, we consider Volterra integro-differential equations using Bergman spaces.

Ahmed Youssfi

On some approximation results in Musielak spaces

We present some new approximation results in Musielak spaces. We give suficient conditions for the continuity in norm of shift operator in Musielak spaces LM. An application to the convergence in norm of approximate identities is given, whereby we prove density results of smooth functions in LM, both in modular and norm topologies. These density results are then applied to get some basic topological properties. We also present an investigation about Poincarée-type integral inequalities in the functional Musielak structure. We give conditions on the Musielak functions under which they hold. An identification with null trace functions space is discussed. The talk is based on the references [1, 2]. [1] A. Youssfi and Y. Ahmida. Some approximation results in Musielak-Orlicz spaces. To appear in Czechoslovak Math. J. [2] A. Youssfi and Y. Ahmida. Poincarée-type inequalities in Musielak spaces. To appear in Ann. Acad. Sci. Fenn. Math. Volumen 44, (2019) 1-14