Processing math: 100%
Research article Special Issues

Mathematical assessment of the role of waning and boosting immunity against the BA.1 Omicron variant in the United States

  • Three safe and effective vaccines against SARS-CoV-2 have played a major role in combating COVID-19 in the United States. However, the effectiveness of these vaccines and vaccination programs has been challenged by the emergence of new SARS-CoV-2 variants of concern. A new mathematical model is formulated to assess the impact of waning and boosting of immunity against the Omicron variant in the United States. To account for gradual waning of vaccine-derived immunity, we considered three vaccination classes that represent high, moderate and low levels of immunity. We showed that the disease-free equilibrium of the model is globally-asymptotically, for two special cases, if the associated reproduction number is less than unity. Simulations of the model showed that vaccine-derived herd immunity can be achieved in the United States via a vaccination-boosting strategy which entails fully vaccinating at least 59% of the susceptible populace followed by the boosting of about 72% of the fully-vaccinated individuals whose vaccine-derived immunity has waned to moderate or low level. In the absence of boosting, waning of immunity only causes a marginal increase in the average number of new cases at the peak of the pandemic, while boosting at baseline could result in a dramatic reduction in the average number of new daily cases at the peak. Specifically, for the fast immunity waning scenario (where both vaccine-derived and natural immunity are assumed to wane within three months), boosting vaccine-derived immunity at baseline reduces the average number of daily cases at the peak by about 90% (in comparison to the corresponding scenario without boosting of the vaccine-derived immunity), whereas boosting of natural immunity (at baseline) only reduced the corresponding peak daily cases (in comparison to the corresponding scenario without boosting of natural immunity) by approximately 62%. Furthermore, boosting of vaccine-derived immunity is more beneficial (in reducing the burden of the pandemic) than boosting of natural immunity. Finally, boosting vaccine-derived immunity increased the prospects of altering the trajectory of COVID-19 from persistence to possible elimination.

    Citation: Salman Safdar, Calistus N. Ngonghala, Abba B. Gumel. Mathematical assessment of the role of waning and boosting immunity against the BA.1 Omicron variant in the United States[J]. Mathematical Biosciences and Engineering, 2023, 20(1): 179-212. doi: 10.3934/mbe.2023009

    Related Papers:

    [1] Yanping Yang, Muhammad Shoaib Saleem, Waqas Nazeer, Ahsan Fareed Shah . New Hermite-Hadamard inequalities in fuzzy-interval fractional calculus via exponentially convex fuzzy interval-valued function. AIMS Mathematics, 2021, 6(11): 12260-12278. doi: 10.3934/math.2021710
    [2] Muhammad Bilal Khan, Pshtiwan Othman Mohammed, Muhammad Aslam Noor, Abdullah M. Alsharif, Khalida Inayat Noor . New fuzzy-interval inequalities in fuzzy-interval fractional calculus by means of fuzzy order relation. AIMS Mathematics, 2021, 6(10): 10964-10988. doi: 10.3934/math.2021637
    [3] Fangfang Shi, Guoju Ye, Dafang Zhao, Wei Liu . Some integral inequalities for coordinated log-h-convex interval-valued functions. AIMS Mathematics, 2022, 7(1): 156-170. doi: 10.3934/math.2022009
    [4] Hongling Zhou, Muhammad Shoaib Saleem, Waqas Nazeer, Ahsan Fareed Shah . Hermite-Hadamard type inequalities for interval-valued exponential type pre-invex functions via Riemann-Liouville fractional integrals. AIMS Mathematics, 2022, 7(2): 2602-2617. doi: 10.3934/math.2022146
    [5] Muhammad Bilal Khan, Gustavo Santos-García, Hüseyin Budak, Savin Treanțǎ, Mohamed S. Soliman . Some new versions of Jensen, Schur and Hermite-Hadamard type inequalities for (p,J)-convex fuzzy-interval-valued functions. AIMS Mathematics, 2023, 8(3): 7437-7470. doi: 10.3934/math.2023374
    [6] Manar A. Alqudah, Artion Kashuri, Pshtiwan Othman Mohammed, Muhammad Raees, Thabet Abdeljawad, Matloob Anwar, Y. S. Hamed . On modified convex interval valued functions and related inclusions via the interval valued generalized fractional integrals in extended interval space. AIMS Mathematics, 2021, 6(5): 4638-4663. doi: 10.3934/math.2021273
    [7] Muhammad Bilal Khan, Muhammad Aslam Noor, Thabet Abdeljawad, Bahaaeldin Abdalla, Ali Althobaiti . Some fuzzy-interval integral inequalities for harmonically convex fuzzy-interval-valued functions. AIMS Mathematics, 2022, 7(1): 349-370. doi: 10.3934/math.2022024
    [8] Muhammad Bilal Khan, Savin Treanțǎ, Hleil Alrweili, Tareq Saeed, Mohamed S. Soliman . Some new Riemann-Liouville fractional integral inequalities for interval-valued mappings. AIMS Mathematics, 2022, 7(8): 15659-15679. doi: 10.3934/math.2022857
    [9] Zehao Sha, Guoju Ye, Dafang Zhao, Wei Liu . On interval-valued K-Riemann integral and Hermite-Hadamard type inequalities. AIMS Mathematics, 2021, 6(2): 1276-1295. doi: 10.3934/math.2021079
    [10] Guoshan Deng, Dafang Zhao, Sina Etemad, Jessada Tariboon . Generalized fractional Hermite-Hadamard-type inequalities for interval-valued s-convex functions. AIMS Mathematics, 2025, 10(6): 14102-14121. doi: 10.3934/math.2025635
  • Three safe and effective vaccines against SARS-CoV-2 have played a major role in combating COVID-19 in the United States. However, the effectiveness of these vaccines and vaccination programs has been challenged by the emergence of new SARS-CoV-2 variants of concern. A new mathematical model is formulated to assess the impact of waning and boosting of immunity against the Omicron variant in the United States. To account for gradual waning of vaccine-derived immunity, we considered three vaccination classes that represent high, moderate and low levels of immunity. We showed that the disease-free equilibrium of the model is globally-asymptotically, for two special cases, if the associated reproduction number is less than unity. Simulations of the model showed that vaccine-derived herd immunity can be achieved in the United States via a vaccination-boosting strategy which entails fully vaccinating at least 59% of the susceptible populace followed by the boosting of about 72% of the fully-vaccinated individuals whose vaccine-derived immunity has waned to moderate or low level. In the absence of boosting, waning of immunity only causes a marginal increase in the average number of new cases at the peak of the pandemic, while boosting at baseline could result in a dramatic reduction in the average number of new daily cases at the peak. Specifically, for the fast immunity waning scenario (where both vaccine-derived and natural immunity are assumed to wane within three months), boosting vaccine-derived immunity at baseline reduces the average number of daily cases at the peak by about 90% (in comparison to the corresponding scenario without boosting of the vaccine-derived immunity), whereas boosting of natural immunity (at baseline) only reduced the corresponding peak daily cases (in comparison to the corresponding scenario without boosting of natural immunity) by approximately 62%. Furthermore, boosting of vaccine-derived immunity is more beneficial (in reducing the burden of the pandemic) than boosting of natural immunity. Finally, boosting vaccine-derived immunity increased the prospects of altering the trajectory of COVID-19 from persistence to possible elimination.



    In convex function theory, the classical Hermite-Hadamard inequality is one of the most well-known inequalities with geometrical interpretation, and it has a wide range of applications, see [1,2].

    Let S:KR+ be a convex function on a convex set K and ρ,ςK with ρς. Then,

    S(ρ+ς2)1ςρςρS(ϖ)dϖS(ρ)+S(ς)2. (1)

    In [3], Fejér looked at the key extensions of HH-inequality which is known as Hermite-Hadamard-Fejér inequality (HH-Fejér inequality).

    Let S:KR+ be a convex function on a convex set K and ρ,ς K with ρς. Then,

    S(ρ+ς2)1ςρD(ϖ)dϖςρS(ϖ)D(ϖ)dϖS(ρ)+S(ς)2ςρD(ϖ)dϖ. (2)

    If D(ϖ)=1, then we obtain (1) from (2). We should remark that Hermite-Hadamard inequality is a refinement of the idea of convexity, and it can be simply deduced from Jensen's inequality. In recent years, the Hermite-Hadamard inequality for convex functions has gotten a lot of attention, and there have been a lot of improvements and generalizations examined. Sarikaya [4] proved the Hadamard type inequality for coordinated convex functions such that

    Let G:ΔR+ be a coordinate convex function on Δ=[ς,ρ]×[μ,ν]. If G is double fractional integrable, then following inequalities hold:

    G(μ+ν2,ς+ρ2)Γ(α+1)4(νμ)α[Iαμ+G(ν,ς+ρ2)+IανG(μ,ς+ρ2)]+Γ(β+1)4(ρς)β[Iβς+G(μ+ν2,ρ)+IβρG(μ+ν2,ς)]Γ(α+1)Γ(β+1)4(νμ)α(ρς)β[Iα,βμ+,ς+G(ν,ρ)+Iα,βμ+,ρG(ν,ς)+Iα,βν,ς+G(μ,ρ)+Iα,βν,ρG(μ,ς)]Γ(α+1)8(νμ)α[Iαμ+G(ν,ς)GIαμ+G(ν,ρ)+IανG(μ,ς)+IανG(μ,ρ)]+Γ(β+1)4(ρς)β[Iβς+G(μ,ρ)˜+IβρG(ν,ς)+Iβς+G(μ,ρ)+IβρG(ν,ς)]G(μ,ς)+G(ν,ς)+G(μ,ρ)+G(ν,ρ)4. (3)

    If α=1, then we obtain the following Dragomir inequality [5] on coordinates:

    G(μ+ν2,ς+ρ2)
    12[1νμνμG(x,ς+ρ2)dx+1ρςρςG(μ+ν2,y)dy]1(νμ)(ρς)νμρςG(x,y)dydx14(νμ)[νμG(x,ς)dx+νμG(x,ρ)dx]+14(ρς)[ρςG(μ,y)dy+ρςG(ν,y)dy]G(μ,ς)+G(ν,ς)+G(μ,ρ)+G(ν,ρ)4. (4)

    For more details related to inequalities, see [6,7,8,9] and reference therein.

    Interval analysis, on the other hand, is a well-known example of set-valued analysis, which is the study of sets in the context of mathematical analysis and general topology. It was created as a way of dealing with the interval uncertainty that can be found in many mathematical or computer models of deterministic real-world phenomena. Archimede's method, which is used to calculate the circumference of a circle, is an old example of an interval enclosure. Moore [10], who is credited with being the first user of intervals in computational mathematics, published the first book on interval analysis in 1966. Following the publication of his book, a number of scientists began to research the theory and applications of interval arithmetic. Interval analysis is now a helpful technique in a variety of fields that are interested in ambiguous data because of its applicability. Computer graphics, experimental and computational physics, error analysis, robotics, and many more fields have applications.

    Furthermore, in recent years, numerous major inequalities (Hermite-Hadamard, Ostrowski and others) have been addressed for interval-valued functions. Chalco-Cano et al. used the Hukuhara derivative for interval-valued functions to construct Ostrowski type inequalities for interval-valued functions in [11,12,13,14]. For interval-valued functions, Román-Flores et al. developed Minkowski and Beckenbach's inequality in [15]. For fuzzy interval-valued function, Khan et al. [16,17,18] derived some new versions of Hermite-Hadamard type inequalities and proved their validity with the help of non-trivial examples. Moreover, Khan et al. [19,20] discussed some novel types of Hermite-Hadamard type inequalities in fuzzy-interval fractional calculus and proved that many classical versions are special cases of these inequalities. Recently, Khan et al. [21] introduced the new class of convexity in fuzzy-interval calculus which is known as coordinated convex fuzzy-interval-valued functions and with the support of these classes, some Hermite-Hadamard type inequalities are obtained via newly defined fuzzy-interval double integrals. We encourage readers to [22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54] for other related results.

    The following is an overview of the paper's structure. Section 2 recalls some preliminary notions and definitions. Moreover, some properties of introduced coordinated LR-convex IVF are also discussed. Section 3 presents some Hermite-Hadamard type inequalities for coordinated LR-convex IVF. With the help of this class, some fractional integral inequalities are also derived for the coordinated LR-convex IVF and for the product of two coordinated LR-convex IVFs. The fourth section, Conclusions and Future Work, brings us to a close.

    Let R be the set of real numbers and RI be the space of all closed and bounded intervals of R, such that URI is defined by

    U=[U,U]={yR|UyU},(U,UR). (5)

    If U=U, then U is said to be degenerate. If U0, then [U,U] is called positive interval. The set of all positive interval is denoted by R+I and defined as R+I={[U,U]:[U,U]RIandU0}.

    Let ϱR and ϱU be defined by

    ϱ.U={[ϱU,ϱU]ifϱ>0,{0}ifϱ=0,[ϱU,ϱU]ifϱ<0. (6)

    Then, the Minkowski difference DU, addition U+D and U×D for U,DRI are defined by

    [D,D][U,U]=[DU,DU],[D,D]+[U,U]=[D+U,D+U], (7)

    and

    [D,D]×[U,U]=[min{DU,DU,DU,DU},max{DU,DU,DU,DU}].

    The inclusion "⊇" means that

    UD if and only if, [U,U][D,D], and if and only if

    UD,DU. (8)

    Remark 1. [36] (ⅰ) The relation "≤p" is defined on RI by

    [D,D]p[U,U]ifandonlyifDU,DU, (9)

    for all [D,D],[U,U]RI, and it is a pseudo order relation. The relation [D,D]p[U,U] coincident to [D,D][U,U] on RI when it is "≤p"

    (ⅱ) It can be easily seen that "p" looks like "left and right" on the real line R, so we call "p" is "left and right" (or "LR" order, in short).

    For [D,D],[U,U]RI, the Hausdorff-Pompeiu distance between intervals [D,D] and [U,U] is defined by

    d([D,D],[U,U])=max{|DU|,|DU|}. (10)

    It is familiar fact that (RI,d) is a complete metric space.

    Theorem 1. [10] If G:[μ,ν]RRI is an I-V-F given by (x) [G(x),G(x)], then G is Riemann integrable over [μ,ν] if and only if, G and G both are Riemann integrable over [μ,ν] such that

    (IR)νμG(x)dx=[(R)νμG(x)dx,(R)νμG(x)dx]. (11)

    The collection of all Riemann integrable real valued functions and Riemann integrable I-V-F is denoted by R[μ,ν] and TR[μ,ν], respectively.

    Definition 1. [31,33] Let G:[μ,ν]RI be interval-valued function and GTR[μ,ν]. Then interval Riemann-Liouville-type integrals of G are defined as

    Iαμ+G(y)=1Γ(α)yμ(yt)α1G(t)dt(y>μ), (12)
    IανG(y)=1Γ(α)νy(ty)α1G(t)dt(y<ν), (13)

    where α>0 and Γ is the gamma function.

    Theorem 2. [20] Let G:[ς,ρ]RI+ be a LR-convex I-V.F such that G(y)=[G(y),G(y)] for all y[ς,ρ]. If GL([ς,ρ],R+I), then

    G(ς+ρ2)pΓ(α+1)2(ρς)α[Iας+G(ρ)+IαρG(ς)]pG(ς)+G(ρ)2. (14)

    Theorem 3. [20] Let G,S:[ς,ρ]R+I be two LR-convex I-V.Fs such that G(x)=[G(x),G(x)] and S(x)=[S(x),S(x)] for all x[ς,ρ]. If G×SL([ς,ρ],R+I) is fuzzy Riemann integrable, then

    Γ(α+1)2(ρς)α[Iας+G(ρ)×S(ρ)+IαρG(ς)×S(ς)]
    p(12α(α+1)(α+2))M(ς,ρ)+(α(α+1)(α+2))N(ς,ρ), (15)

    and

    G(ς+ρ2)×S(ς+ρ2)
    pΓ(α+1)4(ρς)α[Iας+G(ρ)×S(ρ)+IαρG(ς)×S(ς)]
    +12(12α(α+1)(α+2))M(ς,ρ)+12(α(α+1)(α+2))N(ς,ρ), (16)

    where M(ς,ρ)=G(ς)×S(ς)+G(ρ)×S(ρ), N(ς,ρ)=G(ς)×S(ρ)+G(ρ)×S(ς),

    and M(ς,ρ)=[M(ς,ρ),M(ς,ρ)] and N(ς,ρ)=[N(ς,ρ),N(ς,ρ)].

    Note that, the Theorem 1 is also true for interval double integrals. The collection of all double integrable I-V-F is denoted TOΔ, respectively.

    Theorem 4. [35] Let Δ=[ς,ρ]×[μ,ν]. If G:ΔRI is interval-valued doubl integrable (ID-integrable) on Δ. Then, we have

    (ID)ρςνμG(x,y)dydx=(IR)ρς(IR)νμG(x,y)dydx.

    Definition 2. [36] Let G:ΔR+I and GTOΔ. The interval Riemann-Liouville-type integrals Iα,βμ+,ς+,Iα,βμ+,ρ, Iα,βν,ς+,Iα,βν,ρ of G order α,β>0 are defined by

    Iα,βμ+,ς+G(x,y)=1Γ(α)Γ(β)xμyς(xt)α1(ys)β1G(t,s)dsdt(x>μ,y>ς), (17)
    Iα,βμ+,ρG(x,y)=1Γ(α)Γ(β)xμρy(xt)α1(sy)β1G(t,s)dsdt(x>μ,y<ρ), (18)
    Iα,βν,ς+G(x,y)=1Γ(α)Γ(β)νxyς(tx)α1(ys)β1G(t,s)dsdt(x<ν,y>ς), (19)
    Iα,βν,ρG(x,y)=1Γ(α)Γ(β)νxρy(tx)α1(sy)β1G(t,s)dsdt(x<ν,y<ρ). (20)

    Definition 3. [38] The I-V.F G:ΔR+I is said to be coordinated LR-convex I-V.F on Δ if

    G(τμ+(1τ)ν,sς+(1s)ρ)
    pτsG(μ,ς)+τ(1s)G(μ,ρ)+(1τ)sG(ν,ς)+(1τ)(1s)G(ν,ρ), (21)

    for all (μ,ν),(ς,ρ)Δ, and τ,s[0,1]. If inequality (21) is reversed, then G is called coordinate LR-concave I-V.F on Δ.

    Lemma 1. [38] Let G:ΔR+I be an coordinated I-V.F on Δ. Then, G is coordinated LR-convex I-V.F on Δ, if and only if there exist two coordinated LR-convex I-V.Fs Gx:[ς,ρ]R+I, Gx(w)=G(x,w) and Gy:[μ,ν]R+I, Gy(z)=G(z,y).

    Theorem 5. [38] Let G:ΔR+I be a I-V.F on Δ such that

    G(x,ϖ)=[G(x,ϖ),G(x,ϖ)], (22)

    for all (x,ϖ)Δ. Then, G is coordinated LR-convex I-V.F on Δ, if and only if, G(x,ϖ) and G(x,ϖ) are coordinated convex functions.

    Example 1. We consider the I-V.Fs G:[0,1]×[0,1]R+I defined by,

    G(x)(σ)={σ2(6+ex)(6+eϖ),σ[0,2(6+ex)(6+eϖ)]4(6+ex)(6+eϖ)σ2(6+ex)(6+eϖ),σ(2(6+ex)(6+eϖ),4(6+ex)(6+eϖ)]0,otherwise,

    Then, for each θ[0,1], we have G(x)=[2θ(6+ex)(6+eϖ),(4+2θ)(6+ex)(6+eϖ)]. Since end point functions G((x,ϖ),θ), G((x,ϖ),θ) are coordinate concave functions for each θ[0,1]. Hence S(x,ϖ) is coordinate LR-concave I-V.F.

    From Lemma 1, we can easily note that each LR-convex I-V.F is coordinated LR-convex I-V.F. But the converse is not true.

    Remark 2. If one takes G(x,ϖ)=G(x,ϖ), then G is known as coordinated function if G satisfies the coming inequality

    G(τμ+(1τ)ν,sς+(1s)ρ)
    τsG(μ,ς)+τ(1s)G(μ,ρ)+(1τ)sG(ν,ς)+(1τ)(1s)G(ν,ρ),

    is valid which is defined by Dragomir [5]

    Let one takes G(x,ϖ)G(x,ϖ), where G(x,ϖ) is affine function and G(x,ϖ) is a concave function. If coming inequality,

    G(τμ+(1τ)ν,sς+(1s)ρ)
    τsG(μ,ς)+τ(1s)G(μ,ρ)+(1τ)sG(ν,ς)+(1τ)(1s)G(ν,ρ),

    is valid, then G is named as coordinated IVF which is defined by Zhao et al. [37, Definition 2 and Example 2]

    In this section, we shall continue with the following fractional HH-inequality for coordinated LR-convex I-V.Fs, and we also give fractional HH-Fejér inequality for coordinated LR-convex I-V.F through fuzzy order relation.

    Theorem 6. Let G:ΔR+I be a coordinate LR-convex I-V.F on Δ such that G(x,y)=[G(x,y),G(x,y)] for all (x,y)Δ. If GTOΔ, then following inequalities holds:

    G(μ+ν2,ς+ρ2)pΓ(α+1)4(νμ)α[Iαμ+G(ν,ς+ρ2)+IανG(μ,ς+ρ2)]
    +Γ(β+1)4(ρς)β[Iβς+G(μ+ν2,ρ)+IβρG(μ+ν2,ς)]
    pΓ(α+1)Γ(β+1)4(νμ)α(ρς)β[Iα,βμ+,ς+G(ν,ρ)+Iα,βμ+,ρG(ν,ς)+Iα,βν,ς+G(μ,ρ)+Iα,βν,ρG(μ,ς)]
    pΓ(α+1)8(νμ)α[Iαμ+G(ν,ς)+Iαμ+G(ν,ρ)+IανG(μ,ς)+IανG(μ,ρ)]
    +Γ(β+1)4(ρς)β[Iβς+G(μ,ρ)+IβρG(ν,ς)+Iβς+G(μ,ρ)+IβρG(ν,ς)]
    pG(μ,ς)+G(ν,ς)+G(μ,ρ)+G(ν,ρ)4. (23)

    If G(x) coordinated LR-concave I-V.F, then

    G(μ+ν2,ς+ρ2)pΓ(α+1)4(νμ)α[Iαμ+G(ν,ς+ρ2)+IανG(μ,ς+ρ2)]
    +Γ(β+1)4(ρς)β[Iβς+G(μ+ν2,ρ)+IβρG(μ+ν2,ς)]
    pΓ(α+1)Γ(β+1)4(νμ)α(ρς)β[Iα,βμ+,ς+G(ν,ρ)+Iα,βμ+,ρG(ν,ς)+Iα,βν,ς+G(μ,ρ)+Iα,βν,ρG(μ,ς)]
    pΓ(α+1)8(νμ)α[Iαμ+G(ν,ς)+Iαμ+G(ν,ρ)+IανG(μ,ς)+IανG(μ,ρ)]
    +Γ(β+1)4(ρς)β[Iβς+G(μ,ρ)+IβρG(ν,ς)+Iβς+G(μ,ρ)+IβρG(ν,ς)]
    pG(μ,ς)+G(ν,ς)+G(μ,ρ)+G(ν,ρ)4. (24)

    Proof. Let G:[μ,ν]R+I be a coordinated LR-convex I-V.F. Then, by hypothesis, we have

    4G(μ+ν2,ς+ρ2)pG(τμ+(1τ)ν,τς+(1τ)ρ)+G((1τ)μ+τν,(1τ)ς+τρ).

    By using Theorem 5, we have

    4G(μ+ν2,ς+ρ2)G(τμ+(1τ)ν,τς+(1τ)ρ)+G((1τ)μ+τν,(1τ)ς+τρ),4G(μ+ν2,ς+ρ2)G(τμ+(1τ)ν,τς+(1τ)ρ)+G((1τ)μ+τν,(1τ)ς+τρ).

    By using Lemma 1, we have

    2G(x,ς+ρ2)G(x,τς+(1τ)ρ)+G(x,(1τ)ς+τρ),2G(x,ς+ρ2)G(x,τς+(1τ)ρ)+G(x,(1τ)ς+τρ), (25)

    and

    2G(μ+ν2,y)G(τμ+(1τ)ν,y)+G((1τ)μ+tν,y),2G(μ+ν2,y)G(τμ+(1τ)ν,y)+G((1τ)μ+tν,y). (26)

    From (25) and (26), we have

    2[G(x,ς+ρ2),G(x,ς+ρ2)]
    p[G(x,τς+(1τ)ρ),G(x,τς+(1τ)ρ)]
    +[G(x,(1τ)ς+τρ),G(x,(1τ)ς+τρ)],

    and

    2[G(μ+ν2,y),G(μ+ν2,y)]
    p[G(τμ+(1τ)ν,y),G(τμ+(1τ)ν,y)]
    +[G(τμ+(1τ)ν,y),G(τμ+(1τ)ν,y)],

    It follows that

    G(x,ς+ρ2)pG(x,τς+(1τ)ρ)+G(x,(1τ)ς+τρ), (27)

    and

    G(μ+ν2,y)pG(τμ+(1τ)ν,y)+G(τμ+(1τ)ν,y). (28)

    Since G(x,.) and G(.,y), both are coordinated LR-convex-IVFs, then from inequality (14), inequalities (27) and (28) we have

    Gx(ς+ρ2)pΓ(β+1)2(ρς)β[Iβς+Gx(ρ)+IβρGx(ς)]pGx(ς)+Gx(ρ)2. (29)

    and

    Gy(μ+ν2)pΓ(α+1)2(νμ)α[Iαμ+Gy(ν)+IανGy(μ)]pGy(μ)+Gy(ν)2 (30)

    Since Gx(w)=G(x,w), the inequality (29) can be written as

    G(x,ς+ρ2)pΓ(β+1)2(ρς)β[Iας+G(x,ρ)+IαρG(x,ς)]pG(x,ς)+G(x,ρ)2. (31)

    That is

    G(x,ς+ρ2)pβ2(ρς)β[ρς(ρs)β1G(x,s)ds+ρς(sς)β1G(x,s)ds]pG(x,ς)+G(x,ρ)2.

    Multiplying double inequality (31) by α(νx)α12(νμ)α and integrating with respect to x over [μ,ν], we have

    α2(νμ)ανμG(x,ς+ρ2)(νx)α1dx
    pνμρς(νx)α1(ρs)β1G(x,s)dsdx+νμρς(νx)α1(sς)β1G(x,s)dsdx
    pα4(νμ)α[νμ(νx)α1G(x,ς)dx+νμ(νx)α1G(x,ρ)dx]. (32)

    Again multiplying double inequality (31) by α(xμ)α12(νμ)α and integrating with respect to x over [μ,ν], we have

    α2(νμ)ανμG(x,ς+ρ2)(νx)α1dx
    pαβ4(νμ)α(ρς)βνμρς(xμ)α1(ρs)β1G(x,s)dsdx
    +αβ4(νμ)α(ρς)βνμρς(xμ)α1(sς)β1G(x,s)dsdx
    pα4(νμ)α[νμ(xμ)α1G(x,ς)dx+νμ(xμ)α1G(x,d)dx]. (33)

    From (32), we have

    Γ(α+1)2(νμ)α[Iαμ+G(ν,ς+ρ2)]
    pΓ(α+1)Γ(β+1)4(νμ)α(ρς)β[Iα,βμ+,ς+G(ν,ρ)+Iα,βν,ς+G(ν,ς)]
    pΓ(α+1)4(νμ)α[Iαμ+G(ν,ς)+Iαμ+G(ν,ρ)]. (34)

    From (33), we have

    Γ(α+1)2(νμ)α[IανG(μ,ς+ρ2)]
    pΓ(α+1)Γ(β+1)4(νμ)α(ρς)β[Iα,βν,ς+G(μ,ρ)+Iα,βν,ρG(μ,ς)]
    pΓ(α+1)4(νμ)α[IανG(μ,ς)+IανG(μ,ρ)]. (35)

    Similarly, since Gy(z)=G(z,y) then, from (34) and (35), (30) we have

    Γ(β+1)2(ρς)β[Iβς+G(μ+ν2,ρ)]
    pΓ(α+1)Γ(β+1)4(νμ)α(ρς)β[Iα,βμ+,ς+G(ν,ρ)+Iα,βν,ς+G(μ,ρ)]
    pΓ(β+1)4(ρς)β[Iβς+G(μ,ρ)+Iβς+G(ν,ρ)], (36)

    and

    Γ(β+1)2(ρς)α[IβρG(μ+ν2,ς)]
    pΓ(α+1)Γ(β+1)4(νμ)α(ρς)β[Iα,βμ+,ρG(ν,ς)+Iα,βν,ρG(μ,ς)]
    pΓ(β+1)4(ρς)β[IβρG(μ,ς)+IβρG(ν,ς)]. (37)

    After adding the inequalities (46), (35), (36) and (37), we will obtain as resultant second, third and fourth inequalities of (23).

    Now, from left part of inequality (14), we have

    G(μ+ν2,ς+ρ2)pΓ(β+1)2(ρς)β[Iβς+G(μ+ν2,ρ)+IβρG(μ+ν2,ς)], (38)

    and

    G(μ+ν2,ς+ρ2)pΓ(α+1)2(νμ)α[Iαμ+G(ν,ς+ρ2)+IανG(μ,ς+ρ2)]. (39)

    Summing the inequalities (38) and (39), we obtain the following inequality:

    G(μ+ν2,ς+ρ2)
    pΓ(α+1)4(νμ)α[Iαμ+G(ν,ς+ρ2)+IανG(μ,ς+ρ2)]+Γ(β+1)4(ρς)β[Iβς+G(μ+ν2,ρ)+IβρG(μ+ν2,ς)], (40)

    this is the first inequality of (23).

    Now, from right part of inequality (14), we have

    Γ(β+1)2(ρς)β[Iβς+G(μ,ρ)+IβρG(μ,ς)]pG(μ,ς)+G(μ,ρ)2, (41)
    Γ(β+1)2(ρς)β[Iβς+G(ν,ρ)+IβρG(ν,ς)]pG(ν,ς)+G(ν,ρ)2, (42)
    Γ(α+1)2(νμ)α[Iαμ+G(ν,ς)+IανG(μ,ς)]pG(μ,ς)+G(ν,ς)2, (43)
    Γ(α+1)2(νμ)α[Iαμ+G(ν,ρ)+IανG(μ,ρ)]pG(μ,ρ)+G(ν,ρ)2. (44)

    Summing inequalities (41), (42), (43) and (44), and then taking multiplication of the resultant with 14, we have

    Γ(α+1)8(νμ)α[Iαμ+G(ν,ς)+IανG(μ,ς)+Iαμ+G(ν,ρ)+IανG(μ,ρ)]
    +Γ(β+1)2(ρς)β[Iβς+G(μ,ρ)+IβρG(μ,ς)+Iβς+G(ν,ρ)+IβρG(ν,ς)]
    pG(μ,ς)+G(μ,ρ)+G(ν,ς)+G(ν,ρ)4. (45)

    This is last inequality of (23) and the result has been proven.

    Remark 3. If one to take α=1 and β=1, then from (23), we achieve the coming inequality, see [38]:

    G(μ+ν2,ς+ρ2)
    p12[1νμνμG(x,ς+ρ2)dx+1ρςρςG(μ+ν2,y)dy]p1(νμ)(ρς)νμρςG(x,y)dydxp14(νμ)[νμG(x,ς)dx+νμG(x,ρ)dx]+14(ρς)[ρςG(μ,y)dy+ρςG(ν,y)dy]
    pG(μ,ς)+G(ν,ς)+G(μ,ρ)+G(ν,ρ)4. (46)

    Let one takes G(x,y) is an affine function and G(x,y) is concave function. If G(x,y)G(x,y), then from Remark 2 and (24), we acquire the coming inequality, see [31]:

    G(μ+ν2,ς+ρ2)Γ(α+1)4(νμ)α[Iαμ+G(ν,ς+ρ2)+IανG(μ,ς+ρ2)]+Γ(β+1)4(ρς)β[Iβς+G(μ+ν2,ρ)+IβρG(μ+ν2,ς)]
    Γ(α+1)Γ(β+1)4(νμ)α(ρς)β[Iα,βμ+,ς+G(ν,ρ)+Iα,βμ+,ρG(ν,ς)+Iα,βν,ς+G(μ,ρ)+Iα,βν,ρG(μ,ς)]
    Γ(α+1)8(νμ)α[Iαμ+G(ν,ς)GIαμ+G(ν,ρ)+IανG(μ,ς)+IανG(μ,ρ)]
    +Γ(β+1)4(ρς)β[Iβς+G(μ,ρ)˜+IβρG(ν,ς)+Iβς+G(μ,ρ)+IβρG(ν,ς)]
    G(μ,ς)+G(ν,ς)+G(μ,ρ)+G(ν,ρ)4. (47)

    Let one takes α=1 and β=1, G(x,y) is an affine function and G(x,y) is concave function. If G(x,y)G(x,y), then Remark 2 and from (24), we acquire the coming inequality, see [37]:

    G(μ+ν2,ς+ρ2)
    12[1νμνμG(x,ς+ρ2)dx+1ρςρςG(μ+ν2,y)dy]1(νμ)(ρς)νμρςG(x,y)dydx
    14(νμ)[νμG(x,ς)dx+νμG(x,ρ)dx]+14(ρς)[ρςG(μ,y)dy+ρςG(ν,y)dy]
    G(μ,ς)+G(ν,ς)+G(μ,ρ)+G(ν,ρ)4. (48)

    Example 2. We consider the I-V-Fs G:[0,1]×[0,1]R+I defined by,

    G(x)=[2,6](6+ex)(6+ey).

    Since end point functions G(x,y), G(x,y) are convex functions on coordinate, then G(x,y) is convex I-V-F on coordinate. Then for α=1 and β=1, we have

    G(μ+ν2,ς+ρ2)=[2(5+e12)2,6(6+e12)2],
    Γ(α+1)4(νμ)α[Iαμ+G(ν,ς+ρ2)+IανG(μ,ς+ρ2)]+Γ(β+1)4(ρς)β[Iβς+G(μ+ν2,ρ)+IβρG(μ+ν2,ς)]
    =[4(6+e12)(5+e),12(6+e12)(5+e)],
    Γ(α+1)Γ(β+1)4(νμ)α(ρς)β[Iα,βμ+,ς+G(ν,ρ)+Iα,βμ+,ρG(ν,ς)+Iα,βν,ς+G(μ,ρ)+Iα,βν,ρG(μ,ς)]
    =[2(5+e)2,6(5+e)2],
    Γ(α+1)8(νμ)α[Iαμ+G(ν,ς)GIαμ+G(ν,ρ)+IανG(μ,ς)+IανG(μ,ρ)]
    +Γ(β+1)4(ρς)β[Iβς+G(μ,ρ)˜+IβρG(ν,ς)+Iβς+G(μ,ρ)+IβρG(ν,ς)]
    =[(5+e)(13+e),3(5+e)(13+e)]
    G(μ,ς)+G(ν,ς)+G(μ,ρ)+G(ν,ρ)4=[(6+e)(20+e)+492,6((6+e)(20+e)+49)2].

    That is

    [2(5+e12)2,6(6+e12)2]p[4(6+e12)(5+e),12(6+e12)(5+e)]
    p[2(5+e)2,6(5+e)2]
    p[(5+e)(13+e),3(5+e)(13+e)]
    p[(6+e)(20+e)+492,3((6+e)(20+e)+49)].

    Hence, Theorem 3.1 has been verified

    Next both results obtain Hermite-Hadamard type inequalities for the product of two coordinate LR-convex I-V.Fs

    Theorem 7. Let G,S:ΔR+I be a coordinate LR-convex I-V.Fs on Δ such that G(x,y)=[G(x,y),G(x,y)] and S(x,y)=[S(x,y),S(x,y)] for all (x,y)Δ. If G×STOΔ, then following inequalities holds:

    Γ(α+1)Γ(β+1)4(νμ)α(ρς)β[Iα,βμ+,ς+G(ν,ρ)×S(ν,ρ)+Iα,βμ+,ρG(ν,ς)×S(ν,ς)+Iα,βν,ς+G(μ,ρ)×S(μ,ρ)+Iα,βν,ρG(μ,ς)×S(μ,ς)]
    p(12α(α+1)(α+2))(12β(β+1)(β+2))K(μ,ν,ς,ρ)+α(α+1)(α+2)(12β(β+1)(β+2))L(μ,ν,ς,ρ)
    +(12α(α+1)(α+2))β(β+1)(β+2)M(μ,ν,ς,ρ)+β(β+1)(β+2)α(α+1)(α+2)N(μ,ν,ς,ρ). (49)

    If G and S both are coordinate LR-concave I-V.Fs on Δ, then above inequality can be written as

    Γ(α+1)Γ(β+1)4(νμ)α(ρς)β[Iα,βμ+,ς+G(ν,ρ)×S(ν,ρ)+Iα,βμ+,ρG(ν,ς)×S(ν,ς)+Iα,βν,ς+G(μ,ρ)×S(μ,ρ)+Iα,βν,ρG(μ,ς)×S(μ,ς)]
    p(12α(α+1)(α+2))(12β(β+1)(β+2))K(μ,ν,ς,ρ)+α(α+1)(α+2)(12β(β+1)(β+2))L(μ,ν,ς,ρ)
    +(12α(α+1)(α+2))β(β+1)(β+2)M(μ,ν,ς,ρ)+β(β+1)(β+2)α(α+1)(α+2)N(μ,ν,ς,ρ). (50)

    Where

    K(μ,ν,ς,ρ)=G(μ,ς)×S(μ,ς)+G(ν,ς)×S(ν,ς)+G(μ,ρ)×S(μ,ρ)+G(ν,ρ)×S(ν,ρ),
    L(μ,ν,ς,ρ)=G(μ,ς)×S(ν,ς)˜+G(ν,ρ)×S(μ,ρ)+G(ν,ς)×S(μ,ς)+G(μ,ρ)×S(ν,ρ),
    M(μ,ν,ς,ρ)=G(μ,ς)×S(μ,ρ)+G(ν,ς)×S(ν,ρ)+G(μ,ρ)×S(μ,ς)+G(ν,ρ)×S(ν,ς),
    N(μ,ν,ς,ρ)=G(μ,ς)×S(ν,ρ)+G(ν,ς)×S(μ,ρ)+G(μ,ρ)×S(ν,ς)+G(ν,ρ)×S(μ,ς).

    and K(μ,ν,ς,ρ), ˜L(μ,ν,ς,ρ), M(μ,ν,ς,ρ) and N(μ,ν,ς,ρ) are defined as follows:

    K(μ,ν,ς,ρ)=[K(μ,ν,ς,ρ),K(μ,ν,ς,ρ)],
    L(μ,ν,ς,ρ)=[L(μ,ν,ς,ρ),L(μ,ν,ς,ρ)],
    M(μ,ν,ς,ρ)=[M(μ,ν,ς,ρ),M(μ,ν,ς,ρ)],
    N(μ,ν,ς,ρ)=[N(μ,ν,ς,ρ),N(μ,ν,ς,ρ)].

    Proof. Let G and S both are coordinated LR-convex I-V.Fs on [μ,ν]×[ς,ρ]. Then

    G(τμ+(1τ)ν,sς+(1s)ρ)
    pτsG(μ,ς)+τ(1s)G(μ,ρ)+(1τ)sG(ν,ς)+(1τ)(1s)G(ν,ρ),

    and

    S(τμ+(1τ)ν,sς+(1s)ρ)
    pτsS(μ,ς)+τ(1s)S(μ,ρ)+(1τ)sS(ν,ς)+(1τ)(1s)S(ν,ρ).

    Since G and S both are coordinated LR-convex I-V.Fs, then by Lemma 1, there exist

    Gx:[ς,ρ]R+I,Gx(y)=G(x,y),Sx:[ς,ρ]R+I,Sx(y)=S(x,y),

    Since Gx, and Sx are I-V.Fs, then by inequality (15), we have

    Γ(β+1)2(ρς)β[Iβς+Gx(ρ)×Sx(ρ)+IβρGx(ς)×Sx(ς)]
    p(12β(β+1)(β+2))(Gx(ς)×Sx(ς)+Gx(ρ)×Sx(ρ))
    +(β(β+1)(β+2))(Gx(ς)×Sx(ρ)+Gx(ρ)×Sx(ς)).

    That is

    β2(ρς)β[ρς(ρy)β1G(x,y)×S(x,y)ρy+ρς(yς)β1G(x,y)×S(x,y)ρy]
    p(12β(β+1)(β+2))(G(x,ς)×S(x,ς)+G(x,ρ)×S(x,ρ))
    +(β(β+1)(β+2))(G(x,ς)×S(x,ρ)+G(x,ρ)×S(x,ς)). (51)

    Multiplying double inequality (51) by α(νx)α12(νμ)α and integrating with respect to x over [μ,ν], we get

    Γ(α+1)Γ(β+1)4(νμ)α(ρς)β[Iα,βμ+,ς+G(ν,ρ)×S(ν,ρ)+Iα,βμ+,ρG(ν,ς)×S(ν,ς)]
    pΓ(α+1)2(νμ)α(12β(β+1)(β+2))(Iαμ+G(ν,ς)×S(ν,ς)+Iαμ+G(ν,ρ)×S(ν,ρ))
    +Γ(α+1)2(νμ)αβ(β+1)(β+2)(Iαμ+G(ν,ς)×S(ν,ρ)+Iαμ+G(ν,ρ)×S(ν,ς)). (52)

    Again, multiplying double inequality (51) by α(xμ)α12(νμ)α and integrating with respect to x over [μ,ν], we gain

    Γ(α+1)Γ(β+1)4(νμ)α(ρς)β[Iα,βν,ς+G(μ,ρ)×S(μ,ρ)+Iα,βν,ρG(μ,ς)×S(μ,ς)]
    pΓ(α+1)2(νμ)α(12β(β+1)(β+2))(IανG(μ,ς)×S(μ,ς)+IανG(μ,ρ)×S(μ,ρ))
    +Γ(α+1)2(νμ)αβ(β+1)(β+2)(IανG(μ,ς)×S(μ,ρ)+IανG(μ,ρ)×S(μ,ς)). (53)

    Summing (52) and (53), we have

    Γ(α+1)Γ(β+1)4(νμ)α(ρς)β[Iα,βμ+,ς+G(ν,ρ)×S(ν,ρ)+Iα,βμ+,ρG(ν,ς)×S(ν,ς)+Iα,βν,ς+G(μ,ρ)×S(μ,ρ)+Iα,βν,ρG(μ,ς)×S(μ,ς)]
    pΓ(α+1)2(νμ)α(12β(β+1)(β+2))(Iαμ+G(ν,ς)×S(ν,ς)+IανG(μ,ς)×S(μ,ς))
    +Γ(α+1)2(νμ)α(12β(β+1)(β+2))(Iαμ+G(ν,ρ)×S(ν,ρ)+IανG(μ,ρ)×S(μ,ρ))
    +Γ(α+1)2(νμ)αβ(β+1)(β+2)(Iαμ+G(ν,ς)×S(ν,ρ)+IανG(μ,ς)×S(μ,ρ))
    +Γ(α+1)2(νμ)αβ(β+1)(β+2)(Iαμ+G(ν,ρ)×S(ν,ς)+IανG(μ,ρ)×S(μ,ς)). (54)

    Now, again with the help of integral inequality (15) for first two integrals on the right-hand side of (54), we have the following relation

    Γ(α+1)2(νμ)α(Iαμ+G(ν,ς)×S(ν,ς)+IανG(μ,ς)×S(μ,ς))
    p(12α(α+1)(α+2))(G(μ,ς)×S(μ,ς)+G(ν,ς)×S(ν,ς))
    +(α(α+1)(α+2))(G(μ,ς)×S(ν,ς)+G(ν,ς)×S(μ,ς)). (55)
    Γ(α+1)2(νμ)α(Iαμ+G(ν,ρ)×S(ν,ρ)+IανG(μ,ρ)×S(μ,ρ))
    p(12α(α+1)(α+2))(G(μ,ρ)×S(μ,ρ)+G(ν,ρ)×S(ν,ρ))
    +(α(α+1)(α+2))(G(μ,ρ)×S(ν,ρ)+G(ν,ρ)×S(μ,ρ)). (56)
    Γ(α+1)2(νμ)α(Iαμ+G(ν,ς)×S(ν,ρ)+IανG(μ,ς)×S(μ,ρ))
    p(12α(α+1)(α+2))(G(μ,ς)×S(μ,ρ)+G(ν,ς)×S(ν,ρ))
    +(α(α+1)(α+2))(G(μ,ς)×S(ν,ρ)+G(ν,ς)×S(μ,ρ)). (57)

    And

    Γ(α+1)2(νμ)α(Iαμ+G(ν,ρ)×S(ν,ς)+IανG(μ,ρ)×S(μ,ς))
    p(12α(α+1)(α+2))(G(μ,ρ)×S(μ,ς)+G(ν,ρ)×S(ν,ς))
    +(α(α+1)(α+2))(G(μ,ρ)×S(ν,ς)+G(ν,ρ)×S(μ,ς)). (58)

    From (55)–(58), inequality (54) we have

    Γ(α+1)Γ(β+1)4(νμ)α(ρς)β[Iα,βμ+,ς+G(ν,ρ)×S(ν,ρ)+Iα,βμ+,ρG(ν,ς)×S(ν,ς)+Iα,βν,ς+G(μ,ρ)×S(μ,ρ)+Iα,βν,ρG(μ,ς)×S(μ,ς)]
    p(12α(α+1)(α+2))(12β(β+1)(β+2))K(μ,ν,ς,ρ)+α(α+1)(α+2)(12β(β+1)(β+2))L(μ,ν,ς,ρ)
    +(12α(α+1)(α+2))β(β+1)(β+2)M(μ,ν,ς,ρ)+β(β+1)(β+2)α(α+1)(α+2)N(μ,ν,ς,ρ).

    Hence, the result has been proven.

    Remark 4. If one to take α=1 and β=1, then from (49), we achieve the coming inequality, see [38]:

    1(νμ)(ρς)νμρςG(x,y)×S(x,y)dydx
    p19K(μ,ν,ς,ρ)+118[L(μ,ν,ς,ρ)+M(μ,ν,ς,ρ)]+136N(μ,ν,ς,ρ). (59)

    Let one takes G(x,y) is an affine function and G(x,y) is concave function. If G(x,y)G(x,y), then by Remark 2 and (50), we acquire the coming inequality, see [36]:

    Γ(α+1)Γ(β+1)4(νμ)α(ρς)β[Iα,βμ+,ς+G(ν,ρ)×S(ν,ρ)+Iα,βμ+,ρG(ν,ς)×S(ν,ς)+Iα,βν,ς+G(μ,ρ)×S(μ,ρ)+Iα,βν,ρG(μ,ς)×S(μ,ς)]
    (12α(α+1)(α+2))(12β(β+1)(β+2))K(μ,ν,ς,ρ)+α(α+1)(α+2)(12β(β+1)(β+2))L(μ,ν,ς,ρ)
    +(12α(α+1)(α+2))β(β+1)(β+2)M(μ,ν,ς,ρ)+β(β+1)(β+2)α(α+1)(α+2)N(μ,ν,ς,ρ). (60)

    Let one takes G(x,y) is an affine function and G(x,y) is concave function. If G(x,y)G(x,y), then by Remark 2 and (50), we acquire the coming inequality, see [37]:

    1(νμ)(ρς)νμρςG(x,y)×S(x,y)dydx
    19K(μ,ν,ς,ρ)+118[L(μ,ν,ς,ρ)+M(μ,ν,ς,ρ)]+136N(μ,ν,ς,ρ). (61)

    If G(x,y)=G(x,y) and S(x,y)=S(x,y), then from (49), we acquire the coming inequality, see [39]:

    Γ(α+1)Γ(β+1)4(νμ)α(ρς)β[Iα,βμ+,ς+G(ν,ρ)×S(ν,ρ)+Iα,βμ+,ρG(ν,ς)×S(ν,ς)+Iα,βν,ς+G(μ,ρ)×S(μ,ρ)+Iα,βν,ρG(μ,ς)×S(μ,ς)]
    (12α(α+1)(α+2))(12β(β+1)(β+2))K(μ,ν,ς,ρ)+α(α+1)(α+2)(12β(β+1)(β+2))L(μ,ν,ς,ρ)
    +(12α(α+1)(α+2))β(β+1)(β+2)M(μ,ν,ς,ρ)+β(β+1)(β+2)α(α+1)(α+2)N(μ,ν,ς,ρ). (62)

    Theorem 8. Let G,S:ΔR+I be a coordinate LR-convex I-V.F on Δ such that G(x,y)=[G(x,y),G(x,y)] and S(x,y)=[S(x,y),S(x,y)] for all (x,y)Δ. If G×STOΔ, then following inequalities holds:

    4G(μ+ν2,ς+ρ2)×S(μ+ν2,ς+ρ2)
    pΓ(α+1)Γ(β+1)4(νμ)α(ρς)β[Iα,βμ+,ς+G(ν,ρ)×S(ν,ρ)+Iα,βμ+,ρG(ν,ς)×S(ν,ς)+Iα,βν,ς+G(μ,ρ)×S(μ,ρ)+Iα,βν,ρG(μ,ς)×S(μ,ς)]+[α2(α+1)(α+2)+β(β+1)(β+2)(12α(α+1)(α+2))]K(μ,ν,ς,ρ)
    +[12(12α(α+1)(α+2))+α(α+1)(α+2)β(β+1)(β+2)]L(μ,ν,ς,ρ)
    +[12(12β(β+1)(β+2))+α(α+1)(α+2)β(β+1)(β+2)]M(μ,ν,ς,ρ)
    +[14α(α+1)(α+2)β(β+1)(β+2)]N(μ,ν,ς,ρ). (63)

    If G and S both are coordinate LR-concave I-V.Fs on Δ, then above inequality can be written as

    4G(μ+ν2,ς+ρ2)×S(μ+ν2,ς+ρ2)pΓ(α+1)Γ(β+1)4(νμ)α(ρς)β[Iα,βμ+,ς+G(ν,ρ)×S(ν,ρ)+Iα,βμ+,ρG(ν,ς)×S(ν,ς)+Iα,βν,ς+G(μ,ρ)×S(μ,ρ)+Iα,βν,ρG(μ,ς)×S(μ,ς)]+[α2(α+1)(α+2)+β(β+1)(β+2)(12α(α+1)(α+2))]K(μ,ν,ς,ρ)
    +[12(12α(α+1)(α+2))+α(α+1)(α+2)β(β+1)(β+2)]L(μ,ν,ς,ρ)+[12(12β(β+1)(β+2))+α(α+1)(α+2)β(β+1)(β+2)]M(μ,ν,ς,ρ)+[14α(α+1)(α+2)β(β+1)(β+2)]N(μ,ν,ς,ρ). (64)

    Where K(μ,ν,ς,ρ), L(μ,ν,ς,ρ), M(μ,ν,ς,ρ) and N(μ,ν,ς,ρ) are given in Theorem 7.

    Proof. Since G,S:ΔR+I be two LR-convex I-V.Fs, then from inequality (16), we have

    2G(μ+ν2,ς+ρ2)×S(μ+ν2,ς+ρ2)pα2(νμ)α[νμ(νx)α1G(x,ς+ρ2)×S(x,ς+ρ2)dx+νμ(xμ)α1G(x,ς+ρ2)×S(x,ς+ρ2)dx]+(α(α+1)(α+2))(G(μ,ς+ρ2)×S(μ,ς+ρ2)+G(ν,ς+ρ2)×S(ν,ς+ρ2))+(12α(α+1)(α+2))(G(μ,ς+ρ2)×S(ν,ς+ρ2)+G(ν,ς+ρ2)×S(μ,ς+ρ2)), (65)

    and

    2G(μ+ν2,ς+ρ2)×S(μ+ν2,ς+ρ2)pβ2(ρς)β[ρς(ρy)β1G(μ+ν2,y)×S(μ+ν2,y)dy+ρς(yς)β1G(μ+ν2,y)×S(μ+ν2,y)dy]+(β(β+1)(β+2))(G(μ+ν2,ς)×S(μ+ν2,ς)+G(μ+ν2,ρ)×S(μ+ν2,ρ))+(12β(β+1)(β+2))(G(μ+ν2,ς)×S(μ+ν2,ρ)+G(μ+ν2,ρ)×S(μ+ν2,ς)), (66)

    Adding (73) and (74), and then taking the multiplication of the resultant one by 2, we obtain

    8G(μ+ν2,ς+ρ2)×S(μ+ν2,ς+ρ2)pα2(νμ)α[νμ2(νx)α1G(x,ς+ρ2)×S(x,ς+ρ2)dx+νμ2(xμ)α1G(x,ς+ρ2)×S(x,ς+ρ2)dx]+β2(ρς)β[ρς2(ρy)β1G(μ+ν2,y)×S(μ+ν2,y)dy+ρς2(yς)β1G(μ+ν2,y)×S(μ+ν2,y)dy]+(α(α+1)(α+2))(2G(μ,ς+ρ2)×S(μ,ς+ρ2)+2G(ν,ς+ρ2)×S(ν,ς+ρ2))+(12α(α+1)(α+2))(2G(μ,ς+ρ2)×S(ν,ς+ρ2)+2G(ν,ς+ρ2)×S(μ,ς+ρ2))+(β(β+1)(β+2))(2G(μ+ν2,ς)×S(μ+ν2,ς)+2G(μ+ν2,ρ)×S(μ+ν2,ρ))+(12β(β+1)(β+2))(2G(μ+ν2,ς)×S(μ+ν2,ρ)+2G(μ+ν2,ρ)×S(μ+ν2,ς)). (67)

    Again, with the help of integral inequality (16) and Lemma 1 for each integral on the right-hand side of (67), we have

    α2(νμ)ανμ2(νx)α1G(x,ς+ρ2)×S(x,ς+ρ2)dxpαβ4(νμ)α(ρς)β[νμρς(νx)α1(ρy)β1G(x,y)dydx+νμρς(νx)α1(yς)β1G(x,y)dydx]+β(β+1)(β+2)α2(νμ)ανμ(νx)α1(G(x,ς)×S(x,ς)+G(x,ρ)×S(x,ρ))dx+(12β(β+1)(β+2))α2(νμ)ανμ(νx)α1(G(x,ς)×S(x,ρ)+G(x,ρ)×S(x,ς))dx,=Γ(α+1)Γ(β+1)4(νμ)α(ρς)β[Iα,βμ+,ς+G(ν,ρ)×S(ν,ρ)+Iα,βμ+,ρG(ν,ς)×S(ν,ς)]+Γ(α+1)2(νμ)α(β(β+1)(β+2))(Iαμ+G(ν,ς)×S(ν,ς)+Iαμ+G(ν,ρ)×S(ν,ρ))+Γ(α+1)2(νμ)α(12β(β+1)(β+2))(Iαμ+G(ν,ς)×S(ν,ρ)+Iαμ+G(ν,ρ)×S(ν,ς)). (68)
    α2(νμ)ανμ2(xμ)α1G(x,ς+ρ2)×S(x,ς+ρ2)dxpαβ4(νμ)α(ρς)β[νμρς(xμ)α1(ρy)β1G(x,y)dydx+νμρς(xμ)α1(yς)β1G(x,y)dydx]+β(β+1)(β+2)α2(νμ)ανμ(xμ)α1(G(x,ς)×S(x,ς)+G(x,ρ)×S(x,ρ))dx+(12β(β+1)(β+2))α2(νμ)ανμ(xμ)α1(G(x,ς)×S(x,ρ)+G(x,ρ)×S(x,ς))dx,=Γ(α+1)Γ(β+1)4(νμ)α(ρς)β[Iα,βν,ς+G(μ,ρ)×S(μ,ρ)+Iα,βν,ρG(μ,ς)×S(μ,ς)]+Γ(α+1)2(νμ)α(β(β+1)(β+2))(IανG(μ,ς)×S(μ,ς)+IανG(μ,ρ)×S(μ,ρ))+Γ(α+1)2(νμ)α(12β(β+1)(β+2))(IανG(μ,ς)×S(μ,ρ)+IανG(μ,ρ)×S(μ,ς)). (69)
    β2(ρς)β[ρς2(ρy)β1G(μ+ν2,y)×S(μ+ν2,y)dy]
    pΓ(α+1)Γ(β+1)4(νμ)α(ρς)β[Iα,βμ+,ς+G(ν,ρ)×S(ν,ρ)+Iα,βν,ς+G(μ,ρ)×S(μ,ρ)]+Γ(β+1)2(ρς)β(α(α+1)(α+2))(Iβς+G(μ,ρ)×S(μ,ρ)+Iβς+G(ν,ρ)×S(ν,ρ))+Γ(β+1)2(ρς)β(12α(α+1)(α+2))(Iβς+G(μ,ρ)×S(ν,ρ)+Iβς+G(ν,ρ)×S(ν,ρ)). (70)
    β2(ρς)β[ρς2(yς)β1G(μ+ν2,y)×S(μ+ν2,y)dy]
    pΓ(α+1)Γ(β+1)4(νμ)α(ρς)β[Iα,βμ+,ρG(ν,ς)×S(ν,ς)+Iα,βν,ρG(ν,ς)×S(ν,ς)]+Γ(β+1)2(ρς)β(α(α+1)(α+2))(IβρG(μ,ς)×S(μ,ς)+IβρG(ν,ς)×S(ν,ς))+Γ(β+1)2(ρς)β(12α(α+1)(α+2))(IβρG(μ,ς)×S(ν,ς)+IβρG(ν,ς)×S(ν,ς)). (71)

    And

    2G(μ+ν2,ς)×S(μ+ν2,ς)pΓ(α+1)2(νμ)α[Iαμ+G(ν,ς)×S(ν,ς)+IανG(μ,ς)×S(μ,ς)]+α(α+1)(α+2)(G(μ,ς)×S(μ,ς)+G(ν,ς)×S(ν,ς))+(12α(α+1)(α+2))(G(μ,ς)×S(ν,ς)+G(ν,ς)×S(μ,ς)), (72)
    2G(μ+ν2,ρ)×S(μ+ν2,ρ)pΓ(α+1)2(νμ)α[Iαμ+G(ν,ρ)×S(ν,ρ)+IανG(μ,ρ)×S(μ,ρ)]+α(α+1)(α+2)(G(μ,ρ)×S(μ,ρ)+G(ν,ρ)×S(ν,ρ))+(12α(α+1)(α+2))(G(μ,ρ)×S(ν,ρ)+G(ν,ρ)×S(μ,ρ)), (73)
    2G(μ+ν2,ς)×S(μ+ν2,ρ)pΓ(α+1)2(νμ)α[Iαμ+G(ν,ς)×S(ν,ρ)+IανG(μ,ς)×S(μ,ρ)]+α(α+1)(α+2)(G(μ,ς)×S(μ,ρ)+G(ν,ς)×S(ν,ρ))+(12α(α+1)(α+2))(G(μ,ς)×S(ν,ρ)+G(ν,ς)×S(μ,ρ)), (74)
    2G(μ+ν2,ρ)×S(μ+ν2,ς)pΓ(α+1)2(νμ)α[Iαμ+G(ν,ρ)×S(ν,ς)+IανG(μ,ρ)×S(μ,ς)]
    +α(α+1)(α+2)(G(μ,ρ)×S(μ,ς)+G(ν,ρ)×S(ν,ς))+(12α(α+1)(α+2))(G(μ,ρ)×S(ν,ς)+G(ν,ρ)×S(μ,ς)), (75)
    2G(μ,ς+ρ2)×S(μ,ς+ρ2)pΓ(β+1)2(ρς)β[Iβς+G(μ,ρ)×S(μ,ρ)+IβρG(μ,ρ)×S(μ,ς)]+β(β+1)(β+2)(G(μ,ς)×S(μ,ς)+G(μ,ρ)×S(μ,ρ))+(12β(β+1)(β+2))(G(μ,ς)×S(μ,ρ)+G(μ,ρ)×S(μ,ς)), (76)
    2G(ν,ς+ρ2)×Sϕ(ν,ς+ρ2)pΓ(β+1)2(ρς)β[Iβς+G(ν,ρ)×S(ν,ρ)+IβρG(ν,ρ)×S(ν,ς)]+β(β+1)(β+2)(G(ν,ς)×S(ν,ς)+G(ν,ρ)×S(ν,ρ))+(12β(β+1)(β+2))(G(ν,ς)×S(ν,ρ)+G(ν,ρ)×S(ν,ς)), (77)
    2G(μ,ς+ρ2)×S(ν,ς+ρ2)pΓ(β+1)2(ρς)β[Iβς+G(μ,ρ)×S(ν,ρ)+IβρG(μ,ρ)×S(ν,ς)]+β(β+1)(β+2)(G(μ,ς)×S(ν,ς)+G(μ,ρ)×S(ν,ρ))+(12β(β+1)(β+2))(G(μ,ς)×S(ν,ρ)+G(μ,ρ)×S(ν,ς)), (78)

    and

    2G(ν,ς+ρ2)×S(μ,ς+ρ2)pΓ(β+1)2(ρς)β[Iβς+G(ν,ρ)×S(μ,ρ)+IβρG(ν,ρ)×S(μ,ς)]+β(β+1)(β+2)(G(ν,ς)×S(μ,ς)+G(ν,ρ)×S(μ,ρ))+(12β(β+1)(β+2))(G(ν,ς)×S(μ,ρ)+G(ν,ρ)×S(μ,ς)), (79)

    From inequalities (68) to (79), inequality (67) we have

    8G(μ+ν2,ς+ρ2)×S(μ+ν2,ς+ρ2)pΓ(α+1)Γ(β+1)2(νμ)α(ρς)β[Iα,βμ+,ς+G(ν,ρ)×S(ν,ρ)+Iα,βμ+,ρG(ν,ς)×S(ν,ς)+Iα,βν,ς+G(μ,ρ)×S(μ,ρ)+Iα,βν,ρG(μ,ς)×S(μ,ς)]+(2α(α+1)(α+2))[Γ(β+1)2(ρς)β(Iβς+G(μ,ρ)×S(μ,ρ)+Iβς+G(ν,ρ)×S(ν,ρ))+Γ(β+1)2(ρς)β(IβρG(μ,ς)×S(μ,ς)+IβρG(ν,ς)×S(ν,ς))]+2(12α(α+1)(α+2))[Γ(β+1)2(ρς)β(Iβς+G(μ,ρ)×S(ν,ρ)+Iβς+G(ν,ρ)×S(μ,ρ))+Γ(β+1)2(ρς)β(IβρG(μ,ς)×S(ν,ς)+IβρG(ν,ς)×S(μ,ς))]+2(β(β+1)(β+2))[Γ(α+1)2(νμ)α(Iαμ+G(ν,ς)×S(ν,ς)+Iαμ+G(ν,ρ)×S(ν,ρ))+Γ(α+1)2(νμ)α(IανG(μ,ς)×S(μ,ς)+IανG(μ,ρ)×S(μ,ρ))]+2(12β(β+1)(β+2))[Γ(α+1)2(νμ)α(Iαμ+G(ν,ς)×S(ν,ρ)+Iαμ+G(ν,ρ)×S(ν,ς))+Γ(α+1)2(νμ)α(IανG(μ,ς)×S(μ,ρ)+IανG(μ,ρ)×S(μ,ς))]
    +2α(α+1)(α+2)β(β+1)(β+2)K(μ,ν,ς,ρ)++(12α(α+1)(α+2))2β(β+1)(β+2)L(μ,ν,ς,ρ)
    +2α(α+1)(α+2)(12β(β+1)(β+2))M(μ,ν,ς,ρ)+2(12α(α+1)(α+2))(12β(β+1)(β+2))N(μ,ν,ς,ρ). (80)

    Again, with the help of integral inequality (15) and Lemma 1, for each integral on the right-hand side of (80), we have

    Γ(β+1)2(ρς)β(Iβς+G(μ,ρ)×S(μ,ρ)+Iβς+G(ν,ρ)×S(ν,ρ))+Γ(β+1)2(ρς)β(IβρG(μ,ς)×S(μ,ς)+IβρG(ν,ς)×S(ν,ς))p(12β(β+1)(β+2))K(μ,ν,ς,ρ)+β(β+1)(β+2)M(μ,ν,ς,ρ). (81)
    Γ(β+1)2(ρς)β(Iβς+G(μ,ρ)×S(ν,ρ)+Iβς+G(ν,ρ)×S(μ,ρ))+Γ(β+1)2(ρς)β(IβρG(μ,ς)×S(ν,ς)+IβρG(ν,ς)×S(μ,ς))p(12β(β+1)(β+2))L(μ,ν,ς,ρ)+β(β+1)(β+2)N(μ,ν,ς,ρ). (82)
    Γ(α+1)2(νμ)α(Iαμ+G(ν,ς)×S(ν,ς)+Iαμ+G(ν,ρ)×S(ν,ρ))+Γ(α+1)2(νμ)α(IανG(μ,ς)×S(μ,ς)+IανG(μ,ρ)×S(μ,ρ))p(12α(α+1)(α+2))K(μ,ν,ς,ρ)+α(α+1)(α+2)L(μ,ν,ς,ρ). (83)
    Γ(α+1)2(νμ)α(IανG(μ,ς)×S(μ,ρ)+IανG(μ,ρ)×S(μ,ς))+Γ(α+1)2(νμ)α(IανG(μ,ς)×S(μ,ρ)+IανG(μ,ρ)×S(μ,ς))p(12α(α+1)(α+2))M(μ,ν,ς,ρ)+α(α+1)(α+2)N(μ,ν,ς,ρ). (84)

    From (77) to (84), (80) we have

    4G(μ+ν2,ς+ρ2)×S(μ+ν2,ς+ρ2)pΓ(α+1)Γ(β+1)4(νμ)α(ρς)β[Iα,βμ+,ς+G(ν,ρ)×S(ν,ρ)+Iα,βμ+,ρG(ν,ς)×S(ν,ς)+Iα,βν,ς+G(μ,ρ)×S(μ,ρ)+Iα,βν,ρG(μ,ς)×S(μ,ς)]+[α2(α+1)(α+2)+β(β+1)(β+2)(12α(α+1)(α+2))]K(μ,ν,ς,ρ)+[12(12α(α+1)(α+2))+α(α+1)(α+2)β(β+1)(β+2)]L(μ,ν,ς,ρ)+[12(12β(β+1)(β+2))+α(α+1)(α+2)β(β+1)(β+2)]M(μ,ν,ς,ρ)+[14α(α+1)(α+2)β(β+1)(β+2)]N(μ,ν,ς,ρ). (85)

    This concludes the proof of Theorem 8 result has been proven.

    Remark 5. If we take α=1 and β=1, then from (63), we achieve the coming inequality, see [38]:

    4G(μ+ν2,ς+ρ2)×S(μ+ν2,ς+ρ2)p1(νμ)(ρς)νμρςG(x,y)×S(x,y)dydx+536K(μ,ν,ς,ρ)+736[L(μ,ν,ς,ρ)+M(μ,ν,ς,ρ)]+29N(μ,ν,ς,ρ). (86)

    Let one takes G(x,y) is an affine function and G(x,y) is convex function. If G(x,y)G(x,y), then from Remark 2 and (64), we acquire the coming inequality, see [37]:

    4G(μ+ν2,ς+ρ2)×S(μ+ν2,ς+ρ2)1(νμ)(ρς)νμρςG(x,y)×S(x,y)dydx+536K(μ,ν,ς,ρ)+736[L(μ,ν,ς,ρ)+M(μ,ν,ς,ρ)]+29N(μ,ν,ς,ρ). (87)

    Let one takes G(x,y) is an affine function and G(x,y) is convex function. If G(x,y)G(x,y), then from Remark 2 and (64) we acquire the coming inequality, see [36]:

    4G(μ+ν2,ς+ρ2)×S(μ+ν2,ς+ρ2)Γ(α+1)Γ(β+1)4(νμ)α(ρς)β[Iα,βμ+,ς+G(ν,ρ)×S(ν,ρ)+Iα,βμ+,ρG(ν,ς)×S(ν,ς)+Iα,βν,ς+G(μ,ρ)×S(μ,ρ)+Iα,βν,ρG(μ,ς)×S(μ,ς)]+[α2(α+1)(α+2)+β(β+1)(β+2)(12α(α+1)(α+2))]K(μ,ν,ς,ρ)+[12(12α(α+1)(α+2))+α(α+1)(α+2)β(β+1)(β+2)]L(μ,ν,ς,ρ)+[12(12β(β+1)(β+2))+α(α+1)(α+2)β(β+1)(β+2)]M(μ,ν,ς,ρ)+[14α(α+1)(α+2)β(β+1)(β+2)]N(μ,ν,ς,ρ). (88)

    If we take G(x,y)=G(x,y) and S(x,y)=S(x,y), then from (63), we acquire the coming inequality, see [39]:

    4G(μ+ν2,ς+ρ2)×S(μ+ν2,ς+ρ2)Γ(α+1)Γ(β+1)4(νμ)α(ρς)β[Iα,βμ+,ς+G(ν,ρ)×S(ν,ρ)+Iα,βμ+,ρG(ν,ς)×S(ν,ς)+Iα,βν,ς+G(μ,ρ)×S(μ,ρ)+Iα,βν,ρG(μ,ς)×S(μ,ς)]+[α2(α+1)(α+2)+β(β+1)(β+2)(12α(α+1)(α+2))]K(μ,ν,ς,ρ)+[12(12α(α+1)(α+2))+α(α+1)(α+2)β(β+1)(β+2)]L(μ,ν,ς,ρ)+[12(12β(β+1)(β+2))+α(α+1)(α+2)β(β+1)(β+2)]M(μ,ν,ς,ρ)+[14α(α+1)(α+2)β(β+1)(β+2)]N(μ,ν,ς,ρ). (89)

    In this study, with the help of coordinated LR-convexity for interval-valued functions, several novel Hermite-Hadamard type inequalities are presented. It is also demonstrated that the conclusions reached in this study represent a possible extension of previously published equivalent results. Similar inequalities may be discovered in the future using various forms of convexities. This is a novel and intriguing topic, and future study will be able to find equivalent inequalities for various types of convexity and coordinated m-convexity by using different fractional integral operators.

    The authors would like to thank the Rector, COMSATS University Islamabad, Islamabad, Pakistan, for providing excellent research. All authors read and approved the final manuscript. This work was funded by Taif University Researchers Supporting Project number (TURSP-2020/345), Taif University, Taif, Saudi Arabia.

    The authors declare that they have no competing interests.



    [1] S. T. Liang, L. T. Liang, J. M. Rosen, COVID-19: A comparison to the 1918 influenza and how we can defeat it, Postgrad Med. J., 97 (2021), 273–274. https://doi.org/10.1136/postgradmedj-2020-139070 doi: 10.1136/postgradmedj-2020-139070
    [2] Worldometer, COVID-19 coronavirus pandemic, available from: https://www.worldometers.info/coronavirus/ (Accessed May 12, 2022).
    [3] E. Dong, H. Du, L. Gardner, An interactive web-based dashboard to track COVID-19 in real time, Lancet Infect. Dis., 20 (2020), 533–534. https://doi.org/10.1016/S1473-3099(20)30120-1 doi: 10.1016/S1473-3099(20)30120-1
    [4] C. N. Ngonghala, E. Iboi, S. Eikenberry, M. Scotch, C. R. MacIntyre, M. H. Bonds, et al., Mathematical assessment of the impact of non-pharmaceutical interventions on curtailing the 2019 novel coronavirus, Math. Biosci., 325 (2020), 108364. https://doi.org/10.1016/j.mbs.2020.108364 doi: 10.1016/j.mbs.2020.108364
    [5] C. N. Ngonghala, E. A. Iboi, A. B. Gumel, Could masks curtail the post-lockdown resurgence of COVID-19 in the US?, Math. Biosci., 329 (2020), 108452. https://doi.org/10.1016/j.mbs.2020.108452 doi: 10.1016/j.mbs.2020.108452
    [6] C. N. Ngonghala, P. Goel, D. Kutor, S. Bhattacharyya, Human choice to self-isolate in the face of the Covid-19 pandemic: a game dynamic modelling approach, J. Theor. Biol., 521 (2021), 110692. https://doi.org/10.1016/j.jtbi.2021.110692 doi: 10.1016/j.jtbi.2021.110692
    [7] S. E. Eikenberry, M. Mancuso, E. Iboi, T. Phan, K. Eikenberry, Y. Kuang, et al., To mask or not to mask: Modeling the potential for face mask use by the general public to curtail the COVID-19 pandemic, Infect. Dis. Model., 5 (2020), 293–308. https://doi.org/10.1016/j.idm.2020.04.001 doi: 10.1016/j.idm.2020.04.001
    [8] C. N. Ngonghala, J. R. Knitter, L. Marinacci, M. H. Bonds, A. B. Gumel, Assessing the impact of widespread respirator use in curtailing COVID-19 transmission in the USA, Roy. Soc. Open Sci., 8 (2021), 210699. https://doi.org/10.1098/rsos.210699 doi: 10.1098/rsos.210699
    [9] Pfizer, Pfizer and Biontech to submit emergency use authorization request today to the US FDA for COVID-19 vaccine, 2020.
    [10] US Food and Drug Administration, FDA briefing document, in: Oncology Drug Advisory Committee Meeting, Silver Spring, MD, 2009.
    [11] E. Mahase, COVID-19: Moderna vaccine is nearly 95% effective, trial involving high risk and elderly people shows, BMJ- Brit. Med. J., 371 (2020), m4471.
    [12] W. H. Self, M. W. Tenforde, J. P. Rhoads, M. Gaglani, A. A. Ginde, D. J. Douin, et al., Comparative effectiveness of Moderna, Pfizer-Biontech, and Janssen (Johnson & Johnson) vaccines in preventing COVID-19 hospitalizations among adults without immunocompromising conditions—United States, March-August 2021, Morb. Mort. Wkly Rep., 70 (2021), 1337–1343. https://doi.org/10.15585/mmwr.mm7038e1 doi: 10.15585/mmwr.mm7038e1
    [13] US Food and Drug Administration, FDA issues emergency use authorization for third COVID-19 vaccine, FSA News Release, 2021.
    [14] J. Sargent, S. Kumar, K. Buckley, J. McIntyre, Johnson & Johnson announces real-world evidence and phase 3 data confirming substantial protection of single-shot COVID-19 vaccine in the US additional data show a booster increases protection1, 2021.
    [15] F. P. Polack, S. J. Thomas, N. Kitchin, J. Absalon, A. Gurtman, S. Lockhart, et al., Safety and efficacy of the BNT162b2 mRNA COVID-19 vaccine, N. Engl. J. Med., 383 (2020), 2603–2615. https://doi.org/10.1056/NEJMoa2034577 doi: 10.1056/NEJMoa2034577
    [16] Y. M. Bar-On, Y. Goldberg, M. Mandel, O. Bodenheimer, L. Freedman, N. Kalkstein, et al., Protection of BNT162b2 vaccine booster against COVID-19 in Israel, N. Engl. J. Med., 385 (2021), 1393–1400. https://doi.org/10.1056/NEJMoa2114255 doi: 10.1056/NEJMoa2114255
    [17] E. Mahase, COVID-19: What new variants are emerging and how are they being investigated?, BMJ-Brit. Med. J., 372 (2021), n158. https://doi.org/10.1136/bmj.n158 doi: 10.1136/bmj.n158
    [18] A. Gómez-Carballa, J. Pardo-Seco, X. Bello, F. Martinón-Torres, A. Salas, Superspreading in the emergence of covid-19 variants, Trends Genet., 37 (2021), 1069–1080. https://doi.org/10.1016/j.tig.2021.09.003 doi: 10.1016/j.tig.2021.09.003
    [19] S. S. A. Karim, Q. A. Karim, Omicron Sars-Cov-2 variant: a new chapter in the COVID-19 pandemic, The Lancet, 398 (2021), 2126–2128. https://doi.org/10.1016/S0140-6736(21)02758-6 doi: 10.1016/S0140-6736(21)02758-6
    [20] D. Duong, What's important to know about the new COVID-19 variants?, CMAJ: Can. Med. Assoc. J., 193 (2021), E141–E142. https://doi.org/10.1503/cmaj.1095915 doi: 10.1503/cmaj.1095915
    [21] T. Koyama, D. Weeraratne, J. L. Snowdon, L. Parida, Emergence of drift variants that may affect COVID-19 vaccine development and antibody treatment, Pathogens, 9 (2020), 324. https://doi.org/10.3390/pathogens9050324 doi: 10.3390/pathogens9050324
    [22] C. Del Rio, S. B. Omer, P. N. Malani, Winter of omicron—the evolving COVID-19 pandemic, JAMA, 327 (2022), 319–320. https://doi.org/10.1001/jama.2021.24315 doi: 10.1001/jama.2021.24315
    [23] E. Callaway, H. Ledford, How bad is Omicron? what scientists know so far, Nature, 600 (2021), 197–199. https://doi.org/10.1038/d41586-021-03614-z doi: 10.1038/d41586-021-03614-z
    [24] Center for Disease Control and Prevention, Omicron Variant: What You Need to Know, available from: https://www.cdc.gov/coronavirus/2019-ncov/variants/omicron-variant.html#., (Accessed May 09, 2022).
    [25] F. Rahimi, A. T. B. Abadi, The Omicron subvariant BA. 2: Birth of a new challenge during the COVID-19 pandemic, Int. J. Surg., 99 (2022), 106261. https://doi.org/10.1016/j.ijsu.2022.106261 doi: 10.1016/j.ijsu.2022.106261
    [26] K. Katella, Omicron and the BA.2 Subvariant: A Guide to What We Know, available from: https://www.yalemedicine.org/news/5-things-to-know-omicron, (Accessed May 09, 2022).
    [27] C. N. Ngonghala, H. B. Taboe, S. Safdar, A. B. Gumel, Unraveling the dynamics of the Omicron and Delta variants of the 2019 coronavirus in the presence of vaccination, mask usage, and antiviral treatment, medRxiv, (2022), 2022.02.23.22271394. https://doi.org/10.1101/2022.02.23.22271394
    [28] A. B. Gumel, E. A. Iboi, C. N. Ngonghala, G. A. Ngwa, Toward achieving a vaccine-derived herd immunity threshold for COVID-19 in the US, Front. Public Health, 9 (2021), 709369. https://doi.org/10.3389/fpubh.2021.709369 doi: 10.3389/fpubh.2021.709369
    [29] H. E. Fast, E. Zell, B. P. Murthy, N. Murthy, L. Meng, L. G. Scharf, et al., Booster and additional primary dose COVID-19 vaccinations among adults aged 65 years—United States, August 13, 2021–November 19, 2021, Morb. Mortal. Wkly Rep., 70 (2021), 1735. https://doi.org/10.15585/mmwr.mm7050e2 doi: 10.15585/mmwr.mm7050e2
    [30] E. A. Iboi, C. N. Ngonghala, A. B. Gumel, Will an imperfect vaccine curtail the COVID-19 pandemic in the US?, Infect. Dis. Model., 5 (2020), 510–524. https://doi.org/10.1016/j.idm.2020.07.006 doi: 10.1016/j.idm.2020.07.006
    [31] A. B. Gumel, E. A. Iboi, C. N. Ngonghala, E. H. Elbasha, A primer on using mathematics to understand Covid-19 dynamics: Modeling, analysis and simulations, Infect. Dis. Model., 6 (2020), 148–168. https://doi.org/10.1016/j.idm.2020.11.005 doi: 10.1016/j.idm.2020.11.005
    [32] H. B. Taboe, M. Asare-Baah, A. Yesmin, C. N. Ngonghala, Impact of age structure and vaccine prioritization on COVID-19 in West Africa, Infect. Dis. Model., (2022). https://doi.org/10.1016/j.idm.2022.08.006
    [33] C. N. Ngonghala, A. B. Gumel, Mathematical assessment of the role of vaccination against COVID-19 in the United States, in Mathematical Modeling, Simulations, and AI for Emergent Pandemic Diseases: Lessons Learned from COVID-19 (eds. Jorge X. Velasco Hernández and Esteban A. Hernandez-Vargas), Elsevier, (2022), 1–30.
    [34] S. A. Rella, Y. A. Kulikova, E. T. Dermitzakis, F. A. Kondrashov, Rates of SARS-Cov-2 transmission and vaccination impact the fate of vaccine-resistant strains, Sci. Rep., 11 (2021), 1–10. https://doi.org/10.1038/s41598-021-95025-3 doi: 10.1038/s41598-021-95025-3
    [35] B. Curley, How long does immunity from COVID-19 vaccination last?, Healthline, (Accessed on July 25, 2021).
    [36] M. Mrityunjaya, V. Pavithra, R. Neelam, P. Janhavi, P. Halami, P. Ravindra, Immune-boosting, antioxidant and anti-inflammatory food supplements targeting pathogenesis of COVID-19, Front. Immunol., 11 (2020), 570122. https://doi.org/10.3389/fimmu.2020.570122 doi: 10.3389/fimmu.2020.570122
    [37] M. Alagawany, Y. A. Attia, M. R. Farag, S. S. Elnesr, S. A. Nagadi, M. E. Shafi, et al., The strategy of boosting the immune system under the COVID-19 pandemic, Front. Vet. Sci., (2021), 712. https://doi.org/10.3389/fvets.2020.570748 doi: 10.3389/fvets.2020.570748
    [38] Food and Drug Administration, FDA briefing document, Pfizer-Biontech COVID-19 vaccine, in: Vaccines and Related Biological Products Advisory Committee Meeting, 2020.
    [39] S. E. Oliver, J. W. Gargano, M. Marin, M. Wallace, K. G. Curran, et al., The Advisory Committee on Immunization Practices' interim recommendation for use of Pfizer-Biontech COVID-19 vaccine - United States, December 2020, Morb. Mortal. Wkly Rep., 69 (2020), 1922–1924. https://doi.org/10.15585/mmwr.mm6950e2 doi: 10.15585/mmwr.mm6950e2
    [40] US Food and Drug Administration and others, Coronavirus (COVID-19) update: FDA issues policies to guide medical product developers addressing virus variants, FDA. February 23, 2021.
    [41] L. Childs, D. W. Dick, Z. Feng, J. M. Heffernan, J. Li, G. Röst, Modeling waning and boosting of covid-19 in canada with vaccination, Epidemics, (2022), 100583. https://doi.org/10.1016/j.epidem.2022.100583 doi: 10.1016/j.epidem.2022.100583
    [42] Centers for Disease Control and Prevention, CDC expands eligibility for COVID-19 booster shots to all adults, 2021.
    [43] W. Pacific, S. A. W. Hasan, Interim statement on booster doses for COVID-19 vaccination, Update, 4 (2021).
    [44] V. Lakshmikantham, A. Vatsala, Theory of differential and integral inequalities with initial time difference and applications, in: Analytic and Geometric Inequalities and Applications, Springer, Dordrecht. 1999, pp. 191–203. https://doi.org/10.1007/978-94-011-4577-0
    [45] H. W. Hethcote, The mathematics of infectious diseases, SIAM Rev., 42 (2000), 599–653. https://doi.org/10.1137/S0036144500371907 doi: 10.1137/S0036144500371907
    [46] P. Van den Driessche, J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29–48. https://doi.org/10.1016/S0025-5564(02)00108-6 doi: 10.1016/S0025-5564(02)00108-6
    [47] O. Diekmann, J. A. P. Heesterbeek, J. A. Metz, On the definition and the computation of the basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations, J. Math. Biol., 28 (1990), 365–382. https://doi.org/10.1007/BF00178324 doi: 10.1007/BF00178324
    [48] V. Lakshmikantham, S. Leela, A. A. Martynyuk, Stability Analysis of Nonlinear Systems, Springer, 1989.
    [49] A. B. Gumel, C. C. McCluskey, P. van den Driessche, Mathematical study of a staged-progression hiv model with imperfect vaccine, Bull. Math. Biol., 68 (2006), 2105–2128. https://doi.org/10.1007/s11538-006-9095-7 doi: 10.1007/s11538-006-9095-7
    [50] R. M. Anderson, The concept of herd immunity and the design of community-based immunization programmes, Vaccine, 10 (1992), 928–935. https://doi.org/10.1016/0264-410X(92)90327-G doi: 10.1016/0264-410X(92)90327-G
    [51] R. M. Anderson, R. M. May, Vaccination and herd immunity to infectious diseases, Nature, 318 (1985), 323–329. https://doi.org/10.1038/318323a0 doi: 10.1038/318323a0
    [52] S. Pearson, What is the difference between the pfizer, moderna, and johnson & johnson covid-19 vaccines?, GoodRx (Accessed on June 25, 2021) (2021).
    [53] M. Mancuso, S. E. Eikenberry, A. B. Gumel, Will vaccine-derived protective immunity curtail covid-19 variants in the US?, Infect. Dis. Model., 6 (2021), 1110–1134. https://doi.org/10.1016/j.idm.2021.08.008 doi: 10.1016/j.idm.2021.08.008
    [54] Center for Disease Control and Prevention, It's Time for a Boost, available from: https://www.cdc.gov/coronavirus/2019-ncov/covid-data/covidview/past-reports/05202022.html, (Accessed July 08, 2022).
    [55] D.-Y. Lin, Y. Gu, B. Wheeler, H. Young, S. Holloway, S.-K. Sunny, et al., Effectiveness of COVID-19 vaccines over a 9-month period in North Carolina, N. Engl. J. Med., 386 (2022), 933–941. https://doi.org/10.1056/NEJMoa2117128 doi: 10.1056/NEJMoa2117128
    [56] N. Andrews, J. Stowe, F. Kirsebom, S. Toffa, R. Sachdeva, C. Gower, et al., Effectiveness of COVID-19 booster vaccines against COVID-19-related symptoms, hospitalization and death in England, Nat. Med., 28 (2022), 831–837. https://doi.org/10.1038/s41591-022-01699-1 doi: 10.1038/s41591-022-01699-1
    [57] S. M. Sidik, Vaccines protect against infection from Omicron subvariant-but not for long, Nature, 2022 Mar. https://doi.org/10.1038/d41586-022-00775-3
    [58] S. H. Tan, A. R. Cook, D. Heng, B. Ong, D. C. Lye, K. B. Tan, Effectiveness of BNT162b2 vaccine against Omicron in children 5 to 11 years of age, N. Engl. J. Med., 387 (2022), 525–532. https://doi.org/10.1056/NEJMoa2203209 doi: 10.1056/NEJMoa2203209
    [59] R. Grewal, S. A. Kitchen, L. Nguyen, S. A. Buchan, S. E. Wilson, A. P. Costa, et al., Effectiveness of a fourth dose of COVID-19 mRNA vaccine against the Omicron variant among long term care residents in Ontario, Canada: test negative design study, BMJ, (2022), e071502. https://doi.org/10.1136/bmj-2022-071502
    [60] L. Jansen, B. Tegomoh, K. Lange, K. Showalter, J. Figliomeni, B. Abdalhamid, et al., Investigation of a SARS-Cov-2 B. 1.1. 529 (Omicron) variant cluster—Nebraska, November–December 2021, Morb. Mortal. Wkly Rep., 70 (2021), 1782–1784. https://doi.org/10.15585/mmwr.mm705152e3 doi: 10.15585/mmwr.mm705152e3
    [61] B. Curley, "How long does immunity from COVID-19 vaccination last?"Healthline, available from: https://www.healthline.com/health-news/how-long-does-immunity-from-covid-19-vaccination-last, (Accessed March 22, 2022).
    [62] N. M. Linton, T. Kobayashi, Y. Yang, K. Hayashi, A. R. Akhmetzhanov, S. Jung, et al., Incubation period and other epidemiological characteristics of 2019 novel coronavirus infections with right truncation: a statistical analysis of publicly available case data, J. Clin. Med., 9 (2020), 538. https://doi.org/10.3390/jcm9020538 doi: 10.3390/jcm9020538
    [63] K. Weintraub, Enormous spread of Omicron may bring 140M new Covid infections to US in the next two months, model predicts, available from: https://www.wusa9.com/article/news/verify/how-long-does-it-take-for-the-vaccine-booster-to-get-to-full-protection/65-aa7344c2-fcd5-4c70-bbcd-046e9f697be7, (Accessed March 22, 2022).
    [64] M. Gregory, M. Salenetri, How long does immunity from COVID-19 vaccination, available from: https://www.wusa9.com/article/news/verify/how-long-does-it-take-for-the-vaccine-booster-to-get-to-full-protection/65-aa7344c2-fcd5-4c70-bbcd-046e9f697be7, (Accessed March 22, 2022).
    [65] M. G. Thompson, Effectiveness of a third dose of mRNA vaccines against COVID-19–associated emergency department and urgent care encounters and hospitalizations among adults during periods of Delta and Omicron variant predominance—VISION Network, 10 States, August 2021–January 2022, Morb. Mortal. Wkly Rep., 71 (2022), 139–145. https://doi.org/10.15585/mmwr.mm7104e3 doi: 10.15585/mmwr.mm7104e3
    [66] J. Bosman, J. Hoffman, M. Sanger-Katz, T. Arango, Who are the unvaccinated in America? there's no one answer, The New York Times, 2021.
    [67] J. K. Tan, D. Leong, H. Munusamy, N. H. Zenol Ariffin, N. Kori, R. Hod, et al., The prevalence and clinical significance of Presymptomatic COVID-19 patients: how we can be one step ahead in mitigating a deadly pandemic, BMC Infect. Dis., 21 (2021), 1–10. https://doi.org/10.1186/s12879-021-05849-7 doi: 10.1186/s12879-021-05849-7
    [68] S. Desmon, COVID and the Heart: It Spares No One, available from: https://publichealth.jhu.edu/2022/covid-and-the-heart-it-spares-no-one, (Accessed August 30, 2022).
    [69] V. Thakur, R. K. Ratho, Omicron (b. 1.1. 529): A new SARS-CoV-2 variant of concern mounting worldwide fear, J. Med. Virol., 94 (2022), 1821–1824. https://doi.org/10.1002/jmv.27541 doi: 10.1002/jmv.27541
    [70] J. M. Dan, J. Mateus, Y. Kato, K. M. Hastie, E. D. Yu, C. E. Faliti, et al., Immunological memory to SARS-Cov-2 assessed for up to 8 months after infection, Science, 371 (2021), eabf4063. https://doi.org/10.1126/science.abf4063 doi: 10.1126/science.abf4063
    [71] J. M. Ferdinands, S. Rao, B. E. Dixon, P. K. Mitchell, M. B. DeSilva, S. A. Irving, et al., Waning 2-dose and 3-dose effectiveness of mRNA vaccines against COVID-19–associated emergency department and urgent care encounters and hospitalizations among adults during periods of Delta and Omicron variant predominance—vision network, 10 states, August 2021–January 2022, Morb. Mortal. Wkly Rep., 71 (2022), 255–263. https://doi.org/10.15585/mmwr.mm7107e2 doi: 10.15585/mmwr.mm7107e2
    [72] Z. Zhongming, L. Linong, Y. Xiaona, Z. Wangqiang, L. Wei, Omicron largely evades immunity from past infection or two vaccine doses, 2021.
    [73] P. Elliott, O. Eales, B. Bodinier, D. Tang, H. Wang, J. Jonnerby, et al., Dynamics of a national Omicron SARS-CoV-2 epidemic during {J}anuary 2022 in England, Nat. Commun., 13 (2022), 1–10. https://doi.org/10.1038/s41467-022-32121-6 doi: 10.1038/s41467-022-32121-6
    [74] P. Elliott, O. Eales, N. Steyn, D. Tang, B. Bodinier, H. Wang, et al., Twin peaks: the Omicron SARS-CoV-2 BA. 1 and BA. 2 epidemics in England, Science, (2022), eabq4411. https://doi.org/10.1126/science.abq4411
    [75] D. Kim, S. T. Ali, S. Kim, J. Jo, J.-S. Lim, S. Lee, et al., Estimation of serial interval and reproduction number to quantify the transmissibility of SARS-CoV-2 Omicron variant in South Korea, Viruses, 14 (2022), 533. https://doi.org/10.3390/v14030533 doi: 10.3390/v14030533
    [76] H. F. Tseng, B. K. Ackerson, Y. Luo, L. S. Sy, C. A. Talarico, Y. Tian, et al., Effectiveness of mRNA-1273 against SARS-CoV-2 Omicron and Delta variants, Nat. Med., 28 (2022), 1063–1071. https://doi.org/10.1038/s41591-022-01753-y doi: 10.1038/s41591-022-01753-y
    [77] H. Chemaitelly, H. H. Ayoub, S. AlMukdad, P. Coyle, P. Tang, H. M. Yassine, et al., Duration of mRNA vaccine protection against SARS-CoV-2 Omicron BA. 1 and BA. 2 subvariants in Qatar, Nat. Commun., 13 (2022), 3082. https://doi.org/10.1038/s41467-022-30895-3 doi: 10.1038/s41467-022-30895-3
  • This article has been cited by:

    1. Waqar Afzal, Khurram Shabbir, Thongchai Botmart, Generalized version of Jensen and Hermite-Hadamard inequalities for interval-valued (h1,h2)-Godunova-Levin functions, 2022, 7, 2473-6988, 19372, 10.3934/math.20221064
    2. Muhammad Bilal Khan, Omar Mutab Alsalami, Savin Treanțǎ, Tareq Saeed, Kamsing Nonlaopon, New class of convex interval-valued functions and Riemann Liouville fractional integral inequalities, 2022, 7, 2473-6988, 15497, 10.3934/math.2022849
    3. Miguel J. Vivas-Cortez, Hasan Kara, Hüseyin Budak, Muhammad Aamir Ali, Saowaluck Chasreechai, Generalized fractional Hermite-Hadamard type inclusions for co-ordinated convex interval-valued functions, 2022, 20, 2391-5455, 1887, 10.1515/math-2022-0477
    4. Tareq Saeed, Eze R. Nwaeze, Muhammad Bilal Khan, Khalil Hadi Hakami, New Version of Fractional Pachpatte-Type Integral Inequalities via Coordinated ℏ-Convexity via Left and Right Order Relation, 2024, 8, 2504-3110, 125, 10.3390/fractalfract8030125
    5. HAIYANG CHENG, DAFANG ZHAO, GUOHUI ZHAO, FRACTIONAL QUANTUM HERMITE–HADAMARD-TYPE INEQUALITIES FOR INTERVAL-VALUED FUNCTIONS, 2023, 31, 0218-348X, 10.1142/S0218348X23501049
    6. Haiyang Cheng, Dafang Zhao, Guohui Zhao, Delfim F. M. Torres, New quantum integral inequalities for left and right log-ℏ-convex interval-valued functions, 2024, 31, 1072-947X, 381, 10.1515/gmj-2023-2088
  • Reader Comments
  • © 2023 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(3332) PDF downloads(170) Cited by(11)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog