S141 A Theretical Framewrk fr Evaluating the Sensitivity f Surveillance fr Detecting Wild Plivirus: II. Factrs Affecting Detectin Sensitivity in a Ppulatin with Circulating Wild Plivirus Hward E. Gary, Jr., Ray Sers, Mark A. Pallansch Respiratry Enteric Viruses Branch, Divisin f Viral Rickettsial Diseases, Natinal Center fr Infectius Diseases, Centers fr Disease Cntrl Preventin, Atlanta, Gergia; Exped Prgramme n Immunizatin. Reginal Office fr the Western Pacific. Wrld Health Organizatin, Manila. Philippines A basic framewrk fr describing sensitivity f surveillance t detect plivirus is extended frm individuals t general ppulatins (ppulatin sensitivity). Using the mathematical frmulatins fr ppulatin sensitivity, the theretical behavir f surveillancesensitivity under varius cnditins is analyzed. As a regin nears the eliminatin f plivirus, ppulatin sensitivity falls t a value lwer than the case-t-infectin rati, regardless f the system prficiency. Als, a secnd stl specimen makes a substantial cntributin t ppulatin sensitivity nly in regins with relatively lw specimen sensitivity then nly ver a narrw range f ppulatin infectin incidence. Estimates f the mean numbers f infected uninfected acute flaccid paralysis cases investigated in a seasn are derived. These may serve as additinal indicatrs f system peratin but require the cllectin f 2 specimens per case. In a cmpanin reprt [1], we delineate five factrs that cntribute t the verall sensitivity fan acute flaccid paralysis (AFP)-based surveillance system fr detecting wild plivirus: labratry sensitivity, cllectin efficiency, specimen sensitivity, persn sensitivity, ppulatin sensitivity. In that reprt, we fcus n specimen sensitivity, which pertains t a single specimen, persn sensitivity, which pertains t a single persn. Because the ultimate purpse fan AFP-based surveillance system is t detect wild plivirus circulating within a ppulatin, the effects f prtcl changes r plicy decisins n surveillance system sensitivityshuldbe cnsideredin terms f ppulatin sensitivity, the prbability that a ppulatin cntaining at least 1 infected persn will be crrectly identified. Therefre, in this reprt, we shift the fcus frm the detectin f plivirus in a single individual t the detectin fthe virus in a general ppulatin. As applicatins f this framewrk, we analyze the cntributin fa secnd stl specimen t ppulatin sensitivity derive estimatrs finfected uninfected AFP incidences. These rates can be useful as additinal indicatrs fthe perating efficiency fan AFP surveillance system. Methds Definitins ntatin. In ur cmpanin reprt [1], we present a general definitin fr ppulatin sensitivity (w): the prbability f detecting at least 1 infected persn in a ppulatin that cntains at least 1 infected persn. Ppulatin sensitivity depends Reprints r crrespndence: Dr. Hward E. Gary, Jr., Divisin f Viral Rickettsial Diseases, MS-G 17, Centers fr Disease Cntrl Preventin, 1600 Cliftn Rd. NE, Atlanta, GA 30333. The Jurnal finfectius Diseases 1997; 175(Suppll):SI41-5 1997 by The University f Chicag. All rights reserved. 0022-1899/97/7581-0024$01.00 n tw factrs that are nt relevant t specimen r persn sensitivity. One is the prbability that an infected persn will develp AFP, the case-t-infectin rati (TJ). The ther is the prbability that an AFP case will be detected investigated, the case identificatin sensitivity (p). Fr the theretical develpment presentatin, we will assume that the specimen sensitivity is the same fr each specimen that the number f specimens cllected frm each case is independent f the persn's infectin status. If the number f infected persns in the ppulatin is knwn t be P, ppulatin sensitivity is calculated as I - (l - TJP(h)p, where Ok is persn sensitivity fr k specimens. The actual number f infected persns in the ppulatin will vary frm year t year cannt be knwn with certainty. Therefre, t fully characterize ppulatin sensitivity, it is necessary t accunt fr this uncertainty by specifying a prbability distributin fr P. Because P is unbservable, we have n data with which t empirically determine the prbability distributin. In the absence f data, we have chsen t use the Pissn distributin fr several reasns. First, the Pissn distributin characterizes a number f sampling situatins in which a relatively large number f persns culd becme infected, but the prbability f infectin is relatively small. Secnd, the Pissn is a singleparameter distributin family, which simplifies many f the theretical develpments. Finally, under the cnditins cnsidered here, the Pissn prvides a very gd apprximatin t anther reasnable chice, the negative binmial distributin. We will dente the expected number f persns infected per year as X-. Using the prperties f the Pissn distributin, the expected number f infected persns wh are identified thrugh AFP surveillance is TJpX-. Ppulatin sensitivity is the prbability that at least I infected, investigated persn will have at least I virus-psitive stl specimen. This prbability is where k is the number f specimens cllected frm each persn.
S142 Gary et al. JID 1997;175 (Supp11) We als evaluated the cntributin f the secnd specimen t ppulatin sensitivity (w = Wz - WI)' Estimating expected number finfected AFP cases, expected number funinfected AFP cases, specimen sensitivity amng cases with 2 specimens. Ideally, a surveillance system shuld prvide a means fr cunting the number f infected persns in the ppulatin. Mst persns infected with plivirus d nt develp AFP, hwever, cannt be identified by the system. Furthermre, nt all infected persns wh develp AFP will be identified. Using available surveillance data, it is nt pssible t estimate the number finfected persns missed by the system. Fr example, a lw ppulatin infectin rate with high case identificatin sensitivity may lead t the same number finfected AFP cases identified by the system as a high infectin rate with a lw case identificatin sensitivity. In the absence f accurate data n the ttal number infected, it is nt pssible t estimate the expected number f infected persns in a ppulatin (A.). Rather than fcus n the entire ppulatin, hwever, we can restrict ur fcus nly t thse AFP cases that are identified. Using surveillance data, it is pssible t estimate the expected number f infected persns amng the AFP cases investigated the baseline expected number funinfectedpersns amng AFP cases. We will dente these expected numbers as A.A A.B' respectively. Unlike the idealized situatin cnsidered in the theretical framewrk, AFP patients actually investigated can prvide 0, 1, r 2 specimens. It therefre is cnvenient t dente the respective expected numbers as A.Ak A.Bk> where k is the number fspecimens prvided. Fr the AFP patients wh prvide 2 specimens, we can specify the prbabilities that 0, 1, r bth f the specimens are psitive. Using these prbabilities, we derived estimatrs fr specimen sensitivity (y), expected number f infected persns amng AFP cases (A.AZ), the expected baseline number f uninfected persns amng AFP cases (A.Bz). Fllwing the derivatin presented in the Appendix, the maximum likelihd estimatrs (MLEs) fr A.AZ, A.Bz, y are l A. AZ - 4N ' I 1 2N II y=--, y where N is the number f case-patients with bth specimens negative, N IO is the number whse first specimen is psitive but the secnd negative, s n. Variance estimatrs fr these values are presented in the Appendix. The estimatr fr specimen sensitivity,, is identical t that presented in ur cmpanin reprt [1]. Frm the prperties fthe Pissn distributin, the prprtin f investigated AFP patients with 2 specimens wh are infected is A.AZ/(A.AZ + A.Bz) = </>, the prevalence f infectin described in ur cmpanin reprt [1]. By substituting x. AZ BZ int this equatin fr d», we get the MLE l </>=-, 4NIITz where T: is the number f AFP patients prviding tw stl specimens. This is the same estimatr that is derived in ur cmpanin reprt [1] using a slightly different prbability frmulatin. The estimates calculated frm these frmulas can als be used t estimate the expected number funinfected persns amng AFP cases wh prvided 1 (T j ) r n (T) specimens. If we assume that the prevalence finfectin is independent f the number fspecimens cllected, then the expected ttal number f infected persns amng AFP cases is estimated by x.a = x.az + M I + (1 - 'Ye)(f> (T - M) + (1 - J.)T. (1 - 'Ye)4> + (1 - (f» 1 1 'f" 0, where M I is the number f persns whse nly specimen was psitive. Given an estimate f the expected number f infected persns amng AFP cases, the expected number f uninfected persns can be estimated by x. B = T - A' The estimates f the expected numbers can be cnverted t incidences, which are mre cmmnly reprted, by dividing by the ppulatin size f the regin. We calculated estimates f A.AI. A.AZ, A.B!' A.Bz using labratry surveillance data frm the three labratry regins. Data. We applied these results t the labratry data frm tw regins fr 1994, labeled here as A B, t data previusly published by the Pan American Health Organizatin frm the Americas fr 1987-1991 [2] as described in ur cmpanin reprt [1]. Results Cntributin fthe secnd specimen t AFPppulatin sensitivity. In figure la, we present the relatinship between the expected number f infected persns (x..) ppulatin sensitivity based n 1 (WI) 2 (wz) stl specimens fr tw levels f specimen sensitivity (v). Fr this figure, we used a case-t-infectin rati (TJ) f 1/200 a case identificatin sensitivity (p) f 0.80. As the expected number infected nears zer, bth WI Wz apprach the prduct f the case-tinfectin rati (TJ), AFP identificatin sensitivity (p), specimensensitivity (y). Because TJ is small, this prductis verynear zer. Ppulatin sensitivity increases rapidly as the expected number f infected cases increases fr all but the lwest levels fspecimen sensitivity. Fr example, WI is >0.85 fr all specimen sensitivities f 0.60 if the expected number f infected cases in the ppulatin is 790. This crrespnds t an infectin incidence f 7.9/100,000 in a ppulatin f 10,000,000. In figure 1B we present a mre detailed view fthe ptential increases in ppulatin sensitivity frm cllecting testing a secnd stl specimen, shwing the relatinship between the sensitivity cntributin f a secnd specimen (w) the expected number f infected persns (X) fr fur levels f specimen sensitivity (y). Fr each level fspecimen sensitivity, the cntributin fthe secnd specimen t ppulatin sensitivity first reaches a maximum as the expected number finfected persns increases, then declines as A. increases further.. With higher specimen sensitivities, the maximum pssible cntributin f the secnd specimen is lwer, the peak ccurs at a lwer number f expected infectins, appreciable increase
JID 1997;175 (Suppll) Wild Plivirus Surveillance Sensitivity S143 in ppulatin sensitivity is pssible within a narrwer range f expected ppulatin infectins. We als examined the effect f varying p, the case identificatin sensitivity (results nt shwn). With higher values f p, the ptimal range fr X. becmes narrwer the maximum sensitivity increase ccurs at lwer values f x.. The ppsite pattern is seen as values f p are made smaller. The maximum pssible sensitivity increase is nt affected by changing p. Estimates fthe expected numbers finfected uninfected persns amng AFP cases. We applied the estimatrs fr the expected numbers f infected persns amng AFP cases with 2 specimens (A.A2) the baseline expected number f uninfected persns amng AFP cases with 2 specimens (A.B2)t data frm regins A B. We als used the estimated prevalence t estimate the mean number f infected uninfected persns wh prvided 1 specimen amng AFP cases. Because data n the number f AFP patients with n specimens (T) were nt available t us fr any f the three regins, we culd nt estimate the expected ttal number f infected uninfected persns. Furthermre, because data n the number f persns with 2 negative specimens were nt reprted fr the Regin A. B. 0.8. 0.6 CIl UJ c :g 0.4 'S Q, Q. 0.2 Q) 111 III e.e 0.15. 0.1 Q) UJ c 0.05 'S Q, Q. Specimen Sensitivity 0.80 0.40 OL..------'----------'---------' 0.2 r------------------------, 1,000 2,000 Expected Number Infected 3,000 Figure 1. A, Relatinship between ppulatin sensitivity using 1 r 2 stl specimens expected n. f infected persns in ppulatin fr specimen sensitivities f 0.80 0.40. Fr each specimen sensitivity, upper curve represents ppulatin sensitivity using 2 stl specimens; lwer curve represents sensitivity using 1 specimen. All sensitivity calculatins used case-t-infectin rati f 1/200 a case identificatin sensitivity f 0.80. B, Relatinship between increase in ppulatin sensitivity expected number f infected persns in ppulatin fr selected values f specimen sensitivity. Table 1. Estimated numbers f infected uninfected persns prviding ne tw stl specimens amng AFP cases. Regin f the N. f specimens, parameter Regin A Regin B Americas 2 n 32 90 76 Expected n. infected (SE) 28.9 (2.6) 32.8 (8.6) 77.2 (NA) Expected n. uninfected (SE) 3.1 (1.1) 57.2 (1.5) NA n 9 33 NA Expected n. infected 7.8 (NA) 11.3 (NA) NA Expected n. uninfected 1.2 (NA) 21.7 (NA) NA NOTE. NA, nt available. f the Americas, we culd nt estimate the uninfected AFP baseline fr this regin. These estimates are summarized in table 1. Estimates f specimen sensitivity prevalence are presented in the cmpanin reprt [1]. In regin A, the expected numbers f infected persns were estimated as 28.9 wh prvided 2 specimens 7.8 wh prvided 1. The estimated expected numbers funinfected persns were 3.1 with 2 specimens 1.2 with 1. In regin B, the mean numbers f infected persns were 32.8 11.3 fr persns prviding 2 1 specimen, respectively. The crrespnding expected numbers funinfected persns were 57.2 21.7. In the Regin fthe Americas, the expected number f infected persns amng AFP cases fr the 4-year perid was 77.2, fr an average f 19.3 infected persns per year. Discussin In ur cmpanin reprt [1], we shwed that sme aspects fthe sensitivity fafp surveillance fr identifying an infected persn can be examined within a general theretical framewrk. In this reprt, we extend the theretical framewrk develped fr describing specimen persn sensitivity t include sensitivity fr identifying virus circulating within a general ppulatin. One practical use f this theretical framewrk is the develpment f estimatrs fr specimen sensitivity f the expected numbers f infected uninfected AFP cases. The utility f estimating specimen sensitivity is discussed in ur cmpanin paper [1]. The estimated AFP rates develped in this paper can be useful indicatrs fthe surveillance system efficacy. Currently, the incidence fafp identified by the surveillance system is used as ne indicatr f system prficiency. Overall AFP incidence, hwever, includes bth infected uninfected persns amng AFP cases. Therefre, changes in incidence ver time r differences in incidence acrss gegraphic areas culd reflect differences in system efficiency, differences in infectin rates, r bth. Using the estimatrs we have develped, it is pssible t mnitr rates f infected uninfected
SI44 Gary et al. 1101997;175 (Suppll) persns separately amng AFP cases. In the absence funusual events, the incidence f uninfected patients with AFP shuld remain relatively cnstant frm year t year be mre similar acrss regins, making this rate a useful indicatr f case identificatin efficiency cnsistency. Our estimates have tw imprtant limitatins. First, they can be calculated nly if 2 stl specimens are cllected frm each patient. Secnd, the estimates becme less stable as the number f AFP patients with psitive specimens becmes small. In the extreme, if n AFP patient has a psitive specimen, neither specimen sensitivity, prevalence, nr infected AFP incidence can be estimated. This suggests that specimen sensitivity mnitring shuld be initiated well befre a regin nears the pint f plivirus eliminatin. A pssible alternative is t calculate specimen sensitivity using vaccine virus islatin data. Because f grwth pattern shedding differences between wild vaccine virus, hwever, we d nt knw hw similar these specimen sensitivity estimates wuld be t thse derived frm wild virus islatin data. The inability t estimate infected AFP incidence as a regin nears virus eliminatin is less prblematic. As the rate f infectin in a ppulatin becmes lw, the cntributin finfected cases t the verall AFP incidence als becmes small, making simple AFP incidence an adequate surrgate fr uninfected AFP incidence. Because the purpse f AFP surveillance is t find wild plivirus within a ppulatin, cnsideratins f ppulatin sensitivity are critical fr system evaluatin plicy r peratinal decisins. Our theretical framewrk prvides a systematic means fr evaluating many f the factrs that can affect ppulatin sensitivity. One issue that has been raised with respect t AFP surveillance is the cntributin fa secnd stl specimen t system sensitivity [2]. In ur cmpanin reprt [1], we addressed the cntributin a secnd specimen makes t persn sensitivity. Here, we extended this analysis t the ptential cntributin f a secnd stl specimen t ppulatin sensitivity. We fund that any substantial cntributin f a secnd specimen is limited t regins with relatively lw specimen sensitivity then nly within a relatively narrw range finfectin incidence rates. Because all regins are prgressing tward plivirus eliminatin, the range f infectin incidence in which a secnd specimen might cntribute t sensitivity als crrespnds t a relatively narrw windw f time, pssibly as little as ne seasn. We als fund that ppulatin sensitivity decreases t near zer as a regin nears the eliminatin f plivirus at all levels f specimen sensitivity, regardless f the number f specimens cllected. The limit f sensitivity is determined by the prduct f persn sensitivity, case identificatin sensitivity, the case-t-infectin rati. The smallest cmpnent f this limit by far is the case-t-infectin rati. Althugh estimates f this rati vary, they are typically in the range f 0.1%-2% f all infectins, depending n a number ffactrs (reviewed in [3]). The case-t-infectin rati may be even lwer than expected in a heavily vaccinatedppulatin. Ppulatin sensitivity, then, will becme very lw as the number f infected persns in the ppulatin becmes small, even if all cmpnents f the AFP surveillance system are 100% sensitive. In summary, as a regin appraches the eliminatin f circulating plivirus, AFP surveillance becmes a very insensitive methd fr identifying the virus regardless fthe prficiency fthe labratry r field peratins regardless f the number f stl specimens cllected. The insensitivity f AFP surveillance arund the time fplivirus eliminatin suggests that the ptential cntributin f methds that supplement AFP surveillance shuld be cnsidered systematically analyzed. Acknwledgments We acknwledge the significant cntributin that all members f the Glbal Plivirus Labratry Netwrk have made t the gal f pli eradicatin the results that have cntributed t this paper thank Walter R. Dwdle Olen M. Kew fr significant discussin cmments n the ideas presented. References I. Gary HE Jr, Sers R, Pallansch MA. A theretical framewrk fr evaluating the sensitivity f surveillance fr detecting wild plivirus: I. Factrs affecting detectin sensitivity in a persn with acute flaccid paralysis. J Infect Dis 1997; 175(suppl 1):S135-40 2. Silveira CM, de Quadrs CA, Hersh BS, Ngueira AC. Pli diagnsis: ne r tw samples? EPI Newslett 1995; 17:1-2. 3. Mrens DM, Pallansch MA, Mre M. Pliviruses ther enterviruses. In: Belshe RB, ed. Textbk fhuman virlgy. St. Luis: Msby Year Bk, 1991:427-97. Appendix Ntatin l' Specimen sensitivityassuming equal sensitivitiesfr the first secnd specimens Ok Persn sensitivity using k specimens Nij Number f AFP cases with result i fr the firststl specimen result j fr the secnd, where i, j = 0 if negative 1 if psitive A. Expected number f infected persns in a general ppulatin in a seasn A.Ak Expected number f infected AFP patients with k specimens investigated in a seasn A.Bk Expected number f uninfected AFP patients with k specimens investigated in a seasn c/j Prprtin f AFP patients wh are infected (prevalence) p Prbability that an AFP case will be investigated(case identificatin sensitivity) 7] Prbability that an infected persn will develp AFP Wk Ppulatin sensitivity using k specimens assuming equal specimen sensitivities T Ttal number f AFP cases investigated in a time perid y Ttal number f psitive specimens P Ttal number f infected persns in a ppulatin
JID 1997; 175 (Suppl 1) Wild Plivirus Surveillance Sensitivity S145 Ppulatin Sensitivity Fr a given seasn, the prprtin f infected persns wh develp AFP are then identified as having AFP is the prduct fthe respective prbabilities fr these events: TIp. The prprtin f these wh are then fund t be infected is persn sensitivity, Ok. Therefre, if the ttal number f infected persns fllws a Pissn distributin, the number wh are ultimately identified by the system als fllws a Pissn distributin but with mean TlPOkA. By definitin, ppulatin sensitivity, Wk, is the prbability f detecting at least 1 infected persn given that at least 1 member fthe ppulatin is infected. This cnditinal prbability, then, is Wk=---- Maximum Likelihd Estimatrs (MLEs) f A2 A B 2 MLEs f AAZ ABZ can be fund by first delineating the prbabilities fr N,N IO + N OI = Ns. Nll. Because the number f infected persns prviding 2 specimens amng AFP cases fllws a Pissn distributin with mean AAZ, it fllws that N I NIl als fllw Pissn distributins with means AAZ weighted by the prbabilities f 1 r 2 psitive specimens, respectively: N I Pissn[2y(1 - Y)AAZ] Nss Pissn(YZAAZ)' In additin, the number funinfected persns amng AFP cases fllws a Pissn distributin with mean ABZ. Because N is the sum f the uninfected persns infected persns with n psitive specimens amng AFP cases, N als fllws a Pissn distributin: N Pissn[(1 - y)zaaz + ABZ]' The lg-likelihd functin fr y, AAZ, ABZ is the lg f the prduct f these distributins:!ley, AAZ, ABZ) = Nln[(1 - y)zaaz + ABZ] - [(1 - y)zaaz + ABZ] + N lln[2y(1 - Y)AAZ] - 2y(1 - Y)AAZ + Nllln(yZAAz) + c. Taking partial derivatives with respect t y, AAZ, ABZ gives the fllwing system fequatins: B.f = N I + 2Nll - 2(N I + Nll)y 2N(1 - Y)AAZ By y(1 - y) (1 - yfaaz + ABl' Slving this system fr y, AAZ, ABZ gives the fllwing estimatrs: 2NII y A y=-. T find the variance-cvariance matrix fr the estimatrs, we first expressed the Pissn means fr N, N I, NIl as functins f AAZ, ABl' y: J-t(AAZ, ABZ, y) = (1 - y)zaaz + ABZ, J-t1(AAl, ABl' y) = 2y(1 - y)aaz, J-tz(AAZ, ABb y) = yzaaz. The MLEs fr the J-tiS are N, N b Nu, respectively, with a diagnal variance-cvariance matrix with entries r = {J-ti}, which are expressed as a functin f AAZ, ABZ, y. We then invert the functins J-ti frm the matrix ffirst partial derivatives with respect t AAZ, AB2' y, D. The variance-cvariance matrix fr x'az, x'bz, y, then, is given by the matrix prduct A = DrD' [(1 - ')')2 + ]AA2 (1 - ')')2AA2 (1 - ')'iaa2 _ (1 - ')')2 (1 - ')')2 (1 - ')')(2 - ')') ')' ')' 2 The MLEs fr the variances cvariances can be btained by replacing the parameters in this matrix by their respective MLEs. (1 - ')')2 AA2 + AB2 _ (1 - ')' ')'i (1 - ')')2 ')'