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Case report

Acute transverse myelitis (ATM) associated with COVID 19 infection and vaccination: A case report and literature review

  • Received: 01 December 2023 Revised: 27 May 2024 Accepted: 03 June 2024 Published: 11 June 2024
  • Acute transverse myelitis (ATM) is an inflammatory disorder caused by many etiologies, from postinfectious to autoimmune. Rarely, ATM cases have been reported after both COVID-19 infection and COVID-19 vaccination. We described our experience with ATM after COVID-19 infection and conducted a literature review.

    Case finding Methods 

    We reported a case of longitudinally extensive ATM after COVID 19 infection, who also received convalescent plasma therapy, and present a comprehensive literature review of ATM cases reported after COVID-19 infection and COVID-19 vaccination. The literature search was done using PubMed and Google scholar with keywords and selected peer-reviewed articles. The search included all cases from Jan 2020 to Sept 2022.

    Results 

    A total of 60 ATM cases reported association with post COVID 19 infection, and 23 ATM cases reported association with post COVID 19 vaccinations. Among post COVID 19 ATM cases, the mean age was 49 years and the youngest reported was 7-month-old. A total of 55% (33) were longitudinally extensive ATM. The most common symptom was lower extremity weakness. One case was reported as necrotizing myelitis on biopsy, and another case overlapped with syndrome of GBS and longitudinal ATM. No cases reported using convalescent plasma therapy after infection. Almost all the ATM cases were treated with steroids, but some cases needed additional treatment since not all responded adequately. Six cases (10%) responded with steroids plus plasmapheresis, and 5 cases (8%) responded with steroids + IVIG, especially in the pediatric age group. One case reported a positive response after treatment with eculizumab, and another with infliximab. Two cases (3%) remained paraparetic. Among post covid-19 vaccine ATM cases, 4 cases (17%) were reported as longitudinally extensive ATM. Five cases (21%) had symptom onset within a week after vaccination. Almost all reported a response to steroids except for one case which reported fatality after the 58th day after vaccination.

    Conclusion 

    ATM, in the setting of acute COVID-19 infection, has been described in multiple cases and is a rare complication of COVID-19 vaccination.

    Citation: Srinivas Medavarapu, Nitasha Goyal, Yaacov Anziska. Acute transverse myelitis (ATM) associated with COVID 19 infection and vaccination: A case report and literature review[J]. AIMS Neuroscience, 2024, 11(2): 178-187. doi: 10.3934/Neuroscience.2024011

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  • Acute transverse myelitis (ATM) is an inflammatory disorder caused by many etiologies, from postinfectious to autoimmune. Rarely, ATM cases have been reported after both COVID-19 infection and COVID-19 vaccination. We described our experience with ATM after COVID-19 infection and conducted a literature review.

    Case finding Methods 

    We reported a case of longitudinally extensive ATM after COVID 19 infection, who also received convalescent plasma therapy, and present a comprehensive literature review of ATM cases reported after COVID-19 infection and COVID-19 vaccination. The literature search was done using PubMed and Google scholar with keywords and selected peer-reviewed articles. The search included all cases from Jan 2020 to Sept 2022.

    Results 

    A total of 60 ATM cases reported association with post COVID 19 infection, and 23 ATM cases reported association with post COVID 19 vaccinations. Among post COVID 19 ATM cases, the mean age was 49 years and the youngest reported was 7-month-old. A total of 55% (33) were longitudinally extensive ATM. The most common symptom was lower extremity weakness. One case was reported as necrotizing myelitis on biopsy, and another case overlapped with syndrome of GBS and longitudinal ATM. No cases reported using convalescent plasma therapy after infection. Almost all the ATM cases were treated with steroids, but some cases needed additional treatment since not all responded adequately. Six cases (10%) responded with steroids plus plasmapheresis, and 5 cases (8%) responded with steroids + IVIG, especially in the pediatric age group. One case reported a positive response after treatment with eculizumab, and another with infliximab. Two cases (3%) remained paraparetic. Among post covid-19 vaccine ATM cases, 4 cases (17%) were reported as longitudinally extensive ATM. Five cases (21%) had symptom onset within a week after vaccination. Almost all reported a response to steroids except for one case which reported fatality after the 58th day after vaccination.

    Conclusion 

    ATM, in the setting of acute COVID-19 infection, has been described in multiple cases and is a rare complication of COVID-19 vaccination.



    Let A indicate an analytic functions family, which is normalized under the condition f (0)= f(0)1=0 in U={z:zC and |z |<1} and given by the following Taylor-Maclaurin series:

    f (z)=z+n=2anzn .      (1.1)

    Further, by S we shall denote the class of all functions in A which are univalent in U.

    With a view to recalling the principle of subordination between analytic functions, let the functions f and g be analytic in U. Then we say that the function f is subordinate to g if there exists a Schwarz function w(z), analytic in U with

    ω(0)=0, |ω(z)|<1, (zU)

    such that

    f (z)=g (ω(z)).

    We denote this subordination by

    fg or f (z)g (z).

    In particular, if the function g is univalent in U, the above subordination is equivalent to

    f (0)=g (0), f (U)g (U).

    The Koebe-One Quarter Theorem [11] asserts that image of U under every univalent function fA contains a disc of radius 14. thus every univalent function f has an inverse  f1  satisfying  f1(f(z))=z and f ( f1 (w))=w (|w|<r 0(f ),r 0(f ) >14 ), where

     f1(w)=wa2w2+(2a22a3)w3(5a325a2a3+a4)w4+. (1.2)

    A function fA is said to be bi-univalent functions in U if both f and  f1 are univalent in U. A function fS is said to be bi-univalent in U if there exists a function gS such that g(z) is an univalent extension of f1 to U. Let Λ denote the class of bi-univalent functions in U. The functions z1z, log(1z), 12log(1+z1z) are in the class Λ (see details in [20]). However, the familiar Koebe function is not bi-univalent. Lewin [17] investigated the class of bi-univalent functions Λ and obtained a bound |a2|1.51. Motivated by the work of Lewin [17], Brannan and Clunie [9] conjectured that |a2|2. The coefficient estimate problem for |an|(nN,n3) is still open ([20]). Brannan and Taha [10] also worked on certain subclasses of the bi-univalent function class Λ and obtained estimates for their initial coefficients. Various classes of bi-univalent functions were introduced and studied in recent times, the study of bi-univalent functions gained momentum mainly due to the work of Srivastava et al. [20]. Motivated by this, many researchers [1], [4,5,6,7,8], [13,14,15], [20], [21], and [27,28,29], also the references cited there in) recently investigated several interesting subclasses of the class Λ and found non-sharp estimates on the first two Taylor-Maclaurin coefficients. Recently, many researchers have been exploring bi-univalent functions, few to mention Fibonacci polynomials, Lucas polynomials, Chebyshev polynomials, Pell polynomials, Lucas–Lehmer polynomials, orthogonal polynomials and the other special polynomials and their generalizations are of great importance in a variety of branches such as physics, engineering, architecture, nature, art, number theory, combinatorics and numerical analysis. These polynomials have been studied in several papers from a theoretical point of view (see, for example, [23,24,25,26,27,28,29,30] also see references therein).

    We recall the following results relevant for our study as stated in [3].

    Let p(x) and q(x) be polynomials with real coefficients. The (p,q) Lucas polynomials Lp,q,n(x) are defined by the recurrence relation

    Lp,q,n(x)=p(x)Lp,q,n1(x)+q(x)Lp,q,n2(x)(n2),

    from which the first few Lucas polynomials can be found as

    Lp,q,0(x)=2,Lp,q,1(x)=p(x),Lp,q,2(x)=p2(x)+2q(x),Lp,q,3(x)=p3(x)+3p(x)q(x),.... (1.3)

    For the special cases of p(x) and q(x), we can get the polynomials given Lx,1,n(x)Ln(x) Lucas polynomials, L2x,1,n(x)Dn(x) Pell–Lucas polynomials, L1,2x,n(x)jn(x) Jacobsthal–Lucas polynomials, L3x,2,n(x)Fn(x) Fermat–Lucas polynomials, L2x,1,n(x)Tn(x) Chebyshev polynomials first kind.

    Lemma 1.1. [16] Let G{L(x)}(z)be the generating function of the (p,q)Lucas polynomial sequence Lp,q,n(x).Then,

    G{L(x)}(z)=n=0Lp,q,n(x)zn=2p(x)z1p(x)zq(x)z2

    and

    G{L(x)}(z)=G{L(x)}(z)1=1+n=1Lp,q,n(x)zn=1+q(x)z21p(x)zq(x)z2.

    Definition 1.2. [22] For ϑ0, δR, ϑ+iδ0 and fA, let B(ϑ,δ) denote the class of Bazilevič function if and only if

    Re[(zf(z)f(z))(f(z)z)ϑ+iδ]>0.

    Several authors have researched different subfamilies of the well-known Bazilevič functions of type ϑ from various viewpoints (see [3] and [19]). For Bazilevič functions of order ϑ+iδ, there is no much work associated with Lucas polynomials in the literature. Initiating an exploration of properties of Lucas polynomials associated with Bazilevič functions of order ϑ+iδ is the main goal of this paper. To do so, we take into account the following definitions. In this paper motivated by the very recent work of Altinkaya and Yalcin [3] (also see [18]) we define a new class B(ϑ,δ), bi-Bazilevič function of Λ based on (p,q) Lucas polynomials as below:

    Definition 1.3. For fΛ, ϑ0, δR, ϑ+iδ0 and let B(ϑ,δ) denote the class of Bi-Bazilevič functions of order t  and type ϑ+iδ if only if

    [(zf(z)f(z))(f(z)z)ϑ+iδ]G{L(x)}(z)(zU) (1.4)

    and

    [(zg(w)g(w))(g(w)w)ϑ+iδ]G{L(x)}(w)(wU), (1.5)

    where GLp,q,n(z)Φ and the function g is described as g(w)=f1(w).

    Remark 1.4. We note that for δ=0 the class R(ϑ,0)=R(ϑ) is defined by Altinkaya and Yalcin [2].

    The class B(0,0)=SΛ is defined as follows:

    Definition 1.5. A function fΛ is said to be in the class SΛ, if the following subordinations hold

    zf(z)f(z)G{L(x)}(z)(zU)

    and

    wg(w)g(w)G{L(x)}(w)(wU)

    where g(w)=f1(w).

    We begin this section by finding the estimates of the coefficients |a2| and |a3| for functions in the class B(ϑ,δ).

    Theorem 2.1. Let the function f(z) given by 1.1 be in the class B(ϑ,δ). Then

    |a2|p(x)2p(x)|{((ϑ+iδ)2+3(ϑ+iδ)+2)2(ϑ+iδ+1)2}p2(x)4q(x)(ϑ+iδ+1)2|.

    and

    |a3|p2(x)(ϑ+1)2+δ2+p(x)(ϑ+2)2+δ2.

    Proof. Let fB(ϑ,δ,x) there exist two analytic functions u,v:UU with u(0)=0=v(0), such that |u(z)|<1, |v(w)|<1, we can write from (1.4) and (1.5), we have

    [(zf(z)f(z))(f(z)z)ϑ+iδ]=G{L(x)}(z)(zU) (2.1)

    and

    [(zg(w)g(w))(g(w)w)ϑ+iδ]=G{L(x)}(w)(wU), (2.2)

    It is fairly well known that if

    |u(z)|=|u1z+u2z2+|<1

    and

    |v(w)|=|v1w+v2w2+|<1.

    then

    |uk|1and|vk|1(kN)

    It follows that, so we have

    G{L(x)}(u(z))=1+Lp,q,1(x)u(z)+Lp,q,2(x)u2(z)+=1+Lp,q,1(x)u1z+[Lp,q,1(x)u2+Lp,q,2(x)u21]z2+ (2.3)

    and

    G{L(x)}(v(w))=1+Lp,q,1(x)v(w)+Lp,q,2(x)v2(w)+=1+Lp,q,1(x)v1w+[Lp,q,1(x)v2+Lp,q,2(x)v21]w2+ (2.4)

    From the equalities (2.1) and (2.2), we obtain that

    [(zf(z)f(z))(f(z)z)ϑ+iδ]=1+Lp,q,1(x)u1z+[Lp,q,1(x)u2+Lp,q,2(x)u21]z2+, (2.5)

    and

    [(zg(w)g(w))(g(w)w)ϑ+iδ]=1+Lp,q,1(x)v1w+[Lp,q,1(x)v2+Lp,q,2(x)v21]w2+, (2.6)

    It follows from (2.5) and (2.6) that

    (ϑ+iδ+1)a2=Lp,q,1(x)u1,, (2.7)
    (ϑ+iδ1)(ϑ+iδ+2)2a22(ϑ+iδ+2)a3=Lp,q,1(x)u2+Lp,q,2(x)u21, (2.8)

    and

    (ϑ+iδ+1)a2=Lp,q,1(x)v1, (2.9)
    (ϑ+iδ+2)(ϑ+iδ+3)2a22+(ϑ+iδ+2)a3=Lp,q,1(x)v2+Lp,q,2(x)v21, (2.10)

    From (2.7) and (2.9)

    u1=v1 (2.11)

    and

    2(ϑ+iδ+1)2a22=L2p,q,1(x)(u21+v21)., (2.12)

    by adding (2.8) to (2.10), we get

    ((ϑ+iδ)2+3(ϑ+iδ)+2)a22=Lp,q,1(x)(u2+v2)+Lp,q,2(x)(u21+v21), (2.13)

    by using (2.12) in equality (2.13), we have

    [((ϑ+iδ)2+3(ϑ+iδ)+2)2Lp,q,2(x)(ϑ+iδ+1)2L2p,q,1(x)]a22=Lp,q,1(x)(u2+v2),
    a22=L3p,q,1(x)(u2+v2)[((ϑ+iδ)2+3(ϑ+iδ)+2)L2p,q,1(x)2Lp,q,2(x)(ϑ+iδ+1)2]. (2.14)

    Thus, from (1.3) and (2.14) we get

    |a2|p(x)2p(x)|{((ϑ+iδ)2+3(ϑ+iδ)+2)2(ϑ+iδ+1)2}p2(x)4q(x)(ϑ+iδ+1)2|.

    Next, in order to find the bound on |a3|, by subtracting (2.10) from (2.8), we obtain

    2(ϑ+iδ+2)a32(ϑ+iδ+2)a22=Lp,q,1(x)(u2v2)+Lp,q,2(x)(u21v21)2(ϑ+iδ+2)a3=Lp,q,1(x)(u2v2)+2(ϑ+iδ+2)a22a3=Lp,q,1(x)(u2v2)2(ϑ+iδ+2)+a22 (2.15)

    Then, in view of (2.11) and (2.12), we have from (2.15)

    a3=L2p,q,1(x)2(ϑ+iδ+2)2(u21+v21)+Lp,q,1(x)2(ϑ+iδ+2)(u2v2).
    |a3|p2(x)|ϑ+iδ+1|2+p(x)|ϑ+iδ+2|=p2(x)(ϑ+1)2+δ2+p(x)(ϑ+2)2+δ2

    This completes the proof.

    Taking δ=0, in Theorem 2.1, we get the following corollary.

    Corollary 2.2. Let the function f(z) given by (1.1) be in the class B(ϑ). Then

    |a2|p(x)2p(x)|{(ϑ2+3ϑ+2)2(ϑ+1)2}p2(x)4q(x)(ϑ+1)2|

    and

    |a3|p2(x)(ϑ+2)2+p(x)ϑ+2

    Also, taking ϑ=0 and δ=0, in Theorem 2.1, we get the results given in [18].

    Fekete-Szegö inequality is one of the famous problems related to coefficients of univalent analytic functions. It was first given by [12], the classical Fekete-Szegö inequality for the coefficients of fS is

    |a3μa22|1+2exp(2μ/(1μ)) for μ[0,1).

    As μ1, we have the elementary inequality |a3a22|1. Moreover, the coefficient functional

    ςμ(f)=a3μa22

    on the normalized analytic functions f in the unit disk U plays an important role in function theory. The problem of maximizing the absolute value of the functional ςμ(f) is called the Fekete-Szegö problem.

    In this section, we are ready to find the sharp bounds of Fekete-Szegö functional ςμ(f) defined for fB(ϑ,δ) given by (1.1).

    Theorem 3.1. Let f given by (1.1) be in the class B(ϑ,δ) and μR. Then

    |a3μa22|{p(x)(ϑ+2)2+δ2,        0|h(μ)|12(ϑ+2)2+δ22p(x)|h(μ)|,             |h(μ)|12(ϑ+2)2+δ2

    where

    h(μ)=L2p,q,1(x)(1μ)((ϑ+iδ)2+3(ϑ+iδ)+2)L2p,q,1(x)2Lp,q,2(x)(ϑ+iδ+1)2.

    Proof. From (2.14) and (2.15), we conclude that

    a3μa22=(1μ)L3p,q,1(x)(u2+v2)[((ϑ+iδ)2+3(ϑ+iδ)+2)L2p,q,1(x)2Lp,q,2(x)(ϑ+iδ+1)2]+Lp,q,1(x)2(ϑ+iδ+2)(u2v2)
    =Lp,q,1(x)[(h(μ)+12(ϑ+iδ+2))u2+(h(μ)12(ϑ+iδ+2))v2]

    where

    h(μ)=L2p,q,1(x)(1μ)((ϑ+iδ)2+3(ϑ+iδ)+2)L2p,q,1(x)2Lp,q,2(x)(ϑ+iδ+1)2.

    Then, in view of (1.3), we obtain

    |a3μa22|{p(x)(ϑ+2)2+δ2,        0|h(μ)|12(ϑ+2)2+δ22p(x)|h(μ)|,             |h(μ)|12(ϑ+2)2+δ2

    We end this section with some corollaries.

    Taking μ=1 in Theorem 3.1, we get the following corollary.

    Corollary 3.2. If fB(ϑ,δ), then

    |a3a22|p(x)(ϑ+2)2+δ2.

    Taking δ=0 in Theorem 3.1, we get the following corollary.

    Corollary 3.3. Let f given by (1.1) be in the class B(ϑ,0). Then

    |a3μa22|{p(x)ϑ+2,        0|h(μ)|12(ϑ+2)2p(x)|h(μ)|,             |h(μ)|12(ϑ+2)

    Also, taking ϑ=0, δ=0 and μ=1 in Theorem 3.1, we get the following corollary.

    Corollary 3.4. Let f given by (1.1) be in the class B. Then

    |a3a22|p(x)2.

    All authors declare no conflicts of interest in this paper.


    Acknowledgments



    None.

    Funding



    No funding or grant support.

    Conflicts of interest



    The following authors have no financial disclosures: NG, SM, SM, IL.

    Authorship



    All authors attest that they meet the current ICMJE criteria for Authorship.

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