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To fight against Ebola virus disease, several measures have been adopted. Among them, isolation, safe burial and vaccination occupy a prominent place. In this paper, we present a model which takes into account these three control strategies as well as the indirect transmission through a polluted environment. The asymptotic behavior of our model is achieved. Namely, we determine a threshold value [Formula: see text] of the control reproduction number [Formula: see text], below which the disease is eliminated in the long run. Whenever the value of [Formula: see text] ranges from [Formula: see text] and 1, we prove the existence of a backward bifurcation phenomenon, which corresponds to the case, where a locally asymptotically stable positive equilibrium co-exists with the disease-free equilibrium, which is also locally asymptotically stable. The existence of this bifurcation complicates the control of Ebola, since the requirement of [Formula: see text] below one, although necessary, is no longer sufficient for the elimination of Ebola, more efforts need to be deployed. When the value of [Formula: see text] is greater than one, we prove the existence of a unique endemic equilibrium, locally asymptotically stable. That is the disease may persist and become endemic. Numerically, we fit our model to the reported data for the 2018–2020 Kivu Ebola outbreak which occurred in Democratic Republic of Congo. Through the sensitivity analysis of the control reproduction number, we prove that the transmission rates of infected alive who are outside hospital are the most influential parameters. Numerically, we explore the usefulness of isolation, safe burial combined with vaccination and investigate the importance to combine the latter control strategies to the educational campaigns or/and case finding.

In this paper, the two dimensional Signorini static contact problem in linear elasticity is presented. We present the weak formulation of frictionless and the frictional contact problems. In the both cases, the boundary integral operators are used to propose a boundary variational formulation whose resolution by the generalized Newton method is presented. Moreover a particular formulation by the fixed point method associated with the augmented Lagrangian is proposed for efficient analysis of contact problems with friction (Tresca and Coulomb) and powerful algorithms are constructed. The discretization is carried over by using the Galerkin method. The resulting linear system is solved by using preconditioned Conjugate Gradient (CG) iterative solver.

The paper is devoted to the homogenization of Richards' type equations in a deterministic multiscale porous medium filled with soft inclusions. The medium consists of a deterministic ensemble of coarse aggregates embedded within a matrix containing a deterministic ensemble of fine aggregates, leading to a network in which the inclusions (of different sizes) are widely distributed. Under various settings involving the structure of the medium and the coefficients of the equation, we obtain homogenized limits of the underlying micro-model. We also provide a suitable approximation scheme for the homogenized coefficients, thereby providing a numerical method for computing these coefficients. To achieve our goal, apply tools of the multiscale sigma-convergence method.

This work deals with the homogenization of two dimensions&rsquo, tidal equations. We study the asymptotic behavior of the sequence of the solutions using the sigma-convergence method. We establish the convergence of the sequence of solutions towards the solution of an equivalent problem of the same type.

Since 1976 many outbreaks of Ebola virus disease have occurred in Africa, and up to now, no treatment is available. Thus, to fight against this illness, several control strategies have been adopted. Among these measures, isolation, safe burial and vaccination occupy a prominent place. In this paper therefore, we present a model which takes into account these three control strategies as well as the difference in immunological responses of infected by distinguishing individuals with moderate and severe symptoms. The sensitivity analysis of the control reproduction number suggests that, the vaccination response works better than the two other control measures at the beginning of the disease, and there is no need to vaccinate people in order to overcome Ebola. The global asymptotic dynamic of the model with controls is completely achieved exhibiting a sharp threshold behavior. Consequently, the global stability of the endemic equilibrium when the control reproduction number is greater than one, calls for the implementation of the three responses optimally. Thus, we define, analyze and implement different optimal control problems which minimize the cost of controls and the number of infected individuals through vaccination, isolation and safe burial. The obtained results highlight that, in order to mitigate Ebola outbreaks, an optimal vaccination strategy has more impact than the combination of isolation and safe burial responses. Therefore, the broad recommendation is that: if for any disease outbreak, the vaccine can be available quickly, one should think of implementing large vaccination programs rather than any other control measures.

In the current work, we are performing the asymptotic analysis, beyond the periodic setting, of the Cahn-Hilliard-Navier-Stokes system. Under the general deterministic distribution assumption on the microstructures in the domain, we find the limit model equivalent to the heterogeneous one. To this end, we use the sigma-convergence concept which is suitable for the passage to the limit., 28 pages

For a family of linear hyperbolic damped stochastic wave equations with rapidly oscillating coefficients, we establish the homogenization result by using the sigma-convergence method. This is achieved under an abstract assumption covering special cases like the periodicity, the almost periodicity and some others.

The paper deals with the homogenization of a linear Boltzmann equation by the means of the sigma-convergence method. Under a general deterministic assumption on the coefficients of the equation, we prove that the density of the particles converges to a solution of a drift-diffusion equation. To achieve our goal, we use the Krein-Rutman theorem for locally convex spaces together with the Fredholm alternative to solve the so-called corrector problem., Comment: 25 pages

In this paper, we study the acoustic properties of porous media saturated by an incompressible viscoelastic fluid. The model considered here consists of a linear deformable porous skeleton having memory that is surrounded by a viscoelastic Oldroyd fluid. Assuming the microstructures to be almost periodically distributed and under the almost periodicity hypothesis on the coefficients of the governing equations, we determine the macroscopic equivalent medium. To achieve our goal, we use some very recent tools about the sigma convergence of convolution sequences. Copyright © 2017 John Wiley & Sons, Ltd.

We carry out the deterministic homogenization of nonlinear elliptic operators beyond periodicity. To proceed with, we prove the existence of nonlinear correctors in the usual distributional sense. This lays the foundation for the study of regularity results in the nonlinear deterministic homogenization theory beyond the periodic setting.

We address the homogenization of a semilinear hyperbolic stochastic partial differential equation with highly oscillating coefficients, in the context of ergodic algebras with mean value. To achieve our goal, we use a suitable variant of the sigma-convergence concept that takes into account both the random and deterministic behaviours of the phenomenon modelled by the underlying problem. We also provide an appropriate scheme for the approximation of the effective coefficients. To illustrate our approach, we work out some concrete problems such as the periodic homogenization problem, the almost periodic and the asymptotically almost periodic ones.

Let A be an introverted algebra with mean value. We prove that its spectrum Δ ( A ) is a compact topological semigroup, and that the kernel K ( Δ ( A ) ) of Δ ( A ) is a compact topological group over which the mean value on A can be identified as the Haar integral. Based on these facts and also on the fact that K ( Δ ( A ) ) is an ideal of Δ ( A ) , we define the convolution over Δ ( A ) . We then use it to derive some new convergence results involving the convolution product of sequences. These convergence results provide us with an efficient method for studying the asymptotics of nonlocal problems. The obtained results systematically establish the connection between the abstract harmonic analysis and the homogenization theory. To illustrate this, we work out some homogenization problems in connection with nonlocal partial differential equations.

In most of the linear homogenization problems involving convolution terms so far studied, the main tool used to derive the homogenized problem is the Laplace transform. Here we propose a direct approach enabling one to tackle both linear and nonlinear homogenization problems that involve convolution sequences without using Laplace transform. To illustrate this, we investigate in this paper the asymptotic behavior of the solutions of a Stokes–Volterra problem with rapidly oscillating coefficients describing the viscoelastic fluid flow in a fixed domain. Under the almost periodicity assumption on the coefficients of the problem, we prove that the sequence of solutions of our ϵ-problem converges in L2 to a solution of a rather classical Stokes system. One important fact is that the memory disappears in the limit. To achieve our goal, we use some very recent results about the sigma-convergence of convolution sequences. Copyright © 2013 John Wiley & Sons, Ltd.

In this paper, we show that the concept of sigma-convergence associated to stochastic processes can tackle the homogenization of stochastic partial differential equations. In this regard, the homogenization problem for a stochastic nonlinear partial differential equation is studied. Using some deep compactness results such as the Prokhorov and Skorokhod theorems, we prove that the sequence of solutions of this problem converges in probability towards the solution of an equation of the same type. To proceed with, we use a suitable version of sigma-convergence method, the sigma-convergence for stochastic processes, which takes into account both the deterministic and random behaviours of the solutions of the problem. We apply the homogenization result to some concrete physical situations such as the periodicity, the almost periodicity, the weak almost periodicity, and others., To appear in: Stochastic Analysis and Applications

Homogenization of Wilson-Cowan type of nonlocal neural field models is investigated. Motivated by the presence of a convolution term in this type of models, we first prove some general convergence results related to convolution sequences. We then apply these results to the homogenization problem of the Wilson-Cowan-type model in a general deterministic setting. Key ingredients in this study are the notion of algebras with mean value and the related concept of sigma-convergence.

We study homogenization for nonstationary Navier-Stokes systems in a fissured medium of general deterministic type. Assuming that the blocks of the porous medium consist of deterministically distributed inclusions and the elasticity tensors satisfy general deterministic hypotheses, we prove that the macroscopic problem is a Navier-Stokes type equation for Newtonian fluid in a fixed domain. Our setting includes the classical periodic framework, the weakly almost periodic one, and some others.

We study the multiscale homogenization of a nonlinear hyperbolic equation in a periodic setting. We obtain an accurate homogenization result. We also show that as the nonlinear term depends on the microscopic time variable, the global homogenized problem thus obtained is a system consisting of two hyperbolic equations. It is also shown that in spite of the presence of several time scales, the global homogenized problem is not a reiterated one.

The purpose of the present work is to introduce a framework which enables us to study nonlinear homogenization problems. The starting point is the theory of algebras with mean value. Very often in physics, from very simple experimental data, one gets complicated structure phenomena. These phenomena are represented by functions which are permanent in mean, but complicated in detail. In addition the functions are subject to the verification of a functional equation which in general is nonlinear. The problem is therefore to give an interpretation of these phenomena using functions having the following qualitative properties: they are functions that represent a phenomenon on a large scale, and which vary irregularly, undergoing nonperiodic oscillations on a fine scale. In this work we study the qualitative properties of spaces of such functions, which we call generalized Besicovitch spaces, and we prove general compactness results related to these spaces. We then apply these results in order to study some new homogenization problems. One important achievement of this work is the resolution of the generalized weakly almost periodic homogenization problem for a nonlinear pseudo-monotone parabolic-type operator. We also give the answer to the question raised by Frid and Silva in their paper [35] [H. Frid, J. Silva, Homogenization of nonlinear pde’s in the Fourier–Stieltjes algebras, SIAM J. Math. Anal, 41 (4) (2009) 1589–1620] as regards whether there exist or do not exist ergodic algebras that are not subalgebras of the Fourier–Stieltjes algebra.

Motivated by the fact that in nature almost all phenomena behave randomly in some scales and deterministically in some other scales, we build up a framework suitable to tackle both deterministic and stochastic homogenization problems simultaneously, and also separately. Our approach, the stochastic Σ-convergence, can be seen either as a multiscale stochastic approach since deterministic homogenization theory can be seen as a special case of stochastic homogenization theory (see Theorem 3), or as a conjunction of the stochastic and deterministic approaches, both taken globally, but also each separately. One of the main applications of our results is the homogenization of a model of rotating fluids.

We redefine the homogenization algebras without requiring the separability assumption. We show that this enables one to treat more complicated homogenization problems than those solved by the previous theory. In particular we exhibit an example of algebra which, contrary to the algebra of almost periodic functions, induces no homogenization algebra. We prove some general compactness results which are then applied to the resolution of some homogenization problems related to the generalized Reynolds type equations and to some nonlinear hyperbolic equations.

In order to widen the scope of the applications of deterministic homogenization, we consider here the homogenization problem for a family of integral functionals. The homogenization procedure tending to be classical, the choice focused on the convex integral functionals is made just to simplify the presentation of the paper. We use a new approach based on the Stepanov type spaces, which approach allows us to solve various problems such as the almost periodic homogenization problem and others without resorting to additional assumptions. We then apply it to obtain a general homogenization result and then we provide a number of physical applications of the result. The convergence method used falls within the scope of two-scale convergence.

We study the reiterated homogenization of nonlinear parabolic differential equations associated with monotone operators. Contrary to what is usually done in the deterministic homogenization theory, we present a new approach based on a deterministic assumption on the coefficients of the operators, which allows us to consider the concrete homogenization problems from a true and natural perspective, taking into account the discontinuities in general. Based on this new approach we obtain very general homogenization results, and we solve several concrete homogenization problems. Our main tool is the theory of homogenization structures, and our homogenization approach falls within the scope of multiscale convergence method.

This paper is devoted to the homogenization beyond the periodic setting, of nonlinear monotone operators in a domain in ℝN with isolated holes of size ɛ2 (ɛ > 0 a small parameter). The order of the size of the holes is twice that of the oscillations of the coefficients of the operator, so that the problem under consideration is a reiterated homogenization problem in perforated domains. The usual periodic perforation of the domain and the classical periodicity hypothesis on the coefficients of the operator are here replaced by an abstract assumption covering a great variety of behaviors such as the periodicity, the almost periodicity and many more besides. We illustrate this abstract setting by working out a few concrete homogenization problems. Our main tool is the recent theory of homogenization structures.

The paper deals with the homogenization problem beyond the periodic setting, for a degenerated nonlinear non-monotone elliptic type operator on a perforated domain ? ? in ? N with isolated holes. While the space variable in the coefficients a 0 and a is scaled with size ? (?>0 a small parameter), the system of holes is scaled with ? 2 size, so that the problem under consideration is a reiterated homogenization problem in perforated domains. The homogenization problem is formulated in terms of the general, so-called deterministic homogenization theory combining real homogenization algebras with the Σ-convergence method. We present a new approach based on the Besicovitch type spaces to solve deterministic homogenization problems, and we obtain a very general abstract homogenization results. We then illustrate this abstract setting by providing some concrete applications of these results to, e.g., the periodic homogenization, the almost periodic homogenization, and others.

Multiscale homogenization of nonlinear non-monotone degenerated parabolic operators is investigated. Under a periodicity assumption on the coefficients of the operators under consideration, we obtain by means of multiscale convergence method, an accurate homogenization result. It is also shown that in spite of the presence of several time scales the global homogenized problem is not a reiterated one.

Deterministic homogenization has been till now applied to the study of monotone operators, the determination of the limiting problem being systematically based on the monotonicity of the operator under consideration. Here we mean to show that deterministic homogenization also tackle non-monotone operators. More precisely, under an abstract general hypothesis, we study the homogenization of non-linear non-monotone degenerate elliptic operators. We obtain some general homogenization result, which result is applied to the resolution of several concrete homogenization problems such as the periodic homogenization and the almost periodic homogenization problems. Our main tool is the theory of homogenization structures.

We discuss the nonstochastic homogenization of nonlinear parabolic differential operators in an abstract setting framed to bridge the gap between periodic and stochastic homogenization theories. Instead of the classical periodicity hypothesis, we have here an abstract assumption covering a great variety of concrete behaviours in both space and time variables, such as the periodicity, the almost periodicity, the convergence at infinity, and others. Our basic approach is the Σ -convergence method generalizing the well-known two-scale convergence technique. Fundamental homogenization theorems are proved and several concrete examples are worked out, which reveal that homogenization beyond the periodic setting may involve difficulties that are beyond imagination.

We study the flow generated by an incompressible viscoelastic fluid in a fractured porous medium. The model consists of a fluid flow governed by Stokes-Volterra equations evolving in a periodic double-porosity medium. Using the multiscale convergence method associated to some recent tools about the convergence of convolution sequences, we show that the equivalent macroscopic model is of the same type as the microscopic one, but in a fixed domain., 26 pages

The paper deals with the existence and almost periodic homogenization of some model of generalized Navier-Stokes equations. We first establish an existence result for non-stationary Ladyzhenskaya equations with a given non constant density. The external force depends nonlinearly on the velocity. Next, to proceed with homogenization, as the density of the fluid is non constant, we combine the sigma-convergence method with some compactness arguments., Comment: To appear in Journal of Mathematical Sciences

In several works, the theory of strongly continuous groups is used to build a framework for solving stochastic homogenization problems. Following this idea, we construct a detailed and comprehensive theory of homogenization. This enables to solve homogenization problems in algebras with mean value, regardless of whether they are ergodic or not, thereby responding affirmatively to the question raised by Zhikov and Krivenko [V.V. Zhikov, E.V. Krivenko, Homogenization of singularly perturbed elliptic operators. Matem. Zametki, 33 (1983) 571-582 (english transl.: Math. Notes, 33 (1983) 294-300)] to know whether it is possible to homogenize problems in nonergodic algebras. We also state and prove a compactness result for Young measures in these algebras. As an important achievement we study the homogenization problem associated with a stochastic Ladyzhenskaya model for incompressible viscous flow, and we present and solve a few examples of homogenization problems related to nonergodic algebras., Comment: 51 pages

We study in this paper the periodic homogenization problem related to a strongly nonlinear reaction-diffusion equation. Owing to the large reaction term, the homogenized equation has a rather quite different form which puts together both the reaction and convection effects. We show in a special case that, the homogenized equation is exactly of a convection-diffusion type. The study relies on a suitable version of the well-known two-scale convergence method., Comment: 19 pages

Homogenization of a stochastic nonlinear reaction-diffusion equation with a large non- linear term is considered. Under a general Besicovitch almost periodicity assumption on the coefficients of the equation we prove that the sequence of solutions of the said problem converges in probability towards the solution of a rather different type of equation, namely, the stochastic non- linear convection-diffusion equation which we explicitly derive in terms of appropriated functionals. We study some particular cases such as the periodic framework, and many others. This is achieved under a suitable generalized concept of sigma-convergence for stochastic processes., 34 pages