Incorporating Time Dose Response Into Shigella flexneri and Shigella sonnei Outbreak Models

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1 Peer Reviewe Incorporating Time Dose Response Into Shigella flexneri an Shigella sonnei Outbreak Moels BIDYA PRASAD AND CHARLES N. HAAS Department of Civil, Architectural, an Environmental Engineering, Drexel University, Philaelphia, Pa. Experimental time-to-infection ata are a useful but often uneruse material for examining the mechanics of in vivo pathogen growth. The authors incorporate a time ose response (TDR) equation into a Shigella flexneri an Shigella sonnei outbreak moel. Dose response an TDR moels were generate for S. flexneri exposure to monkeys. The TDR equation that best fit the monkey ata the beta-poisson with exponential epenency moel was chosen for incorporation into the outbreak moel. The outbreak moel is a probability moel that convolutes an assume incubation istribution of the infectious agent with an exposure istribution. Since the beta-poisson with exponential epenency moels the time-to-infection ensity istribution, it is entere as the incubation istribution, along with Weibull, lognormal, gamma, an uniform functions. Each of these five incubation functions is convolute with Weibull, lognormal, gamma, an uniform functions (which serve as exposure istributions) for a total of moels. The timeepenent probability istribution yiele best-fit results for the S. sonnei outbreak scenario. Keywors: ose response, epiemiology, outbreak moel, Shigella flexneri, Shigella sonnei, time ose response Shigellosis, or bacillary ysentery, is an acute enteric infection sprea irectly or inirectly through the fecal oral route by the genus Shigella (Scallan et al., Heymann ). Shigella is a genus of gram-negative, nonmotile bacilli belonging to the family Enterobacteriaceae (WHO, Jin et al. ), which inclues four species, of which S. flexneri (serogroup B) an S. sonnei (serogroup D) are the most wiesprea (Bhunia, O Brien & Holmes 97). S. sonnei infections occur most frequently in inustrialize countries an account for about % of Shigella infections in the Unite States (Nygren et al. 3, Pon ). S. flexneri is the secon most common species in the nation, accounting for.% of Shigella isolates submitte to the Centers for Disease Control an Prevention between 99 an (Gupta et al. ). Shigella invaes the colonic epithelium, leaing to micro-ulcers an inflammation. After an incubation perio of one to four ays, ill patients typically experience loose, blooy stools containing organisms per gram, often accompanie with mucus (Gaurav et al. 3, Bhunia ), while other patients have iarrhea without visible bloo or mucus (Shrotriya, Heymann ). Abominal cramps, tenesmus (unprouctive, painful straining), fever, an anorexia are also common in infections (Shrotriya ). Although most patients recover within seven to ays, serious complications may arise, incluing sepsis, hyponatremia, hypoglycemia, seizures an encephalopathy, hemolytic uremic synrome, pneumonia, malnutrition, an eath (Shrotriya, Michael 99). Shigella spp. can be sprea through a common source, such as contaminate foo or water, or through irect contact with an infecte person, usually through the fecal oral route. Shigellosis is often enemic in areas with inaequate sewage isposal or where ineffectively treate water supplies are common, such as eveloping nations (Pon ). Once excrete, the organism is very sensitive to environmental conitions an ies rapily, especially when rie or expose to irect sunlight (Gaurav et al. 3). The number of probable annual cases in the Unite States is,, of which approximately, are laboratory-confirme (Pon ). Since mil an asymptomatic cases often remain unreporte, the actual number of infections is preicte to be times greater than the probable case estimate (Pon ). Between 97 an, 33 waterborne outbreaks were recore in the Unite States, an the secon most common E PRASAD & HAAS DECEMBER 7 9: JOURNAL AWWA 7 American Water Works Association

2 etiologic agent trace to these outbreaks was Shigella, which was foun to be responsible for cases (Craun et al. ). Of the total non-legionella bacterial rinking water outbreaks, Shigella was responsible for.9% (n = ) of outbreaks, an of the total, non-legionella bacterial cases cause by rinking water,.9% (n = 9,77) were cause by Shigella (Craun et al. ). Aitionally, more than million cases of shigellosis are globally reporte each year, resulting in. million eaths (Pon ). Of these infections, 99% occur in eveloping countries, with % of eaths occurring among chilren younger than five years of age (WHO ). Although Shigella has been responsible for numerous outbreaks in the past ecae, few outbreak moels have been generate for the pathogen. A novel metho of moeling outbreaks woul be to assume an incubation-time istribution that closely moels the actual time-to-response for Shigella, which takes into account the time span from ingestion to clinical illness. The time ose response (TDR) moeling methoology, which quantifies a relationship between pathogen kinetics an host response through the time-to-response, provies the means by which ata from osing stuies can be incorporate into an outbreak moel (Huang & Haas, 9; Huang et al. 9). Infection is a time-epenent process because pathogen populations in the host vary with time; an investigation of in vivo kinetic effects on outbreaks coul therefore prove valuable. The objectives of the current stuy were to generate an iscuss the results stemming from a TDR moel for Shigella an to incorporate this moel into the incubation istribution of an outbreak moel. The ensity functions that fit the moel best an parameter values were also investigate. DATA AND METHODS This stuy was broken into three parts: a literature search of TDR an outbreak ata sets was performe; TDR moels were generate using stuies from the literature search; an outbreak ata sets from the initial literature search were fit using a selecte TDR equation, with non-time-epenent moels for comparison. This outbreak ata set was moele by convoluting an assume exposure ensity function with an incubation ensity function. The exposure function efines the infectivity of the pathogen by assuming how infection travels through a susceptible population. The Weibull, gamma, lognormal, an uniform exposure ensity functions were explore. The incubation ensity functions use here were the best-fit TDR moel (in the form of a ensity function) an the non-time-epenent Weibull, gamma, lognormal, an uniform functions. This work buils on the TDR moels evelope by Huang an colleagues (, 9) an the outbreak moel evelope by Gupta (999). Data sources. A literature search of TDR ata sets was conucte first to ientify an collect appropriate caniate ata for eveloping a Shigella TDR moel. The following inclusion criteria were applie to the selection of an appropriate TDR ata set: a clear escription of osing methos, a reporte moe of exposure, a reporte osing for each subject, a reporte number of subjects experiencing averse responses on each respective ay, state criteria use to efine a positive en point, an at least one ose with an intermeiate response between an (Haas et al. ). The literature was then searche to fin an appropriate outbreak ata set that coul be fit to a moel. The inclusion criteria for choosing an appropriate outbreak for the Shigella outbreak moel inclue the following: The size of population susceptible to infection was known or coul be estimate. The members of this susceptible population who became symptomatically ill were confirme cases. The beginning an en of the pathogen contamination were known. Incubation ata for the pathogen were available. The outbreak occurre over a efine perio. The exposure perio was short compare with the outbreak. Each confirme case stemme from the original source, an there was no seconary sprea of isease. All outbreak cases that were ientifie ha the same case efinition. Dose response methos. Cochran Armitage tests of tren were use to etermine whether there were associations between increasing osages an averse responses an an increase in illnesses in the time span following osing (Haas et al., Neuhäuser & Hothorn 999). The null hypothesis of lack of tren is rejecte if the Cochran Armitage test statistic Z CA is above the upper fifth percentile of the normal istribution, which is. for a one-taile test. If positive associations are etermine, the ata can be moele using the ose response an TDR moels. The TDR moels are moifie forms of the traitional exponential an beta-poisson ose response moels that inclue the time post-inoculation. This aitional time parameter quantifies the onset of timeto-response associate with the kinetics of in vivo bacterial growth. TDR moels (Table ) have an empirical basis (Prasa et al. ; Huang & Haas, 9; Huang et al. 9). P(t,) is the probability of averse response at time t post-exposure with aministere ose. The k value is a parameter of the exponential TDR moel; j, j,, an a are parameters of the beta-poisson TDR moels. The TDR moels were fit to the ose response ata from infection stuies, an moel parameters were etermine using maximum likelihoo estimation (MLE) using the R programming language (R Founation ). The PRASAD & HAAS DECEMBER 7 9: JOURNAL AWWA E9 7 American Water Works Association

3 TABLE Time-epenent ose response moels (CDF) Moel Name Expression a Exponential moel with exponential reciprocal time epenency P(t,) = e e k t, where k > Beta-Poisson moel with exponential reciprocal time epenency P(t,) = + e t + j Beta-Poisson moel with exponential time epenency P(t,) = + e (j t+j ) Beta-Poisson moel with inverse time epenency P(t,) = + j t j a a, where j >> j a a + a a CDF cumulative istribution function, TDR time ose response a Develope by Huang et al. (9) P(t,) is the probability of averse response at time t post-exposure with an aministere ose,. The k value is a parameter of the exponential TDR moel; j, j,, an a are parameters of the beta-poisson TDR moels. likelihoo ratio (LR) for the incience occurring on ay j for i ose groups is shown in Eq : LR = mtimes j = i = ( ni,j moses p )p pi,j ( p i,j i,j i,j )ni,j pi,j ( ni,j p )(pi,j o )p i,j ( (p o pi,j i,j )ni,j i,j = mtimes moses j = i = pˆ i,j p o i,j p i,j pˆ i,j p o i,j n i,j p i,j where m oses is the total number of ose groups, m times is the number of time perios uring which observations were mae (generally recore in aylong segments), p i,j is the number of positive responses observe uring time perio j, an n i,j is the number of surviving animals at the beginning of time perio j. The preicte response is π i, an the response for each set base on observations is π i. FIGURE Susceptible X(t) Moifie SIR moel ß(t) Source: Gupta (999) Latent Y(t) SIR susceptible infective recovere Q(t) Symptomatically Ill Z(t) The three states in this moel are the susceptible populations X(t), latent iniviuals Y(t) who have been expose to the pathogen an will ultimately become infecte, an symptomatically ill iniviuals Z(t). The SIR moel was first propose by Kermack an McKenrick (97) an then moifie by Gupta (999). () The corresponing eviance (Haas et al. ) is expresse by Eq : m Y = ln LR = times m j = oses i = pi,j ln pˆ i,j p o i,j + (n i,j p i,j ) ln pˆ i,j p o i,j Outbreak moels. The susceptible infective recovere (SIR) compartment moel is a eterministic moel that escribes a close population of size N consisting of S(t) susceptibles, I(t) infectives, an R(t) removals i.e., iniviuals who either ha ie or ha recovere an were immune at time t (Kermack & McKenrick 97). Gupta (999) propose an SIR moel for water an fooborne infections that omits a recovere compartment an inclues a latent perio, an accounts for iniviuals being either symptomatically or asymptomatically ill (Figure ). In this moel, populations are compartmentalize into groups, with iniviuals moving through the compartments as they become infecte. The three states in this moel are the susceptible populations X(t), latent iniviuals Y(t) who have been expose to the pathogen an will ultimately become infecte, an symptomatically ill iniviuals Z(t). The rates in these equations are given by the force of infectivity b(t), which escribes the rate of iniviuals going from the susceptible population to the latent population, an the rate of newly symptomatic cases is escribe by Q(t). The number of symptomatically ill patients Z(t) per ay is given by Eq 3: () Z(t) = t b(t)x exp [ B (t)] f ( t) t (3) E PRASAD & HAAS DECEMBER 7 9: JOURNAL AWWA 7 American Water Works Association

4 TABLE Cumulative infection percentages of monkeys expose to Shigella Dose mg Viable Agent Count a Cumulative Infection of Monkeys/ Post-inoculation % Total , Source: Takasaka (97). The enpoint in the stuy was clinical illness. a As estimate by Takasaka (97) The full exposure time istribution is represente by the expression b(t)x exp[ B(t)], an the incubation istribution is represente by the expression f( t). The force of infection b(t) has units of inverse time an is a versatile parameter. In the case of a rectangular istribution for infectivity, b(t) woul be a constant for the exposure uration. However, common source outbreaks are frequently cause by time-variable microbial exposures, which means that the parameter takes the form of a ensity istribution, generally of the form foun in Eq : b(t) = b + g(t) () where t(t) is the force of infection, b is the backgroun infectivity level (which may account for any enemic cases in the population), is the scaling factor for increase infectivity above backgroun, an g(t) is the ratio of susceptible persons ultimately becoming infecte at time t. Here g(t) is moele as a probability ensity function for the exposure case. The ensity functions use to moel the exposures are the Weibull, gamma, lognormal, an uniform all of which are stanar functions that have historically been use to fit observe epiemic curves. Each of these ensity functions has parameter values that etermine the shape an scale of the function. The expression B(t) in Eq 3 is the integral of the force of infection t(t). The incubation functions f( t) that are use here are the Weibull, gamma, lognormal, uniform, an the best-fit TDR equation. For the Weibull, gamma, lognormal, an uniform incubation scenarios, there are six parameters for Z(t): the exposure parameters b an, two probability ensity function (PDF) parameters for the exposure istribution, an two PDF parameters for the incubation istribution (Weibull, gamma, lognormal, an uniform istributions all contain two parameters). The number of parameters in the TDR moels varies. For example, in the case of the beta-poisson moel with exponential time epenency (BPwE), there are eight parameters in all, as this moel contains four parameters: a,, j, an j. RESULTS AND DISCUSSION Literature search. A literature search for ose response ata sets an outbreak ata sets was conucte to ientify appropriate caniate moels for Shigella. Although human challenge stuies were preferre, none coul be obtaine that fit the inclusion criteria. Several osing stuies have been performe by inoculating animal hosts with Shigella pathogens, incluing guinea pigs (Formal et al. 9), monkeys (Formal et al. 97, 9, 9), an mice (Rorigues et al. 99). A series of stuies publishe by Takasaka an colleagues an Honjo et al. met the inclusion criteria for the time-epenent ose response moels (Takasaka et al. 97, 99; Honjo et al. 9, 9; Takasaka 97). Cynomolgus monkeys (Macaca irus) were orally inoculate with varying oses of S. flexneri a strain 3 to investigate the minimum ose responsible for clinical illnesses (Takasaka et al. 97, Takasaka 97). The monkeys were from the Philippines, Camboia, or South Vietnam; weighe between. an 3. kg; an were ose with between. an, mg of ysenteric content, corresponing to an 9 viable S. flexneri a counts (Table ). The number of viable agents was estimate by the authors, but since the estimations fluctuate by several orers of magnitue, they are presente as an illustrative measure only. Despite the clear ifferences between animal an human hosts, animal moels have long been use to unerstan the pathogenesis of human isease, specifically through osing stuies (Reagan-Shaw et al. ). Mammalian hosts have often been use in moeling human iseases because of similarities in anatomy an physiology (Lieschke & Currie 7), an macaques specifically have been successfully use to moel more than 7 human etiologic agents, incluing bacteria, viruses, fungi, parasites, an prions (Garner & Luciw PRASAD & HAAS DECEMBER 7 9: JOURNAL AWWA E 7 American Water Works Association

5 ). By inoculating pathogens into a primate that is closely relate to humans, researchers have been able to gain valuable insight into the mechanisms of the host pathogen effects in humans (Garner & Luciw ). The current work followe the preceent of using animal hosts to moel human isease. Two separate outbreak ata sets were chosen to moel shigellosis. S. flexneri was moele using an outbreak that occurre at an elementary school in Sichuan Province, China, where out of 3 stuents fell ill June 7, 9 (He et al. ). The investigation foun S. flexneri type b to be the cause of the outbreak, which was associate with rinking well water. An environmental investigation showe that the school-owne wells were locate m from a pon, where sewage was irectly ischarge without treatment. The untreate pon water was store in towers an pipe to the school without treatment, an stuents reportely rank from this source frequently. The S. sonnei case was moele using a primary wave affecting villagers in Crete, Greece (Samonis et al. 99). Illnesses were attribute to a spring water source contaminate with fecal coliforms from a nearby sewage facility. Samonis an colleagues efine a primary case as the first case to evelop symptoms of Shigella infection in each househol, with five to six loose stools per ay for at least two ays. Dose response. Since the ose response ata sets came from two ifferent stuies (Takasaka et al. 97, Takasaka 97), the ata were poole an teste for lack of fit as escribe by Haas an colleagues (). The eviance of the poole ata set was then compare with the iniviual fits. In this case, beta-poisson of the poole moel was the best-fit moel, with a eviance value of. (critical.9,m npar = 9.9). Because the summation of the iniviual eviances was greater than the poole eviance value ( Y i > Y T ), however, the conclusion was that the ata sets coul not be poole. Yet further consieration of the ata showe what appeare to be an outlier. This was confirme because values of N (the ose at which % of the population is expecte to be infecte) for one ose group (at 3 viable counts) were significantly higher than other values of S. flexneri ( 3 cfu) previously publishe (Crockett et al. 99). The poole ose group was then jackknife, with the ata refit by removing a single ose at a time. When the last ose point was remove, the eviance was reuce by for the beta- Poisson moel, an the N roppe to. mg, which the authors estimate to be approximately 7 viable counts. This perhaps is still at the higher en of the previously publishe values but much closer than the parameters given before removal of the outlier. The higher N value in the monkey ata cite by Takasaka an colleagues (Takasaka et al. 97, Takasaka 97) coul be attributable to several factors. In their series of experiments, the enpoint was loose stools with a measurable amount of bloo an mucus. The authors acknowlege that there were monkeys who ha loose stools at lower ose groups, but because the stools i not contain bloo, they were omitte from what the authors efine as clinical ysentery. For their analysis, Crockett an colleagues (99) use a series of experiments run by DuPont et al. with healthy human male subjects (DuPont et al. 97, 99). The enpoint in these experiments was illness characterize by a fever of at least F, severe abominal cramping, iarrhea (at least two loose stools in a h perio), or blooy mucoi stools. Dose response ata showe positive trens when subjecte to the Cochran Armitage test of tren. The critical Z value was greater than. (Z cr >.) at.7, so the null hypothesis of lack of tren was rejecte an ata coul TABLE 3 Dose response moeling results for Shigella inoculation Dose Remove Moel Type Parameter Value Deviance f.9,f a Conclusions Best-Fit Moel b None Exponential k Beta-Poisson a N 3. 3, mg Exponential k Beta-Poisson a N. Beta-Poisson provies acceptable fit Beta-Poisson provies acceptable fit Beta-Poisson Beta-Poisson f egree of freeom for ata set an moel, N ose at which % of the population is expecte to be infecte, TDR time ose response a Critical values of the chi-square istribution at a 9% confience level (f = m n), where f is the egree of freeom, m is the number of oses, an n is the number of parameters. b Accoring to the best-fit moel criteria, the selecte best fit was the caniate moel with the lowest eviance compare with the critical values of chi-square istribution. N an a are parameters of the beta-poisson TDR moel; k is a parameter of the exponential TDR moel. E PRASAD & HAAS DECEMBER 7 9: JOURNAL AWWA 7 American Water Works Association

6 FIGURE Dose response ata (A) an best-fit bootstrappe beta-poisson moel with confience intervals (B) A. Best-fit moel 9% confience 99% confience Dosage from ata set Corresponing an N values B. Response. N Dose N the ose at which % of the population is expecte to be infecte, TDR time ose response N an are parameters of the beta-poisson TDR moel. TABLE Summary of fits of time-epenent ose response ata Time Depenence Moel Best-Fit Parameters Minimize Deviance.9,f Acceptable Fit Beta-Poisson moel with exponential time epenency a = Yes j =. j =. Beta-Poisson moel with exponential reciprocal time epenency a =. 3.. No j = 3.3 j =.9 Beta-Poisson moel with inverse time epenency a = No j = Exponential moel with exponential reciprocal time epenency k =.9.. Yes f egree of freeom, TDR time ose response The k value is a parameter of the exponential TDR moel; j, j, an a are parameters of the beta-poisson TDR moels. be analyze further. Table 3 shows the best parameter values of the fits an optimization of the poole ata sets, with an without the outlier ose. In the moel in which the outlier is remove, the beta-poisson was the best-fitting mechanistic moel because it was the only moel whose minimize eviance of. was lower than the critical chi-square istribution for 3 egrees of freeom, at 7.. The bootstrappe ata set with, iterations from this monkey moel was plotte with optimize parameters for the beta-poisson moel in Figure. The TDR moels in Table were fitte to the osing stuy ata presente in Table using MLE. The parameter estimates an the minimize eviances are summarize in Table. The one-parameter exponential with exponential reciprocal time epenency an the three-parameter BPwE were foun to have acceptable PRASAD & HAAS DECEMBER 7 9: JOURNAL AWWA E3 7 American Water Works Association

7 FIGURE 3 9 Daily cumulative ata fitte by the beta-poisson moel with exponential time epenency TDR moel Response (Illness) Dose mg TDR time ose response Arrow inicates the irection of the curves from the first ay to the last. fits to the time-epenent ata. The ifference in the minimize eviance between these two moels was 3., which was less than the.9,f of.99 for egrees of freeom. Therefore, the best fit for these moels was the BPwE because it provie the lowest eviance an gave a statistically significant improvement in fit over the other moels. Figure 3 plots the best-fit TDR moel. Outbreak moels. The outbreak moel was next fit to two outbreak ata sets: the primary sprea of S. flexneri that originate from a school well in Sichuan Province an was investigate by He et al. () an the primary S. sonnei outbreak in a Crete village that was linke to contaminate spring water an was investigate by Samonis an colleagues (99). Since the BPwE was the best-fit moel, this was the time-epenency kinetic moel that was use in the incubation istributions. The non-time-epenent Weibull, gamma, lognormal, an uniform istributions were also generate for comparison. In each case, eviance, Akaike information criterion (AIC), an Bayesian information criterion (BIC) output were compare to etermine which convolution of istributions best fit the ata. For the S. flexneri Sichuan Province school case, star plots of the parameter values are presente in Figure. The star plots are rea clockwise. The b an are first, followe by the two exposure istribution parameters an the incubation istribution parameters. For the Weibull incubation or exposure functions, the parameters are the a an b. For the gamma functions, the two parameters are the mean an variance. In the lognormal istributions, the two parameters are the log transform of the mean an variance. The uniform function parameters are the lower enpoint a an the upper enpoint b. For the BPwE, the calculate parameters were the a, j, j, an, respectively. A comparison of the visual representation of the parameter values through the star plots showe what appear to be significant ifferences in the istributions of the exposure an incubation istributions. Selecte criteria values for the S. flexneri moels with the lowest eviance values are presente in Table. Here, the gamma Weibull exposure/incubation convolution ha the lowest eviance at.7. The next best-fit moels were the lognormal Weibull at.73 an the Weibull Weibull at.. In this outbreak, the BPwE incubation moels i not provie any benefit, given the extra parameters, with eviances at.73,.9, an 3. for the Weibull BPwE, gamma BPwE, an lognormal BPwE exposures, respectively. E PRASAD & HAAS DECEMBER 7 9: JOURNAL AWWA 7 American Water Works Association

8 TABLE Summary of parameter an criteria values for Shigella flexneri an Shigella sonnei exposure an incubation perios Distribution Moel Parameters Criteria Values Outbreak Exposure Incubation Exposure parameter b Exposure parameter Exposure istribution mean Exposure istribution SD Incubation istribution mean Incubation istribution SD Deviance AIC BIC S. flexneri a Weibull Weibull S. sonnei b Weibull BPwE Gamma Weibull BPwE Lognormal Weibull BPwE Uniform Weibull Gamma Lognormal BPwE Gamma Uniform BPwE Lognormal BPwE Weibull BPwE Uniform BPwE AIC Akaike information criterion, BIC Bayesian information criterion, BPwE beta-poisson with exponential time epenency, SD stanar eviation a The S. flexneri outbreak occurre in a school in Sichuan Province, China, an was attribute to contaminate well water. b The S. sonnei outbreak occurre in a Crete village in Greece an was attribute to a contaminate spring water source. Dashes inicate not applicable. E PRASAD & HAAS DECEMBER 7 9: JOURNAL AWWA 7 American Water Works Association

9 FIGURE Star plots for the best-fit Shigella flexneri parameters a Weibull Weibull Weibull BPwE b j j b b Gamma Weibull Gamma BPwE b 3 j b 7 3 j Lognormal Weibull Lognormal BPwE 3 b j j 3 b mu parameter mu parameter sigma parameter sigma parameter Uniform Weibull Uniform BPwE b 3 j j b b a parameter b parameter b parameter a parameter BPwE beta-poisson with exponential time epenency a The S. flexneri outbreak occurre in a school in Sichuan Province, China, an was attribute to contaminate well water. Selecte parameter values are shown in Table. E PRASAD & HAAS DECEMBER 7 9: JOURNAL AWWA 7 American Water Works Association

10 Table contains several moel parameters for S. flexneri. The first an secon columns in the table section labele Moel Parameters show the b an exposure rate values. The baseline parameter b preicts the enemic levels of the pathogen in a community before the occurrence of an outbreak. In this outbreak, b ranges from 9. 3 to., with units of inverse time in ays. This means that in this community of susceptible iniviuals, there were between 9 an, cases per, population each ay. The parameter is the infectivity level, or the total proportion of the susceptible population that will become infecte. In the moels selecte here, range from. to.3. The thir an fourth columns of the moel parameters section in Table show the mean an stanar eviation values corresponing to the respective exposure istributions, while the fifth an sixth columns are the mean an stanar eviation values corresponing to the respective incubation istributions. These values were calculate from the parameter values of the initial moels, presente in the star plots of Figure. The mean values of the exposure istribution range from. to.7 ays, with an outlier of ays. He an colleagues () i not inicate the number of ays the schoolchilren were expose to the S. flexneri pathogen. However, it was foun that between June an June, 9, % of affecte stuents rank untreate well water. The outbreak laste June 7 7, so the range of potential exposures from the well coul be between an ays. The mean values for the incubation perio were between 3. an. ays. The incubation perio of Shigella is between one an three ays (Heymann ), so the authors estimates are slightly higher than this range. Because of the complexities of the BPwE moels, the incubation mean an stanar eviation coul not be calculate. Figure plots the observe outbreak ata (number ill) an the preicte moel for the S. flexneri outbreaks in the Sichuan Province school. Each part of the figure groups the respective incubation perios with the corresponing exposure istributions. The x-axis shows the time in ays, an the y-axis inicates the number of cases reporte. The ifferent parts of the figure exhibit a peak at the height of the outbreak that is characteristic of a primary wave in which there are no seconary cases. Each exposure/incubation combination prouces a unique fit to the outbreak ata, with significant ifferences in the preiction moel of the assume exposure an incubation functions in the various combinations. The outbreak fit is highly epenent on the functions use to moel the ata set. As outline previously, the gamma Weibull moel gives the best fit. Moels for fitting S. sonnei outbreak ata sets were generate using the same metho of convoluting istributions. Star plots of the parameter values for S. sonnei are presente in Figure. As with the S. flexneri case, there appear to be significant ifferences in the istributions of the exposure an incubation istributions. Selecte moel parameters an eviance values are shown in Table. Here, the BPwE convolutions provie the best fits. The uniform BPwE convolution has the lowest eviance value at 3., followe by the Weibull BPwE, Weibull uniform, an Weibull gamma, at 7.9,.3, an., respectively. For the uniform BPwE, Weibull BPwE, Weibull uniform, an Weibull gamma cases, AIC values were 9., 3.9,.3, an., respectively, an BIC values were 3.97,.7, 3.73, an 3.9, respectively. Although the BPwE moels ha aitional parameters, the AIC an BIC penalize for the aitional parameters. The uniform BPwE ha the lowest AIC values, an the Weibull BPwE ha the fifth lowest AIC values. If the BIC values were use to rank the moels, the Weibull uniform, Weibull gamma, an uniform BPwE woul be the best fit. Table also shows selecte parameter values for the S. sonnei outbreak that occurre in the Crete village because of contaminate spring water. In this outbreak, the b values range from. to.3, which inicates that there were between an,3 cases per, population each ay. The values for range between. an.. The mean values of the exposure istribution range from. to 3.7 ays. Samonis et al. (99) were not able to etermine a range of exposure times for the resients expose to S. sonnei through contaminate rinking water. The first case occurre on December, an water chlorination an other environmental control measures were initiate on December, presumably ening the Shigella exposures. The mean values for the incubation perio were between. an., while the accepte incubation perio of Shigella range between one an three ays. Figure 7 plots the observe ata an preicte moels for the S. sonnei outbreak in Crete. Each part of the figure groups the respective incubation perios with the corresponing exposure istributions. The istinctions between the iniviual moels are more prominent here. The tren in the exposure istributions appears to be that the uniform group provie the flattest peak, followe by the lognormal series. The Weibull group was more rightskewe, whereas the gamma series seeme to be more centere. There may be a few reasons why the BPwE moel i not provie lowest eviance in the China outbreak but i in the Crete outbreak. The first is that the matrixbase optimization routine may have ha challenges in fitting the ata sets; reliably fitting eight parameters with a egree of reproucibility, given an observe epiemic outbreak, was not easy. The routine was run several times with varying initial guesses, an a local minimum value was calculate uring some of the runs, which may suggest a substitution effect among the parameters. The parameters in the moel themselves have inherent variability. Outbreaks are subject to the pathogenicity, which in turn epens on the features of the host immune PRASAD & HAAS DECEMBER 7 9: JOURNAL AWWA E7 7 American Water Works Association

11 FIGURE Curve fits for the primary exposure incubation istribution moels of the Shigella flexneri outbreak a Number ill Gamma exposure Weibull exposure Lognormal exposure Uniform exposure A Gamma incubation B Weibull incubation C Lognormal incubation D Uniform incubation E BPwE incubation BPwE beta-poisson with exponential time epenency a The S. flexneri outbreak occurre in a school in Sichuan Province, China, an was attribute to contaminate well water. Each part of the figure groups the respective incubation perios with the corresponing exposure istributions: gamma incubation istributions (A); Weibull incubation istributions (B); lognormal incubation istributions (C); uniform incubation istributions (D); an BPwE incubation istributions (E). The x-axis shows time of symptom onset in ays, the y-axis shows the number of cases reporte. Selecte parameter values are shown in Table. E PRASAD & HAAS DECEMBER 7 9: JOURNAL AWWA 7 American Water Works Association

12 FIGURE Star plots for the best-fit Shigella sonnei parameters a Weibull Gamma Weibull Uniform b b parameter " b 3 a parameter Weibull BPwE Gamma Lognormal j j b b sigma parameter b mu parameter Gamma BPwE Lognormal Weibull j b b j sigma parameter mu parameter Lognormal BPwE Uniform BPwE b j j b b a parameter mu parameter b parameter sigma parameter BPwE beta-poisson with exponential time epenency a The S. sonnei outbreak occurre in a school in Crete, Greece, an was attribute to contaminate spring water. Selecte parameter values are shown in Table. system (such as immunity) an the features of the pathogen (such as replication an sprea). Infectivity, virulence factors, rug sensitivity, an route of transmission may also play a role in this variability. Other variables inclue the seasonal variability of certain iseases, which may play a role in the price, eman, an subsequent use of interventions. The global maximum that best captures these complexities may not have been reache. Future stuies woul benefit from further testing the parameter values to ensure that global minimums ha been reache in the optimization routine. Despite this performance, however, there is much that can be gleane from the general outbreak moel. A moel that consiers changes in infectivity in such pathogens may be especially useful for public health officials who are searching for control strategies. Communication with PRASAD & HAAS DECEMBER 7 9: JOURNAL AWWA E9 7 American Water Works Association

13 FIGURE 7 Curve fits for the exposure incubation istribution moels of the Shigella sonnei outbreak a Number ill Gamma exposure Weibull exposure Lognormal exposure Uniform exposure A Gamma incubation B Weibull incubation C Lognormal incubation D Uniform incubation E BPwE incubation 3 3 BPwE beta-poisson with exponential time epenency a The S. sonnei outbreak occurre in a school in Crete, Greece, an was attribute to contaminate spring water. Each part of the figure groups the respective incubation perios with the corresponing exposure istributions: gamma incubation istributions (A); Weibull incubation istributions (B); lognormal incubation istributions (C); uniform incubation istributions (D); BPwE incubation istributions (E). The x-axis shows time of symptom onset in ays; the y-axis shows the number of cases reporte. Selecte parameter values are shown in Table. E PRASAD & HAAS DECEMBER 7 9: JOURNAL AWWA 7 American Water Works Association

14 local public health agencies is an essential part of preictive isease moeling, an outbreak moels that reflect the result of ifferent control measures coul be generate using several ifferent combinations of exposures an incubation functions. By stuying these preictive moels, a best-case control strategy coul be selecte, an public health officials coul make informe ecisions regaring isease control strategies. CONCLUSION In this stuy, a TDR moel was incorporate into an incubation istribution an convolute with an exposure istribution. An exposure istribution was convolute with an incubation istribution in an SIR moel to get a fitte epiemic curve. To this en, ose response an TDR moels were generate for monkey exposure to S. flexneri via the oral route an subsequently were incorporate into the incubation perio of two outbreaks (S. flexneri an S. sonnei) in orer to capture the in vivo ynamics between pathogen an host. In the ose response moel, the meian infectious ose was estimate to be mg of ysenteric content (approximately viable S. flexneri a counts), an the best-fit TDR moel was the beta-poisson with exponential epenency, which ha a minimize eviance of This TDR moel was chosen to be incorporate into two Shigella outbreak moels: a primary wave of S. flexneri resulting from an outbreak attribute to contaminate well water in China an a primary wave of S. sonnei that broke out in Greece because of a contaminate village spring water source. In the primary wave of the S. flexneri outbreak in China, the convolution of the gamma exposure istribution with the Weibull incubation istribution was the best fit for the moel, with a minimize eviance value of.7. In the case of the primary S. sonnei outbreak in Greece, the uniform BPwE istribution performe the best, with a minimize eviance value of 3.. The BPwE moel worke quite well in the S. sonnei outbreak. The incorporation of a time factor into outbreak moels provies a novel approach to moeling that attempts to escribe the in vivo ynamics of the host pathogen system. ENDNOTE MATLAB, MathWorks, Natick, Mass. ABOUT THE AUTHORS Biya Prasa is a PhD grauate of the Department of Civil, Architectural an Environmental Engineering, Drexel University, Philaelphia, Pa.; she can be reache at bp3@rexel. eu. Her main research interest inclues environmental risk assessment using R an MATLAB (Mathworks, Natick, Mass.). She has BS an MS egrees in chemical engineering from Rutgers University, New Brunswick, N.J., an a octorate in environmental engineering from Drexel University, Philaelphia, Pa. Charles N. Haas is the LD Betz professor of environmental engineering an hea of the Department of Civil, Architectural, an Environmental Engineering Department at Drexel University. PEER REVIEW Date of submission: //7 Date of acceptance: //7 REFERENCES Bhunia, A.,. Shigella Species. Fooborne Microbial Pathogens. 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