We consider a nonlinear equation in a reflexive Banach space X, governed by a history-dependent operator, over the time interval [0, T], with T > 0 . We prove that, under appropriate assumptions, the equation admits a unique solution u:[0,T] → X . We then provide necessary and sufficient conditions that guarantee the uniform convergence of any arbitrary sequence of continuous functions, defined on [0, T] with values in X, to the solution u. The proof relies on the fixed point structure of the problem. Our results are applicable to a wide range of nonlinear problems. As an example, we consider a mathematical model describing the contact of a viscoelastic body with a foundation. The contact is frictionless and is modeled by a unilateral condition. The weak formulation of the model takes the form of a history-dependent variational inequality for the displacement field. We apply our abstract results to this problem and, in particular, obtain a continuous dependence result.

This paper deals with the study of a special class of systems which couple an implicit differential equation with an evolutionary variational inequality. Following the terminology used in the literature, we refer to such systems as differential variational inequalities (DVI). Inequalities of this form arise in a large number of mathematical models in Solid and Contact Mechanics. We start with a one-dimensional rheological example which leads to a DVI. Inspired by this example, we formulate the problem in the abstract framework of a real Hilbert space X, and then we state and prove an existence and uniqueness result. The proof is based on a result of evolutionary variational inequalities combined with a fixed-point argument for history-dependent operators. We proceed with a convergence criterion, that is, we identify necessary and sufficient conditions which guarantee the convergence of a sequence of elements of X to the solution. Then, we introduce a well-posedness concept for the DVI we study and we state and prove the corresponding well-posedness result. Finally, we apply these abstract results in the study of the one-dimensional rheological model and provide the corresponding mechanical interpretations.

We use a variational approach based on the mountain-pass theorem to prove the existence of weak solutions for nonlinear variable exponent systems with homogeneous Dirichlet boundary conditions. The class of systems discussed here is governed by Leray-Lions type operators which, under sufficiently relaxed hypotheses, are covering as particular cases generalized Laplace, mean curvature, and capillarity operators, as well as various combinations of them. Thus, our main result yields a unified framework for several nonlinear models connected to a large number of potential applications.

We consider a mathematical model, which describes the quasistatic contact process of an elastic-viscoplastic body with a foundation. The process is quasistatic, the contact is bilateral and the friction is modeled with Tresca's friction law. We provide a new variational formulation of the model which is in the form of a system coupling a time-dependent equation for the stress field, a nonlinear integral equation for the transformation strain field, and an evolutionary variational inequality for the displacement field. We use arguments of variational inequalities, fixed point, and various estimates in order to prove unique weak solvability of the model, together with a continuous dependence result of the weak solution with respect to the yield limit. Next, we use a finite element scheme to approximate the problem, based on the primal-dual active set (PDAS) method. We implement it on the computer and provide numerical simulations which validate the theoretical continuous dependence result.

We consider a history-dependent hemivariational inequality in a reflexive Banach space X, stated on the interval of time [0,T] with T>0 and governed by a time-dependent set of constraints. We use arguments of pseudomonotonicity, Mosco convergence and fixed point in order to provide the existence of a unique solution u is an element of C([0,T];X) of the inequality, together with a pointwise convergence result. Next, under additional assumptions, we provide necessary and sufficient conditions which guarantee the uniform convergence of a sequence of functions {u(n)} subset of C([0,T];X) to the solution u. We then introduce and study two well-posedness concepts for the corresponding inequality. Our results give rise to various applications. To provide an example, we illustrate their use in the study of a mathematical model which describes the equilibrium of a viscoelastic rod in contact with a rigid-deformable obstacle, the so-called foundation.

The vaginal microbiota is associated with the health of women and newborns alike. Despite its comparatively simple composition relative to other human microbiota systems, the mechanisms underpinning the dynamics and stability of vaginal microbial communities remain elusive. A crucial, yet so far underexplored, aspect of vaginal microbiota ecology is the role played by nutritional resources. Glycogen and its derivatives, produced by vaginal epithelia, are accessible to all bacterial constituents of the microbiota. Concurrently, free sialic acid and fucose offer supplementary nutritional resources to bacterial strains capable of cleaving them from glycans, which are structurally integral to mucus. Notably, bacteria adept at sialic acid exploitation are often correlated with adverse clinical outcomes and are frequently implicated in bacterial vaginosis (BV). In this study, we introduce a novel mathematical model tailored to human vaginal microbiota dynamics to explore the interactions between bacteria and their respective nutritional landscape. Our resource-based model examines the impact of the relative availability of glycogen derivatives (accessible to all bacterial species) and sialic acid (exclusive to some BV-associated bacteria) on the composition of the vaginal microbiota. Our findings elucidate that the success of BV-associated bacteria is intricately linked to their exclusive access to specific nutritional resources. This private access fortifies communities dominated by BV-associated bacteria, rendering them resilient to compositional transitions. We empirically corroborate our model prediction with longitudinal clinical data on microbiota composition and previously unpublished metabolomic profiles obtained from a North American cohort. The insights gleaned from this study shed light on potential pathways for BV prevention and treatment.

We consider a mathematical model which describes the equilibrium of an elastic body in contact with a rigid-plastic foundation. The contact is frictionless and the memory effects of the foundations are taken into account. The weak formulation of the model is in a form of a history-dependent variational inequality for the displacement field which, under appropriate assumptions on the data, has a unique solution. Here, we are interested on the continuous dependence of the solution with respect to various data and parameters. To this end, we state and prove an abstract convergence criterion for a class of history-dependent variational inequalities in Hilbert spaces. We apply this criterion in the study of our contact problem and derive two convergence results that we present together with their mechanical interpretation. Finally, we use a finite element scheme to approximate the problem, implement it on the computer, and provide numerical simulations which validate the theoretical convergence results.

We consider an inclusion in a real Hilbert space governed by a time-dependent set of constraints and a history-dependent operator. We introduce the concept of T- well-posedness for this inclusion, associated to a given Tykhonov triple T. Next, we provide a T-well-posedness result that we use in order to deduce the continuous dependence of the solution with respect to the data. Then, we state and prove a convergence criterion to the solution of the inclusion that we use to prove a convergence result for an associate penalty problem. Moreover, we show that this criterion allows us to construct a Tykhonov triple T-which give rise to an optimal well-posedness concept for the corresponding inclusion. Finally, we use these abstract results in the study of a nonlinear viscoelastic constitutive law with long memory term and unilateral constraints.

We consider two real Hilbert spaces X and Y and a nonlinear elliptic system governed by two sets of constraints K subset of X and W subset of Y. We prove that, under appropriate assumptions, the system has a unique solution (u, phi) is an element of K x W. Then, we provide necessary and sufficient conditions which guarantee the convergence of an arbitrary sequence (u(n), phi(n)) is an element of X x Y to the solution (u, phi). The proofs are based on standard results on elliptic variational inequalities, various estimates and arguments of convex analysis. Next, we introduce two concepts of well-posedness for the system, compare them, and derive the corresponding well-posedness results. Our results can be applied in the study of various problems arising and Mechanics and Physics. To provide an example we consider a mathematical model which describes the frictionless unilateral contact of a piezoelectric body with an insulated foundation.

We consider a mathematical model which describes the contact of an elastic-viscoplastic body with a foundation. The contact is frictionless and is modeled with normal compliance and unilateral condition. The weak formulation of the model is in a form of a system which couples a variational inequality for the displacement field with an implicit nonlinear equation for the stress field. We prove the unique solvability of the problem by using a fixed point argument for history-dependent operators. Next, we provide necessary and sufficient conditions which guarantee the convergence to the solution of this elastic-viscoplastic contact model. We use these results in order to obtain a continuous dependence result and to introduce two well-posedness concepts for the problem. Finally, we consider a one-dimensional example which allows us to compare the two well-posedness concepts employed.

We consider an evolutionary inclusion associated with a time-dependent convex in an abstract Hilbert space. We recall a unique solvability result obtained based on arguments of nonlinear equations with maximal monotone operators combined with a penalty method. Then, we state and prove two well-posedness results. Next, we provide three examples of such inclusions that arise in mechanics. The first one concerns an elastic–perfectly plastic constitutive law, while the last two examples are mathematical models that describe the equilibrium of an elastic body and an elastic–perfectly plastic body, respectively, in frictional contact with an obstacle. The contact is bilateral and the friction is modeled with the Tresca friction law. We use our abstract results in the study of these examples to provide the convergence of the solution with respect to the data.

Mathematical models tend to oversimplify the biological details of vaccine or infection-derived immunity effectiveness. Yet, epidemiological outcomes may diverge when assuming polarised immunity-individuals are either fully susceptible or completely immune-compared to leaky immunity-where all individuals are partially protected. We explore the differences between the two by taking advantage of a non-Markovian framework, which allows us to explicitly record the 'age' of the immunity and vary its effectiveness accordingly. A basic scenario reveals that leaky immunity leads to a shorter time between reinfections. A more data-driven scenario based on SARS-CoV-2 data finds that leaky immunity yields substantially more reinfections than polarised immunity and a higher number of infected individuals, yet with a lower probability of hospitalisation. Our findings emphasize the critical role of immune memory modelling assumptions, especially for long-term epidemiological dynamics and public health policies.

A key question in evolutionary epidemiology is to determine differences in the conditions that may allow some mutant strains to spread in a population where a resident strain is already circulating. Evolutionary invasion analyses assume that the immunity is long-lasting for previously infected individuals making it difficult to study traits such as immune escape. We relax this last assumption and allow the environment faced by the mutant to fluctuate outside of any epidemiological equilibrium. We introduce an original two-strains non-Markovian model that accounts for realistic immunity waning and cross-immunity, inspired by the case of SARS-CoV-2 variants. We show that mutants with increased contagiousness or with some immune escape abilities are more likely to invade the population. We also show that the timing of the introduction of mutant strain in the population is key because it is associated with the population's immunisation status. Our results underline the importance of immune waning and non-equilibrium dynamics on infectious disease evolution.

Human papillomavirus (HPV) infections drive one in 20 new cancer cases, exerting a particularly high burden on women. Most anogenital HPV infections are cleared in less than two years, but the underlying mechanisms that favour persistence in around 10% of women remain largely unknown. Notwithstanding, it is precisely this information that is crucial for improving treatment, screening, and vaccination strategies. To understand viral and immune dynamics in non-persisting HPV infections, we set up an observational longitudinal cohort study with frequent on-site visits for biological sample collection. We enrolled 189 women aged from 18 to 25 and living in the area of Montpellier (France) between 2016 and 2020. We performed 974 on-site visits for a total of 1,619 months of follow-up. We collected data on virus load, local immune cell populations, local concentrations of cytokines, and circulating antibody titres. Using hierarchical Bayesian statistical modelling to simultaneously analyse the data from 164 HPV infections from 76 participants, we show that in two months after infection, HPV viral load in non-persisting infections reaches a plateau that lasts on average for 13 to 20 months (95% credibility interval) and is then followed by a rapid clearance phase. This first description of the dynamics of HPV infections comes with the identification of immune correlates associated with infection clearance, especially gamma-delta T cells and CXCL10 concentration. A limitation of this study on HPV kinetics is that many infection follow-ups are censored. Furthermore, some immune cell populations are difficult to label because cervical immunity is less well characterised than systemic immunity. These results open new perspectives for understanding the frontier between acute and chronic infections, and for controlling HPV-associated diseases, as well as for research on human cancers of infectious origin. Trial Registration: This trial was registered is registered at ClinicalTrials.gov under the ID NCT02946346. This study has been approved by the Comité de Protection des Personnes (CPP) Sud Méditerranée I (reference number 2016-A00712-49); by the Comité Consultatif sur le Traitement de l'Information en matière de Recherche dans le domaine de la Santé (reference number 16.504); by the Commission Nationale Informatique et Libertés (reference number MMS/ABD/ AR1612278, decision number DR-2016-488), by the Agence Nationale de Sécurité du Médicament et des Produits de Santé (reference 20160072000007).

We consider a class of elliptic variational inequalities in a reflexive Banach space X for which we recall a convergence criterion obtained in [10]. Each inequality P in the class is governed by a set of constraints K and has a unique solution u is an element of K. The criterion provides necessary and sufficient conditions which guarantee that an arbitrary sequence {un} subset of X converges to the solution u. Then, we consider a sequence {Pn} of unconstrained variationalhemivariational inequalities governed by a sequence of parameters {lambda n} subset of R+. We use our criterion to deduce that, if for each n is an element of N the term un represents a solution of Problem Pn, then the sequence {un} converges to u as lambda n -> 0. We apply our abstract results in the study of an elastic frictional contact problem with unilateral constraints and provide the corresponding mechanical interpretations. We also present numerical simulation in the study of a two-dimensional example which represents an evidence of our convergence results.

We consider a history-dependent variational inequality P which models the frictionless contact between a viscoelastic body and a rigid obstacle covered by a layer of soft material. The inequality is expressed in terms of the displacement field, is governed by the data f (related to the applied body forces and surface tractions) and, under appropriate assumptions, it has a unique solution, denoted by u. Our aim in this paper is to perform a sensitivity analysis of the inequality P, including the study of the regularity of the solution operator f bar right arrow u. To this end, we start by proving the equivalence of P with a fixed point problem, denoted by Q (Theorem 2). We then consider an associated optimal control problem, for which we present an existence result (Theorem 6). Then, we prove the directional differentiability of the solution operator and show that the directional derivative at f in direction delta f is characterized by a history-dependent variational inequality with time-dependent constraints (Theorem 14). Finally, we prove two well-posedness results in the study of Problems P and Q, respectively (Theorem 17), and compare the two well-posedness concepts employed.

BACKGROUNDIn France, cervical cancer screening for females aged 30--65 years primarily tests for high-risk (HR) human papillomavirus (HPV) infections.AIMWe aimed to map the prevalence of cervical infections caused by HPV16 and/or 18, or by any of 12 other carcinogenic HPV genotypes and compare prevalence estimates from tests from spontaneous medical visits (opportunistic screening) or the national screening programme (organised screening).METHODSWe extracted data from a large network of biology laboratories, containing all available results from HR HPV tests performed between 1 January 2020 and 30 November 2023 in metropolitan France. A full hierarchical Bayesian model was used to compute spatially resolved expected prevalence maps at the postcode level.RESULTSThe analytic sample contained results of 362,963 HR HPV tests. Among samples positive for HPV16 and/or 18, 2.9% and 3.8% were from organised and opportunistic screening, respectively. Samples positive for other genotypes were 6.9% and 9.4%, respectively.During the last week of the study (week 48 2023), among females aged 30 years, opportunistic screening was associated with a greater expected prevalence of HPV16 and/or 18 and other genotypes in 97.2% and 99.9% of postcodes, respectively. The probability this percentage was lower among females aged 66 years was below 95% for both genotype groups.For organised screening, a pronounced north-west/south-east gradient in infection prevalence was found across France for both genotype groups, with hotspots located at the border with Italy, Spain and Switzerland.CONCLUSIONOpportunistic screening is associated with systematic inflation of HR HPV infection prevalence.

Since winter 2019, SARS-CoV-2 has emerged, spread, and evolved all around the globe. We explore 4 y of evolutionary epidemiology of this virus, ranging from the applied public health challenges to the more conceptual evolutionary biology perspectives. Through this review, we first present the spread and lethality of the infections it causes, starting from its emergence in Wuhan (China) from the initial epidemics all around the world, compare the virus to other betacoronaviruses, focus on its airborne transmission, compare containment strategies (“zero-COVID” vs. “herd immunity”), explain its phylogeographical tracking, underline the importance of natural selection on the epidemics, mention its within-host population dynamics. Finally, we discuss how the pandemic has transformed (or should transform) the surveillance and prevention of viral respiratory infections and identify perspectives for the research on epidemiology of COVID-19.

We consider an elliptic variational inequality with unilateral constraints in a Hilbert space $X$ which, under appropriate assumptions on the data, has a unique solution $u$. We formulate a convergence criterion to the solution $u$, i.e., we provide necessary and sufficient conditions on a sequence $\{u_n\}\subset X$ which guarantee the convergence $u_n\to u$ in the space $X$. Then, we illustrate the use of this criterion to recover well-known convergence results and well-posedness results in the sense of Tykhonov and Levitin-Polyak. We also provide two applications of our results, in the study of a heat transfer problem and an elastic frictionless contact problem, respectively.

BackgroundInformation related to herpes simplex virus 1 and 2 (HSV-1 and 2), varicella-zoster virus (VZV), Epstein-Barr virus (EBV), and cytomegalovirus (CMV) seroprevalence in France is either lacking, incomplete, or outdated, despite their public health burden.MethodWe used routinely collected serological data between 2018 and 2022 to estimate HSV-1, HSV-2, VZV, EBV, and CMV seroprevalence in France. To account for demographic differences between our analytic samples and the French population and get estimates for sparsely sampled districts and age classes, we used a multilevel regression and poststratification approach combined with Bayesian model averaging via stacking weights.ResultsThe observed seroprevalence (number of positive tests/number of tests) were 64.6% (93,294/144,424), 16.9% (24,316/144,159), 93.0% (141,419/152,084), 83.4% (63,199/75, 781), and 49.0% (23,276/47,525), respectively, for HSV-1, HSV-2, VZV, EBV, and CMV. Between 2018 and 2022, France had a model-based average (equal-tailed interval at 95%) expected seroprevalence equal to 61.1% (60.7,61.5), 14.5% (14.2,14.81), 89.5% (89.3,89.8), 85.6% (85.2,86.0), and 50.5% (49.3,51.7), respectively, for HSV-1, HSV-2, VZV, EBV, and CMV infections. We found an almost certain lower expected seroprevalence in Metropolitan France than in overseas territories for all viruses but VZV, for which it was almost certainly greater. The expected seroprevalences were likely greater among females for all viruses.LimitationsOur results relied on the assumption that individuals were sampled at random conditionally to variables used to build the poststratification table.ImplicationsThe analysis highlights spatial and demographic patterns in seroprevalence that should be considered for designing tailored public health policies.