Identifiability problem for recovering the mortality rate in an age-structured population dynamics model

In this article is studied the identifiability of the age-dependent mortality rate of the Von Foerster-Mc Kendrick model, from the observation of a given age group of the population. In the case where there is no renewal for the population, translated by an additional homogeneous boundary condition to the Von Foerster equation, we give a necessary and sufficient condition on the initial density that ensures the mortality rate identifiability. In the inhomogeneous case, modeled by a non local boundary condition, we make explicit a sufficicent condition for the identifiability property, and give a condition for which the identifiability problem is ill-posed. We illustrate this latter case with numercial simulation

Infection load structured SI model with exponential velocity and external source of contamination

International audienceA mathematical SI model is developed for the dynamics of a contagious disease in a closed population with an external source of contamination. We prove existence and uniqueness of a non-negative mild solution of the problem using semigroup theory. We finally illustrate the model with numerical simulations

Criterion of positivity for semilinear problems with applications in biology

International audienceThe goal of this article is to provide an useful criterion of positivity and well-posedness for a wide range of infinite dimensional semilinear abstract Cauchy problems. This criterion is based on some weak assumptions on the non-linear part of the semilinear problem and on the existence of a strongly continuous semigroup generated by the differential operator. To illustrate a large variety of applications, we exhibit the feasibility of this criterion through three examples in mathematical biology: epidemiology, predator-prey interactions and oncology

Asymptotic behavior and numerical simulations for an infection load-structured epidemiological model; Application to the transmission of prion pathologies

In this article is studied an infection load-structured SI model with exponential growth of the infection, that incorporates a potential external source of contamination. We perform the analysis of the time asymptotic behavior of the solution by exhibiting epidemiological thresholds, such as the basic reproduction number, that ensure extinction or persistence of the disease in the contagion process. Moreover, a numerical scheme adapted to the model is developped and analyzed. This scheme is then used to illustrate the model with simulations, applying this last to the transmission of prion pathologies

Predicting the Evolution of Gene ura3 in the Yeast Saccharomyces Cerevisiae

International audienceSince the late '60s, various genome evolutionary models have been proposed to predict the evolution of a DNA sequence as the generations pass. Essentially, two main categories of such models can be found in the literature. The first one, based on nucleotides evolution, uses a mutation matrix of size 4x4. It encompasses for instance the well-known models of Jukes and Cantor, Kimura, and Tamura. In the second category, exclusively studied by Bahi and Michel, the evolution of trinucleotides is studied through a matrix of size 64x64. By essence, all of these models relate the evolution of DNA sequences to the computation of the successive powers of a mutation matrix. To make this computation possible, particular forms for the mutation matrix are assumed, which are not compatible with mutation rates that have been recently obtained experimentally on gene $ura3$ of the Yeast \textit{Saccharomyces cerevisiae}. Using this experimental study, authors of this paper have deduced a simple mutation matrice, compute the future evolution of the rate purine/pyrimidine for ura3, investigate the particular behavior of cytosines and thymines compared to purines, and simulate the evolution of each nucleotide

Asymptotic behavior of age-structured and delayed Lotka-Volterra models

In this work we investigate some asymptotic properties of an age-structured Lotka-Volterra model, where a specific choice of the functional parameters allows us to formulate it as a delayed problem, for which we prove the existence of a unique coexistence equilibrium and characterize the existence of a periodic solution. We also exhibit a Lyapunov functional that enables us to reduce the attractive set to either the nontrivial equilibrium or to a periodic solution. We then prove the asymptotic stability of the nontrivial equilibrium where, depending on the existence of the periodic trajectory, we make explicit the basin of attraction of the equilibrium. Finally, we prove that these results can be extended to the initial PDE problem.Comment: 29 page

Relaxing the Hypotheses of Symmetry and Time-Reversibility in Genome Evolutionary Models

International audienceVarious genome evolutionary models have been proposed these last decades to predict the evolution of a DNA sequence over time, essentially described using a mutation matrix. By essence, all of these models relate the evolution of DNA sequences to the computation of the successive powers of the mutation matrix. To make this computation possible, hypotheses are assumed for the matrix, such as symmetry and time-reversibility, which are not compatible with mutation rates that have been recently obtained experimentally on genes ura3 and can1 of the Yeast Saccharomyces cerevisiae. In this work, authors investigate systematically the possibility to relax either the symmetry or the time-reversibility hypothesis of the mutation matrix, by investigating all the possible matrices of size 2 and 3. As an application example, the experimental study on the Yeast Saccharomyces cerevisiae has been used in order to deduce a simple mutation matrix, and to compute the future evolution of the rate purine/pyrimidine for ura3 on the one hand, and of the particular behavior of cytosines and thymines compared to purines on the other hand

Identifiabilité de paramètres pour des systèmes décrits par des équations aux dérivées partielles. Application à la dynamique des populations

This thesis aims at studying the identifiability of an epidemiological model described by semilinear integro-differential partial differential equations (PDE) of reaction-transport type. To achieve this goal, we start with a literature survey of inverse problems devoted to parameter identifiability. We study the mathematical bases of the existing techniques, outlining the systems to which they are applied or could be extended. In finite dimension, the three main methods for systems of ordinary differential equations are based on: Taylor series expansion, algebro-differential elimination, and the state isomorphism theorem. In infinite dimension, for PDE systems, two methods are generally applied in the linear case: a spectral approach, and another one based on Carleman estimates. The latter is also applied to some semilinear PDE systems in particular situations where the identifiability problem amounts to studying a linear system. However, this method cannot be, or can hardly be applied to our system owing to the complexity of its nonlinearity. We then perform the identifiability analysis of the epidemiological model. We first build a formal identifiability framework adapted to semilinear PDE systems. This framework requires that a solution space is defined for the PDE problem. Therefore, we determine a functional framework compatible with the biological conditions imposed by the model and prove the existence and uniqueness of the solution. Second, we perform the identifiability analysis of the model by adapting the algebro-differential elimination method. We obtain sufficient identifiability conditions for given parameter classes. We finally discuss and interpret the results we obtain, and provide numerical simulations.L'objectif de cette thèse est d'effectuer une étude d'identifiabilité d'un modèle épidémiologique décrit par un système d'équations aux dérivées partielles (EDP) intégro-différentiel semi-linéaire de type réaction-transport. Dans ce but, nous effectuons tout d'abord une synthèse de la littérature relative aux problèmes inverses d'identifiabilité paramétrique. Nous étudions les fondements mathématiques des différentes techniques employées, en mettant en avant les natures des systèmes auxquels ces méthodes s'appliquent ou se généralisent. En dimension finie, trois méthodes se dégagent pour les systèmes d'équations différentielles ordinaires : par développement en série de Taylor, par élimination algébro-différentielle et par le biais du théorème de l'isomorphisme d'état. En dimension infinie, pour les systèmes d'EDP, deux méthodes sont couramment utilisées dans le cas linéaire : une approche spectrale et une autre reposant sur les inégalités de Carleman. Cette dernière est aussi appliquée à quelques systèmes d'EDP semi-linéaires, dans des cas particuliers où le problème d'identifiabilité peut se ramener à l'étude d'un système linéaire. Cependant, cette méthode n'est pas, ou alors difficilement, applicable à notre système du fait de la complexité de sa non-linéarité. Dans un deuxième temps, nous effectuons l'analyse d'identifiabilité du modèle épidémiologique. Nous commençons par bâtir un cadre formel d'étude d'identifiabilité s'appliquant aux systèmes d'EDP semi-linéaires. Ce cadre nécessite la connaissance d'un espace de vie de la solution du problème d'EDP. En conséquence, nous déterminons un cadre fonctionnel respectant les conditions biologiques imposées par le modèle, puis nous prouvons existence et unicité de la solution. Nous effectuons ensuite l'analyse d'identifiabilité du modèle en adaptant la méthode d'élimination algébro-différentielle. Nous obtenons des conditions suffisantes d'identifiabilité pour des classes de paramètres données. Nous discutons, interprétons et simulons numériquement les résultats obtenus

Parameter identifiability for systems described by partial differential equations. Application to population dynamics

L'objectif de cette thèse est d'effectuer une étude d'identifiabilité d'un modèle épidémiologique décrit par un système d'équations aux dérivées partielles (EDP) intégro-différentiel semi-linéaire de type réaction-transport. Dans ce but, nous effectuons tout d'abord une synthèse de la littérature relative aux problèmes inverses d'identifiabilité paramétrique. Nous étudions les fondements mathématiques des différentes techniques employées, en mettant en avant les natures des systèmes auxquels ces méthodes s'appliquent ou se généralisent. En dimension finie, trois méthodes se dégagent pour les systèmes d'équations différentielles ordinaires : par développement en série de Taylor, par élimination algébro-différentielle et par le biais du théorème de l'isomorphisme d'état. En dimension infinie, pour les systèmes d'EDP, deux méthodes sont couramment utilisées dans le cas linéaire : une approche spectrale et une autre reposant sur les inégalités de Carleman. Cette dernière est aussi appliquée à quelques systèmes d'EDP semi-linéaires, dans des cas particuliers où le problème d'identifiabilité peut se ramener à l'étude d'un système linéaire. Cependant, cette méthode n'est pas, ou alors difficilement, applicable à notre système du fait de la complexité de sa non-linéarité. Dans un deuxième temps, nous effectuons l'analyse d'identifiabilité du modèle épidémiologique. Nous commençons par bâtir un cadre formel d'étude d'identifiabilité s'appliquant aux systèmes d'EDP semi-linéaires. Ce cadre nécessite la connaissance d'un espace de vie de la solution du problème d'EDP. En conséquence, nous déterminons un cadre fonctionnel respectant les conditions biologiques imposées par le modèle, puis nous prouvons existence et unicité de la solution. Nous effectuons ensuite l'analyse d'identifiabilité du modèle en adaptant la méthode d'élimination algébro-différentielle. Nous obtenons des conditions suffisantes d'identifiabilité pour des classes de paramètres données. Nous discutons, interprétons et simulons numériquement les résultats obtenus.This thesis aims at studying the identifiability of an epidemiological model described by semilinear integro-differential partial differential equations (PDE) of reaction-transport type. To achieve this goal, we start with a literature survey of inverse problems devoted to parameter identifiability. We study the mathematical bases of the existing techniques, outlining the systems to which they are applied or could be extended. In finite dimension, the three main methods for systems of ordinary differential equations are based on: Taylor series expansion, algebro-differential elimination, and the state isomorphism theorem. In infinite dimension, for PDE systems, two methods are generally applied in the linear case: a spectral approach, and another one based on Carleman estimates. The latter is also applied to some semilinear PDE systems in particular situations where the identifiability problem amounts to studying a linear system. However, this method cannot be, or can hardly be applied to our system owing to the complexity of its nonlinearity. We then perform the identifiability analysis of the epidemiological model. We first build a formal identifiability framework adapted to semilinear PDE systems. This framework requires that a solution space is defined for the PDE problem. Therefore, we determine a functional framework compatible with the biological conditions imposed by the model and prove the existence and uniqueness of the solution. Second, we perform the identifiability analysis of the model by adapting the algebro-differential elimination method. We obtain sufficient identifiability conditions for given parameter classes. We finally discuss and interpret the results we obtain, and provide numerical simulations

An Introduction to The Basic Reproduction Number in Mathematical Epidemiology

This article introduces the notion of basic reproduction number R0 in mathematical epi-demiology. After an historic reminder describing the steps leading to the statement of its mathematical definition, we explain the next-generation matrix method allowing its calculation in the case of epidemic models described by ordinary differential equations (ODEs). The article then focuses, through four ODEs examples and an infection load structured PDE model, on the usefulness of the R0 to address biological as well mathematical issues