deun~ puod urwj seloa Xlerro-a puo emappul o-polow 6ugout!gs3

276 BLRTOS SINGER ASD JOEL E. COHES

splenomzgaly (enlargement of the spleen). anemia and, in chddren, fre- quently fatal complications. Four species of Plasmodium occur naturally in man: ?. falciparum, P. malariae, P. oc:alc. 2nd P. cicax. The first three of these species are present in the Garki dsriict of northern Nigeria This papse concerns only id-ectio~s to ::?e most prevalent spgies, P. - .-

fofc$amm. To c lb fy how parasi~ological =sasu2ments-tSie raw materid - - --

for the estimation of conversion and recoLsq rates-relate to the infection history of an individual, we describe briefly the life cycle of P. fdcipam. For details and scientific history, see [26] 2nd [25]. For clinical aspects, see

[301- The life cycle of maIvlal parasites coimsts of a sexual phase in certain

anopheles mosquitoes-notably Anopheler g d i a e in Garki-and an asex- d phase with multiplication in man- The a n o p h b phase of the cycle begins when the mosquito ingests human blood containing the sexual form of the parasite, called gametocytes. Once inside the mosquito's s t m the male cell extends and then detaches flagellum-Wre s t rnc t~~es which migrate to and then f e r t i h the female &Is. The fertifized cdk pen-* the wall of the mosquito's stomach and there grow into wcysts containing fdamS- like structures called sporozoites. Upon reaching d t y , the m s t s mptme, releasing up to several hundred th-d sporwites to migrate throughout the mosquito's body cavity. Some of these sporomites reach the salivary g k d s and there remain dormant until in- into man For P. fakipmum this ampheline phase of the life cycle lasts between 7 and 14 days

The W t e r disappear from the peripheral blood witbin 30 mimttes after injection into man, initiating the exoerq- stage of the life cyde The parasites develop in the liver parenchymal cells and reach maturity after about six days. The mature parasit= are about 60 in thkt long&

diameter and release approximately 40.000 merozoites into the blood to invade erythrocytes At this stage of &el-t can

fint he &e&d in the H o b A minimum of 10 pasasits per d is . . d y rcq+ far detection by o r d i m micrmcupk emmumhm of a thick blood smear.

'Ihe invasion of red Mood cells by merozoites i n i h t e ~ the p t a d the next stage of asexaal rqnduction, m p h m i t e r In about 48 hours, each tropbzaite rebases from 8 to 24 new d t e s for further invasicn~ of red blood ceils. After saved perations of this ~SUCCS, the pasitbd Mood cells release male and f d gametocytes capable of infecting

moscphs A series af t r o p f i o z o i t e - m t e waves typically follows As the gamsocyte count rises rhe trophaote count falls, and clinical improvement or remission of symptoms frequently occurs. Parasite

.MALARIA INCIDEUCE rLVD RECOVERY R-\TES 177

counts in falciparum mataria fluctuate markedly. They often show a!ttrmat- ing high and low densities on successive days.

2 2 nELD SURVEYS

-- For at I- y&tfi&Save been internationat eRorts to control ----- malaria in Africa, Asia, and the Americas. Important ingdirnts in rhe planning of antimalarial program are baseline data and quantitative esti- mates of the key factors governing transmission. A good quantitative understanding of the natural course of malaria requires longitudind field surveys of the human population synchronized with mosquito surveys. Such investigatious are very costly and difficult to organize and implement Despite the widespread recognition that such studies arc essential to esti- mate human incidence rates of infection and recovery and entomological inoculation rates, malaria surveys on humans from 1920 to 1970 were almost excIusively emssse&onal prevalence surveys. For a review of malaria prevalence surveys see [XI.

The WHO surveys in Garki are the F i t to make possible direct estimates d the age-specific rates, r, and p,, where A may be either apparent new infection a apparent re-cc~~ery- In the Garki project, sixteen villages were surveyed wery ten we& from the end of the wet sea~an in Nave 1970 to the end of the dry sea~cln in May 1972. There were no attempts at treatment or ccartrol of m a l h duriug this period The surveys axe therefore Oged baseline surveys The surveys aimed at conpiete coveragr of each villagrandinduded thecollectionof a thick WoodEilm,tinked byacode m m k to the penom's identity. Blood f h stained with Gientsa stain and examjned, under oil h m e n i q with 7x oculars and IOOX objective, for 200 fidk Each person was classified as positive for P. fd+m pzmdemia if traphazoites and/or gametocytes were observed in any of the 20 fiehis

IhtaontbtmaPqui tovedor~col lec tedinsorneof the16vi~ ererg 5 weeks in the dry seasoa and every 2 weeks in the wet season.

Man-biting rata and entomologicd inoculation rat- as the amage numkr of qmmzaite-positive bites per man per night, were ob- tained using hmaa bait For furtk details on the entomological surveys in Garid, Dl].

E s t a m o l ~ mey data provide eshwtes of inomlaion rates, which have~aspmme&zsinpqmsedll~3~ofmalariatransrmssl . .

on dating from 1911 [BJ to the present. Despite the substantial theoretical litmmre on the ephbmkdogy d tmmni&cn [lS,m, l9,2], the lack of any data o~mpvabL to tbDv of the Gard surveys has prevented systematic field terrting d these proposal. DieQ et al. [lo] attempreed to model the

BURTON SINGER rtVD JOEL E COHEN

transmission of malaria using data from the baseline surve>s in Garki. They explicitly took account of the effect of immunity on transmission, of superinfection, and of recovery rates (expected number of recoveries per unit time at time s per person positive at tune s).

The deveIopment of an integrated stochastic model of the infection - - histories i n a u p o p & h E d i p r e d taa m6dd of tb~dynarmarmd-&& --

mosquito vector, l i s in the future. Our aim here is much more modest. Using a time series of two-wave panel suneys (Sec. 4) we ascertain whether or not the observations could have been generated by continuous-time Markov chains or certain mixtures of such chains. Having accepted a restricted cIass of processes (e.g. the time-homogeneous Markov chains) as consistent with the panel data, we estimate the rates rA and pA within this class. We then interpret their age- and season -WIC values in the context of malaria dynamics in an unprotected population.

Although we analyze only the baseline nweys in Garki, the age- and season-spes~c rates p,, and rA can also be compared with the com%pd- ing quantities estimated in some of the same villages during 1; years of intexventions 'Ihoug.. we shall not attempt to use incidence rates to assess the impact of interventions, Sec. 3-6 sketch strategies basic to such evalua- tions. For data from the intervention and follow-up phases in Gad& see

[211-

3. SGiPLING ASPECTS OF THE SIMPLEST MARXOV Eh4BEDDING PROBLEM

Consider a two-parameter family of finite r X r stocbstk. ma& fis,t), 0 6s 6 t < t a, in which (a) each dement pi,(s7t) of P(s, t) is continuous in (s.t), (b) P(s,t)= I if and only if s= t, and (c) P(s,t)= fis,u)P(u,t) whenever s < u 6 t .

Such a family P(s7t) may be thought of as *bing a nonstationarJ( can-e f i i Marlcov chain. p,,(s,t) gives the transition pdabilily from state i at time s to state j at time t. We call a finite sto&& matrix P ededdaMe in a c&owtime Markov chain if there exists a two-param- eter family af s tockt ic matrices P(s.t) s a w g (a), (b), (cX and (4 P(O, 1)- P.

Goodman 1121 proved that any embeddatrle matrix P could, by a c h g e of time scale, be edxddd in a continuo- -ter family of transition 6 c e s P(s,t) satisfying. for almost all s and t,

.MALARIA MCIDENCE AND RECOVERY RATES

and

wk-ef&3>sa-hded rneasurabk fiinction such that for all s > 0,

Q is the set of all "intensity matrices." Equations (3. Ia) and (3.1 b) are known as the Kolmogorov forward and backward equations, respectively.

THEOREM [I21

A 2 x 2 stocAartie nmhrx P is embeddabie in a continuous-time Mark00 *ifdpdyiftraccP>l. '

This simpde and vgy usefd characterization was established by D, G. KemM [16, p 15) for homogeneous chains, i.e., b e for wbi& P(s,t)= P(t-sX amd extended to i n h o v e o u s chains by Goodman [12].

Albugh a 2 X 2 stock& matrix P with traceP > 1 can be embedded in manmbUy many bhomgeneous Markov &aim, such a matrix is e m b e d d a M e i n a u u @ e ~ c h a i n T o s e e t h i s , o b ~ e ~ v e ~ t h e

matrku for chains solve (3.la) and (3.lb) with Q(s)=consbnt The unique solution is

P(~,t)=P(t-s)--e'~-")Q, O<s< t< tao . ( 3 4 Tbur

P-eQ for some Q EQ, (3-3)

'Goaknan proved that dCt~->tI is a necesq condition for -ty of a s t o c b d c matrir P d auy 6. Far 2x2 makbs, this is equivaIcat to traccP>l.

is an bmcdk& of (3.4). since traceP> 1 implies that P is ~ i o a ~ c h ? i P .

,230 BCRTON SINGER IWD JOEL E COHM MALARIA ISCIDEVCE AXD RECOVERY RATES

with p,, +pa= rraceP > 1. Since the principal branch of the loga~-ithm in (3.4) is rhe only determination of logP -*hich is a member of Q, this establishes the uniqueness of ernbeddabitity for 2x2 homogneous chains. The uniqueness breaks down and the emkddability criteria become more

- - complicated - - for matncero&hornogeneoui - ~ o d i Z k - i i ~ -

In practical applications where it is natural to consider embedding a stochastic matrix P in a continuous-time family of such matrices, P must be estimated from sampled data The estimated matrix P may fail to be embeddable even when P is. Moreover, log P is then only an estimate of log P, and it is important to know how much variation should-be expected in log given P and the sample size. We now develop a formal test for embeddabiiity, assuming binomial sampling based on the criterim P>I.

3.2 TWO- WAVE PANEL DA W AND BINOLWA L SAMPLING

For individuals evolving independently and observed at two points in we may describe their location in one of two states--i:.g., not infecred

or infected in makuia parasitemia surveys--according to the followkg probability d d .

Let %+ be the number of individuals in state i at the initial obser*ation time t=O, i=1,2, and introduce the independent, identically random variables Y,, . . ., Ynlc where

yi= I with probabilityp,,, 0 otherwise.

Hen pl, is the a priori unknown conditional probability Prob(w : 4A)= l(w(O)=I), where 4 is the time when the second wave of panel data is dected a d {~(t), t > 0) is a parasitemia history, ideuNied as a path afsomestadmstlcprocesf on the state space (1,2).

L.et 2,. . . . , q+ be independent, identically dishitwed random variahies, whicharealsoindependent of Y ,,..., YnI, andset

T = { 1 with p robab i l i ty~~ 0 othenvke-

and

n, may be interpreted as the number of individuals observed in state I at time t =A who were also observed in state i at rime t =O. The conditional probabilities p,,, i=1,2, are not assumed to arise from

any restricted class of stochastic processes such as the continuous-time Markov chains. The above formulation assumes only that individuals evolve independently and defers the specification of the underlying ~ C S In our procedure, a @cation of the dycamics follows tests of hypotheses applied to the data. A similar approach is taken by Anderson and Good- man [11, who test multinomid frequen- from multiwave panel dara-for con&tency with conditional probabilities arising in Markov pmceses of order I,& ..., . However, the sampling tkeory of the embedQbility criterion for contiuwus-time proceoseg has not been sys- tematically dkcussd before.

By co- to this qpmach, it is usually assumed at the outset that the &emations arc generated by independent ~ t i c m s of a continuous-time Mafk# chain, or mixture of Markov chains, semi-M&ov processes, etc. (cg, [5), [15L and their refemnces).

In orQr to create a formal test of hypotheses based on the criterion traceP > 1, we introduce the md hypothesis

and the alternative hypothe&

W e gropase a decision ruIe based on 8, and 8, to be defied as follows:

1f t ~ w e ~ > 1 + 8 ~ , theaaccept~,, If 1 - 8,6 t r a d 6 1 + 4, then the evidence is inadequate to distin,ouish

between If, and H,.' If traces < 1 - 8, then accept HI.

- .

%isaninttanrrofa- - - mle of the kind disflrroed - - by - i17.P- 5 4 9 3 , T b c ~ ~ t a o B c e r a o a ~ w h i c h r r q a i r c a c e e p t a m c e o r r e j e d i o n o f a pro@ hyp&&s, art M d y stringent They fail to formdizc the Scotch verdict "nat pwen" when velua of a test stahtic lie dose to the boumtarJ of the set of v h that arc amkteut with the models.

ESZ

BURTOX SISGER XXD JOEL E COHES I - w m INCIDBCE &iD RECOVERY UTE=

actually generate the data. For example, transition matrices for mktuces of I= TABLE 2

continuous-time Biakov chains

PJO. r) = j- P'Q'(0.r) du( Q). Q

- --. - - --. - -

where p is an arbitrary probability S-kGe on-the-space Q of3m-Ssur&Ie --

intensity matrices Q, must satisfy tracePp(O,r) > L. Here P(Q)(O. r) a x transi- tion mobabilities for a 2-state continuous-time Markov chain with intensity matrix function Q(t), t > 0. In Sec. 7.2 we discuss biases in event-rate estimates which can arise from the use of a simple Markov-chain mode1 rather than a nondegenerate mixture (3.10).

In contrast with the weakoess of inferences associated with the axep tan- of Hb a rejection of HJeads to strong conclusions If t raceeo,~) < 1 - &, then no mixture of the form (3.10)-dqenerate or not--could de- 6 the data. S ~ ~ y , this procedm e h b t e s large classes of mdeIs from contention as candidates to describe an evolving proceso, without directly testing dynamic characteristics such as the Narkw prop erty. Statisticid tests b a d on c h a r a c t ~ o ~ z s of the conditional probabili- ties that can be generated by special models exhibit their greatest powa as t d far 4 rejection.

Onr decision rule allows the posibii of no daision when vafep is f i e d y close to 1. The intuitivdy appdhg propertis of the pr-ty . . d no decision, ProbJl-&<traap<1+6,), for tbe e d d b b h @ test are: (i) For each (p,,,p$) such hat pll+pp+l, and each (n,+,++X if q,i = 1,2 decrease, then the probability of no decision imJegsea (ii) For each a i~ (O ,$ ) andp,, such thatp,,+pp#l, if min(nl+,n2+) incrrasa, then the probabWy of no decision deaeass (ii) For each qE(0,;) a d (q+,n~+), if \pl,+pn-11 incl.easeq then the probability of no decision deawset

To provide further insight, Table 2 illustrates numerircaly tbe rrlation- ship a, p, (nl+,n2+), and the probability d no deckh.

3.3. HOMOGENEOUS CHAINS

a Estimation of Intensib Matrice% Following the of HO in the m t y t s t of Set 3.2, the simplest stochastic ~~ wfiid descrb two-wave panel data are the homogeneous Markov chains. We present t h e w e l I ~ ~ - 1 i k e E b o d &tor of the unique 2 x 2 intensity matrix

f'robabthues of No Deaslon and Cnttcd Valus of 6= 6, = 6Za

Sample& Main diagonal (pI I ,p& of P

Rl + nz+ (.9..2) (99-9) f.R.4) (-4.6) 6 -- A -- - - FTdQbtQe-0 Cte&i-ii--&-&:_ - --- -

10 11 -995 .015 .936 .904 .SO82

Robability of no &cision using normal approxinmhn 10 11 -9% .013 .W2 ,925 5082

-951 0 .789 "767 7 546

-004 0 0 0 .04BS

~ ~ l a c ~ O m 0 5 ~ s h o v n ~ a I n o c h c e l Z c h t ~ , m i d d t , ~ ) o w c r a m i e ~ to U, -%-U-ao1.o.a.a l a F O ~ esch a aad fample si25 8 aar OW

f r o m ( ~ - ~ ~ o f ~ d t c i r i o q P r o b ( l - 8 < a a ~ e ~ < 1 + 8 ~ ~ o b t a i n s d ~ bfmialprabzrbrhtgs .. . for ~ , + = 1 0 . n2+=tl and frnm a wmai -don for all -SiDC*

I which specifies the CWS transition pmbabiitia by (32). We a h asses the variability of e.

" " - - w o o d ertimata of m ~ ) . 1f (3.1 1) satisfies > 1, fh. A-IlagP the maximum-likelihood es&imator of Q in the r-ta- tian (3.2) with A=t-s.

I Thus the 6 c - l h U m d lrdihoafors of q, and q, for embeddable matrbs 2 arc

These &ent but not unbiased, asymptoticdy n o d estimators of Q.

.-;

188 BURTOX SISGER XXD JOEL E. COHEN

Malaria Parasit0103 Tronsirion Data: Observed Xumbers n,, of Persons in State i at First Wave of Two-Wave Panel Survey and Sute j at the Second Wave'

Age cia

.For each two-wave panel survey and age class, the entries in the 2 x 2 table arc

, where lanegative. 2=poSitive. Thus of individuals 5 to 8. 52 ptaple

s u m y 5 were infected at u w e y 6. and 68 infected at m n q 5 were at survey 6. Source: Bekeacy et a1 [q.

results p r h d y from some peapie being &seat f m a village on the survey dates Among infants, new births and a- from < 1 to 1-4 d b u t e to the variation.

We apply the Markov em-ty test of Sec 3.2 to each 2 x 2 array in Tabk 3. As indicated in Table 4, all but t h e of the 2 x 2 arrays are cornistent with some continuom-time Markov procesg The thraeexceptimrr justify no decision. One possible explanation for exce@ms is mPnnlmnent error due to the daily fluctuations in paxasite dexdy in an infected individual (see Set= 7.1). Table 5 ~ t h e ~ ~ o o d e s t i m a t e s ~ , i=l,&oftheparame-

ten in the unique time- chain that embeds e z d Markov- e d d d d d e 2 x 2 table t o w with the varianaxw- matrix corn- ptlted via the Monte Carlo method (Sec. 3.3) and the iwene of the Fish- idomation matrix (3- 13).

W e in- 1/Q, as the expected duration of a spell without pntcnt -& that is s a t e d during the I@wek period between the 3uCCCs-

sive surveys. Sbhr ly , we intap& I/& a expected duration of -1 ?.!FIl

.MALARIA INCIDENCE AND RJECOVERY U T E S

Embeddability Tests for Malaria Infection and Rscorery Dais in Table 3'

Sample sizes 6 for

Surveys Age class nr-nb -ea0-01 a=0.05 a 4 1 0 - -- - - - -- -35r---<-iF - 76 54 .m .I46 114

1 4 5 9 4 9 4 160 -113 088 5-8 74 542 144 -102 .079 9-18 181 357 106 .075 .058 19-28 478 191 .I00 .070 055 2 9 4 3 101 1 270 .080 .056 044 44+ 622 160 .I03 073 057

7-8 < I 69 71 -197 -139 -108 1.674 1 4 44 487 -185 -131 -102 1.426 5-8 0 534 .I39 .099 -077 1.239

9-18 179 322 -108 -077 -060 1.49 19-28 436 179 -103 -073 Lk57 1.328 29-43 1014 262 -081 -057 W 1.219 44+ 636 136 -110 .078 -061 1.161

~*raLmSoo.32warimpt-edhereby m a t = a 2 a n d -% ap- (3.8). bMmt CractPc 1 +8 for the 8 comes- to at least one va& of a. ~ i t h t h i s v a h c a f a ~ & e d e c i s i m m * w o u t d r e ~ ~ ~ n o de+iPion qprdiq tdm&Wq. There were no caw whne u a c r p c i .

294 BURTOS SISGER X\D JOEL E. COH EN

' Bekessy et al. 111) is .sell known to be q,/(q, iqz) , and the mean duiation of infection is I;'+ The quotieilt qlq2/(q,fqz)=rl2(ra) is therefore the inci- dence rate. Sixce a steady state is assumed. r12(ra)= rZI(=)-

6. RESULTS FROM G.\RKI BASELISE SLRVEYS - - -

-- - - Within each a s clais, recovery rates per positive iiidivid-kil t rh ib t

seasonal variation. The minimum occurs in the early part of the wet season. A maximum occurs toward the middle of the dry season. The seasonal patterns are parallel across age classes, and the iecovery rates increase with increasing age (Table 5). The only exception to this pattern is the infants (age < I), who have hi&er conversion and recovery rates than persons a s d 1-4. The increase in level of cj2 with age can be attributed to a correspond- ing increase in immuni~. As Bekessy et aI. [4] point out, the recovery rate may decline from the infants to age goup 1-4 because of the loss of maternal immunity [6] or because of superinfection, which has more of an opportunity to occur in persons at least one year old.

The conversion rates 4, are also roughly paralld across age classes. However, as indicated in Table 5, age class 5-8 has a high= rate of canversion than age class 1-4. Above age 9, these rates t&d to decrease with increasing age, as already suggested by the increasing immunity.

The recovery rates per individual in the survey, i,,, exhibit the same seasonal patterns as & within each age dass and across age c l a s s when

to a common base. For example, Table 8 reflects the same patterns under both s t a n d a r ~ o n s as cji for the same age classes in Table 5. The rates o(0) in Table 6 are not suitable for &g comgarisons across age classes in a given two-wave panel survq or across surveys for given age cIasses, bezause the rates depend on the initial distributions (5.7). -bong the irm&e initial distributions which could be used for standardiza- tioq we apply three to the data from the first two two-wave panel surveys in Table 3, for all 7 age classes (Table 7). The standardidng initial ~~ are the initial counts of the age class with the lowat popor- ticm negative, the initial counts of the age class with the hi&sst proportion negative, and the initial distribution between positive and n s ~ t i v e of the entire pophtim counting each individual equally. To comp2re time series of rates for age classes indexed by a variable a. let n,+[tr.s) be the initial counts fot survey s to s + 1 in a 9 class rz. and let p,(a) =

zz-fi+(a,s)]/Z,Z,~+(a.s). For the age c l ~ a 1 -4 2nd 29-43. we calm- lated the rates (5.6) usingp,(a) in place of P[ D.) (T~ble s)- The d a l feature of the rates *,(0) md the vnrious st.lndxdizatiolls is

W they reflect the influence of the relat~ve sizes ~ ) f the rervoirs of &ye and positive persons at the initid sune!: on the conversion and ramvery rates per individual.

MA- INCIDENCE AXD RECOkERY RATES

.Malaria Conversion rind Recovery Rates at Eqdbri- m d .I\ssuming r =O at rhe Earlier of Each Pair of Swejs*

Equilibrium Convmion ? . s ~ v s ~ Sweys _- . ,L\gt clay_ - .~ ~irk--- -- -

- - . ~~- - ~ ~A

;z,(ol r4!) -- . - 3-4 < I 1.995 1.745 4-5 6-72 24-018 2179 5-6 2673 9.960 1.842 6-7 1.699 1.619 1.741 7-8 1.098 0.739 2.079

sEntrics are 1000xratueL Rater are calculated using intensities in Table 5, %+ in T a b 4 4 (5.6). (5.7).

During the wet season, conversion rates are higher than recovery rates in every age class, becaw of the high mosquito man-biting rate then. How-

BURTOX SlSGER .ASD JOEL E COHE4

TABLE 7

Standzcked Malzria Conversion (r,3 and Reco:-c~ (rrl) Rates px 1OOO Days'

hgc Group -

S w s ) s Rates < 1 1-4 5-8 9-13 19-23 29-43 J.S+

Smdzudlzed b) the m&nims pro-x-5on uninfected - 3--; ( 0305 01747 1.8%- 0.960 &533 0.629 0.576

( 0 3.752 1.608 2.591 5 WZ 9.648 15.990 16.338 4-5 ( 0 4.386 2.987 3.164 1.752 1.208 1.385 1.697

21(0) 7.346 1.556 1.815 2.852 8.210 12.185 15.383

Standardized by the maximum proponion uninfeded 3-4 ( 0 3.023 5.568 13.601 7.159 3.977 4.693 4.295

i,(O) 0.859 0.368 0.593 1.106 2.210 5.662' 3.744 4-5 12(0) 25.087 17.087 18.097 10.019 6.913 7.922 9.709

( 0 ) 1.898 0.402 0.469 0.737 2.121 3.149 3975

Standardized by the weighted mean propomon uninfected 3 4 FI2(0) 2080 3.832 9.360 4.926 2.737 3.230 2956

( 0 ) 1.901 0.815 1313 2580 4.888 8.102 8.283 4-5 0 ) 17.508 11.925 12630 6.992 A824 5.529 a776

( 0 ) 3.893 0.824 0.%2 1.511 4.351 6.457 8152

'In surrey 3, the &urn ratio n, +/%+ = a 1 19 occumd in the age Loup 1 4 , in sumg 4 the minimum ratio nl+/n2+ 50.157 occurred in the age p u p 5-8. In

3, the maxhmn ratio nl+/n2+ ~3.888 occurred in the age goup 44+; in slwq. 4 the maxhnm ratio n, + / %+ =3-478 alro occnmd in the age grwp 44+ . The Pieightcd mean p p r t b n uninfected in nwey 3 was 0.547. and in wwey 4 was a54t

ever, when the rates are standar&ed to a minimum proportion n&e (Table 7), the conversion rates are higher than the recovery rates only in age goup 1-4 and 5-8. The older persons, who also have grater resistance to mabria and shorter veils of patent parasitemi6 then tend to have hi* recovery rates than conversion rates if they are memben of a population with a high initial percentage positive. On the other hand, with a large -& Of initidly d e c t e d persons and the same 4, i= l , t , all dasges have higher (stan*) conversion than recovery rates.

TIte steady-state event rates G(m) (Table 6) may also be interpreied as classical epidemiological incidence rates. We w3l refer to an approximate equilibrium condition as one for which F l ~ O ) ~ F , ( 0 ) ~ i , , o , ( ) , i#j- As indicated in TabIe 6, approximate equilibrium ofcurred far age goups 1-4 through 29-43 at surveys 7-8. These surveys o c c d near the end of a dry season, where superinfection was minimal This approximate equilibrium is temporary, since a new generation of mosquitos arises at the met of the

season. For the age group 1-4 there is also an approximate equilibrium

MALARIA ISCIDESCE AND RECOVERY RATES

i - ~

TABLE 8 ! Grnparisons Across Surveys of Malaria Conversion (r ,d and

Recovery (rz,) Rates per 1020 Days"

;I,(o) F2,CO)

Surveys 1-4 29-43 1-4 29-43 -. ~.~ -~

~ . *&cdanert~.29=4f-s_!1mEEXtci Ti* cEi: La-

Age c I w 1-4 and 29-43 standardized to age class 29-43 3 4 5.026 4.236 0.508 5.W 4-5 15.7% 7.323 0508 3.976 5-6 12565 6.390 a479 4-68 1

.AS d d k d in tbc t e a pi (age dart 1-4)=0-11Q, PI (agt 2943)-0.718

at surveys 5-6, in the latter part of a wet season. This apparent equilibrium may d t from the enomtow imbalance in the ratio of positive to negative pasons, 484/50, at survey 5. A high ntau-biting rate might lead to enough oaayersiom among the 50 negative individuals to balance the recoveries amrmgtkem4~lypositivepersons.

7.1. MEASUREMENT ERROR DUE TO DAILY FLUCNAlTONS IN PARASITE DENSITY

I An individual with patent P. fa.k@mm parasitemia exhibits large daily . . osdamm in parasite density. A blood smear taken at a time of low density in that individual could be as negative, The distribu- tion of phases of these oscillations in a population of infected persons was not recorded in the Garki surveys. Thus we have no direct measmernent d the -don of infected persons who would be i n d y classified by blood smeass.

Despite this lack of numerical information, we describe qualitatively the influeace of this misclasiftcation error on embidability tests, estimates of the parameterr % i= 1,Zand the event rates r;i(s)-

Consider a m d e l of -blood-smear measurement in a two-wave panel study where (w(O),w(A)) represent the actual infection states a t times 0 and

I

298 BURTON SISGER XXD JOEL E. COHM MALARIA IXCIDE?CE XL'D RECOVERY RATES 299

A and (Y(0). Y ( 1 ) ) represent the observed states of infection. For k=O. 1, with x=tracePm- 1, p,, =p*,, +(. and i=i, +& [By (7.6), for large sam- suppose misdasification errors satisfy ples, 6 % ~ > O and il, >O.J Thus, ro within sampIinz iariability, the ssti-

mated expected durations in a c h state implied b) Q* ire longer than those P r o b ( Y ( k l ) = l [ , v ( k 1 ) = l ) = l . ( 7 - 1 ) , implied by Q, i.e. Prob( Y( kl) = lLw(kl) =2)= a. O < r (1, (7-2) - -. -

- -

~rijb(w(kl)=21 ~ ( k A ) = 2 ) = 1. T<T- 1 1 4, 4 i

(7.9)

Suppose also that rniscIassification does not alter the underlying dynamics of infection, so that

The conditional probabilities p6 are assumed only to be associated with some twostate continuous-time stochastic process.-A routine dadation shows that the conditional probabilities for the observed process, Prob(Y(A) = il Y(0) 3 i) =P+,~, obey

Williams and Mallows [29] study lnan elaborate models of erron in the &temktion of p, due to differential nonresponse in two-wave, two-state panels surveys.

The importance o f (7.6) for Markov embeddability tests is that the trace of an &mat.& transition matrix ~ ( o , A ) , based on independent observa- tions (Y,(CI), x(h)), I < i < N, can lead to no decision regard- or rejection of, the null hypothesis H,:trau=P>l, when the unobserplabk matrix f i 0 , ~ ) s&Kes t r a c e ~ ( 4 ~ ) > 1 + 4 in the f d test of Sec 3. The d i a p nal entries in the tbrce matrices ia Tabie 3 leading to no decision might very d -te p, because some infected penons with law parasite density at the sumey times were misd&ed as d e c t e d

If b 0 , A ) is accepted as erdddable in a homogeneous Markov c h k then the mwimum-Wood estimator 6, or ( I / Q ~ O ~ ~ ( O , A ) is dated to 6, up to tennsoforder2 in theeshatedmeasurementermri?, via

logx E'(l-@+;,.) + - 1 [ - 1

The error in 0 in large samples is of smaller order of magnitude than (7.8). From (7.9), the estimated equilibrium event rates satisfy, for i+j.

However, there is no analogous inequality between the nonequilibrium rates ~ ( s ) and rLi,(s) valid for all s < + m. I n d e sgn[r,,(s) - r+,(s)] depends on the magnitude of $=is -j+,, and on the proportions of individuals in each state at the first wave of observation.

72. MODEL MISSPECZFIIGQ~ON

An important source o f heterogeneity among individuals in Garki is differential immuniw, even within the age classes used in the WHO malaria surveys. Homogeneous Madcov chains treat these heteromeous popula- tions as though they were homogeneous

To understand the biases which can arise from treating heterogeneous populations as if they were homoogeous, we describe some mixtures of Markov chains which have { P : trace P > 1) as their reachable set of condi- tional probabilities and cornpart expected event rates uncier these modeis with the same quaatities calculated under a hamogmeous Markov modd

Let t=O be the time of initiation of a mixtun of wntim~~us-time hffarlrov chains Then fqr every intensity-matrix-valued function a t ) , t > 0, let dQ? i= 1.2, be the probability distribution of type-Q individuals be- tween states at the initial survey, t=O. Let pf?) be transition probabilities arising from (3.la,b) with intensity-matrix function Q(t), t > 0. The transi- tian pmbddities E,(O,t) for a mixture of such Markov chains are defined by

im BURTON SINGER AND JOEL E COHEY

where do is a probab~lity msasurz on the spacz Q of measurable intensity- matrix-~alued funciions Q(1). t 20. We rewiite (7.11) in term of the measures

The two subclasses below of this family of hetemgeneolls population models have the same reachable set of conditional probabilities as would arise if do were a point mass at one intensity-matrb function. These mixtures are indistinguishable from the continuow+th.m Markov models using only twewave panel data:

Class (i) contains those mixtures for which 4, = &- Here traceq0, t) > 1 for every probability measure dp, on Q. Class (ii) contains those intensity-matrix-valued functions Q(t) such that

p$QXO,t)>; for t>O. Here m c e q ~ , t ) > l for every pair of probarn ( d h , w which are nonzero d y at such a t ) .

Arbitrary mixtures (7.13) of stochastic matrices P with trace P > 1 need not satisfy traceF > 1 . For example, let

With the measares a$A,(Q,)=s,=l -+,(a) and 4XQJ=s2=1 - 4 4 Q 3 , O<s,<l, i=1,2, we have traceqo,A)<l whenever s t - s ,<- ; . This ex- ample p'ovides +er possible expiadon of the matrices in Table 3 which led to no decision in the Markov embeddabilitg test, &j, that these data arose from mixtures of homopeous M* chains

To illuurate the biases in event rates due to model consider the homo,-us Markov chains with intensity matrices

For each h, the equilibrium event rates are q,(m)=A/Z. Let dp,Q= &(A) be any probability measure on [O, i- m). Then

f MALARIA INCIDESCE :\XD RECOVERY %.A iES 30 1 L

: and for the mixture. :he equilibrium ctvenr r a t s are

an estimated transition matrix actually ~iises from a mixrure of the form (7.15) but. following an embsddability :-st. ?he hfarkov models Snssd on

' (7.14) are used to calculate <,(x). [hen :he actual event rate fy( x) will

- . always be ovnestimatod - Tins -- k-duc ~ -- - ~ ro.~.qo&l ~~~ rnjjrpqcjf ic.tion F6Il6us from Jensen's inequality. If p$*=(eQ-S),; and Q is given by (7.14). rhsn

I

1 For general homogeneous chains the went-rate biases depend strondy on the mixing measures d b In the Garki study. if heterogeneity arises from diFfeaw.tces in immunity, these mixing measures are estimable only from serological data [8].

I % REVIEW AND CONCLUSlOSS The Garki project is the f i i longitudinal field survey of malaria with

enwgh parasitological measurements to make possible direct estimates of age- aad seasan-spcdic incidence rates of conversion and recovery. Formal e d d w t y tests showed that for each of seven age classes a time series of wo-wave panel surveys were indistinguishable from samples of time- - h4arkov chains The two estimated parameters of the inten- Gty matrix associated with each two-wave pawl were interpreted as cower- Gon and recovery rates per person at risk of conversion or recovery. respectively. The conversion and recovery rates per person surveyed were Wmputed from the intensitier

Thus, for each age class, the two-state (positivqnegative) parasitemia histories (X( t , w), 2A< t < 7 4 A- 10 weeks), which evolve during the time span of the basdine surveys in Table 3, are modeled by a time series of twrostate continuowtime Markov chains (&(t, w), kh G t < ( k + 1)A, 2 < k

, < 7 ) . Here q ( t , -) is a Markov chain that represents the parasitological h h i i e r only during the 10-week interval between sutiels k and k + 1.

I QIdLcuaanof- . . in Sec. 3 serves to clarify the statement of k k e s q et a1 [q: .'if traceP < 1 it could be suspected that the model did not fit well: either the process was not Markovian or the parameters were not constant between subsequent observations" If traceP< 1 , then the mdexiying colltirmau-time gochasdc process must be non-Markovian. If the widedying process is inhomoo~neous Markov, with he-varying inten- sities, then trace P > 1 . Thus the second proposed explanation is incorrect if the rumconstant parameters occur in a Markov chain.

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