Processing math: 83%
Research article Special Issues

Mathematical modeling of infectious diseases and the impact of vaccination strategies

  • Received: 26 June 2024 Revised: 20 August 2024 Accepted: 06 September 2024 Published: 19 September 2024
  • Mathematical modeling plays a crucial role in understanding and combating infectious diseases, offering predictive insights into disease spread and the impact of vaccination strategies. This paper explored the significance of mathematical modeling in epidemic control efforts, focusing on the interplay between vaccination strategies, disease transmission rates, and population immunity. To facilitate meaningful comparisons of vaccination strategies, we maintained a consistent framework by fixing the vaccination capacity to vary from 10 to 100% of the total population. As an example, at a 50% vaccination capacity, the pulse strategy averted approximately 45.61% of deaths, while continuous and hybrid strategies averted around 45.18 and 45.69%, respectively. Sensitivity analysis further indicated that continuous vaccination has a more direct impact on reducing the basic reproduction number R0 compared to pulse vaccination. By analyzing key parameters such as R0, pulse vaccination coefficients, and continuous vaccination parameters, the study underscores the value of mathematical modeling in shaping public health policies and guiding decision-making during disease outbreaks.

    Citation: Diana Bolatova, Shirali Kadyrov, Ardak Kashkynbayev. Mathematical modeling of infectious diseases and the impact of vaccination strategies[J]. Mathematical Biosciences and Engineering, 2024, 21(9): 7103-7123. doi: 10.3934/mbe.2024314

    Related Papers:

    [1] Hiroshi Nishiura . Joint quantification of transmission dynamics and diagnostic accuracy applied to influenza. Mathematical Biosciences and Engineering, 2011, 8(1): 49-64. doi: 10.3934/mbe.2011.8.49
    [2] Qingling Zeng, Kamran Khan, Jianhong Wu, Huaiping Zhu . The utility of preemptive mass influenza vaccination in controlling a SARS outbreak during flu season. Mathematical Biosciences and Engineering, 2007, 4(4): 739-754. doi: 10.3934/mbe.2007.4.739
    [3] Kasia A. Pawelek, Anne Oeldorf-Hirsch, Libin Rong . Modeling the impact of twitter on influenza epidemics. Mathematical Biosciences and Engineering, 2014, 11(6): 1337-1356. doi: 10.3934/mbe.2014.11.1337
    [4] Xiaomeng Wang, Xue Wang, Xinzhu Guan, Yun Xu, Kangwei Xu, Qiang Gao, Rong Cai, Yongli Cai . The impact of ambient air pollution on an influenza model with partial immunity and vaccination. Mathematical Biosciences and Engineering, 2023, 20(6): 10284-10303. doi: 10.3934/mbe.2023451
    [5] Boqiang Chen, Zhizhou Zhu, Qiong Li, Daihai He . Resurgence of different influenza types in China and the US in 2021. Mathematical Biosciences and Engineering, 2023, 20(4): 6327-6333. doi: 10.3934/mbe.2023273
    [6] Eunha Shim . Prioritization of delayed vaccination for pandemic influenza. Mathematical Biosciences and Engineering, 2011, 8(1): 95-112. doi: 10.3934/mbe.2011.8.95
    [7] Fangyuan Chen, Rong Yuan . Dynamic behavior of swine influenza transmission during the breed-slaughter process. Mathematical Biosciences and Engineering, 2020, 17(5): 5849-5863. doi: 10.3934/mbe.2020312
    [8] Dennis L. Chao, Dobromir T. Dimitrov . Seasonality and the effectiveness of mass vaccination. Mathematical Biosciences and Engineering, 2016, 13(2): 249-259. doi: 10.3934/mbe.2015001
    [9] Junyuan Yang, Guoqiang Wang, Shuo Zhang . Impact of household quarantine on SARS-Cov-2 infection in mainland China: A mean-field modelling approach. Mathematical Biosciences and Engineering, 2020, 17(5): 4500-4512. doi: 10.3934/mbe.2020248
    [10] Sherry Towers, Katia Vogt Geisse, Chia-Chun Tsai, Qing Han, Zhilan Feng . The impact of school closures on pandemic influenza: Assessing potential repercussions using a seasonal SIR model. Mathematical Biosciences and Engineering, 2012, 9(2): 413-430. doi: 10.3934/mbe.2012.9.413
  • Mathematical modeling plays a crucial role in understanding and combating infectious diseases, offering predictive insights into disease spread and the impact of vaccination strategies. This paper explored the significance of mathematical modeling in epidemic control efforts, focusing on the interplay between vaccination strategies, disease transmission rates, and population immunity. To facilitate meaningful comparisons of vaccination strategies, we maintained a consistent framework by fixing the vaccination capacity to vary from 10 to 100% of the total population. As an example, at a 50% vaccination capacity, the pulse strategy averted approximately 45.61% of deaths, while continuous and hybrid strategies averted around 45.18 and 45.69%, respectively. Sensitivity analysis further indicated that continuous vaccination has a more direct impact on reducing the basic reproduction number R0 compared to pulse vaccination. By analyzing key parameters such as R0, pulse vaccination coefficients, and continuous vaccination parameters, the study underscores the value of mathematical modeling in shaping public health policies and guiding decision-making during disease outbreaks.



    Stochastic homogenization is a subject broadly studied starting from '80 since the seminal papers by Kozlov [11] and Papanicolaou-Varadhan [18] who studied boundary value problems for second order linear PDEs. We prove here an abstract homogenization result for the graph of a random maximal monotone operator

    v(x,ω)αε(x,ω,u(x,ω)),

    where xRn and ω is a parameter in a probability space Ω. In the spirit of [10,Chapter 7], the random operator αε is obtained from a stationary operator α via an ergodic dynamical system Tx:ΩΩ

    αε(x,ω,):=α(Tx/εω,). (1)

    The aim of this paper is to extend existing results where α is the subdifferential of a convex function [21,24] to general maximal monotone operators and to provide a simple proof based on Tartar's oscillating test function method. The crucial ingredient in our analysis is the scale integration/disintegration theory introduced by Visintin [25]. Moreover, relying on Fitzpatrick's variational formulation of monotone graphs [8], which perfectly suits the scale integration/disintegration setting [26], in the proof we can directly exploit the maximal monotonicity, without turning to (stochastic) two-scale convergence [2,9,20], Γ-convergence [5,6], G-convergence [16,17], nor epigraph convergence [13,14]. An advantage of our approach is that we do not need to assume an additional compact metric space structure on the probability space Ω. Moreover, the effective relation is obtained directly, i.e. without an intermediate two-scale problem, which often needs to be studied separately. We also obtain the existence of the oscillating test functions as a byproduct of scale disintegration, without having to study the auxiliary problem (see, e.g., [17,Section 3.2]).

    The outline of the proof is the following: Let X be a separable and reflexive Banach space, with dual X, let An:XX be a sequence of monotone operators, and for every nN let (xn,yn)X×X be a point in the graph of An, i.e., such that yn=Anxn for all nN. Assuming that (xn,yn)(x,y) in X×X and that we already know the limit maximal monotone operator A:XX, a classical question of functional analysis is:

    Under   which  assumptions  can  we  conclude  that   y=Ax?

    A classical answer (see, e.g., [3]) is: If we can produce an auxiliary sequence of points on the graph of An, and we know that they converge to a point on the graph of A, that is, if there exist

    (ξn,ηn)X×X such that ηn=Anξn(ξn,ηn)(ξ,η) and η=Aξ, (2)

    then, denoting by y,x the duality pairing between xX and yX, by monotonicity of An

    ynηn,xnξn0.

    In order to pass to the limit as n, since the duality pairing of weakly converging sequences in general does not converge to the duality pairing of the limit, we need the additional hypothesis

    lim supngn,fng,f(fn,gn)(f,g) in X×X, (3)

    which, together with the weak convergence of (xn,yn) and (ξn,ηn), yields

    yη,xξ0.

    By maximal monotonicity of A, if the last inequality is satisfied for all (ξ,η) such that η=Aξ, then we can conclude that (x,y) is a point of the graph of A, i.e., y=Ax. Summarizing, this procedure is based on

    1. Existence and weak compactness of solutions (xn,yn) such that yn=Anxn and (xn,yn)(x,y);

    2. A condition for the convergence of the duality pairing (3);

    3. Existence of a recovery sequence (2) for all points in the limit graph.

    The first step depends on the well-posedness of the application; the second step is ensured, e.g., by compensated compactness (in the sense of Murat-Tartar [15,23]), and, like the first one, it depends on the character of the differential operators that appear in the application, rather than on the homogenization procedure. In the present paper we focus on the third step: in the context of stochastic homogenization, we prove that the scale integration/disintegration idea introduced by Visintin [25], combined with Birkhoff's ergodic theorem (Theorem 2.4) yields the desired recovery sequence. We obtain an explicit formula for the limit operator A through the scale integration/disintegration procedure with Fitzpatrick's variational formulation. With the notation introduced in (1), the outline of this procedure is the following:

    α a) f b) f0 c) α0,

    where a) the random operator α(ω) is represented through a variational inequality involving Fitzpatrick's representation f(ω); b) the representation is "scale integrated" to a ω-independent effective f0; c) a maximal monotone operator α0 is associated to f0. In Theorem 3.8 we prove that α0 is the homogenised limit of αε.

    In Section 2.1 we review the properties of maximal monotone operators and their variational formulation due to Fitzpatrick. In Section 2.2 we recall the basis of ergodic theory that we need in order to state our first main tool: Birkhoff's Ergodic Theorem. Section 3 is devoted to the translation to the stochastic setting of Visintin's scale integration-disintegration theory, which paves the way to our main result, Theorem 3.8. The applications we provide in the last section are: Ohmic electric conduction with Hall effect (Section 4.1), and nonlinear elasticity, (Section 4.2).

    We use the notation ab for the standard scalar product for vectors in Rn. The arrows and denote weak and weak convergence, respectively. As usual, D(D) stands for the space of C-functions with compact support in DRn; its dual is denoted by D(D).

    In this section we summarize the variational representation of maximal monotone operators introduced in [8]. Further details and proofs of the statements can be found, e.g., in [27]. Let B be a reflexive, separable and real Banach space; we denote by B its dual, by P(B) the power set of B, and by y,x the duality pairing between xB and yB. Let α:BP(B) be a set-valued operator and let

    Gα:={(x,y)B×B:yα(x)}

    be its graph. (We will equivalently write yα(x) or (x,y)Gα.) The operator α is said to be monotone if

    (x,y)Gαyy0,xx00,(x0,y0)Gα (4)

    and strictly monotone if there is θ>0 such that

    (x,y)Gαyy0,xx0θxx02,(x0,y0)Gα. (5)

    We denote by α1 the inverse operator in the sense of graphs, that is

    xα1(y)yα(x).

    The monotone operator α is said to be maximal monotone if the reverse implication of (4) is also fulfilled, namely if

    yy0,xx00(x0,y0)Gα(x,y)Gα.

    An operator α is maximal monotone if and only if α1 is maximal monotone. For any operator α:BP(B), which is not identically , we introduce the Fitzpatrick function of α as the function fα:B×BR defined by

    fα(x,y):=y,x+sup{yy0,x0x:(x0,y0)Gα}=sup{y,x0+y0,xy0,x0:(x0,y0)Gα}.

    As a supremum of a family of linear functions, the Fitzpatrick function fα is convex and lower semicontinuous. Moreover, the following lemma holds true (see [8]).

    Lemma 2.1. An operator α is monotone if and only if

    (x,y)Gαfα(x,y)=y,x,

    while α is maximal monotone if and only if

    {fα(x,y)y,x (x,y)B×Bfα(x,y)=y,x(x,y)Gα.

    In the case B=B=R, it is easy to compute some simple examples of Fitzpatrick function of a monotone operator:

    1. Let α(x):=ax+b, with a>0,bR. A straightforward computation shows that

    fα(x,y)=(yb+ax)24a+bx.

    2. Let

    α(x)={1if x>0,[0,1]if x=0,1if x<0.

    Then

    fα(x,y)={|x|if |y|1,+if |y|>1.

    and in both cases fα coincides with g(x,y)=xy exactly on the graph of α.

    We define F=F(B) to be the class of all proper, convex and lower semicontinuous functions f:B×BR{+} such that

    f(x,y)y,x(x,y)B×B.

    We call F(B) the class of representative functions. The above discussion shows that given a monotone operator α, one can construct its representative function in F(B), and viceversa, given a function fF(B), we define the operator represented by f, which we denote αf, by:

    (x,y)Gαff(x,y)=y,x. (6)

    A crucial point is whether αf is monotone (or maximal monotone, see also [26,Theorem 2.3]).

    Lemma 2.2. Let fF(B), then

    (i) the operator αf defined by (6) is monotone;

    (ii) the class of maximal monotone operators is strictly contained in the class of operators representable by functions in F(B).

    Proof. (ⅰ) If Gαf is empty or reduced to a single element, then the statement is trivially satisfied. Let x1,x2B, yiαf(xi) for i=1,2, and assume, by contradiction, that y2y1,x2x1<0. Define Pi:=(xi,yi)B×B and g(x,y):=y,x. We compute

    g(P1+P22)g(P1)+g(P2)2=14(y1+y2,x1+x2)12(y1,x1+y2,x2)=14(y1,x2+y2,x1y1,x1y2,x2)=14(y2y1,x2x1)>0.

    Since fg and f(xi,yi)=g(xi,yi) for i=1,2, the last inequality implies

    f(P1+P22)>f(P1)+f(P2)2,

    which contradicts the convexity of f.

    (ⅱ) Maximal monotone operators are representable by Lemma 2.1. To see that the inclusion is strict, assume that αf is maximal monotone. Let (x0,y0)Gαf; since f is proper there exists cR such that c>f(x0,y0). The function

    h(x,y)=max{c,f(x,y)}

    clearly belongs to F(B), but Gαh is strictly contained in Gαf. Indeed by the maximality of αf

    h(x0,y0)c>f(x0,y0)=y0,x0,

    and thus αh is not maximal.

    Remark 1. When α=φ for some proper, convex, lower semicontinuous function φ:BR{+}, the definition of Fenchel transform yields

    φ(x)+φ(y)y,x(x,y)B×B,
    yα(x)φ(x)+φ(y)=y,x.

    Thus, Fitzpatrick's representative function fα generalizes φ+φ to maximal monotone operators which are not subdifferentials. Remark that, even if α=φ, in general fαφ+φ: for example, let α(x):=x in R, then

    fα(x,y)=(x+y)24x22+y22=φ(x)+φ(y).

    We need to introduce also parameter-dependent operators. For any measurable space X we say that a set-valued mapping g:XP(B) is measurable if for any open set RB the set

    g1(R):={xX:g(x)R}

    is measurable.

    Let B(B) be the σ-algebra of the Borel subsets of the separable and reflexive Banach space B, let (Ω,A,μ) be a probability space equipped with the σ-algebra A and the probability measure μ. We define a random (maximal) monotone operator as α:B×ΩP(B) such that

    α  is  B(B)A-measurable, (7)
    α(x,ω)  is  closed  for  any xB  and  for μ-a.e.  ωΩ, (8)
    α(,ω)  is (maximal) monotone  for μ-a.e.  ωΩ. (9)

    If α fulfills (7) and (8) then for any A-measurable mapping v:ΩB, the multivalued mapping ωα(v(ω),ω) is closed-valued and measurable. We denote by F(Ω;B) the set of all measurable representative functions f:B×B×ΩR{+} such that

    (a) f:B×B×ΩR{+} is B(B×B)A -measurable,

    (b) f(,,ω) is convex and lower semicontinuous for μ-a.e. ωΩ,

    (c) f(x,y,ω)y,x for all (x,y)B×B and for μ-a.e. ωΩ.

    As above, fF(Ω;B) represents the operator α=αf(,ω) in the following sense:

    yα(x,ω)  f(x,y,ω)=y,x(x,y)B×B,for μ-a.e. ωΩ. (10)

    Precisely, any measurable representative function f:B×B×ΩR{+} represents a closed-valued, measurable, monotone operator α:B×ΩP(B), while any closed-valued, measurable, maximal monotone operator α can be represented by a measurable representative function f, for instance, by its Fitzpatrick function [26,Proposition 3.2].

    In this subsection we review the basic notions and results of stochastic analysis that we need in Section 3. For more details see [10,Chapter 7]. Let (Ω,A,μ) be a probability space, where A is a σ-algebra of subsets of Ω and μ is a probability measure on Ω. Let nN with n1. An n-dimensional dynamical system Tx on Ω is a family of mappings Tx:ΩΩ, with xRn, such that

    (a) T0 is the identity and Tx+y=TxTy for any x,yRn;

    (b) for every xRn and every set EA we have TxEA and

    μ(TxE)=μ(E) (11)

    (c) for any measurable function f:ΩRm, the function ˜f:Rn×ΩRm given by

    ˜f(x,ω)=f(Txω)

    is measurable.

    Given an n-dimensional dynamical system T on Ω, a measurable function f defined on Ω is said to be invariant if f(Txω)=f(ω) μ-a.e. in Ω, for each xRn. A dynamical system is said to be ergodic if the only invariant functions are the constants. The expected value of a random variable f:ΩRn is defined as

    E(f):=Ωfdμ.

    In the context of stochastic homogenization, it is useful to provide an orthogonal decomposition of L2(Ω) into functions, whose realizations are curl-free and divergence-free, in the sense of distributions (see, e.g., [10,Section 7]). For p[1,+[, Peter-Weyl's decomposition theorem [19] can be generalized to a relation of orthogonality between subspaces of Lp(Ω) and Lp(Ω), where p denotes the conjugate exponent of p. Let vLploc(Rn;Rn). We say that v is potential (or curl-free) in Rn if

    (viφxjvjφxi)dx=0,    i,j=1,,n,φD(Rn)

    and we say that v is solenoidal (or divergence-free) in Rn if

    ni=1viφxidx=0,   φD(Rn).

    Next we consider a vector field on (Ω,A,μ). We say that fLp(Ω;Rn) is potential if μ-almost all its realizations xf(Txω) are potential. We denote by Lppot(Ω;Rn) the space of all potential fLp(Ω;Rn). In the same way, fLp(Ω;Rn) is said to be solenoidal if μ-almost all its realizations xf(Txω) are solenoidal and we denote by Lpsol(Ω;Rn) the space of all solenoidal fLp(Ω;Rn). In the following Lemma we collect the main properties of potential and solenoidal Lp spaces (see [10,Section 15]).

    Lemma 2.3. Define the spaces

    Vppot(Ω;Rn):={fLppot(Ω;Rn):E(f)=0},Vpsol(Ω;Rn):={fLpsol(Ω;Rn):E(f)=0}.

    The spaces Lppot(Ω;Rn),Lpsol(Ω;Rn),Vppot(Ω;Rn),Vpsol(Ω;Rn) are closed subspaces of Lp(Ω;Rn). If uLpsol(Ω;Rn) and vLppot(Ω;Rn) with 1/p+1/p=1, then

    E(uv)=E(u)E(v) (12)

    and the relations

    (Vpsol(Ω;Rn))=Vppot(Ω;Rn)Rn,(Vppot(Ω;Rn))=Vpsol(Ω;Rn)Rn

    hold in the sense of duality pairing between the spaces Lp(Ω) and Lp(Ω).

    One of the most important results regarding stochastic homogenization is Birk-hoff's Ergodic Theorem. We report the statement given in [10,Theorem 7.2].

    Theorem 2.4. (Birkhoff's Ergodic Theorem) Let fL1(Ω;Rm) and let T be an n-dimensional ergodic dynamical system on Ω. Then

    E(f)=limε01|K|Kf(Tx/εω)dx

    for μ-a.e. ωΩ, for any KRn bounded, measurable, with |K|>0.

    Remark 2. Birkhoff's theorem implies that μ-almost every realization ˜fε(x,ω)=f(Tx/εω) satisfies

    limε01|K|K˜fε(x,ω)dx=E(f).

    Since this holds for every measurable bounded set KRn, it entails in particular that if fLp(Ω), then

    ˜fεE(f)  weakly  in Lploc(Rn;Rm) for μ-a.e.   ωΩ. (13)

    In what follows, the dynamical system Tx is assumed to be ergodic and KRn is bounded, measurable and |K|>0.

    Let be given a probability space (Ω,A,μ) endowed with an n-dimensional ergodic dynamical system Tx:ΩΩ, xRn. Let p(1,+), p=pp1 and let α be a random maximal monotone operator, as in (7)-(9).

    We rephrase here Visintin's scale integration/disintegration [25,26] to the stochastic homogenization setting.

    Remark 3. While most of this subsection's statements are Visintin's results written in a different notation, some others contain a small, but original contribution. Namely: Lemma 3.1 can be found in [26,Lemma 4.1], where the assumption of boundedness for K is used to obtain the lower semicontinuity of the inf function. Since we prefer not to impose this condition, we independently proved the lower semicontinuity part, making use of the coercivity of g. Lemma 3.2 and Proposition 1 were taken for granted in [26], but we decided to write a proof for sake of clarity. Lemma 3.3 and Lemma 3.4 are essentially [26,Theorem 4.3] and [26,Theorem 4.4], cast in the framework of stochastic homogenization in the probability space (Ω,A,μ), instead of periodic homogenization on the n-dimensional torus. Theorem 3.5 collects other results of [26]. Lemma 3.6 is an original remark.

    For every fixed ωΩ let f(,,ω):Rn×RnR{+} be the Fitzpatrick representation of the operator α(,ω). We assume the following coercivity condition on f: there exist c>0 and kL1(Ω) such that for any ξ,ηRn, for any ωΩ it holds

    f(ξ,η,ω)c(|ξ|p+|η|p)+k(ω). (14)

    We define the homogenised representation f0:Rn×RnR{+} as

    f0(ξ,η):=inf{Ωf(ξ+v(ω),η+u(ω),ω)dμ:uVppot(Ω;Rn),vVpsol(Ω;Rn)}. (15)

    Lemma 3.1. Let X be a reflexive Banach space, let K be a weakly closed subset of a reflexive Banach space. Let the function g:X×KR{+} be weakly lower semicontinuous and bounded from below. If g is coercive1, then the function h:XR{+} given by

    1i.e., for all M>0 the set {(x,y)X×K:g(x,y)M} is bounded.

    h(x):=infyKg(x,y)

    is weakly lower semicontinuous and coercive. Moreover, if g and K are convex then h is convex.

    Proof. Let xjxX, we must show that

    lim infj+h(xj)h(x). (16)

    Let

    :=lim infj+h(xj).

    If =+, then (16) is trivially satisfied. On the other hand, since g is bounded from below, then >, and we can assume that R. By definition of inferior limit, there exists a subsequence of {xj} (not relabeled), such that limjh(xj)=. Up to extracting another subsequence, we can also assume that h(xj)2 for all jN. Let ε>0 be fixed, by definition of infimum, for all jN, there exists yjK such that

    h(xj)=infyKg(xj,y)g(xj,yj)ε. (17)

    Therefore

    g(xj,yj)2+εjN.

    By the coercivity assumption on g, we deduce that yj is bounded, we can therefore extract a subsequence {yjk}K such that yjky. Since K is weakly closed, then yK. We can now pass to the inferior limit in (17), using the lower semicontinuity of g

    lim infk+h(xjk)lim infk+g(xjk,yjk)εg(x,y)εh(x)ε. (18)

    By arbitrariness of ε>0, this proves the weak lower semicontinuity of h. Assume now that K is convex. Take λ[0,1], x1,x2X and y1,y2K. By convexity of g

    h(λx1+(1λ)x2)g(λx1+(1λ)x2,λy1+(1λ)y2)λg(x1,y1)+(1λ)g(x2,y2).

    Passing to the infimum with respect to y1,y2K we conclude

    h(λx1+(1λ)x2)λh(x1)+(1λ)h(x2).

    Regarding the coercivity of h, denote

    Bt:={xX:h(x)t},At:={xX:g(x,y)t, for some yK}.

    Let M,ε>0, for all xBM there exists yK such that g(x,y)h(x)+εM+ε, therefore BMAM+ε. Since g is coercive, AM+ε is bounded and thus BM is bounded, i.e., h is coercive.

    In the proof of Proposition 1 we need the following estimate

    Lemma 3.2. For all p[1,+[ there exists C>0 such that

    Ω|ξ+u(ω)|pdμCΩ|ξ|p+|u(ω)|pdμ

    for all ξRn, for all uLp(Ω;Rn) such that E(u)=0.

    Proof. Consider the operator

    Φ:Lp(Ω;Rn)Lp(Ω;Rn)×Lp(Ω;Rn)u(E(u),uE(u)).

    Clearly, Φ is linear and continuous. Therefore, there exists C>0 such that

    Ω|E(u)|pdμ+Ω|u(ω)E(u)|pdμ(E(u)Lp+uE(u)Lp)p2p/2(E(u)2Lp+uE(u)2Lp)p/2=2p/2Φ(u)pLp×LpCupLp=CΩ|u(ω)|pdμ.

    Apply now the last inequality to u(ω)=ξ+˜u(ω), with E(˜u)=0:

    Ω|ξ|p+|˜u(ω)|pdμCΩ|ξ+˜u(ω)|pdμ.

    Proposition 1. For all (ξ,η)Rn×Rn there exists a couple (˜u,˜v)Vpsol(Ω;Rn)×Vppot(Ω;Rn) such that the infimum on the right-hand side of (15) is attained. Moreover, f0F(Rn). In particular, it holds

    f0(ξ,η)ξη(ξ,η)Rn×Rn. (19)

    Proof. Let K:=Vpsol(Ω;Rn)×Vppot(Ω;Rn). Then K is weakly closed in Lp(Ω;Rn)×Lp(Ω;Rn) since it is closed and convex. Let ξ,ηRn be fixed, for any (u,v)K let

    Fξ,η(u,v):=Ωf(ξ+v(ω),η+u(ω),ω)dμ.

    We prove that the problem infKFξ,η has a solution applying the direct method of the Calculus of Variations. First, by (14), infKFξ,η>. Then, if (uh,vh)K is a minimizing sequence for Fξ,η, by the coercivity assumption (14), up to subsequences, (uh,vh)(u,v) weakly in Lp(Ω;Rn)×Lp(Ω;Rn), therefore (u,v)K, since K is weakly closed. Finally, Fξ,η is Lp×Lp-weakly lower semicontinuous since f(,,ω) is convex, lower semicontinuous, and bounded from below by an integrable function (14), therefore

    Fξ,η(u,v)lim infhFξ,η(uh,vh)=infKFξ,η.

    This concludes the first part of the statement. We now want to show that f0F(Rn). Owing to (14) and Lemma 3.2, for all ξ,η,(u,v)Rn×Rn×K, there exists a constant C>0 such that

    Fξ,η(u,v)cΩ|ξ+v(ω)|p+|η+u(ω)|p+k(ω)dμCΩ|ξ|p+|u(ω)|p+|η|p+|v(ω)|pdμ+E(k)C(|ξ|p+upLp(Ω)+|η|p+vpLp(Ω))kL1(Ω).

    Thus, for any M0, the set

    {(ξ,η,(u,v))Rn×Rn×K:Fξ,η(u,v)M}

    is bounded in Rn×Rn×Lp(Ω;Rn)×Lp(Ω;Rn). We are therefore in a position to apply Lemma 3.1 and to conclude that f0 is convex and lower semicontinuous. Furthermore, let (˜u,˜v)K be a minimizer of Fξ,η, using (12)

    f0(ξ,η)=Ωf(ξ+˜u(ω),η+˜v(ω),ω)dμΩ(ξ+˜u(ω))(η+˜v(ω))dμ=E(ξ+˜u)E(η+˜v)=ξη,

    which yields the conclusion.

    We denote by α0 the operator on Rn represented by f0 through the usual relation

    ηα0(ξ)f0(ξ,η)=ξη.

    We refer to α0 as the scale integration of α, since it is obtained through f0, which is the scale integration of the Fitzpatrick representative f of α.

    Lemma 3.3. Let ξ,ηRn be such that ηα0(ξ). Then, there exist uLpsol(Ω;Rn) and vLppot(Ω;Rn) such that

    v(ω)α(u(ω),ω),forμa.e.ωΩ. (20)

    Moreover, E(u)=ξ and E(v)=η, that is

    E(v)α0(E(u)). (21)

    Proof. Since f0 represents α0, ηα0(ξ) implies

    f0(ξ,η)=ξη. (22)

    Take now ˜uVpsol(Ω;Rn) and ˜vVppot(Ω;Rn) such that

    f0(ξ,η)=Ωf(ξ+˜u(ω),η+˜v(ω),ω)dμ. (23)

    Since E(˜u)=E(˜v)=0,

    ξη=E(ξ+˜u)E(η+˜v)(12)=Ω(ξ+˜u(ω))(η+˜v(ω))dμfF(Rn)Ωf(ξ+˜u(ω),η+˜v(ω),ω)dμ(23)=f0(ξ,η)(22)=ξη

    from which we obtain

    (ξ+˜u(ω))(η+˜v(ω))=f(ξ+˜u(ω),η+˜v(ω),ω),μ-a.e. ωΩ. (24)

    Let u(ω):=ξ+˜u(ω) and v(ω):=η+˜v(ω). Since f represents α, (24) is equivalent to (20). Moreover, since E(u)=ξ and E(v)=η, ηα0(ξ) implies also (21).

    Lemma 3.3 is also referred to as scale disintegration (see [26,Theorem 4.4]), as it shows that given a solution (ξ,η) to the integrated problem ηα0(ξ), it is possible to build a solution to the original problem v(ω)α(u(ω),ω). The converse, known as scale integration (see [26,Theorem 4.3]) is provided by the next Lemma.

    Lemma 3.4. Let uLpsol(Ω;Rn) and vLppot(Ω;Rn) satisfy

    v(ω)α(u(ω),ω),forμa.e.ωΩ, (25)

    then

    E(v)α0(E(u)). (26)

    Proof. By (25) and (12)

    Ωf(u(ω),v(ω),ω)dμ=Ωu(ω)v(ω)dμ=E(u)E(v).

    On the other hand, by definition of f0,

    Ωf(u(ω),v(ω),ω)dμf0(E(u),E(v))E(u)E(v).

    We conclude that f0(E(u),E(v))=E(u)E(v), which yields (26).

    How the properties of α and f reflect on α0 and f0 was thoroughly studied in [26]:

    Theorem 3.5. If

    fF(Ω,Rn) is uniformly bounded from below,

    there exists (u,v)Lp(Ω;Rn)×Lp(Ω;Rn) such that

    Ωf(u(ω),v(ω),ω)dμ<+,

    f represents a maximal monotone operator for μ-a.e. ωΩ then f0 represents a (proper) maximal monotone operator [26,Theorem 5.3]. Moreover, if f is strictly convex, then

    f0 is strictly convex [26,Lemma 5.4],

    the operators α0 and α10 are both strictly monotone [26,Proposition 5.5] and if Dom(α0) and Dom(α10) are unbounded, then α0 and α10 are coercive [26,Proposition 5.6].

    In order to obtain strict monotonicity of α0 and α10, by the next Lemma we provide an alternative to strict convexity of the Fitzpatrick function.

    Lemma 3.6. Let α(,ω):BB be maximal and strictly monotone, and assume that its Fitzpatrick representation f is coercive, in the sense of (14). Then its scale integration α0 is strictly monotone.

    Proof. For all ηiα0(ξi), i=1,2, by Lemma 3.3, there exist uiLpsol(Ω;Rn) and viLppot(Ω;Rn) such that

    vi(ω)α(ui(ω),ω),for μ-a.e. ωΩ (27)

    and E(ui)=ξi, E(vi)=ηi. By (12), strict monotonicity of α, and Jensen's inequality

    (η2η1)(ξ2ξ1)=Ω(v2(ω)v1(ω))(u2(ω)u1(ω))dμθΩ|u2(ω)u1(ω)|2dμθ|Ωu2(ω)u1(ω)dμ|2=θ|ξ2ξ1|2.

    Let DRn be a Lipschitz and bounded domain with |D|>0. We recall the following classical result.

    Lemma 3.7 (Div-Curl lemma, [15]). Let p]1,+[, let vn,vLp(D;Rm) and un,uLp(D;Rm) be such that

    vnvweaklyinLp(D;Rm),unuweaklyinLp(D;Rm).

    In addition, assume that

    {curlvn} is compact in W1,p(D;Rm×m),     {div un} is compact in W1,p(D).

    Then

    vnunvuin D(D).

    We are now ready to prove our main result concerning the stochastic homogenization of a maximal monotone relation.

    Theorem 3.8. Let (Ω,A,μ) be a probability space with an n-dimensional ergodic dynamical system Tx:ΩΩ, xRn. Let DRn be a bounded domain, let p]1,+[. Let α:Rn×ΩP(Rn) be a closed-valued, measurable, maximal monotone random operator, in the sense of (7)-(9).

    Let f be a Fitzpatrick representation of α, as in (10). Let f satisfy (14) and assume that for μ-a.e. ωΩ and ε0 there exists a couple

    (Jεω,Eεω)Lp(D;Rn)×Lp(D;Rn)

    such that

    {divJεω}ε0 is compact in W1,p(D),{curlEεω}ε0 is compact in W1,p(D;Rn×n), (28a)
    limε0Jεω=J0ωweaklyinLp(D;Rn),limε0Eεω=E0ωweaklyinLp(D;Rn), (28b)
    Eεω(x)α(Jεω(x),Tx/εω)a.e.inD. (28c)

    Then, for μ-a.e. ωΩ

    E0ω(x)α0(J0ω(x))a.e.inD, (29)

    where α0 is the maximal monotone operator represented by the homogenized representation f0:Rn×RnR{+} defined by

    f0(ξ,η):=inf{Ωf(ξ+u(ω),η+v(ω),ω)dμ:uVpsol(Ω;Rn),vVppot(Ω;Rn)}.

    Proof. By Lemma 3.3 for all ξ,ηRn such that ηα0(ξ) there exist uLpsol(Ω;Rn) and vLppot(Ω;Rn) such that E(u)=ξ, E(v)=η, and

    v(ω)α(u(ω),ω),for μ-a.e. ωΩ. (30)

    Define the stationary random fields uε,vε:D×ΩRn as

    uε(x,ω):=u(Tx/εω),vε(x,ω):=v(Tx/εω).

    For μ-a.e. ωΩ

    xuε(x,ω)Lploc(Rn;Rn),xvε(x,ω)Lploc(Rn;Rn).

    Equation (30) implies

    vε(x,ω)α(uε(x,ω),Tx/εω),for a.e. xD, μ-a.e. ωΩ. (31)

    By Birkhoff's Theorem (and (13), in particular), for μ-a.e. ωΩ we have

    uε(,ω)E(u)weakly in Lp(D;Rn),vε(,ω)E(v)weakly in Lp(D;Rn). (32)

    Since α is monotone, by (28c) and (31), for μ-a.e. ωΩ

    D(Eεω(x)vε(x,ω))(Jεω(x)uε(x,ω))ϕ(x)dx0, (33)

    for any ϕCc(D) with ϕ0. Since uε is solenoidal and vε is potential, by (28a)

    {curl(Eεωvε(,ω))}ε is compact in W1,p(D;Rn×n),{div(Jεωuε(,ω))}ε is compact in W1,p(D).

    By (28b), (32), and Lemma 3.7, we can thus pass to the limit as ε0 in (33):

    D(E0ω(x)E(v))(J0ω(x)E(u))ϕ(x)dx0,for μ-a.e. ωΩ.

    Since the last inequality holds for all nonnegative ϕCc(D), it holds also pointwise, for almost every xD:

    (E0ω(x)E(v))(J0ω(x)E(u))0,for μ-a.e. ωΩ.

    To conclude, since E(u)=ξ, E(v)=η are arbitrary vectors in Gα0, the maximal monotonicity of α0 implies that

    E0ω(x)α0(J0ω(x))

    for a.e. xD, μ-a.e. ωΩ.

    Remark 4. In this section's results, the function spaces Lpsol(Ω) and Lppot(Ω) can be generalized to a couple of nonempty, closed and convex sets

    ULp(Ω;Rn),VLp(Ω;Rn)

    such that

    E(uv)=E(u)E(v),(u,v)U×V.

    Furthermore, Proposition 1 and Lemma 3.3 remain valid if the previous equality is replaced by the inequality

    E(uv)E(u)E(v),(u,v)U×V.

    In this subsection we address the homogenization problem for the Ohm-Hall model for an electric conductor. For further information about the Ohm-Hall effect we refer the reader to [1,pp. 11-15], [12,Section 22] and we also follow [26] for the suitable mathematical formulation in terms of maximal monotone operators. We consider a non-homogeneous electric conductor, that occupies a bounded Lipschitz domain DR3 and is subjected to a magnetic field. We assume that the electric field E and the current density J fulfill the constitutive law

    E(x)α(J(x),x)+h(x)J(x)×B(x)+Ea(x)in D, (34)

    where α(,x):R3R3 is a maximal monotone mapping for a.e. xD, B is the magnetic induction field, h is the (material-dependent) Hall coefficient, and Ea is an applied electromotive force. We deal with a stationary system and thus we couple (34) with the Faraday law and with the stationary law of charge-conservation:

    curlE=g,divJ=0,

    where the vector field g:DR3 is prescribed. Following [26], we assume that h,B,Ea are given and we define the maximal monotone operator β(,x):R3R3 as

    β(J,x):=α(J,x)+h(x)J×B(x)+Ea(x).

    A single-valued parameter-dependent operator β is strictly monotone uniformly in x, if there exists θ>0 such that for a.e. xD

    (β(v1,x)β(v2,x))(v1v2)θv1v22v1,v2R3. (35)

    The following existence and uniqueness result is a classical consequence of the maximal monotonicity of α (see, e.g., [22,26]).

    Theorem 4.1. Let DR3 be a bounded Lipschitz domain. Let {β(,x)}xD be a family of single-valued maximal monotone operators on R3. Assume moreover that there exist constants a,c>0 and b0 such that for a.e. xD,vR3

    |β(x,v)|c(1+|v|), (36)
    β(x,v)va|v|2b. (37)

    Let gL2(D;R3) be given, such that divg=0, distributionally. Then, there exists E,JL2(D;R3) such that

    EL2+JL2C(1+gL2) (38)

    and, denoting by ν the outward unit normal to D,

    E(x)=β(J(x),x) inD, (39)
    curlE(x)=g(x) inD, (40)
    divJ(x)=0 inD, (41)
    E(x)×ν(x)=0 onD. (42)

    Moreover, if β is strictly monotone uniformly in xD, then the field J is uniquely determined, while if β1 is strictly monotone uniformly in xD, then the field E is uniquely determined.

    Remark 5. Conditions (40)-(41) have to be intended in the weak sense -see below -while (42) holds in H1/2(D;R3). Note that, according to (40), the divergence of g also vanishes.

    Let (Ω,A,μ) be a probability space endowed with a 3-dimensional ergodic dynamical system Tx:ΩΩ, with xR3. Let {α(,ω)}ωΩ be a family of maximal monotone operators on R3, and let

    hL(Ω),BL(Ω;R3),EaL2(Ω;R3). (43)

    For any JR3 and for any (x,ω)D×Ω let

    β(J,ω):=α(J,ω)+h(ω)J×B(ω)+Ea(ω). (44)

    In order to apply the scale integration procedure, we assume that

    the representative function f of β is coercive, in the sense of (14),  (45)

    moreover, to ensure uniqueness of a solution (E,J), we assume that

    β and β1 are strictly monotone, uniformly with respect to xD. (46)

    As in the previous section β0 stands for the maximal monotone operator represented by f0 given by (15). For any ε>0 define

    βε(,x,ω):=β(,Tx/εω).

    Then {βε(,x,ω)}(x,ω)D×Ω is a family of maximal monotone operators on R3. Let gεL2(D×Ω;R3) with gεg in L2(D;R3) for some gL2(D;R3), for μ-a.e. ωΩ; assume that

    divgε=0,in D(D), for μ-a.e. ωΩ. (47)

    We are ready to state and prove the homogenization result for the Ohm-Hall model.

    Theorem 4.2. Assume that (43)-(47) are fulfilled. Then

    1. For μ-a.e. ωΩ, for any ε>0 there exists (E_\omega^\varepsilon, J_\omega^\varepsilon)\in L^2(D;{\mathbb{R}}^3) \times L^2(D;{\mathbb{R}}^3) such that

    \begin{align} & E_\omega^\varepsilon(x) = \beta_\varepsilon (J_\omega^\varepsilon(x), x, \omega) & &in\;\;\;D, \label{P:incl-eps}\end{align} (48)
    \begin{align}& {\rm{curl}}\, E_\omega^\varepsilon(x) = g_\varepsilon (x, \omega) & &in\;\;\;D, \label{P:ele-eps}\end{align} (49)
    \begin{align}& {\rm{div}}\, J_\omega^\varepsilon(x) = 0 & &in\;\;\;D, \label{P:magn-eps}\end{align} (50)
    \begin{align}&E_\omega^\varepsilon(x) \times \nu(x) = 0 & &on \;\;\;\partial D. \label{P:bound-eps} \end{align} (51)

    2. There exists (E, J)\in L^2(D;{\mathbb{R}}^3) \times L^2(D;{\mathbb{R}}^3) such that, up to a subsequence,

    \label{eq:conv} E_\omega^\varepsilon \rightharpoonup E \;\;\;\;\;and\;\;\;\;\; J_\omega^\varepsilon \rightharpoonup J (52)

    as \varepsilon \to0, weakly in L^2(D;{\mathbb{R}}^3).

    3. The limit couple (E, J) is a weak solution of

    \begin{align} & E(x) = \beta_0(J(x)) \;\;\;\;\; & &in\;\;\; D, \label{P:incl-hom} \end{align} (53)
    \begin{align}& {\rm{curl}}\, E(x) = g(x)\;\;\;\;\; & &in\;\;\; D, \label{P:ele-hom} \end{align} (54)
    \begin{align}& {\rm{div}}\, J(x) = 0 \;\;\;\;\; & &in \;\;\; D, \label{P:magn-hom} \end{align} (55)
    \begin{align}& E(x) \times \nu(x) = 0 \;\;\;\;\; & &on\;\;\; \partial D. \label{P:bound-hom} \end{align} (56)

    Proof. 1. Assumption (46) implies that \beta is single valued and that almost every realization (x, v)\mapsto \beta(v, T_x\omega) satisfies the boundedness and coercivity assumptions (36) and (37). Therefore, by Theorem 4.1 for almost any \omega\in\Omega and for any \varepsilon >0 problem (48)-(51) has a unique solution.

    2. Let \omega\in \Omega be fixed. By (38) the families \{E_\omega^\varepsilon\}_\varepsilon and \{J_\omega^\varepsilon\}_\varepsilon are weakly relatively compact in L^2(D;{\mathbb{R}}^3), therefore, there exist a subsequence \varepsilon_n \to 0 and a couple (E_\omega, J_\omega)\in L^2(D;{\mathbb{R}}^3) \times L^2(D;{\mathbb{R}}^3) satisfying (52). A priori, (E_\omega, J_\omega) depends on \omega\in \Omega.

    3. The weak formulation of (49)-(51) is:

    \label{eq:weak} \int_D E_\omega^\varepsilon \cdot \text{curl}\, \phi + J_\omega^\varepsilon \cdot \nabla \psi\, dx = \int_D g_\varepsilon \cdot \phi\, dx, (57)

    for all \phi\in \{H^1(D;{\mathbb{R}}^3): \phi \times \nu = 0 \text{ on }\partial D\}, for all \psi\in H^1_0(D). Passing to the limit in (57), one gets

    \int_D E_\omega \cdot \text{curl}\, \phi + J_\omega \cdot \nabla \psi\, dx = \int_D g \cdot \phi\, dx,

    which is exactly the weak formulation of (54)-(56). Equations (49) and (50) imply that \{E_\omega^\varepsilon\}_\varepsilon and \{J_\omega^\varepsilon\}_\varepsilon satisfy also the div-curl compactness condition (28a). Therefore, we can apply the abstract stochastic homogenization Theorem 3.8, which yields

    E_\omega(x) = \beta_0(J_\omega(x)).

    We have thus proved that (E_\omega, J_\omega) is a weak solution of (53)-(56). In order to conclude we have to eliminate the dependence on \omega\in \Omega.

    4. By Lemma 3.6 and assumption (46), \beta_0 and \beta_0^{-1} are strictly monotone, and therefore (53)-(56) admits a unique solution. Thus, (E, J): = (E_\omega, J_\omega) is independent of \omega \in \Omega.

    Another straightforward application of the homogenization theorem 3.8 is given in the framework of deformations in continuum mechanics (see, e.g., [4,Chapter 3]). Elastic materials are usually described through the deformation vector u:D\times(0, T) \to {\mathbb{R}}^3 and the stress tensor \sigma:D\times(0, T) \to {\mathbb{R}}^{3 \times 3}_s. Here D\subset {\mathbb{R}}^3 is the spatial domain and {\mathbb{R}}^{3 \times 3}_s the space of symmetric 3\times 3 matrices. We assume the following constitutive relation relating stress and deformation:

    \label{eq:nlelastic} \sigma(x, t) = \beta(\nabla u(x, t), x), (58)

    where \beta(\cdot, x):{\mathbb{R}}^{3\times 3} \mapsto {\mathbb{R}}^{3\times 3} is a (single-valued) maximal monotone mapping for a.e. x\in D. We couple (58) with the conservation of linear momentum:

    \rho \partial _{t}^{2}u-\text{div}\sigma =F,

    where \rho is the density and F represents the external forces. For sake of simplicity, we choose to deal with the stationary system only and we set \rho\partial_t^2 u = 0.

    The following existence and uniqueness result is a classical consequence of the maximal monotonicity of \beta (see, e.g., [7,22]).

    Theorem 4.3. Let D \subset {\mathbb{R}}^3 be a bounded Lipschitz domain. Let \{\beta(\cdot, x)\}_{x\in D} be a family of single-valued maximal monotone operators on {\mathbb{R}}^{3\times 3} that satisfy (36) and (37). Let F \in L^2(D;{\mathbb{R}}^3) be given. Then, there exists \sigma\in L^2(D;{\mathbb{R}}^{3\times 3}) and u\in H^1_0(D;{\mathbb{R}}^3) such that

    \label{Q:estimates} {\|u\|}_{H^1} +{\|\sigma\|}_{L^2}\leq C\left(1+{\|F\|}_{L^2}\right) (59)

    and, denoting by \nu the outward unit normal to \partial D,

    \begin{align} \sigma(x) & = \beta(\nabla u(x), x)\;\;\;\;\; in\;\;\; D, \label{Q:incl}\end{align} (60)
    \begin{align} -div\, \sigma(x) & = F(x) \;\;\;\;\; in\;\;\; D, \label{Q:ele}\end{align} (61)
    \begin{align} u(x) & = 0 \;\;\;\;\; on\;\;\; \partial D. \label{Q:bound} \end{align} (62)

    Moreover, if \beta is strictly monotone uniformly in x\in D, then u is uniquely determined, while if \beta^{-1} is strictly monotone uniformly in x\in D, then \sigma is uniquely determined.

    As above, we consider a family of maximal monotone operators \{\beta(\cdot, \omega)\}_{\omega\in \Omega} on {\mathbb{R}}^{3\times 3}, \beta_0 stands for the maximal monotone operator represented by f_0, and for any \varepsilon >0

    \beta_\varepsilon (\cdot, x, \omega): = \beta(\cdot, T_{x/\varepsilon }\omega)

    defines a family of maximal monotone operators on {\mathbb{R}}^{3\times 3}. Let F_\varepsilon \in L^2(D\times \Omega;{\mathbb{R}}^3) with F_\varepsilon \rightharpoonup F in L^2(D;{\mathbb{R}}^3) for some F \in L^2(D;{\mathbb{R}}^3), for \mu-a.e. \omega\in\Omega. The correspondent homogenization theorem is the following.

    Theorem 4.4. Assume that (45) and (46) are fulfilled. Then

    1. For \mu-a.e. \omega\in\Omega, for any \varepsilon >0 there exist (u_\omega^\varepsilon, \sigma_\omega^\varepsilon)\in H^1_0(D;{\mathbb{R}}^3) \times L^2(D;{\mathbb{R}}^3) such that

    \begin{align} & \sigma_\omega^\varepsilon(x) = \beta_\varepsilon (\nabla u_\omega^\varepsilon(x), x, \omega) & &in\;\;\; D, \label{Q:incl-eps}\end{align} (63)
    \begin{align}& -{\rm{div}}\, \sigma_\omega^\varepsilon(x) = F_\varepsilon (x, \omega) & &in \;\;\;D, \label{Q:ele-eps}\end{align} (64)
    \begin{align}&u_\omega^\varepsilon(x) = 0 & &on\;\;\; \partial D. \label{Q:bound-eps} \end{align} (65)

    2. There exist (u, \sigma)\in H^1_0(D;{\mathbb{R}}^3) \times L^2(D;{\mathbb{R}}^3) such that, up to a subsequence,

    \label{Q:conv} u_\omega^\varepsilon \rightharpoonup u \;\;\;\;\;and\;\;\;\;\; \sigma_\omega^\varepsilon \rightharpoonup \sigma (66)

    as \varepsilon \to0, weakly in H^1(D;{\mathbb{R}}^3) and L^2(D;{\mathbb{R}}^3), respectively.

    3. The limit couple (u, \sigma) is a weak solution of

    \begin{align} & \sigma(x) = \beta_0(\nabla u(x)) & &in\;\;\; D, \label{Q:incl-hom}\end{align} (67)
    \begin{align}& -{\rm{div}}\, \sigma(x) = F(x) & &in \;\;D, \label{Q:ele-hom}\end{align} (68)
    \begin{align}& u(x) = 0 & &on\;\;\; \partial D. \label{Q:bound-hom} \end{align} (69)

    Proof. Steps 1. and 2. follow exactly as in the proof of Theorem 4.2.

    3. The weak formulation of (64)-(65) is the following:

    \label{Q:weak} \int_D \sigma_\omega^\varepsilon \cdot \nabla \phi\, dx = \int_D F_\varepsilon \phi\, dx, (70)

    for all \phi\in H^1_0(D). Passing to the limit as \varepsilon \to 0, one gets

    \int_D \sigma_\omega \cdot \nabla \phi\, dx = \int_D F \phi\, dx,

    which is exactly the weak formulation of (68)-(69). Equation (64) and estimate (59) imply that \{\sigma_\omega^\varepsilon\}_\varepsilon and \{\nabla u_\omega^\varepsilon\}_\varepsilon satisfy also the div-curl compactness condition (28a)

    {{\{\text{div}\sigma _{\omega }^{\varepsilon }\}}_{\varepsilon \ge 0}}\text{ is compact in }{{W}^{-1,2}}(D;{{\mathbb{R}}^{3}}),
    {{\{\text{curl}\nabla u_{\omega }^{\varepsilon }\}}_{\varepsilon \ge 0}}\text{ is compact in }{{W}^{-1,2}}(D;{{\mathbb{R}}^{3\times 3}}).

    Therefore, we can apply the abstract stochastic homogenization Theorem 3.8, (with \sigma in place of J and \nabla u in place of E), which yields

    \sigma_\omega(x) = \beta_0(\nabla u_\omega(x)).

    Finally, the strict monotonicity of the limit operators \beta_0 and \beta_0^{-1} yields uniqueness and therefore independence of \omega for the solution (u, \sigma).

    We would like to thank the anonymous referees for their valuable comments and remarks.



    [1] W. O. Kermack, A. G. Mckendrick, A contribution to the mathematical theory of epidemics, Proc. R. Soc. Math. Phys. Eng. Sci., 115 (1927), 700–721. https://doi.org/10.1098/rspa.1927.0118 doi: 10.1098/rspa.1927.0118
    [2] A. L. Lloyd, S. Valeika, Network models in epidemiology: an overview, in Complex Population Dynamics, World Scientific, (2007), 189–214. https://doi.org/10.1142/9789812771582_0008
    [3] W. M. Getz, R. M. Salter, L. L. Vissat, Simulation platforms to support teaching and research in epidemiological dynamics, preprint, medRxiv: 2022.02.09.22270752.
    [4] G. Huang, Y. Takeuchi, W. Ma, D. Wei, Global stability for delay SIR and SEIR epidemic models with nonlinear incidence rate, Bull. Math. Biol., 72 (2010), 1192–1207. https://doi.org/10.1007/s11538-009-9487-6 doi: 10.1007/s11538-009-9487-6
    [5] K. Hattaf, A. A. Lashari, Y. Louartassi, N. Yousfi, A delayed SIR epidemic model with a general incidence rate, Electron. J. Qual. Theory Differ. Equations, 2013 (2013), 1–9. http://dx.doi.org/10.14232/ejqtde.2013.1.3 doi: 10.14232/ejqtde.2013.1.3
    [6] M. Naim, Y. Sabbar, M. Zahri, B. Ghanbari, A. Zeb, N. Gul, et al., The impact of dual time delay and Caputo fractional derivative on the long-run behavior of a viral system with the non-cytolytic immune hypothesis, Phys. Scr., 97 (2022). https://doi.org/10.1088/1402-4896/ac9e7a
    [7] I. Al-Darabsah, A time-delayed SVEIR model for imperfect vaccine with a generalized nonmonotone incidence and application to measles, Appl. Math. Modell., 91 (2020), 74–92. https://doi.org/10.1016/j.apm.2020.08.084 doi: 10.1016/j.apm.2020.08.084
    [8] U. Ghosh, S. Chowdhury, D. K. Khan, W. Bengal, Mathematical modelling of epidemiology in presence of vaccination and delay, Comput. Sci. Inf. Technol., 2013 (2013), 91–98. http://dx.doi.org/10.5121/csit.2013.3209 doi: 10.5121/csit.2013.3209
    [9] K. E. Church, X. Liu, Analysis of a SIR model with pulse vaccination and temporary immunity: Stability, bifurcation and a cylindrical attractor, Nonlinear Anal. Real World Appl., 50 (2019), 240–266. https://doi.org/10.1016/j.nonrwa.2019.04.015 doi: 10.1016/j.nonrwa.2019.04.015
    [10] A. Kashkynbayev, D. Koptleuova, Global dynamics of tick-borne diseases, Math. Biosci. Eng., 17 (2020), 4064–4079. https://doi.org/10.3934/mbe.2020225 doi: 10.3934/mbe.2020225
    [11] A. Kashkynbayev, F. A. Rihan, Dynamics of fractional-order epidemic models with general nonlinear incidence rate and time-delay, Mathematics, 9 (2021), 1829. https://doi.org/10.3390/math9151829 doi: 10.3390/math9151829
    [12] A. Sow, C. Diallo, H. Cherifi, Interplay between vaccines and treatment for dengue control: An epidemic model, Plos One, 19 (2024), e0295025. https://doi.org/10.1371/journal.pone.0295025 doi: 10.1371/journal.pone.0295025
    [13] S. Gao, L. Chen, Z. Teng, Pulse vaccination of an SEIR epidemic model with time delay, Nonlinear Anal. Real World Appl., 9 (2008), 599–607. https://doi.org/10.1016/j.nonrwa.2006.12.004 doi: 10.1016/j.nonrwa.2006.12.004
    [14] W. Wei, M. Li, Pulse vaccination strategy in the SEIR epidemic dynamics model with latent period, in 2010 2nd International Conference on Information Engineering and Computer Science, (2010), 1–4. https://doi.org/10.1109/ICIECS.2010.5677692
    [15] G. Bolarin, O. M. Bamigbola, Pulse vaccination strategy in a SVEIRS epidemic model with two-time delay and saturated incidence, Univ. J. Appl. Math., 2014 (2014). http://dx.doi.org/10.13189/ujam.2014.020505
    [16] D. J. Nokes, J. Swinton, Vaccination in pulses: a strategy for global eradication of measles and polio?, Trends Microbiol., 5 (1997), 14–19. https://doi.org/10.1016/s0966-842x(97)81769-6 doi: 10.1016/s0966-842x(97)81769-6
    [17] O. Diekmann, J. A. P. Heesterbeek, J. A. J. Metz, On the definition and the computation of the basic reproduction ratio R_0 in models for infectious diseases in heterogeneous populations, J. Math. Biol., 28 (1990), 365–382. http://dx.doi.org/10.1007/BF00178324 doi: 10.1007/BF00178324
    [18] H. R. Thieme Global asymptotic stability in epidemic models, in Equadiff 82: Proceedings of the international conference held in Würzburg, (2006), 608–615. http://dx.doi.org/10.1007/BFb0103284
    [19] S. Gao, L. Chen, Z. Teng, Impulsive vaccination of an SEIRS model with time delay and varying total population size, Bull. Math. Biol., 69 (2007), 731–745. https://doi.org/10.1007/s11538-006-9149-x doi: 10.1007/s11538-006-9149-x
    [20] Z. Bai, Threshold dynamics of a time-delayed SEIRS model with pulse vaccination, Math. Biosci., 269 (2015), 178–185. https://doi.org/10.1016/j.mbs.2015.09.005 doi: 10.1016/j.mbs.2015.09.005
    [21] H. L. Smith, An introduction to delay differential equations with applications to the life sciences, Springer-Verlag, New York, 2011. https://doi.org/10.1007/978-1-4612-0873-0
    [22] A. Kashkynbayev, M. Yeleussinova, S. Kadyrov, An SIRS pulse vaccination model with nonlinear incidence rate and time delay, Lett. Biomath., 10 (2023), 133–148.
    [23] N. Bacaër, S. Guernaoui, The epidemic threshold of vector-borne diseases with seasonality: the case of cutaneous leishmaniasis in Chichaoua, Morocco J. Math. Biol., 53 (2006), 521–436. http://dx.doi.org/10.1007/s00285-006-0015-0
    [24] N. S. Al-Shimari, A. S. Al-Jilawi, Using second-order optimization algorithms approach for solving the numerical optimization problem with new software technique, in American Institute of Physics Conference Series, 2591 (2023), 050004. https://doi.org/10.1063/5.0119641
    [25] Z. Bai, Z. Ma, L. Jing, Y. Li, S. Wang, B. G. Wang, et al., Estimation and sensitivity analysis of a COVID-19 model considering the use of face mask and vaccination, Sci. Rep., 13 (2023), 6434. https://doi.org/10.1038/s41598-023-33499-z doi: 10.1038/s41598-023-33499-z
    [26] G. Albi, L. Pareschi, M. Zanella, Modelling lockdown measures in epidemic outbreaks using selective socio-economic containment with uncertainty, Math. Biosci. Eng., 18 (2021), 7161–7191. https://doi.org/10.3934/mbe.2021355 doi: 10.3934/mbe.2021355
    [27] G. Ledder, Incorporating mass vaccination into compartment models for infectious diseases, Math. Biosci. Eng., 19 (2022), 9457–9480. https://doi.org/10.3934/mbe.2022440 doi: 10.3934/mbe.2022440
    [28] H. B. Saydaliyev, S. Kadyrov, L. Chin Attitudes toward vaccination and its impact on economy, Int. J. Econ. Manage., 16 (2022). http://dx.doi.org/10.47836/ijeamsi.16.1.009
  • Reader Comments
  • © 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(1333) PDF downloads(128) Cited by(0)

Figures and Tables

Figures(16)  /  Tables(4)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog