The COVID-19 pandemic (caused by SARS-CoV-2) has introduced significant challenges for accurate prediction of population morbidity and mortality by traditional variable-based methods of estimation. Challenges to modelling include inadequate viral physiology comprehension and fluctuating definitions of positivity between national-to-international data. This paper proposes that accurate forecasting of COVID-19 caseload may be best preformed non-parametrically, by vector autoregression (VAR) of verifiable data regionally.
A non-linear VAR model across 7 major demographically representative New York City (NYC) metropolitan region counties was constructed using verifiable daily COVID-19 caseload data March 12–July 23, 2020. Through association of observed case trends with a series of (county-specific) data-driven dynamic interdependencies (lagged values), a systematically non-assumptive approximation of VAR representation for COVID-19 patterns to-date and prospective upcoming trends was produced.
Modified VAR regression of NYC area COVID-19 caseload trends proves highly significant modelling capacity of observed patterns in longitudinal disease incidence (county R2 range: 0.9221–0.9751, all p < 0.001). Predictively, VAR regression of daily caseload results at a county-wide level demonstrates considerable short-term forecasting fidelity (p < 0.001 at one-step ahead) with concurrent capacity for longer-term (tested 11-week period) inferences of consistent, reasonable upcoming patterns from latest (model data update) disease epidemiology.
In contrast to macroscopic variable-assumption projections, regionally-founded VAR modelling may substantially improve projection of short-term community disease burden, reduce potential for biostatistical error, as well as better model epidemiological effects resultant from intervention. Predictive VAR extrapolation of existing public health data at an interdependent regional scale may improve accuracy of current pandemic burden prognoses.
Citation: Aaron C Shang, Kristen E Galow, Gary G Galow. Regional forecasting of COVID-19 caseload by non-parametric regression: a VAR epidemiological model[J]. AIMS Public Health, 2021, 8(1): 124-136. doi: 10.3934/publichealth.2021010
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[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 |
The COVID-19 pandemic (caused by SARS-CoV-2) has introduced significant challenges for accurate prediction of population morbidity and mortality by traditional variable-based methods of estimation. Challenges to modelling include inadequate viral physiology comprehension and fluctuating definitions of positivity between national-to-international data. This paper proposes that accurate forecasting of COVID-19 caseload may be best preformed non-parametrically, by vector autoregression (VAR) of verifiable data regionally.
A non-linear VAR model across 7 major demographically representative New York City (NYC) metropolitan region counties was constructed using verifiable daily COVID-19 caseload data March 12–July 23, 2020. Through association of observed case trends with a series of (county-specific) data-driven dynamic interdependencies (lagged values), a systematically non-assumptive approximation of VAR representation for COVID-19 patterns to-date and prospective upcoming trends was produced.
Modified VAR regression of NYC area COVID-19 caseload trends proves highly significant modelling capacity of observed patterns in longitudinal disease incidence (county R2 range: 0.9221–0.9751, all p < 0.001). Predictively, VAR regression of daily caseload results at a county-wide level demonstrates considerable short-term forecasting fidelity (p < 0.001 at one-step ahead) with concurrent capacity for longer-term (tested 11-week period) inferences of consistent, reasonable upcoming patterns from latest (model data update) disease epidemiology.
In contrast to macroscopic variable-assumption projections, regionally-founded VAR modelling may substantially improve projection of short-term community disease burden, reduce potential for biostatistical error, as well as better model epidemiological effects resultant from intervention. Predictive VAR extrapolation of existing public health data at an interdependent regional scale may improve accuracy of current pandemic burden prognoses.
coronavirus disease 2019;
Severe acute respiratory syndrome coronavirus 2;
vector autoregression;
New York City;
susceptible, infectious, removed (immune) framework of compartmental disease modelling;
susceptible, unquarantined infected, quarantined infected, confirmed infected framework of compartmental disease modelling;
Akaike Information Criteria;
confirmed positive COVID-19 case;
mean absolute error;
Centers for Disease Control and Prevention
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
αε(x,ω,⋅):=α(Tx/εω,⋅). | (1) |
The aim of this paper is to extend existing results where
The outline of the proof is the following: Let
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
(ξn,ηn)∈X×X′ such that ηn=Anξn, (ξn,ηn)⇀(ξ,η) and η=Aξ, | (2) |
then, denoting by
⟨yn−ηn,xn−ξn⟩≥0. |
In order to pass to the limit as
lim supn→∞⟨gn,fn⟩≤⟨g,f⟩∀(fn,gn)⇀(f,g) in X×X′, | (3) |
which, together with the weak convergence of
⟨y−η,x−ξ⟩≥0. |
By maximal monotonicity of
1. Existence and weak compactness of solutions
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
α a)⟶ f b)⟶ f0 c)⟶ α0, |
where a) the random operator
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
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
Gα:={(x,y)∈B×B′:y∈α(x)} |
be its graph. (We will equivalently write
(x,y)∈Gα⇒⟨y−y0,x−x0⟩≥0,∀(x0,y0)∈Gα | (4) |
and strictly monotone if there is
(x,y)∈Gα⇒⟨y−y0,x−x0⟩≥θ‖x−x0‖2,∀(x0,y0)∈Gα. | (5) |
We denote by
x∈α−1(y)⇔y∈α(x). |
The monotone operator
⟨y−y0,x−x0⟩≥0∀(x0,y0)∈Gα⇔(x,y)∈Gα. |
An operator
fα(x,y):=⟨y,x⟩+sup{⟨y−y0,x0−x⟩:(x0,y0)∈Gα}=sup{⟨y,x0⟩+⟨y0,x⟩−⟨y0,x0⟩:(x0,y0)∈Gα}. |
As a supremum of a family of linear functions, the Fitzpatrick function
Lemma 2.1. An operator
(x,y)∈Gα⇒fα(x,y)=⟨y,x⟩, |
while
{fα(x,y)≥⟨y,x⟩ ∀(x,y)∈B×B′fα(x,y)=⟨y,x⟩⟺(x,y)∈Gα. |
In the case
1. Let
fα(x,y)=(y−b+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
We define
f(x,y)≥⟨y,x⟩∀(x,y)∈B×B′. |
We call
(x,y)∈Gαf⇔f(x,y)=⟨y,x⟩. | (6) |
A crucial point is whether
Lemma 2.2. Let
(i) the operator
(ii) the class of maximal monotone operators is strictly contained in the class of operators representable by functions in
Proof. (ⅰ) If
g(P1+P22)−g(P1)+g(P2)2=14(⟨y1+y2,x1+x2⟩)−12(⟨y1,x1⟩+⟨y2,x2⟩)=14(⟨y1,x2⟩+⟨y2,x1⟩−⟨y1,x1⟩−⟨y2,x2⟩)=−14(⟨y2−y1,x2−x1⟩)>0. |
Since
f(P1+P22)>f(P1)+f(P2)2, |
which contradicts the convexity of
(ⅱ) Maximal monotone operators are representable by Lemma 2.1. To see that the inclusion is strict, assume that
h(x,y)=max{c,f(x,y)} |
clearly belongs to
h(x0,y0)≥c>f(x0,y0)=⟨y0,x0⟩, |
and thus
Remark 1. When
φ(x)+φ∗(y)≥⟨y,x⟩∀(x,y)∈B×B′, |
y∈α(x)⇔φ(x)+φ∗(y)=⟨y,x⟩. |
Thus, Fitzpatrick's representative function
fα(x,y)=(x+y)24≠x22+y22=φ(x)+φ∗(y). |
We need to introduce also parameter-dependent operators. For any measurable space
g−1(R):={x∈X:g(x)∩R≠∅} |
is measurable.
Let
α is B(B)⊗A-measurable, | (7) |
α(x,ω) is closed for any x∈B and for μ-a.e. ω∈Ω, | (8) |
α(⋅,ω) is (maximal) monotone for μ-a.e. ω∈Ω. | (9) |
If
(a)
(b)
(c)
As above,
y∈α(x,ω) ⇔ f(x,y,ω)=⟨y,x⟩∀(x,y)∈B×B′,for μ-a.e. ω∈Ω. | (10) |
Precisely, any measurable representative function
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
(a)
(b) for every
μ(TxE)=μ(E) | (11) |
(c) for any measurable function
˜f(x,ω)=f(Txω) |
is measurable.
Given an
E(f):=∫Ωfdμ. |
In the context of stochastic homogenization, it is useful to provide an orthogonal decomposition of
∫(vi∂φ∂xj−vj∂φ∂xi)dx=0, ∀i,j=1,…,n,∀φ∈D(Rn) |
and we say that
n∑i=1∫vi∂φ∂xidx=0, ∀φ∈D(Rn). |
Next we consider a vector field on
Lemma 2.3. Define the spaces
Vppot(Ω;Rn):={f∈Lppot(Ω;Rn):E(f)=0},Vpsol(Ω;Rn):={f∈Lpsol(Ω;Rn):E(f)=0}. |
The spaces
E(u⋅v)=E(u)⋅E(v) | (12) |
and the relations
(Vpsol(Ω;Rn))⊥=Vp′pot(Ω;Rn)⊕Rn,(Vppot(Ω;Rn))⊥=Vp′sol(Ω;Rn)⊕Rn |
hold in the sense of duality pairing between the spaces
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
E(f)=limε→01|K|∫Kf(Tx/εω)dx |
for
Remark 2. Birkhoff's theorem implies that
limε→01|K|∫K˜fε(x,ω)dx=E(f). |
Since this holds for every measurable bounded set
˜fε⇀E(f) weakly in Lploc(Rn;Rm) for μ-a.e. ω∈Ω. | (13) |
In what follows, the dynamical system
Let be given a probability space
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
For every fixed
f(ξ,η,ω)≥c(|ξ|p+|η|p′)+k(ω). | (14) |
We define the homogenised representation
f0(ξ,η):=inf{∫Ωf(ξ+v(ω),η+u(ω),ω)dμ:u∈Vppot(Ω;Rn),v∈Vp′sol(Ω;Rn)}. | (15) |
Lemma 3.1. Let
1i.e., for all
h(x):=infy∈Kg(x,y) |
is weakly lower semicontinuous and coercive. Moreover, if
Proof. Let
lim infj→+∞h(xj)≥h(x). | (16) |
Let
ℓ:=lim infj→+∞h(xj). |
If
h(xj)=infy∈Kg(xj,y)≥g(xj,yj)−ε. | (17) |
Therefore
g(xj,yj)≤2ℓ+ε∀j∈N. |
By the coercivity assumption on
lim infk→+∞h(xjk)≥lim infk→+∞g(xjk,yjk)−ε≥g(x,y)−ε≥h(x)−ε. | (18) |
By arbitrariness of
h(λx1+(1−λ)x2)≤g(λx1+(1−λ)x2,λy1+(1−λ)y2)≤λg(x1,y1)+(1−λ)g(x2,y2). |
Passing to the infimum with respect to
h(λx1+(1−λ)x2)≤λh(x1)+(1−λ)h(x2). |
Regarding the coercivity of
Bt:={x∈X:h(x)≤t},At:={x∈X:g(x,y)≤t, for some y∈K}. |
Let
In the proof of Proposition 1 we need the following estimate
Lemma 3.2. For all
∫Ω|ξ+u(ω)|pdμ≥C∫Ω|ξ|p+|u(ω)|pdμ |
for all
Proof. Consider the operator
Φ:Lp(Ω;Rn)→Lp(Ω;Rn)×Lp(Ω;Rn)u↦(E(u),u−E(u)). |
Clearly,
∫Ω|E(u)|pdμ+∫Ω|u(ω)−E(u)|pdμ≤(‖E(u)‖Lp+‖u−E(u)‖Lp)p≤2p/2(‖E(u)‖2Lp+‖u−E(u)‖2Lp)p/2=2p/2‖Φ(u)‖pLp×Lp≤C‖u‖pLp=C∫Ω|u(ω)|pdμ. |
Apply now the last inequality to
∫Ω|ξ|p+|˜u(ω)|pdμ≤C∫Ω|ξ+˜u(ω)|pdμ. |
Proposition 1. For all
f0(ξ,η)≥ξ⋅η∀(ξ,η)∈Rn×Rn. | (19) |
Proof. Let
Fξ,η(u,v):=∫Ωf(ξ+v(ω),η+u(ω),ω)dμ. |
We prove that the problem
Fξ,η(u,v)≤lim infh→∞Fξ,η(uh,vh)=infKFξ,η. |
This concludes the first part of the statement. We now want to show that
Fξ,η(u,v)≥c∫Ω|ξ+v(ω)|p+|η+u(ω)|p′+k(ω)dμ≥C∫Ω|ξ|p+|u(ω)|p+|η|p′+|v(ω)|p′dμ+E(k)≥C(|ξ|p+‖u‖pLp(Ω)+|η|p′+‖v‖p′Lp′(Ω))−‖k‖L1(Ω). |
Thus, for any
{(ξ,η,(u,v))∈Rn×Rn×K:Fξ,η(u,v)≤M} |
is bounded in
f0(ξ,η)=∫Ωf(ξ+˜u(ω),η+˜v(ω),ω)dμ≥∫Ω(ξ+˜u(ω))⋅(η+˜v(ω))dμ=E(ξ+˜u)⋅E(η+˜v)=ξ⋅η, |
which yields the conclusion.
We denote by
η∈α0(ξ)⇔f0(ξ,η)=ξ⋅η. |
We refer to
Lemma 3.3. Let
v(ω)∈α(u(ω),ω),forμ−a.e.ω∈Ω. | (20) |
Moreover,
E(v)∈α0(E(u)). | (21) |
Proof. Since
f0(ξ,η)=ξ⋅η. | (22) |
Take now
f0(ξ,η)=∫Ωf(ξ+˜u(ω),η+˜v(ω),ω)dμ. | (23) |
Since
ξ⋅η=E(ξ+˜u)⋅E(η+˜v)(12)=∫Ω(ξ+˜u(ω))⋅(η+˜v(ω))dμf∈F(Rn)≤∫Ωf(ξ+˜u(ω),η+˜v(ω),ω)dμ(23)=f0(ξ,η)(22)=ξ⋅η |
from which we obtain
(ξ+˜u(ω))⋅(η+˜v(ω))=f(ξ+˜u(ω),η+˜v(ω),ω),μ-a.e. ω∈Ω. | (24) |
Let
Lemma 3.3 is also referred to as scale disintegration (see [26,Theorem 4.4]), as it shows that given a solution
Lemma 3.4. Let
v(ω)∈α(u(ω),ω),forμ−a.e.ω∈Ω, | (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
∫Ωf(u(ω),v(ω),ω)dμ≥f0(E(u),E(v))≥E(u)⋅E(v). |
We conclude that
How the properties of
Theorem 3.5. If
∫Ωf(u(ω),v(ω),ω)dμ<+∞, |
In order to obtain strict monotonicity of
Lemma 3.6. Let
Proof. For all
vi(ω)∈α(ui(ω),ω),for μ-a.e. ω∈Ω | (27) |
and
(η2−η1)⋅(ξ2−ξ1)=∫Ω(v2(ω)−v1(ω))⋅(u2(ω)−u1(ω))dμ≥θ∫Ω|u2(ω)−u1(ω)|2dμ≥θ|∫Ωu2(ω)−u1(ω)dμ|2=θ|ξ2−ξ1|2. |
Let
Lemma 3.7 (Div-Curl lemma, [15]). Let
vn⇀vweaklyinLp′(D;Rm),un⇀uweaklyinLp(D;Rm). |
In addition, assume that
{curlvn} is compact in W−1,p′(D;Rm×m), {div un} is compact in W−1,p(D). |
Then
vn⋅un∗⇀v⋅uin D′(D). |
We are now ready to prove our main result concerning the stochastic homogenization of a maximal monotone relation.
Theorem 3.8. Let
Let
(Jεω,Eεω)∈Lp(D;Rn)×Lp′(D;Rn) |
such that
{divJεω}ε≥0 is compact in W−1,p(D),{curlEεω}ε≥0 is compact in W−1,p′(D;Rn×n), | (28a) |
limε→0Jεω=J0ωweaklyinLp(D;Rn),limε→0Eεω=E0ωweaklyinLp′(D;Rn), | (28b) |
Eεω(x)∈α(Jεω(x),Tx/εω)a.e.inD. | (28c) |
Then, for
E0ω(x)∈α0(J0ω(x))a.e.inD, | (29) |
where
f0(ξ,η):=inf{∫Ωf(ξ+u(ω),η+v(ω),ω)dμ:u∈Vpsol(Ω;Rn),v∈Vp′pot(Ω;Rn)}. |
Proof. By Lemma 3.3 for all
v(ω)∈α(u(ω),ω),for μ-a.e. ω∈Ω. | (30) |
Define the stationary random fields
uε(x,ω):=u(Tx/εω),vε(x,ω):=v(Tx/εω). |
For
x↦uε(x,ω)∈Lploc(Rn;Rn),x↦vε(x,ω)∈Lp′loc(Rn;Rn). |
Equation (30) implies
vε(x,ω)∈α(uε(x,ω),Tx/εω),for a.e. x∈D, μ-a.e. ω∈Ω. | (31) |
By Birkhoff's Theorem (and (13), in particular), for
uε(⋅,ω)⇀E(u)weakly in Lp(D;Rn),vε(⋅,ω)⇀E(v)weakly in Lp′(D;Rn). | (32) |
Since
∫D(Eεω(x)−vε(x,ω))⋅(Jεω(x)−uε(x,ω))ϕ(x)dx≥0, | (33) |
for any
{curl(Eεω−vε(⋅,ω))}ε is compact in W−1,p′(D;Rn×n),{div(Jεω−uε(⋅,ω))}ε is compact in W−1,p(D). |
By (28b), (32), and Lemma 3.7, we can thus pass to the limit as
∫D(E0ω(x)−E(v))⋅(J0ω(x)−E(u))ϕ(x)dx≥0,for μ-a.e. ω∈Ω. |
Since the last inequality holds for all nonnegative
(E0ω(x)−E(v))⋅(J0ω(x)−E(u))≥0,for μ-a.e. ω∈Ω. |
To conclude, since
E0ω(x)∈α0(J0ω(x)) |
for a.e.
Remark 4. In this section's results, the function spaces
U⊂Lp(Ω;Rn),V⊂Lp′(Ω;Rn) |
such that
E(u⋅v)=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(u⋅v)≥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
E(x)∈α(J(x),x)+h(x)J(x)×B(x)+Ea(x)in D, | (34) |
where
curlE=g,divJ=0, |
where the vector field
β(J,x):=α(J,x)+h(x)J×B(x)+Ea(x). |
A single-valued parameter-dependent operator
(β(v1,x)−β(v2,x))⋅(v1−v2)≥θ‖v1−v2‖2∀v1,v2∈R3. | (35) |
The following existence and uniqueness result is a classical consequence of the maximal monotonicity of
Theorem 4.1. Let
|β(x,v)|≤c(1+|v|), | (36) |
β(x,v)⋅v≥a|v|2−b. | (37) |
Let
‖E‖L2+‖J‖L2≤C(1+‖g‖L2) | (38) |
and, denoting by
E(x)=β(J(x),x) inD, | (39) |
curlE(x)=g(x) inD, | (40) |
divJ(x)=0 inD, | (41) |
E(x)×ν(x)=0 on∂D. | (42) |
Moreover, if
Remark 5. Conditions (40)-(41) have to be intended in the weak sense -see below -while (42) holds in
Let
h∈L∞(Ω),B∈L∞(Ω;R3),Ea∈L2(Ω;R3). | (43) |
For any
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
β and β−1 are strictly monotone, uniformly with respect to x∈D. | (46) |
As in the previous section
βε(⋅,x,ω):=β(⋅,Tx/εω). |
Then
divgε=0,in D′(D), for μ-a.e. ω∈Ω. | (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
Eεω(x)=βε(Jεω(x),x,ω)inD, | (48) |
curlEεω(x)=gε(x,ω)inD, | (49) |
divJεω(x)=0inD, | (50) |
Eεω(x)×ν(x)=0on∂D. | (51) |
2. There exists
Eεω⇀EandJεω⇀J | (52) |
as
3. The limit couple
E(x)=β0(J(x))inD, | (53) |
curlE(x)=g(x)inD, | (54) |
divJ(x)=0inD, | (55) |
E(x)×ν(x)=0on∂D. | (56) |
Proof. 1. Assumption (46) implies that
2. Let
3. The weak formulation of (49)-(51) is:
∫DEεω⋅curlϕ+Jεω⋅∇ψdx=∫Dgε⋅ϕdx, | (57) |
for all
∫DEω⋅curlϕ+Jω⋅∇ψdx=∫Dg⋅ϕdx, |
which is exactly the weak formulation of (54)-(56). Equations (49) and (50) imply that
Eω(x)=β0(Jω(x)). |
We have thus proved that
4. By Lemma 3.6 and assumption (46),
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
σ(x,t)=β(∇u(x,t),x), | (58) |
where
ρ∂2tu−divσ=F, |
where
The following existence and uniqueness result is a classical consequence of the maximal monotonicity of
Theorem 4.3. Let
‖u‖H1+‖σ‖L2≤C(1+‖F‖L2) | (59) |
and, denoting by
σ(x)=β(∇u(x),x)inD, | (60) |
−divσ(x)=F(x)inD, | (61) |
u(x)=0on∂D. | (62) |
Moreover, if
As above, we consider a family of maximal monotone operators
βε(⋅,x,ω):=β(⋅,Tx/εω) |
defines a family of maximal monotone operators on
Theorem 4.4. Assume that (45) and (46) are fulfilled. Then
1. For
σεω(x)=βε(∇uεω(x),x,ω)inD, | (63) |
−divσεω(x)=Fε(x,ω)inD, | (64) |
uεω(x)=0on∂D. | (65) |
2. There exist
uεω⇀uandσεω⇀σ | (66) |
as
3. The limit couple
σ(x)=β0(∇u(x))inD, | (67) |
−divσ(x)=F(x)inD, | (68) |
u(x)=0on∂D. | (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:
∫Dσεω⋅∇ϕdx=∫DFεϕdx, | (70) |
for all
∫Dσω⋅∇ϕdx=∫DFϕdx, |
which is exactly the weak formulation of (68)-(69). Equation (64) and estimate (59) imply that
{divσεω}ε≥0 is compact in W−1,2(D;R3), |
{curl∇uεω}ε≥0 is compact in W−1,2(D;R3×3). |
Therefore, we can apply the abstract stochastic homogenization Theorem 3.8, (with
σω(x)=β0(∇uω(x)). |
Finally, the strict monotonicity of the limit operators
We would like to thank the anonymous referees for their valuable comments and remarks.
[1] | World Health Organization Coronavirus disease (COVID-19): Weekly Epidemiological Report, 27 January 2021 (2021) .Available from: https://www.who.int/publications/m/item/weekly-epidemiological-update---27-january-2021. |
[2] |
Bai Y, Yao L, Wei T, et al. (2020) Presumed asymptomatic carrier transmission of COVID-19. JAMA 323: 1406-1407. doi: 10.1001/jama.2020.2565
![]() |
[3] | Bastos ML, Tavaziva G, Abidi SK, et al. (2020) Diagnostic accuracy of serological tests for covid-19: systematic review and meta-analysis. BMJ 1: 370. |
[4] |
Roda WC, Varughese MB, Han D, et al. (2020) Why is it difficult to accurately predict the COVID-19 epidemic? Infect Dis Modell 5: 271-281. doi: 10.1016/j.idm.2020.03.001
![]() |
[5] | Naudé W (2020) Artificial intelligence vs COVID-19: limitations, constraints and pitfalls. AI Soc 35. |
[6] |
Volpert V, Banerjee M, Petrovskii S (2020) On a quarantine model of coronavirus infection and data analysis. Math Modell Nat Phenom 15: 24. doi: 10.1051/mmnp/2020006
![]() |
[7] | Zhao S, Chen H (2020) Modeling the epidemic dynamics and control of COVID-19 outbreak in China. Quant Biol 11: 1-9. |
[8] | Shen CY (2020) A logistic growth model for COVID-19 proliferation: experiences from China and international implications in infectious diseases. Int J Infect Dis . |
[9] |
Elliott G, Stock JH (2001) Confidence intervals for autoregressive coefficients near one. J Econometrics 103: 155-181. doi: 10.1016/S0304-4076(01)00042-2
![]() |
[10] |
Hsiao WC, Huang HY, Ing CK (2018) Interval Estimation for a First-Order Positive Autoregressive Process. J Time Ser Anal 39: 447-467. doi: 10.1111/jtsa.12297
![]() |
[11] | Branas CC, Rundle A, Pei S, et al. (2020) Flattening the curve before it flattens us: hospital critical care capacity limits and mortality from novel coronavirus (SARS-CoV2) cases in US counties. medRxiv . |
[12] | Biswas K, Khaleque A, Sen P (2003) Covid-19 spread: Reproduction of data and prediction using a SIR model on Euclidean network. arXiv preprint arXiv:2003.07063 2020 Mar 16. |
[13] |
Postnikov EB (2020) Estimation of COVID-19 dynamics “on a back-of-envelope”: Does the simplest SIR model provide quantitative parameters and predictions? Chaos, Solitons Fractals 135: 109841. doi: 10.1016/j.chaos.2020.109841
![]() |
[14] |
Metcalf CJ, Lessler J (2017) Opportunities and challenges in modeling emerging infectious diseases. Science 357: 149-152. doi: 10.1126/science.aam8335
![]() |
[15] |
Funk S, Camacho A, Kucharski AJ, et al. (2018) Real-time forecasting of infectious disease dynamics with a stochastic semi-mechanistic model. Epidemics 22: 56-61. doi: 10.1016/j.epidem.2016.11.003
![]() |
[16] |
He ZL, Li JG, Nie L, et al. (2017) Nonlinear state-dependent feedback control strategy in the SIR epidemic model with resource limitation. Adv Differ Equ 2017: 1-8. doi: 10.1186/s13662-016-1057-2
![]() |
[17] | Dubey B, Dubey P, Dubey US (2015) Dynamics of an SIR Model with Nonlinear Incidence and Treatment Rate. Appl Appl Math 10: 718-737. |
[18] |
Harjule P, Tiwari V, Kumar A (2021) Mathematical models to predict COVID-19 outbreak: An interim review. J Interdiscip Math 13: 1-26. doi: 10.1080/09720502.2020.1848316
![]() |
[19] |
Eker S (2020) Validity and usefulness of COVID-19 models. Humanit Soc Sci Commun 7: 1-5. doi: 10.1057/s41599-020-00553-4
![]() |
[20] | Iwasaki A, Yang Y (2020) The potential danger of suboptimal antibody responses in COVID-19. Nat Rev Immunol 21: 1-3. |
[21] | To KK, Tsang OT, Leung WS, et al. (2020) Temporal profiles of viral load in posterior oropharyngeal saliva samples and serum antibody responses during infection by SARS-CoV-2: an observational cohort study. Lancet Infect Dis . |
[22] | Bertozzi AL, Franco E, Mohler G, et al. (2020) The challenges of modeling and forecasting the spread of COVID-19. arXiv preprint arXiv:2004.04741 . |
[23] | Nakamura G, Grammaticos B, Deroulers C, et al. (2020) Effective epidemic model for COVID-19 using accumulated deaths. arXiv preprint arXiv:2007.02855 . |
[24] | Bogg T, Milad E Slowing the Spread of COVID-19: Demographic, personality, and social cognition predictors of guideline adherence in a representative US sample (2020) .Available from: https://www.researchgate.net/publication/340427042_Slowing_the_Spread_of_COVID-19_Demographic_Personality_and_Social_Cognition_Predictors_of_Guideline_Adherence_in_a_Representative_US_Sample. |
[25] |
Dowd JB, Andriano L, Brazel DM, et al. (2020) Demographic science aids in understanding the spread and fatality rates of COVID-19. P Natl Acad Sci USA 117: 9696-9698. doi: 10.1073/pnas.2004911117
![]() |
![]() |
![]() |