
The number of daily new cases of an epidemic is assumed to evolve as the exponential of a Wiener process with Poissonian jumps that are exponentially distributed. The model parameters can be estimated by using the method of moments. In an application to the COVID-19 pandemic in the province of Québec, Canada, the proposed model is shown to be acceptable. General formulas for the probability that a given increase in the number of daily new cases is due to the normal variations of the continuous part of the process or rather to a jump of this process are given. Based on these formulas, the probability of observing the likely start of a new wave of infections is calculated for the application to the COVID-19 pandemic.
Citation: Mario Lefebvre. A Wiener process with jumps to model the logarithm of new epidemic cases[J]. AIMS Biophysics, 2022, 9(3): 271-281. doi: 10.3934/biophy.2022023
[1] | Guohui Zhang, Jinghe Sun, Xing Liu, Guodong Wang, Yangyang Yang . Solving flexible job shop scheduling problems with transportation time based on improved genetic algorithm. Mathematical Biosciences and Engineering, 2019, 16(3): 1334-1347. doi: 10.3934/mbe.2019065 |
[2] | Ruiping Yuan, Jiangtao Dou, Juntao Li, Wei Wang, Yingfan Jiang . Multi-robot task allocation in e-commerce RMFS based on deep reinforcement learning. Mathematical Biosciences and Engineering, 2023, 20(2): 1903-1918. doi: 10.3934/mbe.2023087 |
[3] | Shixuan Yao, Xiaochen Liu, Yinghui Zhang, Ze Cui . An approach to solving optimal control problems of nonlinear systems by introducing detail-reward mechanism in deep reinforcement learning. Mathematical Biosciences and Engineering, 2022, 19(9): 9258-9290. doi: 10.3934/mbe.2022430 |
[4] | Kongfu Hu, Lei Wang, Jingcao Cai, Long Cheng . An improved genetic algorithm with dynamic neighborhood search for job shop scheduling problem. Mathematical Biosciences and Engineering, 2023, 20(9): 17407-17427. doi: 10.3934/mbe.2023774 |
[5] | Jianguo Duan, Mengting Wang, Qinglei Zhang, Jiyun Qin . Distributed shop scheduling: A comprehensive review on classifications, models and algorithms. Mathematical Biosciences and Engineering, 2023, 20(8): 15265-15308. doi: 10.3934/mbe.2023683 |
[6] | Zilong Zhuang, Zhiyao Lu, Zizhao Huang, Chengliang Liu, Wei Qin . A novel complex network based dynamic rule selection approach for open shop scheduling problem with release dates. Mathematical Biosciences and Engineering, 2019, 16(5): 4491-4505. doi: 10.3934/mbe.2019224 |
[7] | Shaofeng Yan, Guohui Zhang, Jinghe Sun, Wenqiang Zhang . An improved ant colony optimization for solving the flexible job shop scheduling problem with multiple time constraints. Mathematical Biosciences and Engineering, 2023, 20(4): 7519-7547. doi: 10.3934/mbe.2023325 |
[8] | Zichen Wang, Xin Wang . Fault-tolerant control for nonlinear systems with a dead zone: Reinforcement learning approach. Mathematical Biosciences and Engineering, 2023, 20(4): 6334-6357. doi: 10.3934/mbe.2023274 |
[9] | Jin Zhang, Nan Ma, Zhixuan Wu, Cheng Wang, Yongqiang Yao . Intelligent control of self-driving vehicles based on adaptive sampling supervised actor-critic and human driving experience. Mathematical Biosciences and Engineering, 2024, 21(5): 6077-6096. doi: 10.3934/mbe.2024267 |
[10] | Lu-Wen Liao . A branch and bound algorithm for optimal television commercial scheduling. Mathematical Biosciences and Engineering, 2022, 19(5): 4933-4945. doi: 10.3934/mbe.2022231 |
The number of daily new cases of an epidemic is assumed to evolve as the exponential of a Wiener process with Poissonian jumps that are exponentially distributed. The model parameters can be estimated by using the method of moments. In an application to the COVID-19 pandemic in the province of Québec, Canada, the proposed model is shown to be acceptable. General formulas for the probability that a given increase in the number of daily new cases is due to the normal variations of the continuous part of the process or rather to a jump of this process are given. Based on these formulas, the probability of observing the likely start of a new wave of infections is calculated for the application to the COVID-19 pandemic.
In this paper, we consider the following diffusion equation on
{−∇⋅(α∇u)=f,inΩ,u=0,on∂Ω. | (1) |
To approximate (1), taking advantage of the adaptive mesh refinement (AMR) to save valuable computational resources, the adaptive finite element method on quadtree mesh is among the most popular ones in the engineering and scientific computing community [20]. Compared with simplicial meshes, quadtree meshes provide preferable performance in the aspects of the accuracy and robustness. There are lots of mature software packages (e.g., [1,2]) on quadtree meshes. To guide the AMR, one possible way is through the a posteriori error estimation to construct computable quantities to indicate the location that the mesh needs to be refined/coarsened, thus to balance the spacial distribution of the error which improves the accuracy per computing power. Residual-based and recovery-based error estimators are among the most popular ones used. In terms of accuracy, the recovery-based error estimator shows more appealing attributes [28,3].
More recently, newer developments on flux recovery have been studied by many researchers on constructing a post-processed flux in a structure-preserving approximation space. Using (1) as an example, given that the data
However, these
More recently, a new class of methods called the virtual element methods (VEM) were introduced in [4,8], which can be viewed as a polytopal generalization of the tensorial/simplicial finite element. Since then, lots of applications of VEM have been studied by many researchers. A usual VEM workflow splits the consistency (approximation) and the stability of the method as well as the finite dimensional approximation space into two parts. It allows flexible constructions of spaces to preserve the structure of the continuous problems such as higher order continuities, exact divergence-free spaces, and many others. The VEM functions are represented by merely the degrees of freedom (DoF) functionals, not the pointwise values. In computation, if an optimal order discontinuous approximation can be computed elementwisely, then adding an appropriate parameter-free stabilization suffices to guarantee the convergence under common assumptions on the geometry of the mesh.
The adoption of the polytopal element brings many distinctive advantages, for example, treating rectangular element with hanging nodes as polygons allows a simple construction of
The major ingredient in our study is an
If
(α∇uT,∇vT)=(f,vT),∀vT∈Qk(T)∩H10(Ω), | (2) |
in which the standard notation is opted.
Qk(T):={v∈H1(Ω):v|K∈Qk(K),∀K∈T}. |
and on
Qk(K):=Pk,k(K)={p(x)q(y),p∈Pk([a,b]),q∈Pk([c,d])}, |
where
On
NH:={z∈N:∃K∈T,z∈∂K∖NK} | (3) |
Otherwise the node
For each edge
{v}γe:=γv−+(1−γ)v+. |
In this subsection, the quadtree mesh
For the embedded element
Subsequently,
On
Vk(K):={τ∈H(div;K)∩H(rot;K):∇⋅τ∈Pk−1(K),∇×τ=0,τ⋅ne∈Pk(e),∀e⊂∂K}. | (4) |
An
Vk:={τ∈H(div):τ|K∈Vk(K),onK∈Tpoly}. | (5) |
Next we turn to define the degrees of freedom (DoFs) of this space. To this end, we define the set of scaled monomials
Pk(e):=span{1,s−mehe,(s−mehe)2,…,(s−mehe)k}, | (6) |
where
Pk(K):=span{mα(x):=(x−xKhK)α,|α|≤k}. | (7) |
The degrees of freedom (DoFs) are then set as follows for a
(e)k≥1∫e(τ⋅ne)mds,∀m∈Pk(e),one⊂Epoly.(i)k≥2∫Kτ⋅∇mdx,∀m∈Pk−1(K)/RonK∈Tpoly. | (8) |
Remark 1. We note that in our construction, the degrees of freedom to determine the curl of a VEM function originally in [8] are replaced by a curl-free constraint thanks to the flexibility to virtual element. The reason why we opt for this subspace is that the true flux
As the data
Consider
On each
{−α∇uT}γee⋅ne:=(γe(−αK−∇uT|K−)+(1−γe)(−αK+∇uT|K+))⋅ne, | (9) |
where
γe:=α1/2K+α1/2K++α1/2K−. | (10) |
First for both
σT⋅ne={−α∇uT}γee⋅ne. | (11) |
In the lowest order case
|K|∇⋅σT=∫K∇⋅σTdx=∫∂KσT⋅n∂Kds=∑e⊂∂K∫eσT⋅n∂K|eds. | (12) |
If
∇⋅σT=Πk−1f+cK. | (13) |
The reason to add
cK=1|K|(−∫KΠk−1fdx+∑e⊂∂K∫e{−α∇uT}γee⋅n∂K|eds), | (14) |
Consequently for
(σT,∇q)K=−(Πk−1f+cK,q)K+∑e⊂∂K({−α∇uT}γee⋅n∂K|e,q)e. | (15) |
To the end of constructing a computable local error indicator, inspired by the VEM formulation [8], the recovered flux is projected to a space with a much simpler structure. A local oblique projection
(Πτ,∇p)K=(τ,∇p)K,∀p∈Pk(K)/R. | (16) |
Next we are gonna show that this projection operator can be straightforward computed for vector fields in
When
(τ,∇p)K=−(∇⋅τ,p)K+(τ⋅n,p)∂K. | (17) |
By definition of the space (4) when
When
(τ,∇p)K=−(∇⋅τ,Πk−1p)K+(τ⋅n,p)∂K=(τ,∇Πk−1p)K+(τ⋅n,p−Πk−1p)∂K, | (18) |
which can be evaluated using both DoF sets
Given the recovered flux σT in Section 3, the recovery-based local error indicator
ηflux,K:=‖α−1/2(σT+α∇uT)‖K,andηres,K:=‖α−1/2(f−∇⋅σT)‖K, | (19) |
then
ηK={ηflux,Kwhenk=1,(η2flux,K+η2res,K)1/2whenk≥2. | (20) |
A computable
ˆηflux,K:=‖α−1/2KΠ(σT+αK∇uT)‖K, | (21) |
with the oblique projection
ˆηstab,K:=|α−1/2K(I−Π)(σT+αK∇uT)|S,K. | (22) |
Here
SK(v,w):=∑e⊂∂Khe(v⋅ne,w⋅ne)e+∑α∈Λ(v,∇mα)K(w,∇mα)K, | (23) |
where
The computable error estimator
ˆη2={∑K∈T(ˆη2flux,K+ˆη2stab,K)=:∑K∈Tˆη2Kwhenk=1,∑K∈T(ˆη2flux,K+ˆη2stab,K+η2res,K)=:∑K∈Tˆη2Kwhenk≥2. | (24) |
In this section, we shall prove the proposed recovery-based estimator
Theorem 4.1. Let
ˆη2flux,K≲osc(f;K)2+η2elem,K+η2edge,K, | (25) |
where
osc(f;K)=α−1/2KhK‖f−Πk−1f‖K,ηelem,K:=α−1/2KhK‖f+∇⋅(α∇uT)‖K,andηedge,K:=(∑e⊂∂KheαK+αKe‖[[α∇uT⋅ne]]‖2e)1/2. |
In the edge jump term,
Proof. Let
ˆη2flux,K=(Π(σT+αK∇uT),∇p)K=(σT+αK∇uT,∇p)K=−(∇⋅(σT+αK∇uT),p)K+∑e⊂∂K∫e(σT+αK∇uT)⋅n∂K|epds. | (26) |
By (11), without loss of generality we assume
(σT+αK∇uT)⋅ne=((1−γe)αK∇uT|K−(1−γe)αKe∇uT|Ke)⋅ne=α1/2Kα1/2K+α1/2Ke[[α∇uT⋅ne]]e. | (27) |
The boundary term in (26) can be then rewritten as
∫e(σT+αK∇uT)⋅nepds=∫e1α1/2K+α1/2Ke[[α∇uT⋅ne]]eα1/2Kpds≲1(αK+αKe)1/2h1/2e‖[[α∇uT⋅ne]]‖eα1/2Kh−1/2e‖p‖e. | (28) |
By a trace inequality on an edge of a polygon (Lemma 7.1), and the Poincaré inequality for
h−1/2e‖p‖e≲h−1K‖p‖K+‖∇p‖K≲‖∇p‖K. |
As a result,
∑e⊂∂K∫e(σT+αK∇uT)⋅nepds≲ηedge,Kα1/2K‖∇p‖e=ηedge,Kˆηflux,K. |
For the bulk term on
−(∇⋅(σT+αK∇uT),p)K≤|∇⋅(σT+αK∇uT)||K|1/2‖p‖K≤1|K|1/2|∫K∇⋅(σT+αK∇uT)dx|‖p‖K=1|K|1/2|∑e⊂∂K∫e(σT+αK∇uT)⋅neds|‖p‖K≤(∑e⊂∂K1α1/2K+α1/2Ke‖[[α∇uT⋅ne]]‖eα1/2Khe)‖∇p‖≲ηedge,Kˆηflux,K. |
When
−(∇⋅(σT+αK∇uT),p)K=−(Πk−1f+cK+∇⋅(αK∇uT),p)K≤(‖f−Πk−1f‖K+‖f+∇⋅(α∇uT)‖K+|cK||K|1/2)‖p‖K. | (29) |
The first two terms can be handled by combining the weights
cK|K|1/2=1|K|1/2(−∫K(Πk−1f−f)dx−∫K(f+∇⋅(α∇uT))dx+∫K∇⋅(α∇uT)dx+∑e⊂∂K∫e{−α∇uT}γee⋅neds)≤‖f−Πk−1f‖K+‖f+∇⋅(α∇uT)‖K+1|K|1/2∑e⊂∂K∫e(αK∇uT−{α∇uT}γee)⋅neds≤‖f−Πk−1f‖K+‖f+∇⋅(α∇uT)‖K+∑e⊂∂Kα1/2Kα1/2K+α1/2Ke‖[[α∇uT⋅ne]]‖e. | (30) |
The two terms on
−(∇⋅(σT+αK∇uT),p)K≲(osc(f;K)+ηelem,K+ηedge,K)α1/2K‖∇p‖ |
and the theorem follows.
Theorem 4.2. Under the same setting with Theorem 4.1, let
ˆη2stab,K≲osc(f;K)2+η2elem,K+η2edge,K, | (31) |
The constant depends on
Proof. This theorem follows directly from the norm equivalence Lemma 7.3:
|α−1/2K(I−Π)(σT+αK∇uT)|S,K≲|α−1/2K(σT+αK∇uT)|S,K, |
while evaluating the DoFs
Theorem 4.3. Under the same setting with Theorem 4.1, on any
ˆηK≲osc(f;K)+‖α1/2∇(u−uT)‖ωK, | (32) |
with a constant independent of
Proof. This is a direct consequence of Theorem 4.1 and 4.2 and the fact that the residual-based error indicator is efficient by a common bubble function argument.
In this section, we shall prove that the computable error estimator
Assumption 1 (
By Assumption 1, we denote the father
Assumption 2 (Quasi-monotonicity of
Denote
πzv={∫ωz∩ωm(z)vϕz∫ωz∩ωm(z)ϕzifz∈Ω,0ifz∈∂Ω. | (33) |
We note that if
Iv:=∑z∈N1(πzv)ϕz. | (34) |
Lemma 4.4 (Estimates for
α1/2Kh−1K‖v−Iv‖K+α1/2K‖∇Iv‖K≲‖α1/2∇v‖ωK, | (35) |
and for
∑K⊂ωzh−2z‖α1/2(v−πzv)ϕz‖2K≲‖α1/2∇v‖2ωz, | (36) |
in which
Proof. The estimate for
Denotes the subset of nodes
osc(f;T)2:=∑z∈N1∩(N∂Ω∪NI)h2z‖α−1/2f‖2ωz+∑z∈N1∖(N∂Ω∪NI)h2z‖α−1/2(f−fz)‖2ωz, | (37) |
with
Theorem 4.5. Let
‖α1/2∇(u−uT)‖≲(ˆη2+osc(f;T)2)1/2. | (38) |
For
‖α1/2∇(u−uT)‖≲ˆη, | (39) |
where the constant depends on
Proof. Let
‖α1/2∇ε‖2=(α∇(u−uT),∇(ε−Iε))=(α∇u+σT,∇(ε−Iε))−(α∇uT+σT,∇(ε−Iε))=(f−∇⋅σT,ε−Iε)−(α∇uT+σT,∇(ε−Iε))≤(∑K∈Tα−1Kh2K‖f−∇⋅σT‖2K)1/2(∑K∈TαKh−2K‖ε−Iε‖2K)1/2(∑K∈Tα−1K‖α∇uT+σT‖2K)1/2(∑K∈TαK‖∇(ε−Iε)‖2K)1/2.≲(∑K∈T(η2res,K+η2flux,K))1/2(∑K∈T‖α1/2∇ε‖ωK)1/2. |
Applying the norm equivalence of
When
(f,ε−Iε)=∑z∈N1∑K⊂ωz(f,(ε−πzε)ϕz)K, | (40) |
in which a patch-wise constant
(f−∇⋅σT,ε−Iε)=(f,ε−Iε)−(∇⋅(σT+αK∇uT),ε−Iε)=∑z∈N∑K⊂ωz(f,(ε−πzε)ϕz)K−(∇⋅(σT+αK∇uT),ε−Iε)≤(osc(f;T)2)1/2(∑z∈N1∑K⊂ωzh−2z‖α1/2(ε−πzε)ϕz‖2K)1/2+(∑K∈Tα−1Kh2K‖∇⋅(σT+αK∇uT)‖2K)1/2(∑K∈TαKh−2K‖ε−Iε‖2K)1/2. |
Applied an inverse inequality in Lemma 7.2 on
The numerics is prepared using the bilinear element for common AMR benchmark problems. The codes for this paper are publicly available on https://github.com/lyc102/ifem implemented using
The adaptive finite element (AFEM) iterative procedure is following the standard
SOLVE⟶ESTIMATE⟶MARK⟶REFINE. |
The linear system is solved by MATLAB
∑K∈Mˆη2K≥θ∑K∈Tˆη2K,forθ∈(0,1). |
Throughout all examples, we fix
η2Residual,K:=α−1Kh2K‖f+∇⋅(α∇uT)‖2K+12∑e⊂∂KheαK+αKe‖[[α∇uT⋅ne]]‖2e, |
Let
effectivityindex:=η/‖α1/2∇ε‖,whereε:=u−uT,η=ηResidualorˆη, |
i.e., the closer to 1 the effectivity index is, the more accurate this estimator is to measure the error of interest. We use an order
lnηn∼−rηlnNn+c1,andln‖α1/2∇(u−uT)‖∼−rerrlnNn+c2, |
where the subscript
In this example, a standard AMR benchmark on the L-shaped domain is tested. The true solution
The solution
This example is a common benchmark test problem introduced in [9], see also [17,12]) for elliptic interface problems. The true solution
μ(θ)={cos((π/2−δ)γ)⋅cos((θ−π/2+ρ)γ)if0≤θ≤π/2cos(ργ)⋅cos((θ−π+δ)γ)ifπ/2≤θ≤πcos(δγ)⋅cos((θ−π−ρ)γ)ifπ≤θ<3π/2cos((π/2−ρ)γ)⋅cos((θ−3π/2−δ)γ)if3π/2≤θ≤2π |
While
γ=0.1,R≈161.4476387975881,ρ=π/4,δ≈−14.92256510455152, |
By this choice, this function is very singular near the origin as the maximum regularity it has is
The AFEM procedure for this problem stops when the relative error reaches
A postprocessed flux with the minimum
However, we do acknowledge that the technical tool involving interpolation is essentially limited to
The author is grateful for the constructive advice from the anonymous reviewers. This work was supported in part by the National Science Foundation under grants DMS-1913080 and DMS-2136075, and no additional revenues are related to this work.
Unlike the identity matrix stabilization commonly used in most of the VEM literature, for
((σ,τ))K:=(Πσ,Πτ)K+SK((I−Π)σ,(I−Π)τ), | (41) |
where
To show the inverse inequality and the norm equivalence used in the reliability bound, on each element, we need to introduce some geometric measures. Consider a polygonal element
Proposition 1. Under Assumption 1,
Lemma 7.1 (Trace inequality on small edges [13]). If Proposition 1 holds, for
h−1/2e‖v‖e≲h−1K‖v‖K+‖∇v‖K,one⊂K. | (42) |
Proof. The proof follows essentially equation (3.9) in [13,Lemma 3.3] as a standard scaled trace inequality on
h−1/2e‖v‖e≲h−1e‖v‖Te+‖∇v‖Te≲h−1K‖v‖K+‖∇v‖K. |
Lemma 7.2 (Inverse inequalities). Under Assumption 1, we have the following inverse estimates for
‖∇⋅τ‖K≲h−1K‖τ‖K,and‖∇⋅τ‖K≲h−1KSK(τ,τ)1/2. | (43) |
Proof. The first inequality in (43) can be shown using a bubble function trick. Choose
‖∇⋅τ‖2K≲(∇⋅τ,pbK)=−(τ,∇(pbK))≤‖τ‖K‖∇(pbK)‖K, |
and then
‖∇(pbK)‖≤‖bK∇p‖K+‖p∇bK‖K≤‖bK‖∞,Ω‖∇p‖K+‖p‖K‖∇bK‖∞,K. |
Consequently, the first inequality in (43) follows above by the standard inverse estimate for polynomials
To prove the second inequality in (43), by integration by parts we have
‖∇⋅τ‖2=(∇⋅τ,p)=−(τ,∇p)+∑e⊂∂K(τ⋅ne,p). | (44) |
Expand
‖p‖2K=p⊤Mp≥p⊤diag(M)p≥minjMjj‖p‖2ℓ2≃h2K‖p‖2ℓ2, | (45) |
since
‖∇⋅τ‖2≤(∑α∈Λ(τ,mα)2K)1/2(∑α∈Λp2α)1/2+(∑e⊂∂Khe‖τ⋅ne‖2e)1/2(∑e⊂∂Kh−1e‖p‖2e)1/2≲SK(τ,τ)1/2(‖p‖ℓ2+h−1K‖p‖K+‖∇p‖K). |
As a result, the second inequality in (43) is proved when apply an inverse inequality for
Remark 2. While the proof in Lemma 7.2 relies on
Lemma 7.3 (Norm equivalence). Under Assumption 1, let
γ∗‖τ‖K≤‖τ‖h,K≤γ∗‖τ‖K, | (46) |
where both
Proof. First we consider the lower bound, by triangle inequality,
‖τ‖K≤‖Πτ‖K+‖(τ−Πτ)‖K. |
Since
‖τ‖2K≤SK(τ,τ),forτ∈Vk(K). | (47) |
To this end, we consider the weak solution to the following auxiliary boundary value problem on
{Δψ=∇⋅τinK,∂ψ∂n=τ⋅n∂Kon∂K. | (48) |
By a standard Helmholtz decomposition result (e.g. Proposition 3.1, Chapter 1[23]), we have
‖τ−∇ψ‖2K=(τ−∇ψ,∇⊥ϕ)=0. |
Consequently, we proved essentially the unisolvency of the modified VEM space (4) and
‖τ‖2K=(τ,∇ψ)K=(τ,∇ψ)K=−(∇⋅τ,ψ)K+(τ⋅n∂K,ψ)∂K≤‖∇⋅τ‖K‖ψ‖K+∑e⊂∂K‖τ⋅ne‖e‖ψ‖e≤‖∇⋅τ‖K‖ψ‖K+(∑e⊂∂Khe‖τ⋅ne‖2e)1/2(∑e⊂∂Kh−1e‖ψ‖2e)1/2 | (49) |
Proposition 1 allows us to apply an isotropic trace inequality on an edge of a polygon (Lemma 7.1), combining with the Poincaré inequality for
h−1/2e‖ψ‖e≲h−1K‖ψ‖K+‖∇ψ‖K≲‖∇ψ‖K. |
Furthermore applying the inverse estimate in Lemma 7.2 on the bulk term above, we have
‖τ‖2K≲SK(τ,τ)1/2‖∇ψ‖K, |
which proves the validity of (47), thus yield the lower bound.
To prove the upper bound, by
he‖τ⋅ne‖2e≲‖τ‖K,and|(τ,∇mα)K|≤‖τ‖K. | (50) |
To prove the first inequality, by Proposition 1 again, consider the edge bubble function
‖∇be‖∞,K=O(1/he),and‖be‖∞,K=O(1). | (51) |
Denote
‖τ⋅ne‖2e≲(τ⋅ne,beqe)e=x(τ⋅ne,beqe)∂K=(τ,qe∇be)K+(∇⋅τ,beqe)K≤‖τ‖K‖qe∇be‖Te+‖∇⋅τ‖K‖qebe‖Te,≤‖τ‖K‖qe‖Te‖∇be‖∞,K+‖∇⋅τ‖K‖qe‖Te‖be‖∞,K. |
Now by the fact that
The second inequality in (50) can be estimated straightforward by the scaling of the monomials (7)
|(τ,∇mα)K|≤‖τ‖K‖∇mα‖K≤‖τ‖K. | (52) |
Hence, (46) is proved.
[1] | Lefebvre M (2007) Applied Stochastic Processes. New York: Springer. https://doi.org/10.1007/978-0-387-48976-6 |
[2] |
Ross SM (2011) An Elementary Introduction to Mathematical Finance. Cambridge: Cambridge University Press. https://doi.org/10.1017/CBO9780511921483 ![]() |
[3] | Kermack WO, McKendrick AG (1927) A contribution to the mathematical theory of epidemics. Proc R Soc A 115: 700-721. https://doi.org/10.1098/rspa.1927.0118 |
[4] |
Lefebvre M (2018) Optimally ending an epidemic. Optimization 67: 399-407. https://doi.org/10.1080/02331934.2017.1397147 ![]() |
[5] |
Premarathna H, Srivastava H, Juman Z, et al. (2022) Mathematical modeling approach to predict COVID-19 infected people in Sri Lanka. AIMS Math 7: 4672-4699. https://doi.org/10.3934/math.2022260 ![]() |
[6] |
Bhattacharyya A, Chakraborty T, Rai SN (2022) Stochastic forecasting of COVID-19 daily new cases across countries with a novel hybrid time series model. Nonlinear Dynam 107: 3025-3040. https://doi.org/10.1007/s11071-021-07099-3 ![]() |
[7] | Ghosh S, Chakraborty A, Bhattacharya S (2022) How surface and fomite infection affect contagion dynamics: a study with self-propelled particles. Eur Phys J Spec Top . https://doi.org/10.1140/epjs/s11734-022-00431-x |
[8] |
Tesfay A, Saeed T, Zeb A, et al. (2021) Dynamics of a stochastic COVID-19 epidemic model with jump-diffusion. Adv Differ Equ 2021: 228. https://doi.org/10.1186/s13662-021-03396-8 ![]() |
[9] | Kharrazi ZEI, Saoud S (2021) Simulation of COVID-19 epidemic spread using stochastic differential equations with jump diffusion for SIR model. 7th International Conference on Optimization and Applications (ICOA) . 20801254. https://doi.org/10.1109/ICOA51614.2021.9442639 |
[10] |
Fabiano N, Radenović S (2021) Geometric Brownian motion and a new approach to the spread of Covid-19 in Italy. Gulf J Math 10: 25-30. https://doi.org/10.56947/gjom.v10i2.516 ![]() |
[11] | Ross SM (2019) Introduction to Probability Models. Amsterdam: Elsevier/Academic Press. https://doi.org/10.1016/C2017-0-01324-1 |
1. | Hengliang Tang, Jinda Dong, Solving Flexible Job-Shop Scheduling Problem with Heterogeneous Graph Neural Network Based on Relation and Deep Reinforcement Learning, 2024, 12, 2075-1702, 584, 10.3390/machines12080584 | |
2. | Chen Han, Xuanyin Wang, TPN:Triple network algorithm for deep reinforcement learning, 2024, 591, 09252312, 127755, 10.1016/j.neucom.2024.127755 | |
3. | Miguel S. E. Martins, João M. C. Sousa, Susana Vieira, A Systematic Review on Reinforcement Learning for Industrial Combinatorial Optimization Problems, 2025, 15, 2076-3417, 1211, 10.3390/app15031211 | |
4. | Tianhua Jiang, Lu Liu, A Bi-Population Competition Adaptive Interior Search Algorithm Based on Reinforcement Learning for Flexible Job Shop Scheduling Problem, 2025, 24, 1469-0268, 10.1142/S1469026824500251 | |
5. | Tianyuan Mao, A Review of Scheduling Methods for Multi-AGV Material Handling Systems in Mixed-Model Assembly Workshops, 2025, 5, 2710-0723, 227, 10.54691/p4x5a536 | |
6. | Peng Zhao, You Zhou, Di Wang, Zhiguang Cao, Yubin Xiao, Xuan Wu, Yuanshu Li, Hongjia Liu, Wei Du, Yuan Jiang, Liupu Wang, 2025, Dual Operation Aggregation Graph Neural Networks for Solving Flexible Job-Shop Scheduling Problem with Reinforcement Learning, 9798400712746, 4089, 10.1145/3696410.3714616 | |
7. | Yuxin Peng, Youlong Lyu, Jie Zhang, Ying Chu, Heterogeneous Graph Neural-Network-Based Scheduling Optimization for Multi-Product and Variable-Batch Production in Flexible Job Shops, 2025, 15, 2076-3417, 5648, 10.3390/app15105648 |