> +*y!`!bjbj\\ 8B>>
8t
p
(
sss
~
~
~
~
~
~
$he
ss
H
H
H

H

H
H
H
kPvH

0
H

"H

H
4sH
Ssss
2
sss
dd
Detecting emerging transmissibility of avian influenza virus
in human households
Text S1
Model Structure
The Methods section of the main text gives an overview of our methodology. Here we give a number of additional technical details. Reference numbers refer to the references in the main text.
The analyses are based on a stochastic SEIR epidemiological model in which individuals are classified as susceptible (S), latently infected (infected but not yet infectious)(E), infectious (I), and recovered and immune (R). No a priori assumptions are made regarding the duration or distribution of the latent and infectious periods. We consider a model with two types of individuals, where ni and ai represent the initial number of susceptible and infected type i individuals in a household, respectively (i=1,2). In the following, type 1 individuals are assumed to have been infected by animals, while type 2 individuals have been infected by humans. There are no individuals that have prior immunity, so that total household size is given by N=n1+n2+a1+a2. The parameter Bi denotes the probability that a type i (i=1,2) individual escapes infection from outside the household (i.e. from the animal reservoir). The final size distribution is determined by triangular equations that can be solved recursively [3435]:
EMBED Equation.3 (A1)
(j1=0,1,, n1, j2=0,1,, n2). The factor EMBED Equation.3 in the numerator of equation (A1) represents the probability that EMBED Equation.3 type 1 and EMBED Equation.3 type 2 individuals are infected in a population with j1 and j2 susceptibles, while EMBED Equation.3 in the denominator is the probability that all initially present susceptible individuals avoid infection. For given ni and ai the final size distribution is fully specified by the escape probabilities EMBED Equation.3 , the Laplace transforms EMBED Equation.3 of the infectious period probability distributions, and the transmission parameters EMBED Equation.3 ( EMBED Equation.3 ).
Scenarios
Equation (A1) is quite general and can be used when only limited information on the nature of the transmission chain in the households is available. The data of the Dutch epidemiological study are more specific and allow a number of simplifications (see Methods for details). In particular, in our analyses we have throughout taken B2=1, a1=1, a2=0, and n1=0.
Four scenarios are considered that are defined by the assumptions regarding the distribution of the infectious period and the mechanism of pathogen transmission. With respect to the infectious period we assume that the infectious period is exponentially distributed (the general stochastic epidemic) or of fixed duration (the ReedFrost model). If the infectious period is exponentially distributed, the Laplace transform of the infectious period probability distribution is given by EMBED Equation.3 . If, on the other hand, the infectious period is of fixed duration it is given by EMBED Equation.3 . For simplicity we assume that the infectious period is independent of the source of infection, i.e. EMBED Equation.3 . Further, by rescaling the time axis we may measure time in units of the infectious period [3132]. Without loss of generality we may therefore take EMBED Equation.3 .
Furthermore, we assume that transmission is frequencydependent or densitydependent [37]. In a frequencydependent model the number of contacts per unit of time is fixed, and the transmission rate is proportional to the relative frequency (prevalence) of infectious individuals. In a densitydependent model the number of contacts per unit of time is proportional to the number of individuals. Equation (A1) describes a densitydependent transmission process. In a frequencydependent transmission model the argument of EMBED Equation.3 in equation (A1) is divided by household size (N), i.e. EMBED Equation.3 has to be replaced by EMBED Equation.3 . Hence, the transmission parameters of the densitydependent model are denoted by EMBED Equation.3 and those of the frequencydependent model are given by EMBED Equation.3 ( EMBED Equation.3 ).
Notice that if the infectious period is fixed the distribution of the number of type i infections caused by a type j infected individual in a large susceptible population (the offspring distribution) would be Poisson with mean EMBED Equation.3 or EMBED Equation.3 (depending on whether transmission is frequency or densitydependent). If, on the other hand, the infectious period is exponentially distributed, there is more variation in the number of infections caused by an infected individual. In this case, the number of infections caused by an infected individual in a large susceptible population has a geometric distribution with parameter EMBED Equation.3 or EMBED Equation.3 [36].
Notice furthermore that if secondary humantohuman transmission is absent (model1 in Tables23 and TablesS1S4), the household final size is binomially distributed if the infectious period is of fixed duration (with parameter EMBED Equation.3 and binomial totals given by the number of susceptibles, EMBED Equation.3 ). If, on the other hand, the infectious period is exponentially distributed, then the final size is given by a compound exponentialbinomial distribution. In case of the former model assumption the final size distribution is clearly unimodal, while for the latter simulations also indicate that the final size distribution remains unimodal.
In Tables S1S4 we focus on scenarios that are defined by the mechanism of pathogen transmission (densitydependent versus frequencydependent), and by the distribution of the infectious period (fixed versus exponentially distributed). Throughout, models denoted by A and B assume densitydependent transmission, while models C and D assume frequencydependent transmission. Furthermore, models denoted by A and C assume a fixed infectious period, while models B and D assume an exponentially distributed infectious period. Hence, model A assumes densitydependent transmission and a fixed infectious period, model B assumes densitydependent transmission and an exponentially distributed infectious period, and so forth.
PAGE 3 DATE \@ "d/M/yy" 5/6/07
<QRYZj
#(+{dL2dL2dLdLd2h6+h36CJH*OJPJQJ^JaJmH sH /h6+h36CJOJPJQJ^JaJmH sH ,h6+h3CJOJPJQJ^JaJmH sH /h6+h35CJOJPJQJ^JaJmH sH ,hdh3CJOJPJQJ^JaJmH sH ,hdhdCJOJPJQJ^JaJmH sH &hdCJ$OJPJQJ^JaJ$mH sH &h3CJ$OJPJQJ^JaJ$mH sH ,h+Ch3CJ$OJPJQJ^JaJ$mH sH >RZj( #
r
./97z!!!!!dhgd3
$dha$gd3
$dha$gdd
$dha$gd3!!+./0UVX]
#
@
A
T
U
V
W
s
t
սսսսՌucJuս1jh6+h3CJEHOJPJQJU^JaJ#jG
h6+h3CJUVaJjh6+h3CJOJPJQJU^JaJ,h6+h3CJOJPJQJ^JaJmH sH 2h6+h36CJH*OJPJQJ^JaJmH sH /h6+h36CJOJPJQJ^JaJmH sH ,h6+h3CJOJPJQJ^JaJmH sH &h3CJOJPJQJ^JaJmH sH t
u
~
)淜m[>9jh6+h3CJEHOJPJQJU^JaJmH sH #jhG
h6+h3CJUVaJ9jh6+h3CJEHOJPJQJU^JaJmH sH #jdG
h6+h3CJUVaJ5jh6+h3CJOJPJQJU^JaJmH sH ,h6+h3CJOJPJQJ^JaJmH sH /h6+h36CJOJPJQJ^JaJmH sH 2h6+h36CJH*OJPJQJ^JaJmH sH )*+,abchij
_`еllZ=ll9jh6+h3CJEHOJPJQJU^JaJmH sH #jpG
h6+h3CJUVaJ2h6+h36CJH*OJPJQJ^JaJmH sH /h6+h36CJOJPJQJ^JaJmH sH ,h6+h3CJOJPJQJ^JaJmH sH 5jh6+h3CJOJPJQJU^JaJmH sH 9j h6+h3CJEHOJPJQJU^JaJmH sH #jlG
h6+h3CJUVaJ`stuv(p^A9jh6+h3CJEHOJPJQJU^JaJmH sH #jG
h6+h3CJUVaJ9jGh6+h3CJEHOJPJQJU^JaJmH sH #jG
h6+h3CJUVaJ5jh6+h3CJOJPJQJU^JaJmH sH 9j@h6+h3CJEHOJPJQJU^JaJmH sH #jvG
h6+h3CJUVaJ,h6+h3CJOJPJQJ^JaJmH sH ()*+.9еnTnnTnnTnnTn@&hdCJOJPJQJ^JaJmH sH 2h6+h36CJH*OJPJQJ^JaJmH sH /h6+h36CJOJPJQJ^JaJmH sH /h6+h35CJOJPJQJ^JaJmH sH ,h6+h3CJOJPJQJ^JaJmH sH 5jh6+h3CJOJPJQJU^JaJmH sH 9jh6+h3CJEHOJPJQJU^JaJmH sH #jG
h6+h3CJUVaJ
qrͻ͊x[I#jG
h6+h3CJUVaJ9jh6+h3CJEHOJPJQJU^JaJmH sH #jG
h6+h3CJUVaJ&h3CJOJPJQJ^JaJmH sH 9j
h6+h3CJEHOJPJQJU^JaJmH sH #jG
h6+h3CJUVaJ,h6+h3CJOJPJQJ^JaJmH sH 5jh6+h3CJOJPJQJU^JaJmH sH 2345p@ATUVWǰǰmǰǰ[>ǰ9j!h6+h3CJEHOJPJQJU^JaJmH sH #jG
h6+h3CJUVaJ9jh6+h3CJEHOJPJQJU^JaJmH sH #j G
h6+h3CJUVaJ&h3CJOJPJQJ^JaJmH sH ,h6+h3CJOJPJQJ^JaJmH sH 5jh6+h3CJOJPJQJU^JaJmH sH 9j&h6+h3CJEHOJPJQJU^JaJmH sH ()<=еУеtWеE#j^VH
h6+h3CJUVaJ9j&h6+h3CJEHOJPJQJU^JaJmH sH #jG
h6+h3CJUVaJ9j$h6+h3CJEHOJPJQJU^JaJmH sH #j0VH
h6+h3CJUVaJ5jh6+h3CJOJPJQJU^JaJmH sH ,h6+h3CJOJPJQJ^JaJmH sH /h6+h36CJOJPJQJ^JaJmH sH =>?xy ǰǰǰǰoRǰ::ǰ/h6+h36CJOJPJQJ^JaJmH sH 9jh6+h3CJEHOJPJQJU^JaJmH sH #jG
h6+h3CJUVaJ9j}+h6+h3CJEHOJPJQJU^JaJmH sH #jVH
h6+h3CJUVaJ,h6+h3CJOJPJQJ^JaJmH sH 5jh6+h3CJOJPJQJU^JaJmH sH 9jJ)h6+h3CJEHOJPJQJU^JaJmH sH ABUVWX\]pеo]@9j74h6+h3CJEHOJPJQJU^JaJmH sH #j1H
h6+h3CJUVaJ9j1h6+h3CJEHOJPJQJU^JaJmH sH #j10H
h6+h3CJUVaJ,h6+h3CJOJPJQJ^JaJmH sH 5jh6+h3CJOJPJQJU^JaJmH sH 9j/h6+h3CJEHOJPJQJU^JaJmH sH #j&0H
h6+h3CJUVaJpqrsxy`atuvwеvvvgJ;jkI
h3CJUVaJ9j9hh3CJEHOJPJQJU^JaJmH sH jkI
h3CJUVaJ&hdCJOJPJQJ^JaJmH sH ,h6+h3CJOJPJQJ^JaJmH sH &h3CJOJPJQJ^JaJmH sH 5jh6+h3CJOJPJQJU^JaJmH sH 9j6h6+h3CJEHOJPJQJU^JaJmH sH #j%2H
h6+h3CJUVaJ)/) !!!G!ǳpUpJ9+hr0JOJPJQJ^J h!Vhr0JOJPJQJ^Jh3h_mH sH 5h6+h3B*CJOJPJQJ^JaJmH phsH /h3B*CJOJPJQJ^JaJmH phsH &hdCJOJPJQJ^JaJmH sH ,h6+h3CJOJPJQJ^JaJmH sH &h3CJOJPJQJ^JaJmH sH 5jh6+h3CJOJPJQJU^JaJmH sH 9jQ;hh3CJEHOJPJQJU^JaJmH sH
G!H!N!O!P!Q!!!!!!!!!!!~zvkh3h_mH sH hhr$h>hrCJOJPJQJ^JaJh0JCJOJPJQJ^JaJmHnHu"hr0JCJOJPJQJ^JaJhr0JCJOJPJQJ^JaJmHnHu(h>hr0JCJOJPJQJ^JaJ1jh>hr0JCJOJPJQJU^JaJ6&P 1h:pd. A!8"#7$%Dd
Lb
c$A??3"`?2rq)
"(oLDD`!rq)
"(oLv h+
wxڥ[HQǿsfh`%~zz+RZFjQ
EPHPP)COQCSԃ/aPtlՂnsffvsw\8"Cƿċ7!B%mNek gdLڧYqޚM[woqcL9v`}蚑RC(J;Zm7jiZ@n~?ֲͧUG
x~)ttL3Oѫb
"R7/9ޗH]9GokPe^럗ϛU`ѫbzb%SggCb^WE^p4tk4Мg;em01ewT]/4{_uu\R3QR/tm(״I}XbWgh~YJL\'zp=P)X4XWgUVC ?%ҏnc1/БvcYf^̫վubjeT;Zi787h,oڬzHU~~XK>$EO&y/q9}QW vuj6W.4$x'w2ů20TDd7]!P
p2(WnzjO$7]=?oű? ֮ ^_cf9OuӜx+ŜRޖ`^Oܫ;5G9G;O*mZg%L5%EH xcdd``> @c112BYL%bpu @c112BYL%bpub8Xt9F%{*!3(4O 27)?a&'v`5D2z.΄]F&&\7@]
U`nzKxDd
p
E=b
c$A??3"`?2p4BJ}(&oD`!p4BJ}(&o@S!dxڭUMhA~6iMjmU"O[ڋ)'Q6&ŰPpX.x=x?Pix:vYfw]q% 3mlg 7.G㳓m)84ǫ})(NK/?~ӨmtxG\ow8oP<
;nO#;3~43YYhI{S]^Xԏ*vt#X_ӯL.JZ/%+@:FZo8UpU$&p4:{>N{ZÛ>xVҾ5}M+ʼxMG:Vw!u5:[2>M=zLyr3B>^Ow;&>{^Ol3!w=x=īҗdw~ з!Ϛw,t_qqY ]]Dd
@E<b
c$A??3"`?2Q
!#$%&'()_,/V0214356798:;<>=?A@BCDFEGHILJKMNOQPRTSUWXZY[]\^wv`abcdefghijklmnopqrstux{}~Root Entry} FP.*Data
"a=WordDocument8BObjectPool'kPP_1207475602FkPkPOle
CompObjfObjInfo
#&'()*258;>ADGJMPSTW\adgjknqty~
FMicrosoft Equation 3.0DS EquationEquation.39qh0
j11
()j22
()
P1
,2
Equation Native _1207475812FkPkPOle
CompObj
f[]Bknk"jk
kk=12
"
*
lk
nl
"jl
()l=12
"
[]k+ak
=12=0j2
"1=0j1
"
FMicrosoft Equation 3.0DS EquationEquation.39q(
j11
()j22
()
P1
,2
[]ObjInfo
Equation Native _1207475816FkPkPOle
CompObjfObjInfoEquation Native 6_1207475820 FkPkP
FMicrosoft Equation 3.0DS EquationEquation.39q flW
1
FMicrosoft Equation 3.0DS EquationEquation.39qOle
CompObjfObjInfoEquation Native 6U
2
FMicrosoft Equation 3.0DS EquationEquation.39q5d
Bknk"jk
kk=12
"
*
lk
_1207475824FkPkPOle
!CompObj"fObjInfo$Equation Native %Q_1207475830"FkPkPOle
+CompObj ,fnl
"jl
()l=12
"
[]k+ak
FMicrosoft Equation 3.0DS EquationEquation.39q4T
BiObjInfo!.Equation Native /6_1207475839$FkPkPOle
0
FMicrosoft Equation 3.0DS EquationEquation.39q.p4T
i
s[]
FMicrosoft Equation 3.0DS EqCompObj#%1fObjInfo&3Equation Native 4J_1207475843;)FkPkPOle
6CompObj(*7fObjInfo+9Equation Native :CuationEquation.39q'R4T
ij*
FMicrosoft Equation 3.0DS EquationEquation.39q_1207475847.FkPkPOle
<CompObj/=fObjInfo0?Equation Native @E_1207475952,63FkPkPOle
BCompObj24Cf)Y4T
i,j=1,2
FMicrosoft Equation 3.0DS EquationEquation.39qK
s[]=1/1+Ts()ObjInfo5EEquation Native Fg_12074759578FkPkPOle
H
FMicrosoft Equation 3.0DS EquationEquation.39qK@lW
s[]=exp"Ts[]
FMicrosoft Equation 3.0DS EqCompObj79IfObjInfo:KEquation Native Lg_12074759741^=FkPkPOle
NCompObj<>OfObjInfo?QEquation Native RuationEquation.39qrxO/
1
s[]=2
s[]=s[]
FMicrosoft Equation 3.0DS EquationEquation.39q_1207475977BFkPkPOle
UCompObjACVfObjInfoDXl
T=1
FMicrosoft Equation 3.0DS EquationEquation.39ql
kEquation Native Y1_1207475987@OGFkPkPOle
ZCompObjFH[fObjInfoI]Equation Native ^6_1217156656hYLFkPkPOle
_CompObjKM`fObjInfoNbEquation Native c{_1207475992QFkPkP
FMicrosoft Equation 3.0DS EquationEquation.39q_`<
lk*
nl
"jl
()
FMicrosoft Equation 3.0DS EquationEquation.39qOle
eCompObjPRffObjInfoShEquation Native igPl
lk
nl
"jl
()N
FMicrosoft Equation 3.0DS EquationEquation.39q_1217156702VFkPkPOle
lCompObjUWmfObjInfoXoEquation Native pC_1217156760Tr[FkPkPOle
rCompObjZ\sf't\
ij*
FMicrosoft Equation 3.0DS EquationEquation.39q#<
ijObjInfo]uEquation Native v?_1209282598EJ`FkPkPOle
wCompObj_axfObjInfobzEquation Native {:_1209282609eFkPkP
FMicrosoft Equation 3.0DS EquationEquation.39qw
ij
FMicrosoft Equation 3.0DS EquationEquation.39qOle
CompObjdf}fObjInfogEquation Native Dw(w
ij*
N
FMicrosoft Equation 3.0DS EquationEquation.39qw3 c
11+ij_1209283069cmjFkPkPOle
CompObjikfObjInfolEquation Native O_1209283109oFkPkPOle
CompObjnpf
FMicrosoft Equation 3.0DS EquationEquation.39qw=`XT
11+ij*
N
FMicrosoft Equation 3.0DS EqObjInfoqEquation Native Y_1235905513wtFkPkPOle
CompObjsufObjInfovEquation Native >_1235905452yFkPkPuationEquation.39q@"yl
1+
FMicrosoft Equation 3.0DS EquationEquation.39q@
n2Ole
CompObjxzfObjInfo{Equation Native 6̜RD`!%̜RH xcdd``> @c112BYL%bpu1,(Dd
0@E<b
c$A??3"`?2p<(a1jD`!p<(a1H xcdd``db``baV d,FYzP1n:&n! KA?H1ZqC0&dT20d
@MdMa`%vÎ%H4kB]j mvot%!"ظq? dDl%Nps^RܫN{%Hq~NݫH{p!w? Q&ǢмT`ܛ6waC_`ĂOuYp{YP>XG
w[=>HNF0O+\mv0o`dbR
,.IeԡRYݏ`t'Dd
E;b
c$A??3"`?2qB1
1j#dn8l1e$ѝs2!*\{&`pX321)W2ԡ"b@3X?_&Dd
,@E<b
c$A??3"`?
2i}YK#p<
/E;"D`!=}YK#p<
/Ȓ xcdd``> @c112BYL%bpub
iq,@u@@ڈkJy!4O 27)?艇B;?M`Uu=؇`!v 0y{aĤ\Y\2C<bb#3XsKDd
E=b
c$A??3"`?2#솨T\* ЦZ$D`!#솨T\* Ц`hnhwxcdd``> @c112BYL%bpub~i+Y
u9F?&VJBqsm#d^6Ps3@01(8e&(e&秤*3p'\F?T̨
p8#Ds0`U c=XpW, #t1"31C@[0:0M䂦Z.p;f=Ĥ\Y\˰d.P"CX]`9;Dd
lE=b
c$A??3"`?2ic,Bj$&D`!ic,Bj$nxcdd``$d@9`,&FF(`Tv! KA?H1Z @=P5<%!vf2w KP 27)?!+}!K>bQ\i6,@@@ڈ+ Uy\icd}2B%dCܰ
X2sB2SR F&32LYp%~WO03Ah{N%lC6oG?8Iy+2BK2mnsǘ272l:.v0o8321)W2ePdk)hx,Āf~23Dd
D=b
c$A??3"`?2}۪0>T1Y)D`!Q۪0>T1`0hxڕPJA}3\(bbc괳0O4OA#%Hsv Xx0o7>B}`,}3RE#&82EJ^#AXB42HܤW\\bFnFI\Pp}6b;FLLJ%@2u(2tA4T}bb#3X $Q1<Dd
H,F;b
c$A??3"`?2%\FMbD`!Z%\FM*@"(xcdd``ed``baV d,FYzP1n:B@?b uu0nĒʂTݿ;aR&ǢмT`%rq,@u@@ڈˁћըƿ`!"~& Ma`Xŵ$32Hcaa.#77lBZ`gVrAC;WLLJ%+A0u(2t5B~b<%d\ Dd
@hF>b
c$A??3"`?2j7mvnU7F"0D`!>7mvnU7@ xcdd``> @c112BYL%bpu`ꁪaM,,H$~1At2Ba:f YX@ʹDY2sB2SR.?
_8 f222rdW 1O?y1nL_L`B7DwBeI^,t3P^Gf+Y3%W&00Աr=p(İ(;2+M~&$sٹq~ApigqAS8b;^+KRs@2u(2t5B~`lu3FDd
0lG=b
c$A??3"`?2VGrhtlO9D`!dVGrht&k2xcdd``f2
ĜL0##0KQ*
WYRcgbR 63$ 憪aM,,He`,@1[%d2i? 201W&00rqy*İc@`\; b&#.#f&ߒ*!DF/FP00 u L`@Gl66sVp,Lr+>wCek2°\PK'F&&\$@]`
υaDd
@E@b
c$A??3"`?2Zoۚ}Y46;D`!.oۚ}Y4H xcdd``> @c112BYL%bpuDetecting emerging transmissibility of avian influenza virus Title
FMicrosoft Office Word Document
MSWordDocWord.Document.89q@`@3NormalCJ_HaJmHsHtHDA@DDefault Paragraph FontRi@RTable Normal4
l4a(k@(No List4 `43Footer
9r .)`.3Page Number B>RZj(#r./9 7
z000000000000000@0@000Z#r9 7
@0@0@0{0q1r{0q100
@0{00@F0FF&
00+t
)`(=pG!! !!@TV
)+_su(*
q
2
4
@TV(<>xAUW\pr`tv:::::::::::::::::::::::::BIK!8@0(
B
S ?s*:A9*urn:schemasmicrosoftcom:office:smarttagsplace*E#$UVXY
>@&([]wy
Q3333333333/3rjrd_@E@{@UnknownGz Times New Roman5Symbol3&z ArialI&??Arial Unicode MS"q*F*F*F,,!87x24 2QKP)?32=Detecting emerging transmissibility of avian influenza virus van boven van boven