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Research article

Prevalence and factors associated with chronic Hepatitis B infection among adults in the Central Highland, Vietnam

  • Objective  The study was conducted to estimate the prevalence of chronic hepatitis B infection and related factors among adult population in Central Highland, 2018.
    Methods  The study applied the cross-sectional descriptive design with the combination of structured interview and serological blood tests. There were 2428 respondents in 03 provinces, namely Kon Tum, Gia Lai, Dak Nong included in the study which was conducted from September to December 2018.
    Results  The prevalence of hepatitis B infection among adults in Central Highland was 11.2% with the positive uniformity among provinces in the region. The findings also showed that male adults, farmers and those who have relatives with liver disease and experienced the kidney dialysis possessed a higher rate of hepatitis B virus infection than the one of other groups. The findings also showed that the respondents had a low uptake of the vaccine among infected group (6.3%). Study subjects with vaccination were less affected by hepatitis B virus infection rate than the ones without vaccination.
    Conclusions  The high prevalence of chronic hepatitis B in the Central Highland indicates that it needs much more attention of Ministry of Health and local authorities for prevention of infection through mother-to-child transmission and early infection due to late immunization and more rescue investment for diagnostic, evaluation, follow up and treatment of people with chronic HBV infection. The consideration regarding occupations, family history of liver diseases is also required for during the intervention implementation.

    Citation: Pham Ngoc Thanh, Nguyen Thi Thi Tho, Tran Dac Phu, Tran Dai Quang, Nguyen Thuy Duong, Vien Chinh Chien, Phan Trong Lan. Prevalence and factors associated with chronic Hepatitis B infection among adults in the Central Highland, Vietnam[J]. AIMS Medical Science, 2020, 7(4): 337-346. doi: 10.3934/medsci.2020023

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  • Objective  The study was conducted to estimate the prevalence of chronic hepatitis B infection and related factors among adult population in Central Highland, 2018.
    Methods  The study applied the cross-sectional descriptive design with the combination of structured interview and serological blood tests. There were 2428 respondents in 03 provinces, namely Kon Tum, Gia Lai, Dak Nong included in the study which was conducted from September to December 2018.
    Results  The prevalence of hepatitis B infection among adults in Central Highland was 11.2% with the positive uniformity among provinces in the region. The findings also showed that male adults, farmers and those who have relatives with liver disease and experienced the kidney dialysis possessed a higher rate of hepatitis B virus infection than the one of other groups. The findings also showed that the respondents had a low uptake of the vaccine among infected group (6.3%). Study subjects with vaccination were less affected by hepatitis B virus infection rate than the ones without vaccination.
    Conclusions  The high prevalence of chronic hepatitis B in the Central Highland indicates that it needs much more attention of Ministry of Health and local authorities for prevention of infection through mother-to-child transmission and early infection due to late immunization and more rescue investment for diagnostic, evaluation, follow up and treatment of people with chronic HBV infection. The consideration regarding occupations, family history of liver diseases is also required for during the intervention implementation.


    We consider a singular no-sign obstacle problem of the type

    {div(xa1u)=xa1f(x)χ{u0}in B+1,u=0on B1{x1=0}, (1.1)

    where a>1, χD is the characteristic function of D, B1Rn is the unit ball and B+1=B1{x1>0}. The equation is considered in the weak form,

    B+1xa1uφdx=B+1xa1f(x)φχ{u0}dx,

    for all φW1,20(B+1). This problem, when the non-negativity assumption u0 is imposed, is already studied in [9]. The above no-sign problem, as a general semilinear PDE with non-monotone r.h.s., introduces certain difficulties and to some extent some challenges. To explain this we shall give a very short review of the existing results and methods for similar type of problems (see also the book [6] and Caffarelli's review of the classical obstacle problem [2]). The general methodology of approaching such problems lies in using the so-called ACF-monotonicity formula (see [8]) or alternatively using John Andersson's dichotomy (see [1] or [3]). Although there are still some chances that both these methods will work for our problem above, we shall introduce a third method here which relies on a softer version of a monotonicity formula (which has a wider applicability) in combination with some elaborated analysis. We refer to this as a Weiss-type monotonicity formula, see (2.1) below.

    For clarity of exposition we shall introduce some notations and definitions here that are used frequently in the paper. Throughout this paper, Rn will be equipped with the Euclidean inner product xy and the induced norm |x|, Br(x0) will denote the open n-dimensional ball of center x0, radius r with the boundary Br(x0). In addition, Br=Br(0) and Br=Br(0). Rn+ stands for half space {xRn:x1>0} as well as B+r=BrRn+. Moreover, in the text we use the n-dimensional Hausdorff measure Hn. For a multi-index μ=(μ1,,μn)Zn+, we denote the partial derivative with μu=μ1x1μnxnu and |μ|1=μ1++μn.

    For a domain ΩRn+ and 1p<, we use the notation Lp(Ω) and Wm,p(Ω) as the standard spaces. However, we need some new notation for the weighted spaces

    Lp(Ω;xθ1):={u:Ωxθ1|u(x)|pdx<},

    where θR. For mN, we define the weighted Sobolev space Wm,p(Ω;xθ1) as the closure of C(¯Ω) with the following norm,

    uWm,p(Ω;xθ1):=uLp(Ω;xθ1)+x1DuLp(Ω;xθ1)++xm1DmuLp(Ω;xθ1).

    It is noteworthy that for θ=0, we have Lp(Ω;1)=Lp(Ω) but Wm,p(Ω;1)Wm,p(Ω). Generally, the trace operator has no sense for θ>1, while functions in W1,p(Ω;xθ1) have zero traces on {x1=0} for θ1. (Theorem 6 in [7]).

    We consider uW1,p(B+1,xθ1) for some θ<n and n<p to be a weak solution of (1.1). This condition provides the continuity of x(θ+n)/p1u up to the boundary according to Sobolev embedding Theorem 3.1 in [5]. First, we prove the following a priori regularity result.

    Proposition 1.1. (Appendix A) Let uW1,p(B+1,xθ1) be a solution of (1.1) for some θ<n, n<p and fL(B+1). Then for each max{0,1+θ+np}<β<1 there exists C=C(β,n,a) such that for r1/2,

    supB+r(x0)|xβ11u|Cr2β,

    for all x0{x1=0}.

    In Appendix A we will prove this proposition. Our main result in this paper concerns the optimal growth rate of solution u of (1.1) at touching free boundary points, which is stated in the following theorem.

    Theorem 1.2. Suppose uW1,p(B+1,xθ1) is a solution of (1.1) for some θ<n, n<p and x0{u=0}{x1=0}B+1/4. Moreover, if fCα(¯B+1) for some α(0,1), then

    |u(x)|Cx21((|xx0|+x1x1)(n+a+4)/2+1),

    for a universal constant C=C(a,n,[f]0,α).

    Our main tool in proving optimal decay for solutions from the free boundary points is Weiss-monotonicity formula, combined with some elaborated techniques. We define the balanced energy functional

    Φx0(r,u)=rn2aB+r(x0)(xa1|u|2+2xa1f(x0)u)dx2rn3aBr(x0)Rn+xa1u2dHn1. (2.1)

    Considering the scaling ur,x0=ur(x)=u(rx+x0)r2, Φx0(r,u)=Φ0(1,ur). In what follows we prove almost-monotonicity of the energy.

    Lemma 2.1 (Almost-Monotonicity Formula). Let u solve (1.1) and be as in Proposition 1.1 and assume that u(x0)=0 for some x0{x1=0} and fCα(¯B+1) for some α(0,1). Then u satisfies, for rr0 such that B+r0(x0)B+1,

    ddrΦx0(r,u)2rB1Rn+xa1(rur)2dHn1Crα+β2,

    where C depends only on fCα(¯B+1) and the constant C(β,n,a) in Proposition 1.1.

    Proof. Let ur(x):=u(rx+x0)r2, then

    =12ddrΦx0(r,u)=12ddr[B+1(xa1|ur|2+2xa1f(x0)ur)dx2B1Rn+xa1u2rdHn1]=12[B+1(2xa1urrur+2xa1f(x0)rur)dx4B1Rn+xa1urrurdHn1]=B+1div(xa1rurur)rurdiv(xa1ur)+xa1f(x0)rurdx2B1Rn+xa1urrurdHn1=B+1(f(x0)f(rx+x0)χ{ur0})xa1rurdx+B+1xa1rururνdHn12B1Rn+xa1urrurdHn1=B+1(f(x0)f(rx+x0)χ{ur0})xa1rurdx+rB1Rn+xa1(rur)2dHn1=B+1(f(x0)f(rx+x0))χ{ur0}xa1rurdx+f(x0)B+1{ur=0}xa1rurdx+rB1Rn+xa1(rur)2dHn1.

    Note that the second integral

    B+1{ur=0}xa1rurdx=0

    as |{ur=0ur0}|=0 and rur=0 on {ur=0ur=0}. Since |rur|Crβ2 we infer that

    B+1(f(x0)f(rx+x0))χ{ur0}xa1rurdxCrα+β2

    and conclude that

    12ddrΦx0(r,u)rB1Rn+xa1(rur)2dHn1Crα+β2.

    Definition 2.2. Let HP2 stand for the class of all two-homogeneous functions PW1,2(B+1;xa21) satisfying div(xa1P)=0 in Rn+ with boundary condition P=0 on x1=0. We also define the operator Π(v,r,x0) to be the projection of vr,x0 onto HP2 with respect to the inner product

    vw=B1n+xa1vwdHn1.

    We will use the following extension of [10,Lemma 4.1].

    Lemma 2.3. Assume that div(xa1w)=0 in B+1 with boundary condition w=0 on x1=0, and w(0)=|w(0)|=0. Then

    B+1xa1|w|2dx2B1Rn+xa1w2dHn10,

    and equality implies that wHP2, i.e., it is homogeneous of degree two.

    Proof. We define an extension of the Almgren frequency,

    rN(w,r):=rB+rxa1|w|2dxB+rxa1w2dHn1,
    12N(w,r)N(w,r)=B+rxa1(νw)2dHn1B+rxa1wνwdHn1B+rxa1wνwdHn1B+rxa1w2dHn10.

    Moreover, if N(w,r)=κ for ρ<r<σ, it implies that w is homogeneous of degree κ in BσBρ.

    Now supposing towards a contradiction that N(w,s)<2 for some s(0,1], and defining wr(x):=w(rx)w(rx)L2(B+1,xb1), we infer from N(w,s)<2 that wr is bounded in L2(B+1;xa1) and so wrmw0 weakly in L2(B+1;xa1) and wrmw0 strongly in L2(B+1;xa1) as a sequence rm0. Consequently, w0 satisfies div(xa1w0)=0 in B+1, w0(0)=|w0(0)|=0 and w0=0 on x1=0 as well as w0L2(B+1;xa1)=1. Furthermore, for all r(0,1) we have

    N(w0,r)=limrm0N(wrm,r)=limrm0N(w,rrm)=N(w,0+),

    and so w0 must be a homogeneous function of degree κ:=N(w,0+)<2. Note that for every multi-index μ{0}×Zn1+, the higher order partial derivative ζ=μw0 satisfies the equation div(xa1ζ)=0 in Rn+. From the integrability and homogeneity we infer that μw00 for κ|μ|1<n2, otherwise

    B+1|μw0|2dx=(10r2(κ|μ|1)+n1dr)B1Rn+|μw0|2dHn1

    can not be bounded. Thus xw0(x1,x) is a polynomial, and we can write w0(x1,x)=xκ1p(xx1). Consider the multi-index μ such that |μ|1=degp, so μw0=xκ|μ|11μp is a solution of div(xa1ζ)=0 in Rn+. Therefore, μw0W1,2(B+1;xθ1) for 1<θ according to Proposition A.1, which implies that 2(κ|μ|1)+θ>1. So, degp<κ+θ+12.

    Substituting w0(x)=xκ11(αx1+x) for κ>1 in the equation and comparing with w0(0)=|w0(0)|=0 we arrive at the only nonzero possible case being κ+a=2, which contradicts a>1. The case κ<1 leads to degp=0 and w0(x)=αxκ1, which implies κ+a=1 and a contradiction to a>1.

    Proposition 3.1. Let fCα(¯B+1) and u be solution of (1.1) satisfying the condition in Proposition 1.1. Then the function

    rrn3aB+r(x0)xa1u2(x)dHn1

    is bounded on (0,1/8), uniformly in x0{u=0}{x1=0}B1/8.

    Proof. Let us divide the proof into steps.

    Step 1 We claim that there exists a constant C1< such that for all x0{u=0}{x1=0}B1/8 and r1/8,

    f(x0)B+1xa1ux0,r(x)dxC1,

    where ux0,r:=u(rx+x0)r2. To prove this we observe that w:=ux0,r satisfies

    div(xa1w)=xa1fr(x):=xa1f(x0+rx)χ{ux0,r0},in B+1.

    Moreover, for ϕ(ρ):=ρna+1B+ρxa1w(x)dHn1 we have

    ϕ(ρ)=B+1xa1w(ρx)xdHn1=ρna+1B+ρdiv(xa1w(x))dx=ρna+1B+ρxa1fr(x)dx.

    If f(x0)18α[f]0,α then fr0 for r1/8. Therefore ϕ is increasing and ϕ(ρ)ϕ(0)=0 (recall that w(0)=0). Similarly, if f(x0)18α[f]0,α, we obtain that ϕ(ρ)0. Therefore the claim is true for C1=0 in these cases.

    In the case |f(x0)|18α[f]0,α, then |fr(x)|28α[f]0,α and then

    |ϕ(ρ)|213α[f]0,αρna+1B+ρxa1dx213α[f]0,αρ.

    So, |ϕ(ρ)|23α[f]0,αρ2 and

    |f(x0)B+1xa1ux0,r(x)dx|=|f(x0)10ρn+a1ϕ(ρ)dρ|26α[f]20,αn+a+2=:C1.

    Step 2 We claim that there exists a constant C2< such that

    distL2(B1Rn+;xa1)(ux0,r,HP2)C2,

    for every x0{u=0}{x1=0}B1/8, r1/8. Suppose towards a contradiction that this is not true, then there exists a sequence um, xmˉx and rm0 such that

    Mm=umxm,rmΠ(um,rm,xm)L2(B1Rn+;xa1),m.

    Let um:=umxm,rm and pm=Π(um,rm,xm) and wm=umpmMm. Then, since um(0)=|um(0)|=0 and by the monotonicity formula and the result of previous step, we find that

    B+1xa1|wm|2dx2B+1xa1w2mdHn1=1M2m[Φxm(rm)2B+1f(xm)xa1umdx]+1M2mB+1xa1(pmpmν2umpmν2p2m+4umpm)dHn11M2m(Φxm(rm)+2C1)1M2m(Φxm(12)+2C1)0,m. (3.1)

    Passing to a subsequence such that wmw in L2(B+1;xa1) as m, the compact embedding on the boundary implies that wL2(B1Rn+;xa1)=1, and

    B+1xa1|w|2dx2B1xa1w2dHn1 (3.2)

    and that

    B1wpdHn1=0,pHP2. (3.3)

    Since div(xa1wm)=x1Mmf(xm+r)χ{um0}, it follows that div(xa1w)=0 in B+1. Moreover, we obtain from Lp-theory that wmw in C1,αloc(B+1) for each α(0,1) as m. Consequently w(0)=|w(0)|=0. Thus we can apply Lemma 2.3 and obtain from (3.2) that w is homogeneous of degree 2, contradicting (3.3) and wL2(B1)=1. This proves the claim.

    Step 3 We will show that there exists constant C2 such that for all x0{u=0}{x1=0} satisfying

    lim infr0+|B+r(x0){u=0}||B+r|=0, (3.4)

    we have

    Φx0(0+)B+1xa1f(x0)ux0,rdxC2rα|f(x0)|.

    In order to see this, we can observe that

    B+1xa1f(x0)ux0,r(x)dx=f(x0)1010s[B+ρ(sx1)aux0,r(sx)dHn1(x)]dsdρ=f(x0)10ρ10B+ρ(sx1)aux0,r(sx)νdHn1dsdρ=f(x0)10ρ10sB+ρdiv((sx1)aux0,r(sx))dxdsdρ=f(x0)10ρ10sB+ρ(sx1)af(rsx)χΩx0,r(sx)dxdsdρ=f(x0)210ρ1+a+n10s1+aB+1xa1dxdsdρ+f(x0)10ρ1+a+n10s1+aB+1xa1(f(rsρx)f(x0))dxdsdρf(x0)2(n+a+2)(a+2)B+1xb1dx+C2rα|f(x0)|.

    Now by condition (3.4), consider a sequence rm0 such that |B+rm(x0){v=0}||B+rm|0 and assume that (ux0,rmpx0,rm)w in L2(B+1;xa1) as m. Observe now div(xa1w)=f(x0) in B+1 and by similar calculation as above we will have

    B+1xa1f(x0)wdx=f(x0)2(n+a+2)(a+2)B+1xa1dx.

    On the other hand,

    Φx0(0+)=limmΦx0(rm)B+1(xa1|w|2+2f(x0)xa1w)dx2B+1xa1w2dHn1=B+1(wdiv(xa1w)+2f(x0)xa1w)dx=B+1xa1f(x0)wdx=f(x0)2(n+a+2)(a+2)B+1xa1dx.

    Step 4 In this step, we prove the proposition for the points satisfying condition (3.4). For these points, we have

    12r[B+1xa1u2x0,rdHn1]=B+1xa1ux0,rrux0,rdHn1=1rB+1xa1ux0,r(ux0,rx2ux0,r)dHn1=1rB+1(xa1|ux0,r|2+ux0,rdiv(xa1ux0,r))dx2rB+1xa1u2x0,rdHn1=1r(Φx0(r)2B+1f(x0)xa1ux0,r(x)dx+B+1xa1f(rx)ux0,r(x)dx)=1r(Φx0(r)B+1f(x0)xa1ux0,rdx)+1r(B+1(f(rx)f(x0))xa1ux0,r(x)dx)1r(Φx0(0+)B+1f(x0)xa1ux0,rdx)Crα+β2C3rα+β2C2rα1Crα+β2C3rα+β2

    Thus rr[B+1xa1u2x0,rdHn1] is integrable and we obtain uniform boundedness of B+1xa1u2x0,rdHn1=rn3bBr(x0)+xa1u2dHn1 for all points with property (3.4). It follows that the boundedness holds uniformly on the closure of those points x0.

    Step 5 We now consider the case

    lim infr0+|B+r(x0){u=0}||B+r|>0.

    Let us assume towards a contradiction that there are sequences um, rm and xm such that and Mm=uxm,rmL2(B+1)+ as m. Setting wm=umxm,rmMm we obtain, as in Step 2, that a subsequence of wm converges weakly in W1,2(B+1;xa21) to a function w, with wL2(B+1;xa1)=1, w(0)=|w(0)|=0, div(xa1w)=0 and

    B+1xa1|w|2dx2B+1xa1w2dHn1.

    According to Lemma 2.3, wHP2. In addition we now know that

    B+1χ{w=0}lim suprm0+B+1χ{ux0,rm=0}>0.

    This however contradicts the analyticity of w inside B+1, knowing that wL2(B1;xa1)=1.

    Now we are ready to prove the main result of the article.

    Proof of Theorem 1.2. From Theorem 8.17 in [4], we know that if div(b(x)w)=g such that 1b(x)5a, then there exists a universal constant C=C(a,n) such that

    wL(BR/2)C(a,n)(Rn/2wL2(BR)+R2gL(BR)).

    Now for x0{u=0}{x1=0}B+1/8 and an arbitrary point yB+r(x0), we apply the above estimate for R=2δ/3, w=(δ/3)au and equation div(b(x)w)=xa1fχ{u0}, where δ=y1 and b(x)=xa1(δ/3)a. Note that 1b(x)5a in B2δ/3(y) and

    |u(y)|C(a,n)((2δ/3)n/2uL2(B2δ/3(y))+(2δ/3)25afL(B2δ/3(y))).

    According to Proposition 3.1,

    u2L2(B2δ/3(y))(3δ)aB2δ/3(y)xa1|u|2dx(3δ)aBr+2δ/3(x0)xa1|u|2dxC(3δ)a(r+2δ/3)n+a+4.

    Hence,

    |u(y)|C(δ(n+a)/2(r+δ)(n+a+4)/2+δ2)Cy21((r+y1y1)(n+a+4)/2+1).

    From this theorem it follows that solutions have quadratic growth inside cones.

    Corollary 3.2. Suppose u is a solution of (1.1) satisfying the condition in Proposition 1.1 and x0{u=0}{x1=0}B+1/8. Then, for every constant τ>0,

    supBr(x0)C|u|C((1τ+1)n+a+42+1)r2,

    where C:={x:x1τ|xx0|}.

    This paper was prepared while M. Fotouhi was visiting KTH Royal Institute of Technology. A. Minne was supported by the Knut and Alice Wallenberg Foundation. H. Shahgholian was supported by Swedish Research Council.

    The authors declare no conflict of interest.

    Let u be a solution of (1.1) for fL(B+1). We are going to show a priori regularity for solutions to (1.1). Consider the operator La,cu:=x21Δu+ax11ucu. The following proposition is the regularity result related to this operator which has been proven by Krylov [5,Theorem 2.7,Theorem 2.8].

    Proposition A.1. i) For any aR,p>1 and θR there exists a constant c0>0 such that for any cc0 the operator La,c is a bounded one-to-one operator from W2,p(Rn+;xθ1) onto Lp(Rn+;xθ1) and its inverse is also bounded, in particular for any uW2,p(Rn+;xθ1)

    uW2,p(Rn+;xθ1)CLa,cuLp(Rn+;xθ1),

    where C is independent of u and c.

    ii) The statement in i) is valid for the operator La,0 when 1<θ<a2 and a>1 or either a2<θ<1 and a<1.

    Now we can deduce a priori regularity result for u as follows.

    Proof of Proposition 1.1. Notice that xβ11uC(¯B+1) due to Sobolev embedding Theorem 3.1 in [5]. Then if the statement of proposition fails, there exists a sequence uj of solutions (1.1), xj{x1=0} and rj0 such that

    supB+r(xj)|xβ11uj|jr1+β/2,rrj,supB+rj(xj)|xβ11uj|=jr1+β/2j.

    In particular, the function ˜uj(x)=uj(xj+rjx)jr1+β/2j, satisfies

    supB+R|xβ11˜uj|R1+β/2,for 1R1rj, (A.1)

    and with equality for R=1, along with

    La,c0˜uj=r1β/2jjf(xj+rjx)c0˜uj, (A.2)

    where c0 is defined in Proposition A.1. According to (A.1), the right hand side of (A.2) is uniformly bounded in Lp(B+R;xθ1) for p(β1)1<θ1. From here and Proposition A.1 we conclude that {˜uj} is bounded in W2,p(B+R;xθ1) for some θ1 and there is a convergent subsequence, tending to a function u0 with properties

    supB+R|xβ11u0|R1+β/2,R1,supB+1|xβ11u0|=1,div(xa1u0)=0, (A.3)

    as well as the condition θ1 insures that the trace operator is well defined and u0 is zero on {x1=0}. The Liouville type theorem in [9,Lemma 20]) implies that

    |D2u0(x0)|CR2supB+R(x0)|u0|C(R+|x0|)2β/2R20.

    Therefore, u0 is a linear function, which contradicts (A.3).



    Conflicts of interest



    All authors declare that there is no conflict of interest.

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