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Effects of exercise, physical activity, and sports on physical fitness in adults with Down syndrome: A systematic review

  • This systematic review aimed to analyze the effects of exercise, physical activity, and sports on physical fitness in adults with Down syndrome (DS). A literature search was conducted across four databases EBSCO, Scopus, Web of Science, and PubMed. The PRISMA guidelines were followed. The PEDro scale and the Cochrane risk of bias tool were used to assess the quality and risk of the studies, respectively. The protocol was registered in PROSPERO (code: CRD42023449627). Of the 423 records initially found, 13 were finally included in the systematic review, in which 349 adults with DS participated. 92% of the articles declared at least one significant difference post-intervention. The available evidence indicates that exercise, physical activity, and sports have a positive effect on some variables of physical fitness, especially strength, balance, body composition, cardiorespiratory fitness, flexibility, and functional capacity. Furthermore, it should be considered as an additional treatment or complementary therapy to improve the functionality and quality of life of adults with DS.

    Citation: Felipe Montalva-Valenzuela, Antonio Castillo-Paredes, Claudio Farias-Valenzuela, Oscar Andrades-Ramirez, Yeny Concha-Cisternas, Eduardo Guzmán-Muñoz. Effects of exercise, physical activity, and sports on physical fitness in adults with Down syndrome: A systematic review[J]. AIMS Public Health, 2024, 11(2): 577-600. doi: 10.3934/publichealth.2024029

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  • This systematic review aimed to analyze the effects of exercise, physical activity, and sports on physical fitness in adults with Down syndrome (DS). A literature search was conducted across four databases EBSCO, Scopus, Web of Science, and PubMed. The PRISMA guidelines were followed. The PEDro scale and the Cochrane risk of bias tool were used to assess the quality and risk of the studies, respectively. The protocol was registered in PROSPERO (code: CRD42023449627). Of the 423 records initially found, 13 were finally included in the systematic review, in which 349 adults with DS participated. 92% of the articles declared at least one significant difference post-intervention. The available evidence indicates that exercise, physical activity, and sports have a positive effect on some variables of physical fitness, especially strength, balance, body composition, cardiorespiratory fitness, flexibility, and functional capacity. Furthermore, it should be considered as an additional treatment or complementary therapy to improve the functionality and quality of life of adults with DS.



    Delay differential equations (DDEs) are used numerously in many applications of engineering sciences and technology. They are used to describe the propagation of transport phenomena in dynamical systems, especially those dynamical systems which are nonlinear in nature. As in DDEs the unknown functions also depend on the history, therefore it is natural to use DDEs in the mathematical modeling of some biological processes (cell growth etc.) and economical system (evaluation of market, investment policy etc.). Functional differential equation also known as pantograph type delay differential equation is an important class of DDEs arises in many application, for example, immunology, physiology, electrodynamics, communication and neural network where signal transmission is carried by time interval (nonzero) between the initial and delivery time of a signal or message, where such systems are often described by functional spaces in mathematical framework. The delay term in these models is related to some hidden processes and therefore one must use a high order numerical technique to capture these hidden processes. A comprehensive list of applications of DDEs can be found in [12,22]. Consider the DDEs of the form

    {u(t)=α(t)u(t)+β(x)u(rt)+γ(x)u(rt),tI:=[0,T]u(0)=u0. (1.1)

    where α(t),β(t) and γ(t) are smooth functions on I:=[0,T] and r(0,1) is a fixed constant known as proportional delay. Equation (1.1), which is a special type DDEs called the general pantograph type DDE with neutral term. Due to the transcendental nature of Eq (1.1) most of the author's used approximate methods to solve it numerically. In the start of twenty's, the researcher uses the application of collocation method and continuous Runge-Kutta (CRK) method [12,15]. The CRK method does not achieve the required accuracy due the insufficient information on the right hand side of Eq (1.1), while using the collocation method with piecewise polynomials having degree n1 with meshes to be uniform does not achieve the classical superconvergence rate of O(h2n)-; for n2 the order (optimal) is only n+2 [13,14]. Thus, it was natural to switch to some methods to avoid these difficulties and get the exponential order of convergence with less computational efforts [10,11]. To this end, the use of orthogonal polynomials and their properties are more useful to achieve the required accuracy. The idea of rational polynomial approximation was introduced in [24], while the Legendre collocation methods with detail convergence analysis results was used for the variety of DDEs and stochastic DDE including the pantograph type in [8,9,25,26,27,28]. Similarly the Tau method based on Chebyshev approximation and their operational matrix is used in [23]. The main aim of this work is to use Bernstein polynomial to find the approximate solution of Eq (1.1). The disadvantage of using Bernstein polynomial is that these polynomials are not orthogonal in nature. For this reason the change of basis function from Bernstein to Legendre polynomial will be used with the help of some matrix transformation [4,5,6]. The Bernstein approximation method is a powerful numerical technique used by a number of authors for the numerical approximation of different type of differential equations[17,18,19,20,29,30,31].

    The rest of the paper is organized as: section 2 of the paper consist of preliminaries, followed by Bernstein collocation method in section 3. Section 4 describe the Bernstein-Legendre basis transformation. The error analysis is presented in section 5. Numerical examples are given in section 6, followed by conclusion in section 7.

    First we will introduce some basic of Bernstein polynomials and their properties. For any ˉt[0,1], Bernstein polynomials are define as [3].

    ˉBˉnˉi(ˉt)=(ˉnˉi)ˉt(1ˉt)ˉnˉi,ˉi=0,...,ˉn,where(ˉnˉi)=ˉn!ˉi!(ˉnˉi)!, (2.1)

    satisfying the following 3-term recurrence relation

    ˉBˉnˉi(ˉt)=ˉBˉn1ˉi(ˉt)ˉtˉBˉn1ˉi(ˉt)+ˉtˉBˉn1ˉi1(ˉt). (2.2)

    First few terms of Bernstein polynomials are given by

    ˉB01=1ˉt,ˉB11=ˉt,ˉB02=1ˉt2,ˉB12=2ˉt(1ˉt),ˉB22=ˉt2,
    ˉB03=(1ˉt)3,ˉB13=3ˉt(1ˉt)2,ˉB23=3ˉt2(1ˉt),ˉB33=ˉt3.

    Bernstein polynomials also satisfying the following properties.

    ⅰ. ˉnˉi=0ˉBˉnˉi(ˉt)1, (Unitary).

    ⅱ. ˉBˉnˉi(ˉt)0,ˉt[0,1] (non negative).

    ⅲ. ˉBˉnˉi(ˉt)=ˉBˉnˉnˉi(1ˉt),(symmetric).

    ⅳ. ˉBˉnˉi(ˉt), has maximum value at ˉt=ˉiˉn (uni-modality).

    There product and integral is given

    ˉBˉnˉi(ˉt)ˉBˉmˉj(ˉt)=(ˉnˉi)(ˉmˉj)(ˉn+ˉmˉi+ˉj)(ˉn+ˉmˉi+ˉj),
    10ˉBˉnˉi(ˉt)dˉt=1ˉn+1.

    Bernstein polynomial form a complete basis over the interval [a,b]. Any unknown function u(t) which is define on [a,b] can be approximated with Bernstein polynomials having n degree basis function as

    u(t)ˉnˉi=0ˉCiˉBˉnˉi(ˉt)=ˉCTˉB(ˉt), (2.3)

    where ˉC and ˉB(ˉt) are (ˉn+1)×1 given as

    ˉC=[ˉc0,ˉc1,ˉc2,...,ˉcˉn]T,
    ˉB(ˉt)=[ˉBˉnˉ0,ˉBˉnˉ1,ˉBˉnˉ2,...,ˉBˉnˉn].

    Since we are interested in the Legendre form of Bernstein polynomials. The Legendre polynomials form orthonormal basis on [1,1], while Bernstein polynomials are define over [0,1]. In order to use the orthogonality properties of Legendre polynomials with very sophisticated geometric properties of Bernstein polynomials, the recurrence relation of Legendre polynomials ˉL(ˉt) on ˉt[0,1] is given by

    ˉLˉn(ˉt)=2ˉn1ˉn(2ˉt1)ˉLˉn1(ˉt)ˉn2ˉnˉLˉn2(ˉt).

    The first few Legendre polynomials on [0,1] are given by

    L0(ˉt)=1,ˉL1(ˉt)=3(2ˉt1), =L2(ˉt)=5(6ˉt26ˉt+1),
    ˉL3(ˉt)=7(20ˉt330ˉt2+12ˉt1).

    The orthonormal properties of the shifted Legendre polynomial is given by

    τf0ˉLˉj(ˉt)Lˉk(ˉt)={τf2ˉk+1,ifˉj=ˉk,0,ifˉj ˉk.

    As we know that for non-negative bases polynomials orthogonality is not possible. To avoid this and in order to fully use the properties of orthogonal polynomials with the geometric properties of Bernstein basis, we will use matrices transformation between Bernstein and Legendre polynomials[1,2].

    Consider ˉPˉn(ˉt), a polynomial of degree ˉn can be expressed in the degree ˉn Bernstein and Legendre basis on ˉt[0,1] in the following form:

    ˉPˉn(ˉt)=ˉnˉj=0ˉcˉjˉBˉnˉj(ˉt)=ˉnˉk=0 =lˉkˉLˉk(ˉt). (3.1)

    The linear transformation that maps the Bernstein coefficients ˉcˉ0,ˉc =1,...,ˉcˉn into the Legendre coefficient ˉlˉ0,ˉlˉ1,...,ˉlˉn is given by Eqs (5) and (7) respectively.

    ˉcˉj=ˉnj=0ˉMˉn(ˉj,ˉk)ˉlˉk,ˉj=0,1,...,ˉn, (3.2)
    ˉlˉk=ˉnˉk=0ˉM1ˉn(ˉj,ˉk)ˉcˉj,ˉk=0,1,...,ˉn, (3.3)

    where

    ˉM=1(ˉnˉk)min(ˉj,ˉk)ˉi=max(ˉ0,ˉj+ˉkˉn)(1)ˉk+ˉi(ˉjˉi)(ˉkˉi)(ˉnˉkˉjˉi),

    and

    ˉM1=2ˉj+1ˉn+ˉj+ˉk(ˉnˉk)ˉjˉi=0(1)ˉj+ˉi(ˉjˉi)(ˉjˉi)(ˉnˉjˉjˉi),

    As we are interested in the Bernstein form of Legendre polynomial, therefore the the Legendre polynomial in Bernstein form are given by

    ˉLˉn(ˉt)=ˉnˉi=0(1)ˉn+ˉi(ˉnˉi)ˉBˉnˉi(ˉt), (3.4)

    where the first few Legendre polynomial in Bernstein form are given by:

    ˉL0(ˉt)=B00(ˉt),ˉL1(ˉt)=B10(ˉt)+B11(ˉt),ˉL2(ˉt)=B20(ˉt)2B21(ˉt)+B22(ˉt).
    ˉL3(ˉt)=B30(ˉt)+3B31(ˉt)3B32(ˉt)+B33(ˉt).

    In order to fully use the properties of orthogonal polynomials, we will apply spectral method to the integrated form of Eq (1.1). For the reason integrating Eq (1.1) from [0,t], we get:

    u(t)=u0+t0α(s)u(s)ds+t0β(s)u(rs)ds+t0γ(s)u(rs)ds. (4.1)

    Let Eq (4.1) holds at tj, where tj=t0+kh are the collocation points with t0=a, h=ba/ ˉn,k=0,1,2,...,ˉn1, we get

    u(tj)=u0+tj0α(s)u(s)ds+tj0β(s)u(rs)ds+tj0γ(s)u(rs)ds, (4.2)

    or

    u(tj)=u0+γ(tj)u(rtj)γ(0)u(0)+tj0α(s)u(s)ds+tj0β(s)u(rs)ds+tj0γ(s)u(rs)ds, (4.3)

    Using the linear transformation

    s=stj/τf,0tjτf,

    we get

    u(tj)=u0+γ(tj)u(rtj)γ(0)u(0)+tj10α(s)u(s)ds+rtj10β(s)u(rs)ds+tj10γ(s)u(rs)ds. (4.4)

    Using Eq (4) in Eq (11), we get

    ˉCTˉB(ˉt)=ˉCTˉB(0)+γ(tj)ˉCTˉB(ˉt)(rtj)γ(0)ˉCT =B(0)+tj10α(s)ˉCT =B(ˉt)(s)ds+rtj10β(s)ˉCTˉB(ˉt)(rs)ds+tj10γ(s)ˉCTˉB(ˉt)(rs)ds. (4.5)

    Thus together with the initial condition we get a linear system of 2 =n+2 equations. As we are more interested in the Legendre form of Bernstein polynomial, therefore using the (N+1)-point Gauss-Legendre points relative to the Legendre weight gives

    u(tj)=u0+tjNk=0α(s)u(s)ωk+rtjNk=0β(s)u(rs)ωk+γ(tj)u(rtj)u(0)γ(0)tjNk=0γ(s)u(rs)ωk. (4.6)

    Let Uiu(tj) and assume that UPN is of the form

    U(t)=Nj=0UjFj(t), (4.7)

    where Fj(t) is Lagrange interpolation polynomials associated with Legendre-Gauss points {tj}Nj=0. The numerical approximation for solving (1.1) is then given by

    Uj=u0(1γ(0))/r+tjNk=0α(s)U(s)ωk+rtjNk=0β(s)U(rs)ωktjNk=0γ(s)U(rs)ωk. (4.8)

    Let U=[U0,,UN]T and FN=[u0(1γ(0)),,uN(1γ(0))]T, we can obtain a matrix form:

    U+AU=FN, (4.9)

    To compute F(s) in efficient way, we express it in terms of the Bernstein form of the Legendre functions given in Eq (5).

    Theorem. If uˉj(ˉt),j=1,2,...ˉn denotes the exact solution to the neutral functional differential equation of pantograph type (1.1), while Uˉj,ˉm(ˉt) denotes its approximate solution, then the error between the exact and approximate solution converge exponentially that is

    uˉj(ˉt)Uˉj,ˉm(ˉt)0,ˉm.

    Proof. Let Uˉj,ˉm(ˉt)=ˉmˉp=0ˉcˉpˉjˉBˉmˉp(ˉt), where Bˉmˉp(ˉt) is the m degree Bernstein polynomial, denote the approximate solution to equation and uˉj(ˉt),j=1,2,...ˉn represent the exact solution. Assume that

    uˉj(ˉt)=limmUˉj,ˉm(ˉt),

    holds. Let

    eˉm(ˉt)=ˉnˉi=0eˉi,ˉm(ˉx), (5.1)

    where eˉm(ˉx) denotes the difference between the exact and approximate solution. From Eq (11), we have

    eˉm(ˉt)ˉni=0eˉi,ˉm(ˉt)ˉni=0uˉj(ˉt)Uˉj,ˉm(ˉt). (5.2)

    Since all the coefficient in (1.1) are smooth function and therefore are all bounded, hence e =m( =t)0,asˉm.

    Example 6.1. Consider the the following constructed example [16]

    {u(t)=αu(t)+βu(rt)+cos(t)αsin(t)βsin(rt),tI:=[0,T]u(0)=0. (6.1)

    The error between numerical solution and exact solution for α=1,β=0.5,r=0.5 and T=5 for different N is shown in Table 1.

    Table 1.  Example 6.1: the point-wise error in L norm.
    N Error N Error N Error
    6 1.174e-001 12 6.204e-006 17 9.730e-1
    7 3.635e-002 13 4.092e-007 18 8.243e-2
    9 1.752e-003 15 1.103e-008 19 6.651e-2
    10 2.291e-004 16 1.075e-009 20 5.508e-3

     | Show Table
    DownLoad: CSV

    Example 6.2. Choose α(t)=sin(t),β(t)=cos(rt),γ(t)=sin(rx) in (1.1). Figure 1 indicates the error behavior between approximate and exact solution for r=0.05 and T=5. The comparison was made with the Legendre spectral method presented in [8]. We found that both the method has a very good agreement with each other.

    Figure 1.  Example 6.2. The error behavior in L norm.

    Example 6.3. Consider the nonlinear equation of the form:

    u(t)=αu(t)+βu(rt)(1u(rt)).

    The error behavior for α=0.25,β=1,r=0.5 and T=1, relative to N is shown in Figure 2.

    Figure 2.  Example 6.3. The error behavior in L norm.

    Example 6.4. Consider the following initial value problem [7].

    {u(t)=u(t)+αy(rt)+u(rt)+cos(t)cos(rt)+sin(t),tI:=[0,T]u(0)=0

    The maximum point wise error for β=0,r=0.5 and T=2 for different N is given in Table 2.

    Table 2.  Example 6.4: the point-wise error in L norm.
    N Error N Error N Error
    8 6.800e004 16 2.817e011 24 2.665e014
    10 1.916e005 18 2.317e012 26 3.442e014
    12 2.198e007 20 3.162e013 28 3.442e014
    14 3.522e009 22 1.998e013 30 3.096e014

     | Show Table
    DownLoad: CSV

    Example 6.5. Consider the the following example

    {u(t)=αu(t)+βu(rt),tI:=[0,T]u(0)=1. (6.2)

    The error between numerical solution and exact solution for α=0.5,β=0.5,r=0.5 and T=3, for different values of N is shown in Table 3. We compare the result with the Bernstein series solution method and found a very good agreement with it [29].

    Table 3.  Example 6.5: the point-wise error in L norm.
    N Error N Error N Error
    6 1.023e002 12 5.345e007 17 8.971e013
    7 2.130e003 13 3.021e009 18 7.209e013
    9 1.567e004 15 2.376e010 19 3.312e015
    10 3.121e006 16 3.218e011 20 4.273e016

     | Show Table
    DownLoad: CSV

    A new method based on the Bernstein polynomials is introduced for the approximate solution of neutral functional differential equation of pantograph type with proportional delay. For better efficiency of the proposed scheme, a transformation from Bernstein to Legendre polynomial is used, which allow us to take the advantage of orthogonality of Legendre polynomials which is not possible in case of Bernstein polynomial directly. An error analysis is provided and a number of numerical experiments were performed to confirm the theoretical justification. The numerical as well as theoretical result shows that the method has a spectral accuracy. It is observed from our numerical experiments that while increasing the number of collocation points that is N one lose the spectral accuracy because of the fact that using Lagrange interpolating polynomials which is bounded by Lebesgue constant grows exponentially while increasing N. This is also because of the oscillating nature of orthogonal polynomials. In our proposed scheme one does not need to increase the number of collocation points as we achieve a spectral accuracy after a few collocations points.

    The author acknowledge the Deanship of Scientific Research at King Faisal University for the financial support under Nasher Track (Grant No. 206106).

    The author declares no competing interest regarding the publication of this paper.


    Acknowledgments



    Beca de Magister Nacional, Año Académico 2022, folio 22220751, ANID, Chile.

    Conflict of interest



    There are no conflicts of interest in this study.

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