デングウイルス伝播に関する基本再生産数の推定(第3回生物数学の理論とその応用)

デングウイルス伝播に関する基本再生産数の推定

Estimation of the basic reproductionnumber for denguevirus transmission

長崎大学熱帯医学研究所・チュービンゲン大学 西浦博 (Hiroshi Nishiura)

Nagasaki University Institute of Tropical Medicine, Japan Department of Medical Biometry, University of T\"ubingen, Germany Resume

.1.

Background

Denguevirus is

a

widelyprevalent pathogen throughout tropical countries. Infection with denguevirus (DENV) results in asymptomatic(inapparent) infection, dengue fever(DF)

or

dengue hemorrhagic fever(DHF) [1]. DHF is themost

severe

form ofclinicalmanifestations which bt leadto death. Thereare 4 closelyrelated serotypes. Pathogenesis ofDHF

is.

associated with secondary infectionwith heterologous serotypes. Whereas epidemiologic risksofDHF have been explored for

more

than 30

years,

transmission dynamics of dengue

areyet to be clarified. In particular, although there have beenvarious theoretical works which stressed out ecological interests [2] (e.g., super-annual cycle ofthe epidemic,co-circulation of heterologous strains inrelationto antibody-dependent enhancement, andspatial spreadand heterogeneity), there

are

few operational researches which offered practical andquantitative epidemiologic

implications

for dengue control. This study

was

aimed

at estimating the

transmissionpotential of dengue using DHF incidence only.

2. Methods

Thisstudyassumesthat dengue is endemic(i.e. endemicsteadystate) and estimates the force of infection only using age-specific profile ofinfection. When the force ofinfection, $\lambda,$ . isage-independent, analytical solution of

a

static SImodel (e.g. age-specificproportion of

infected individualsatage a) is givenby:

$I(a)=1-\exp(-\lambda a)$ (1)

where$I(O)=0$

.

Usually, eqn ₍₁₎ isappliedto age-specific seroprevalence data[3].This study

estimated $\lambda$from incidence data using the similar analytical solution ofSISImodel:

$\frac{dS_{0}}{da}=-\lambda S_{0}$ $\frac{dS_{1}}{da}=\delta I_{1}-p_{1}\lambda S_{1}$

(2)

$\frac{d\Gamma_{1}}{da}=\lambda S_{0}-\delta I_{1}$ $\frac{dI_{2}}{da}=.p_{\iota}\lambda S_{l}$

where $\delta$istherate to loose cross-protective immunity and$p_{1}$ is thereduction ofthe forceof 数理解析研究所講究録

infection during secondaryinfection. Assuming that the force ofinfection is identical between serotypes,$p_{1}$ is 3/4 (sinceimmunity against the

same

serotype islife-long). Thecumulative

distribution ofthose experiencing secondary infection until age$a$ is:

$I_{2}(a)= \frac{p_{1}\lambda^{2}\delta}{\delta-\lambda}[\frac{]}{p_{1}\lambda-\lambda}(\frac{\exp(-p_{1}\lambda a)-1}{p_{I}\lambda}-\frac{\exp(-\lambda a)-1}{\lambda})$

(3)

$- \frac{1}{p_{1}\lambda-\delta}(\frac{\exp(-p_{1}\lambda a)-1\exp(-\delta a)-1}{\dot{p}_{1}\lambda\delta})]$

Maximumlikelihood estimates of$\lambda$and _{$\delta were$}

obtained by minimizing binomial deviance between theeqn(3) andobserved data. With regard toobserved age-specificdataofDHFand

’those

experiencing secondaryinfection, observed DHF incidencereportedto surveillanceand age-specific probability of secondary infection (amongall DHF) inBangkok

were

used.

The effective reproduction number,$R$, is given by:

R=Iら$s$ (4)

where$R_{0}.is$the basic reproductionnumber and$s^{*}$ isthe proportion ofsusceptible. In

an

endemic equilibrium,$R=1$ andthus$R_{0}$ isgiven by

an

inverseofthe proportion of susceptible

individuals [4]. I

assume

that Bangkok populationapproximatelyfollows

so

called “Type-II survivorship’ hnction, exponentiallydistributedage-specific survivorship,$l(a)$

:

$l(a)=\exp(-\mu a)$ ₍₅₎

where$\mu$is theforce ofdeath. Under this assumption,thenumber ofsusceptibleindividuals,

$S^{*}$, and total population,$N^{*}$, inan endemicequilibrium

are

given by:

$S^{\cdot}= \frac{N(0)}{\lambda+\mu}$

(6)

$N^{*}=\underline{N(0)}$

$\mu$

where$N(O)$ isthe populationsize atbirth. $s^{*}$ in eqn (4) is givenby_$S^{*}/W[5]$

.

Thus,_{$R_{0}$} is:

$R_{0}=1+\lambda L$ (7)

where$L$ isthe

average

life expectancyatbirth and equals to$\mu^{-1}$

.

Using the estimated $\lambda$and_$L$,

estimate

of$h$

was

obtained.

3. Results and Conclusion

Using DHF incidenceduring $1990s,$$R_{0}$(andthe corresponding95% confidence intervals

(CI))

was

estimated to be 15.4 (95% CI: 14.3, 29.6). Themaximum likelihood estimateof$\delta$

was

3.12 (2.65, 3.89) per year.The

same

model

was

applied to theincidence during $1980s.R_{0}$

was

estimated

as

18.4 (16.3, 31.4). In addition, this model permittedadual estimation ofthe

age-specific risk of DHF followingsecondaryinfection, the$o$

ualitative-

patternofwhich

was

consistent

withaprevious observation on innatesusceptibility to DHF in Cuba.

Thus,serotype-unspecific$R_{0}$ should be assumed to be larger than 15 inendemic

areas

(where4serotypes

are

co-circulating).The discrepancy

seen

indifferentestimates$ofR_{0}$ (in

previous studies using DF epidemicdata) mightbe largelyattributabletothedifferentvector ecologyandvirulence, but, mostimportantly, this also reflects that substantial proportion of asymptomatic infections and unreportedDFwould exist.Itwould be appropriate to

assume

that serotype-unspecific $R_{0}$fordengue isapproximately 16, at least,to designthe monovalent

vaccination strategies. It is difficulttoeradicate dengue byvectorcontrol only. Comparedto

the difficultyoferadicationusing monovalentmassvaccination only(which necessitatesto

cover>94%), itis easierto eradicate dengue iftetravalentvaccine equally

covers

four

differentserotypes. If this isthe case,the critical coverage ofvaccination should be assumed to beone-fourthoftheaboveserotype-unspecific $\backslash R_{0}$(i.e. approximately4) since $\lambda$inthe

above model should have been $4\lambda$in

a

serotype-specific

manner

[6].

It is interestingtonotethatthe force ofinfection significantly decreased $\theta om$ 1980sto

$1990s$

.

Indeed, elevatedaverage ageatcontracting DHFin 1990scomparedto 1980s

reasonablyreflects the decrease inthe force

of

infection. In $B$angkok, thehabitation ofAedes

spp, vector ofdengue, mayhave been reduced.Moreover,themodel confirmed that the qualitativepattem ofage-specific risk of DHF following secondary infection

was

consistent $\dot{w}$

ith previous suggestion

on

the innatesusceptibilitytoDHF. Duringsecondaryinfections,

smallchildren

are more

vulnerabletoDHFthan adolescents andadults.

4.Aeknowledgements

Thanks

are

due to Scott Halstead and KlausDietz. HNreceived ffindingsupportfrom the Banyu Life Science Foundation Intemational.

5. References(Note: original

paper

describing the abovework isunderreview)

[1]Nishiura$H$ and Halstead SB. TheJournal

ofInfectious

Diseases2007;195(7):

1007-1013.

[2]NishiuraH. Dengue Bulletin2006;30:in

press

(to

appear

online without

access

restriction).

[3] Farrington CP,Kanaan MN andGayNJ.Journal

_ofthe

RoyalStatistical Society: Series$C$,

$2001;50(3):251- 292$

.

[4]Anderson RM andMay RM. InfectiousDiseases ofHumans: Dynamics and Control. . OxfordUniversityPress, 1991.

[5]Dietz K. Transmission and control ofarbovirus diseases. In: Ludwig$D$, Cooke KL(ed.),

Epidemiology. SIAM,Philadelphia, 1975, 104-121.

[6]FergusonNM, Donnelly CA and Anderson RM. Philosophical Transactions

_of

theRoyal Society

_of

LondonSeries$B$_, BiologicalSciences, 1$999;354(1384):757- 768$

.