arXiv subjects

Nicolas Vauchelet

Publications and source records attributed to Nicolas Vauchelet.

At least 19 recordsLinked to original sources

arXiv2026

Optimal Dirac controls for time-periodic bistable ODEs, application to population replacement

This work addresses an optimal control problem on a dynamics governed by a nonlinear differential equation with a bistable time-periodic nonlinearity. This problem, relevant in population dynamics, models the strategy of replacing a population of A-type individuals by a population of B-type individuals in a time-varying environment, focusing on the evolution of the proportion of B-type individuals among the whole population. The control term accounts for the instant release of B-type individuals. Our main goal, after noting some interesting properties on the differential equation, is to determine the optimal time at which this release should be operated to ensure population replacement while minimizing the release effort. The results establish that the optimal release time appears to be the minimizer of a function involving the carrying capacity of the environment and the threshold periodic solution of the dynamics; they also describe the convergence of the whole optimal release strategy. An application to the biocontrol of mosquito populations using Wolbachia-infected individuals illustrates the relevance of the theoretical results. Wolbachia is a bacterium that helps preventing the transmission of some viruses from mosquitoes to humans, making the optimization of Wolbachia propagation in a mosquito population a crucial issue.

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arXiv2025

Rolling carpet strategy to reduce mosquito populations in two-dimensional space

Mosquitoes are vectors of numerous diseases; a strategy to fight the spread of these diseases is to control the vector population. In this article, we focus on the use of the sterile insect technique. Starting from a reaction-diffusion system, we show the existence of 'forced' traveling waves obtained by translating the intervention zone at constant speed. This result is proved in a two-dimensional space by using the radial symmetry.

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arXiv2025

On a reaction-diffusion system modeling strong competition between two mosquito populations

This paper is devoted to the analysis of a reaction-diffusion system with strong competition and spatial heterogeneities modelling the interaction between two species of mosquitoes. In particular, we propose a mathematical model that accounts for the spatial segregation observed between two species of mosquito vectors of numerous viruses. Indeed, it has been observed that, in tropical regions, Aedes aegypti mosquitoes are well established in urban areas whereas Aedes albopictus mosquitoes spread widely in forest regions. Moreover, these species of mosquitoes compete with each other in the larval stage. Based on these observations, we introduce a simple mathematical model to account for this phenomenon. This model consists of a system of reaction-diffusion equations describing the dynamics of the aquatic and aerial phases of each species in a spatially heterogeneous environment. The competition takes place at the aquatic phase and is assumed to be strong which allows us to reduce the dimensionality of the system. We first establish a sufficient condition on the parameters to prevent one species from invading another in a homogeneous environment. Next, using this sufficient condition, we show that spatial segregation may be observed in a spatially heterogeneous environment. Our theoretical results are also illustrated by some numerical simulations.

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arXiv2025

The Filippov characteristic flow for the aggregation equation with mildly singular potentials

Existence and uniqueness of global in time measure solution for the multidimensional aggregation equation is analyzed. Such a system can be written as a continuity equation with a velocity field computed through a self-consistent interaction potential. In Carrillo et al. (Duke Math J (2011)), a well-posedness theory based on the geometric approach of gradient flows in measure metric spaces has been developed for mildly singular potentials at the origin under the basic assumption of being lambda-convex. We propose here an alternative method using classical tools from PDEs. We show the existence of a characteristic flow based on Filippov's theory of discontinuous dynamical systems such that the weak measure solution is the pushforward measure with this flow. Uniqueness is obtained thanks to a contraction argument in transport distances using the lambda-convexity of the potential. Moreover, we show the equivalence of this solution with the gradient flow solution. Finally, we show the convergence of a numerical scheme for general measure solutions in this framework allowing for the simulation of solutions for initial smooth densities after their first blow-up time in Lp-norms.

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arXiv2025

Analysis of a toy model for optimal crop protection

In this paper we investigate an optimal control problem involving a toy model for the protection on a crop field. Precisely, we consider a protection on a crop field and we want to place intervention zones represented by a control, in order to maximise the protection on the field during a given period. Using a relaxation method, we prove that there exists a control which maximises the protection and, moreover, it must be a bang-bang control. Furthermore, with additional assumptions on the crop field geometry, some results on the shape of the optimal intervention are proved using comparison results for elliptic equations via Schwarz and Steiner symmetrizations. Finally, some numerical simulations are performed in order to illustrate those results.

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arXiv2025

Optimal strategies for Wolbachia mosquito replacement technique: influence of the carrying capacity on spatial releases

This work is devoted to the mathematical study of an optimization problem regarding control strategies of mosquito population in a heterogeneous environment. Mosquitoes are well-known vectors of diseases. For some diseases, such as dengue, it has been found that mosquitoes have a reduced vector capacity when carrying the endosymbiotic bacterium Wolbachia. We consider a mathematical model of a replacement technique consisting in rearing and releasing Wolbachia-infected mosquitoes to replace the wild population. Our goal is to optimize the release protocol to maximize replacement effectiveness in a spatially inhomogeneous environment. Using a scalar model with space-dependent carrying capacity, we explore the existence and properties of an optimal release profile maximizing the replacement across the domain. In particular, neglecting mosquito mobility and under some assumptions on the biological parameters, we characterize the optimal releasing strategy for a short time horizon, and we reduce the case of a long time horizon to a one-dimensional optimization problem. Our theoretical results are illustrated with several numerical simulations.

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arXiv2024

Efficacy of the Sterile Insect Technique in the presence of inaccessible areas: A study using two-patch models

The Sterile Insect Technique (SIT) is one of the sustainable strategies for the control of disease vectors, which consists of releasing sterilized males that will mate with the wild females, resulting in a reduction and, eventually a local elimination, of the wild population. The implementation of the SIT in the field can become problematic when there are inaccessible areas where the release of sterile insects cannot be carried out directly, and the migration of wild insects from these areas to the treated zone may influence the efficacy of this technique. However, we can also take advantage of the movement of sterile individuals to control the wild population in these unreachable places. In this paper, we derive a two-patch model for Aedes mosquitoes where we consider the discrete diffusion between the treated area and the inaccessible zone. We investigate two different release strategies (constant and impulsive periodic releases), and by using the monotonicity of the model, we show that if the number of released sterile males exceeds some threshold, the technique succeeds in driving the whole population in both areas to extinction. This threshold depends on not only the biological parameters of the population but also the diffusion between the two patches.

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arXiv2024

Influence of the competition in the spatial dynamics of a population of Aedes mosquitoes

In this article, we investigate a competitive reaction-diffusion system modelling the interaction between several species of mosquitoes. In particular, it has been observed that in tropical regions, Aedes aegypti mosquitoes are well established in urban area whereas Aedes albopictus mosquitoes spread widely in forest region. The aim of this paper is to propose a simple mathematical system modeling this segregation phenomenon. Moreprecisely, after modeling the dynamics by a competitive reaction-diffusion system with spatial heterogeneity, we prove that when there is a strong competition between these two species of mosquitoes, solutions to this system converge in long time to segregated stationary solutions. Then we study the influence of this strong competition on the success of a population replacement strategy using Wolbachia bacteria. Our theoretical results are also illustrated by some numerical simulations.

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arXiv2023

Final size and convergence rate for an epidemic in heterogeneous population

We formulate a general SEIR epidemic model in a heterogenous population characterized by some trait in a discrete or continuous subset of a space R d. The incubation and recovery rates governing the evolution of each homogenous subpopulation depend upon this trait, and no restriction is assumed on the contact matrix that defines the probability for an individual of a given trait to be infected by an individual with another trait. Our goal is to derive and study the final size equation fulfilled by the limit distribution of the population. We show that this limit exists and satisfies the final size equation. The main contribution is to prove the uniqueness of this solution among the distributions smaller than the initial condition. We also establish that the dominant eigenvalue of the next-generation operator (whose initial value is equal to the basic reproduction number) decreases along every trajectory until a limit smaller than 1. The results are shown to remain valid in presence of diffusion term. They generalize previous works corresponding to finite number of traits (including metapopulation models) or to rank 1 contact matrix (modeling e.g. susceptibility or infectivity presenting heterogeneity independently of one another).

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arXiv2022

Steady-state solutions for a reaction-diffusion equation with Robin boundary conditions: Application to the control of dengue vectors

In this paper, we investigate an initial-boundary-value problem of a reaction-diffusion equation in a bounded domain with a Robin boundary condition and introduce some particular parameters to consider the non-zero flux on the boundary. This problem arises in the study of mosquito populations under the intervention of the population replacement method, where the boundary condition takes into account the inflow and outflow of individuals through the boundary. Using phase-plane analysis, the present paper studies the existence and properties of non-constant steady-state solutions depending on several parameters. Then, we use the principle of linearized stability to prove some sufficient conditions for their stability. We show that the long-time efficiency of this control method depends strongly on the size of the treated zone and the migration rate. To illustrate these theoretical results, we provide some numerical simulations in the framework of mosquito population control.

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arXiv2021

Optimal control strategies for the sterile mosquitoes technique

Mosquitoes are responsible for the transmission of many diseases such as dengue fever, zika or chigungunya. One way to control the spread of these diseases is to use the sterile insect technique (SIT), which consists in a massive release of sterilized male mosquitoes. This strategy aims at reducing the total population over time, and has the advantage being specific to the targeted species, unlike the use of pesticides.In this article, we study the optimal release strategies in order to maximize the efficiency of this technique.We consider simplified models that describe the dynamics of eggs, males, females and sterile males in order to optimize the release protocol. We determine in a precise way optimal strategies, which allows us to tackle numerically the underlying optimization problem in a very simple way. We also present some numerical simulations to illustrate our results.

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arXiv2021

Optimization of spatial control strategies for population replacement, application to Wolbachia

In this article, we are interested in the analysis and simulation of solutions to an optimal control problem motivated by population dynamics issues. In order to control the spread of mosquito-borne arboviruses, the population replacement technique consists in releasing into the environment mosquitoes infected with the Wolbachia bacterium, which greatly reduces the transmission of the virus to the humans. Spatial releases are then sought in such a way that the infected mosquito population invades the uninfected mosquito population. Assuming very high mosquito fecundity rates, we first introduce an asymptotic model on the proportion of infected mosquitoes and then an optimal control problem to determine the best spatial strategy to achieve these releases. We then analyze this problem, including the optimality of natural candidates and carry out first numerical simulations in one dimension of space to illustrate the relevance of our approach.

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arXiv2021

Analysis of the "Rolling carpet" strategy to eradicate an invasive species

In order to prevent the propagation of human diseases transmitted by mosquitoes (such as dengue or zika), one possible solution is to act directly on the mosquito population. In this work, we consider an invasive species (the mosquitoes) and we study two strategies to eradicate the population in the whole space by a local intervention. The dynamics of the population is modeled through a bistable reaction diffusion equation in an one dimensional setting and both strategies are based on the same idea : we act on a moving interval. The action of the first strategy is to kill as many individuals as we can in this moving interval. The action of the second strategy is to release sterile males in this moving interval. For both strategies, we manage to generate traveling waves that propagate in the opposite direction relative to the one of the natural invasive traveling wave. These cases correspond to succeeding in eradicating the invasive species. Furthermore, for the first strategy, we fully characterize the minimal size of the interval. All the results are illustrated by numerical simulations.

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arXiv2021

Relaxation limit of the aggregation equation with pointy potential

This work is devoted to the study of a relaxation limit of the so-called aggregation equation with a pointy potential in one dimensional space. The aggregation equation is by now widely used to model the dynamics of a density of individuals attracting each other through a potential. When this potential is pointy, solutions are known to blow up in final time. For this reason, measure-valued solutions have been defined. In this paper, we investigate an approximation of such measure-valued solutions thanks to a relaxation limit in the spirit of Jin and Xin. We study the convergence of this approximation and give a rigorous estimate of the speed of convergence in one dimension with the Newtonian potential. We also investigate the numerical discretization of this relaxation limit by uniformly accurate schemes.

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arXiv2020

Numerical scheme for kinetic transport equation with internal state *

We investigate the numerical discretization of a two-stream kinetic system with an internal state, such system has been introduced to model the motion of cells by chemotaxis. This internal state models the intracellular methylation level. It adds a variable in the mathematical model, which makes it more challenging to simulate numerically. Moreover, it has been shown that the macroscopic or mesoscopic quantities computed from this system converge to the Keller-Segel system at diffusive scaling or to the velocity-jump kinetic system for chemotaxis at hyperbolic scaling. Then we pay attention to propose numerical schemes uniformly accurate with respect to the scaling parameter. We show that these schemes converge to some limiting schemes which are consistent with the limiting macroscopic or kinetic system. This study is illustrated with some numerical simulations and comparisons with Monte Carlo simulations.

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arXiv2020

Optimal immunity control by social distancing for the SIR epidemic model

Until a vaccine or therapy is found against the SARS-CoV-2 coronavirus, reaching herd immunity appears to be the only mid-term option. However, if the number of infected individuals decreases and eventually fades only beyond this threshold, a significant proportion of susceptible may still be infected until the epidemic is over. A containment strategy is likely the best policy in the worst case where no vaccine or therapy is found. In order to keep the number of newly infected persons to a minimum, a possible strategy is to apply strict containment measures, so that the number of susceptible individuals remains close to herd immunity. Such an action is unrealistic since containment can only last for a finite amount of time and is never total. In this article, using a classical SIR model, we determine the (partial or total) containment strategy on a given finite time interval that maximizes the number of susceptible individuals over an infinite horizon, or equivalently that minimizes the total infection burden during the curse of the epidemic. The existence and uniqueness of the optimal strategy is proved and the latter is fully characterized. If applicable in practice, such a strategy would lead theoretically to an increase by 30% of the proportion of susceptible on an infinite horizon, for a containment level corresponding to the sanitary measures put in place in France from March to May 2020. We also analyze the minimum intervention time to reach a fixed distance from herd immunity, and show the relationship with the previous problem. Simulations are provided that illustrate and validate the theoretical results.

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arXiv2020

The sterile insect technique used as a barrier control against reinfestation

The sterile insect technique consists in massive release of sterilized males in the aim to reduce the size of mosquitoes population or even eradicate it. In this work, we investigate the feasability of using the sterile insect technique as a barrier against reinvasion. More precisely, we provide some numerical simulations and mathematical results showing that performing the sterile insect technique on a band large enough may stop reinvasion.

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arXiv2020

Reduction of a model for sodium exchanges in kidney nephron

This work deals with a mathematical analysis of sodium's transport in a tubular architecture of a kidney nephron. The nephron is modelled by two counter-current tubules. Ionic exchange occurs at the interface between the tubules and the epithelium and between the epithelium and the surrounding environment (interstitium). From a mathematical point of view, this model consists of a 5x5 semi-linear hyperbolic system. In the literature similar models neglect the epithelial layers. In this paper, we show rigorously that such models may be obtained by assuming that the permeabilities between lumen and epithelium are large. Indeed we show that when these grow, solutions of the 5x5 system converge in a certain way to solutions of a reduced 3x3 system where no epithelial layer is present. The problem is dened on a bounded spacial domain with initial and boundary data. Establishing BV compactness forces to introduce initial layers and to handle carefully the presence of lateral boundaries.

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