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Authors:Luis Gómez Nava, Thierry Goudon, Fernando Peruani Pages: 1 - 27 Abstract: Mathematical Models and Methods in Applied Sciences, Ahead of Print. A large number of biological systems — from bacteria to sheep — can be described as ensembles of self-propelled agents (active particles) with a complex internal dynamic that controls the agent’s behavior: resting, moving slow, moving fast, feeding, etc. In this study, we assume that such a complex internal dynamic can be described by a Markov chain, which controls the moving direction, speed, and internal state of the agent. We refer to this Markov chain as the Navigation Control System (NCS). Furthermore, we model that agents sense the environment by considering that the transition rates of the NCS depend on local (scalar) measurements of the environment such as e.g. chemical concentrations, light intensity, or temperature. Here, we investigate under which conditions the (asymptotic) behavior of the agents can be reduced to an effective convection–diffusion equation for the density of the agents, providing effective expressions for the drift and diffusion terms. We apply the developed generic framework to a series of specific examples to show that in order to obtain a drift term three necessary conditions should be fulfilled: (i) the NCS should possess two or more internal states, (ii) the NCS transition rates should depend on the agent’s position, and (iii) transition rates should be asymmetric. In addition, we indicate that the sign of the drift term — i.e. whether agents develop a positive or negative chemotactic response — can be changed by modifying the asymmetry of the NCS or by swapping the speed associated to the internal states. The developed theoretical framework paves the way to model a large variety of biological systems and provides a solid proof that chemotactic responses can be developed, counterintuitively, by agents that cannot measure gradients and lack memory as to store past measurements of the environment. Citation: Mathematical Models and Methods in Applied Sciences PubDate: 2021-07-10T07:00:00Z DOI: 10.1142/S0218202521500366

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Authors:Olena Burkovska, Max Gunzburger Pages: 1 - 38 Abstract: Mathematical Models and Methods in Applied Sciences, Ahead of Print. A nonlocal Cahn–Hilliard model with a non-smooth potential of double-well obstacle type that promotes sharp interfaces in the solution is presented. To capture long-range interactions between particles, a nonlocal Ginzburg–Landau energy functional is defined which recovers the classical (local) model as the extent of nonlocal interactions vanish. In contrast to the local Cahn–Hilliard problem that always leads to diffuse interfaces, the proposed nonlocal model can lead to a strict separation into pure phases of the substance. Here, the lack of smoothness of the potential is essential to guarantee the aforementioned sharp-interface property. Mathematically, this introduces additional inequality constraints that, in a weak formulation, lead to a coupled system of variational inequalities which at each time instance can be restated as a constrained optimization problem. We prove the well-posedness and regularity of the semi-discrete and continuous in time weak solutions, and derive the conditions under which pure phases are admitted. Moreover, we develop discretizations of the problem based on finite element methods and implicit–explicit time-stepping methods that can be realized efficiently. Finally, we illustrate our theoretical findings through several numerical experiments in one and two spatial dimensions that highlight the differences in features of local and nonlocal solutions and also the sharp interface properties of the nonlocal model. Citation: Mathematical Models and Methods in Applied Sciences PubDate: 2021-07-05T07:00:00Z DOI: 10.1142/S021820252150038X

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Authors:Hai-Yang Jin, Tian Xiang Pages: 1 - 45 Abstract: Mathematical Models and Methods in Applied Sciences, Ahead of Print. In this work, we rigorously study chemotaxis effect versus haptotaxis effect on boundedness, blow-up and asymptotical behavior of solutions for a chemotaxis-haptotaxis model in 2D settings. It is well-known that the corresponding Keller–Segel chemotaxis-only model possesses a striking feature of critical mass blowup phenomenon, namely, subcritical mass ensures boundedness, whereas, supercritical mass induces the existence of blow-ups. Herein, we show that this critical mass blow-up phenomenon stays almost the same in the full chemotaxis-haptotaxis model and that any global-in-time haptotaxis solution component vanishes exponentially and the other two solution components converge exponentially to that of chemotaxis-only model in a global sense for suitably large chemo-sensitivity and in the usual sense for suitably small chemo-sensitivity. Therefore, haptotaixs is neither good nor bad than chemotaxis, showing negligibility of haptotaxis effect in the underlying chemotaxis-haptotaxis model. Citation: Mathematical Models and Methods in Applied Sciences PubDate: 2021-06-28T07:00:00Z DOI: 10.1142/S0218202521500287

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Authors:G. K. Duong, N. I. Kavallaris, H. Zaag Pages: 1 - 35 Abstract: Mathematical Models and Methods in Applied Sciences, Ahead of Print. In this paper, we provide a thorough investigation of the blowing up behavior induced via diffusion of the solution of the following non-local problem: {∂tu=Δu−u+up(−∫Ωurdr)γin Ω×(0,T),∂u∂ν=0on Γ=∂Ω×(0,T),u(0)=u0, where [math] is a bounded domain in [math] with smooth boundary [math] such problem is derived as the shadow limit of a singular Gierer–Meinhardt system, Kavallaris and Suzuki [On the dynamics of a non-local parabolic equation arising from the Gierer–Meinhardt system, Nonlinearity (2017) 1734–1761; Non-Local Partial Differential Equations for Engineering and Biology: Mathematical Modeling and Analysis, Mathematics for Industry, Vol. 31 (Springer, 2018)]. Under the Turing type condition r p − 1 < N 2 ,γr≠p − 1,p> 1, we construct a solution which blows up in finite time and only at an interior point [math] of [math] i.e. u(x0,t) ∼ (ðœƒ∗)− 1 p−1κ(T − t)−1 p−1, where θ∗:=limt→T(−∫Ωurdr)−γandκ=(p−1)−1p−1. More precisely, we also give a description on the final asymptotic profile at the blowup point u(x,T) ∼ (ðœƒ∗)− 1 p−1 (p − 1)2 8p x − x0 2 ln x − x0 − 1 p−1as x → 0, and thus we unveil the form of the Turing patterns occurring in that case due to driven-diffusion instability. The applied technique for the construction of the preceding blowing up solution mainly relies on the approach developed in [F. Merle and H. Zaag, Reconnection of vortex with the boundary and finite time quenching, Nonlinearity 10 (1997) 1497–1550] and [G. K. Duong and H. Zaag, Profile of a touch-down solution to a nonlocal MEMS model, Math. Models Methods Appl. Sci. 29 (2019) 1279–1348]. Citation: Mathematical Models and Methods in Applied Sciences PubDate: 2021-06-23T07:00:00Z DOI: 10.1142/S0218202521500305

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Authors:Marta D’Elia, Max Gunzburger, Christian Vollmann Pages: 1 - 63 Abstract: Mathematical Models and Methods in Applied Sciences, Ahead of Print. The implementation of finite element methods (FEMs) for nonlocal models with a finite range of interaction poses challenges not faced in the partial differential equations (PDEs) setting. For example, one has to deal with weak forms involving double integrals which lead to discrete systems having higher assembly and solving costs due to possibly much lower sparsity compared to that of FEMs for PDEs. In addition, one may encounter nonsmooth integrands. In many nonlocal models, nonlocal interactions are limited to bounded neighborhoods that are ubiquitously chosen to be Euclidean balls, resulting in the challenge of dealing with intersections of such balls with the finite elements. We focus on developing recipes for the efficient assembly of FEM stiffness matrices and on the choice of quadrature rules for the double integrals that contribute to the assembly efficiency and also posses sufficient accuracy. A major feature of our recipes is the use of approximate balls, e.g. several polygonal approximations of Euclidean balls, that, among other advantages, mitigate the challenge of dealing with ball-element intersections. We provide numerical illustrations of the relative accuracy and efficiency of the several approaches we develop. Citation: Mathematical Models and Methods in Applied Sciences PubDate: 2021-06-19T07:00:00Z DOI: 10.1142/S0218202521500317

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Authors:N. Bellomo, F. Brezzi Pages: 1 - 6 Abstract: Mathematical Models and Methods in Applied Sciences, Ahead of Print. This editorial paper presents the articles published in a special issue devoted to active particle methods applied to modeling, qualitative analysis, and simulation of the collective dynamics of large systems of interacting living entities in science and society. The modeling approach refers to the mathematical tools of behavioral swarms theory and to the kinetic theory of active particles. Applications focus on classical problems of swarms theory, on crowd dynamics related to virus contagion problems, and to multiscale problems related to the derivation of models at a large scale from the mathematical description at the microscopic scale. A critical analysis of the overall contents of the issue is proposed, with the aim to provide a forward look to research perspectives. Citation: Mathematical Models and Methods in Applied Sciences PubDate: 2021-06-17T07:00:00Z DOI: 10.1142/S0218202521020012

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Authors:Ruiwen Shu, Eitan Tadmor Pages: 1 - 25 Abstract: Mathematical Models and Methods in Applied Sciences, Ahead of Print. We investigate the large time behavior of multi-dimensional aggregation equations driven by Newtonian repulsion, and balanced by radial attraction and confinement. In case of Newton repulsion with radial confinement we quantify the algebraic convergence decay rate toward the unique steady state. To this end, we identify a one-parameter family of radial steady states, and prove dimension-dependent decay rate in energy and 2-Wassertein distance, using a comparison with properly selected radial steady states. We also study Newtonian repulsion and radial attraction. When the attraction potential is quadratic it is known to coincide with quadratic confinement. Here, we study the case of perturbed radial quadratic attraction, proving that it still leads to one-parameter family of unique steady states. It is expected that this family to serve for a corresponding comparison argument which yields algebraic convergence toward steady repulsive-attractive solutions. Citation: Mathematical Models and Methods in Applied Sciences PubDate: 2021-06-17T07:00:00Z DOI: 10.1142/S0218202521500263

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Authors:Stanislav Sysala, Jaroslav Haslinger, B. Daya Reddy, Sergey Repin Pages: 1 - 31 Abstract: Mathematical Models and Methods in Applied Sciences, Ahead of Print. This paper is concerned with an abstract inf-sup problem generated by a bilinear Lagrangian and convex constraints. We study the conditions that guarantee no gap between the inf-sup and related sup-inf problems. The key assumption introduced in the paper generalizes the well-known Babuška–Brezzi condition. It is based on an inf-sup condition defined for convex cones in function spaces. We also apply a regularization method convenient for solving the inf-sup problem and derive a computable majorant of the critical (inf-sup) value, which can be used in a posteriori error analysis of numerical results. Results obtained for the abstract problem are applied to continuum mechanics. In particular, examples of limit load problems and similar ones arising in classical plasticity, gradient plasticity and delamination are introduced. Citation: Mathematical Models and Methods in Applied Sciences PubDate: 2021-06-17T07:00:00Z DOI: 10.1142/S0218202521500330

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Authors:Pierluigi Colli, Hector Gomez, Guillermo Lorenzo, Gabriela Marinoschi, Alessandro Reali, Elisabetta Rocca Pages: 1 - 50 Abstract: Mathematical Models and Methods in Applied Sciences, Ahead of Print. Prostate cancer can be lethal in advanced stages, for which chemotherapy may become the only viable therapeutic option. While there is no clear clinical management strategy fitting all patients, cytotoxic chemotherapy with docetaxel is currently regarded as the gold standard. However, tumors may regain activity after treatment conclusion and become resistant to docetaxel. This situation calls for new delivery strategies and drug compounds enabling an improved therapeutic outcome. Combination of docetaxel with antiangiogenic therapy has been considered a promising strategy. Bevacizumab is the most common antiangiogenic drug, but clinical studies have not revealed a clear benefit from its combination with docetaxel. Here, we capitalize on our prior work on mathematical modeling of prostate cancer growth subjected to combined cytotoxic and antiangiogenic therapies, and propose an optimal control framework to robustly compute the drug-independent cytotoxic and antiangiogenic effects enabling an optimal therapeutic control of tumor dynamics. We describe the formulation of the optimal control problem, for which we prove the existence of at least a solution and determine the necessary first-order optimality conditions. We then present numerical algorithms based on isogeometric analysis to run a preliminary simulation study over a single cycle of combined therapy. Our results suggest that only cytotoxic chemotherapy is required to optimize therapeutic performance and we show that our framework can produce superior solutions to combined therapy with docetaxel and bevacizumab. We also illustrate how the optimal drug-naïve cytotoxic effects computed in these simulations may be successfully leveraged to guide drug production and delivery strategies by running a nonlinear least-square fit of protocols involving docetaxel and a new design drug. In the future, we believe that our optimal control framework may contribute to accelerate experimental research on chemotherapeutic drugs for advanced prostate cancer and ultimately provide a means to design and monitor its optimal delivery to patients. Citation: Mathematical Models and Methods in Applied Sciences PubDate: 2021-06-14T07:00:00Z DOI: 10.1142/S0218202521500299

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Authors:Walter Boscheri, Giacomo Dimarco, Lorenzo Pareschi Pages: 1 - 39 Abstract: Mathematical Models and Methods in Applied Sciences, Ahead of Print. In this paper, we propose a novel space-dependent multiscale model for the spread of infectious diseases in a two-dimensional spatial context on realistic geographical scenarios. The model couples a system of kinetic transport equations describing a population of commuters moving on a large scale (extra-urban) with a system of diffusion equations characterizing the non-commuting population acting over a small scale (urban). The modeling approach permits to avoid unrealistic effects of traditional diffusion models in epidemiology, like infinite propagation speed on large scales and mass migration dynamics. A construction based on the transport formalism of kinetic theory allows to give a clear model interpretation to the interactions between infected and susceptible in compartmental space-dependent models. In addition, in a suitable scaling limit, our approach permits to couple the two populations through a consistent diffusion model acting at the urban scale. A discretization of the system based on finite volumes on unstructured grids, combined with an asymptotic preserving method in time, shows that the model is able to describe correctly the main features of the spatial expansion of an epidemic. An application to the initial spread of COVID-19 is finally presented. Citation: Mathematical Models and Methods in Applied Sciences PubDate: 2021-04-29T07:00:00Z DOI: 10.1142/S0218202521400017

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Authors:Seung-Yeal Ha, Shi Jin, Doheon Kim, Dongnam Ko Pages: 1 - 37 Abstract: Mathematical Models and Methods in Applied Sciences, Ahead of Print. We present a uniform-in-time (and in particle numbers as well) error estimate for the random batch method (RBM) [S. Jin, L. Li and J.-G. Liu, Random batch methods (RBM) for interacting particle systems, J. Comput. Phys. 400 (2020) 108877] to the Cucker–Smale (CS) model. The uniform-in-time error estimates of the RBM have been obtained for various interacting particle systems, when corresponding flow generates a contraction semigroup. In this paper, we derive a uniform-in-time error estimate for RBM-approximation to the CS model in which the corresponding flow does not generate contractive semigroup. To derive uniform error estimate, we use asymptotic flocking estimate of the RBM-approximated CS model which yields the decay of relative velocities to zero, at least in the order of [math], while velocities of the original system decay exponentially. Here, [math] is the decay rate of the communication weight with respect to the distance between particles in the CS model. We also provide several numerical simulations to confirm the analytical results. Citation: Mathematical Models and Methods in Applied Sciences PubDate: 2021-04-29T07:00:00Z DOI: 10.1142/S0218202521400029

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Authors:Jeongho Kim, David Poyato, Juan Soler Pages: 1 - 73 Abstract: Mathematical Models and Methods in Applied Sciences, Ahead of Print. In this paper, we present the hydrodynamic limit of a multiscale system describing the dynamics of two populations of agents with alignment interactions and the effect of an internal variable. It consists of a kinetic equation coupled with an Euler-type equation inspired by the thermomechanical Cucker–Smale (TCS) model. We propose a novel drag force for the fluid-particle interaction reminiscent of Stokes’ law. While the macroscopic species is regarded as a self-organized background fluid that affects the kinetic species, the latter is assumed sparse and does not affect the macroscopic dynamics. We propose two hyperbolic scalings, in terms of a strong and weak relaxation regime of the internal variable towards the background population. Under each regime, we prove the rigorous hydrodynamic limit towards a coupled system composed of two Euler-type equations. Inertial effects of momentum and internal variable in the kinetic species disappear for strong relaxation, whereas a nontrivial dynamics for the internal variable appears for weak relaxation. Our analysis covers both the case of Lipschitz and weakly singular influence functions. Citation: Mathematical Models and Methods in Applied Sciences PubDate: 2021-04-29T07:00:00Z DOI: 10.1142/S0218202521400042

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Authors:Nicolás Barnafi, Gabriel N. Gatica, Daniel E. Hurtado, Willian Miranda, Ricardo Ruiz-Baier Pages: 1 - 42 Abstract: Mathematical Models and Methods in Applied Sciences, Ahead of Print. This work introduces and analyzes new primal and dual-mixed finite element methods for deformable image registration, in which the regularizer has a nontrivial kernel, and constructed under minimal assumptions of the registration model: Lipschitz continuity of the similarity measure and ellipticity of the regularizer on the orthogonal complement of its kernel. The aforementioned singularity of the regularizer suggests to modify the original model by incorporating the additional degrees of freedom arising from its kernel, thus granting ellipticity of the former on the whole solution space. In this way, we are able to prove well-posedness of the resulting extended primal and dual-mixed continuous formulations, as well as of the associated Galerkin schemes. A priori error estimates and corresponding rates of convergence are also established for both discrete methods. Finally, we provide numerical examples confronting our formulations with the standard ones, which prove our finite element methods to be particularly more efficient on the registration of translations and rotations, in addition for the dual-mixed approach to be much more suitable for the quasi-incompressible case, and all the above without losing the flexibility to solve problems arising from more realistic scenarios such as the image registration of the human brain. Citation: Mathematical Models and Methods in Applied Sciences PubDate: 2021-04-29T07:00:00Z DOI: 10.1142/S021820252150024X

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Authors:Benedetto Piccoli, Francesco Rossi Pages: 1 - 40 Abstract: Mathematical Models and Methods in Applied Sciences, Ahead of Print. Bounded-confidence models in social dynamics describe multi-agent systems, where each individual interacts only locally with others. Several models are written as systems of ordinary differential equations (ODEs) with discontinuous right-hand side: this is a direct consequence of restricting interactions to a bounded region with non-vanishing strength at the boundary. Various works in the literature analyzed properties of solutions, such as barycenter invariance and clustering. On the other side, the problem of giving a precise definition of solution, from an analytical point of view, was often overlooked. However, a rich literature proposing different concepts of solution to discontinuous differential equations is available. Using several concepts of solution, we show how existence is granted under general assumptions, while uniqueness may fail even in dimension one, but holds for almost every initial conditions. Consequently, various properties of solutions depend on the useddefinition and initial conditions. Citation: Mathematical Models and Methods in Applied Sciences PubDate: 2021-04-22T07:00:00Z DOI: 10.1142/S0218202521400054

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Authors:Daewa Kim, Kaylie O’Connell, William Ott, Annalisa Quaini Pages: 1 - 26 Abstract: Mathematical Models and Methods in Applied Sciences, Ahead of Print. In this paper, we present a computational modeling approach for the dynamics of human crowds, where the spreading of an emotion (specifically fear) has an influence on the pedestrians’ behavior. Our approach is based on the methods of the kinetic theory of active particles. The model allows us to weight between two competing behaviors depending on fear level: the search for less congested areas and the tendency to follow the stream unconsciously (herding). The fear level of each pedestrian influences their walking speed and is influenced by the fear levels of their neighbors. Numerically, we solve our pedestrian model with emotional contagion using an operator splitting scheme. We simulate evacuation scenarios involving two groups of interacting pedestrians to assess how domain geometry and the details of fear propagation impact evacuation dynamics. Further, we reproduce the evacuation dynamics of an experimental study involving distressed ants. Citation: Mathematical Models and Methods in Applied Sciences PubDate: 2021-04-17T07:00:00Z DOI: 10.1142/S0218202521400030

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Authors:Franco Flandoli, Eleonora La Fauci, Martina Riva Pages: 1 - 33 Abstract: Mathematical Models and Methods in Applied Sciences, Ahead of Print. A Markov chain individual-based model for virus diffusion is investigated. Both the virus growth within an individual and the complexity of the contagion within a population are taken into account. A careful work of parameter choice is performed. The model captures very well the statistical variability of quantities like the incubation period, the serial interval and the time series of infected people in Tuscany towns. Citation: Mathematical Models and Methods in Applied Sciences PubDate: 2021-04-17T07:00:00Z DOI: 10.1142/S0218202521500226

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Authors:Xiaoming Fu, Quentin Griette, Pierre Magal Pages: 1 - 45 Abstract: Mathematical Models and Methods in Applied Sciences, Ahead of Print. In this work, we describe a hyperbolic model with cell–cell repulsion with a dynamics in the population of cells. More precisely, we consider a population of cells producing a field (which we call “pressure”) which induces a motion of the cells following the opposite of the gradient. The field indicates the local density of population and we assume that cells try to avoid crowded areas and prefer locally empty spaces which are far away from the carrying capacity. We analyze the well-posedness property of the associated Cauchy problem on the real line. We start from bounded initial conditions and we consider some invariant properties of the initial conditions such as the continuity, smoothness and monotony. We also describe in detail the behavior of the level sets near the propagating boundary of the solution and we find that an asymptotic jump is formed on the solution for a natural class of initial conditions. Finally, we prove the existence of sharp traveling waves for this model, which are particular solutions traveling at a constant speed, and argue that sharp traveling waves are necessarily discontinuous. This analysis is confirmed by numerical simulations of the PDE problem. Citation: Mathematical Models and Methods in Applied Sciences PubDate: 2021-04-13T07:00:00Z DOI: 10.1142/S0218202521500214

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Authors:Parveena Shamim Abdul Salam, Wolfgang Bock, Axel Klar, Sudarshan Tiwari Pages: 1 - 19 Abstract: Mathematical Models and Methods in Applied Sciences, Ahead of Print. Modeling and simulation of disease spreading in pedestrian crowds have recently become a topic of increasing relevance. In this paper, we consider the influence of the crowd motion in a complex dynamical environment on the course of infection of the pedestrians. To model the pedestrian dynamics, we consider a kinetic equation for multi-group pedestrian flow based on a social force model coupled with an Eikonal equation. This model is coupled with a non-local SEIS contagion model for disease spread, where besides the description of local contacts, the influence of contact times has also been modeled. Hydrodynamic approximations of the coupled system are derived. Finally, simulations of the hydrodynamic model are carried out using a mesh-free particle method. Different numerical test cases are investigated, including uni- and bi-directional flow in a passage with and without obstacles. Citation: Mathematical Models and Methods in Applied Sciences PubDate: 2021-04-09T07:00:00Z DOI: 10.1142/S0218202521400066

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Authors:Guoqiang Ren, Bin Liu Pages: 1 - 38 Abstract: Mathematical Models and Methods in Applied Sciences, Ahead of Print. In this work, we consider the two-species chemotaxis system with Lotka–Volterra competitive kinetics in a bounded domain with smooth boundary. We construct weak solutions and prove that they become smooth after some waiting time. In addition, the asymptotic behavior of the solutions is studied. Our results generalize some well-known results in the literature. Citation: Mathematical Models and Methods in Applied Sciences PubDate: 2021-04-09T07:00:00Z DOI: 10.1142/S0218202521500238

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Authors:Luis Almeida, Pierre-Alexandre Bliman, Grégoire Nadin, Benoît Perthame, Nicolas Vauchelet Pages: 1 - 31 Abstract: Mathematical Models and Methods in Applied Sciences, Ahead of Print. We formulate a general SEIR epidemic model in a heterogeneous population characterized by some trait in a discrete or continuous subset of a space [math]. The incubation and recovery rates governing the evolution of each homogeneous 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 of this work 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 the presence of a diffusion term. They generalize previous works corresponding to finite number of traits (including metapopulation models) or to rank 1 contact matrices (modeling e.g. susceptibility or infectivity presenting heterogeneity independently of one another). Citation: Mathematical Models and Methods in Applied Sciences PubDate: 2021-04-09T07:00:00Z DOI: 10.1142/S0218202521500251