The Science of the Total Environment, 4 (1975) 165-175 © Elsevier Scientific Publishing Company, Amsterdam - Printed in Belgium
A PROBABILISTIC MODEL OF BATHING BEACH SAFETY
G. W O L F G A N G F U H S Division of Laboratories and Research, New York State Department of Health, Albany, N.Y. 12201
(U.S.A.) (Received February 8th, 1975)
ABSTRACT
An improved mathematical model for bathing beach safety is proposed. It is derived by joining the probability of infection from a given dose (Poisson distribution and the probability of acquiring such a dose (lognormal distribution). Even in the absence of better clinical and epidemiological data, the model permits an assessment of relative risk from certain hazards and the design of more meaningful bacteriological standards for individual beaches.
INTRODUCTION
Bacteriological standards for bathing beaches have for some time been the subject of controversy 1-3. Most critics contend that the standards are overly restrictive, citing the extremely favorable epidemiological record of beaches considered of marginal bacteriological quality and even of areas used by bathers despite poor bacteriological quality. Other critics point out that bacteriological standards are arbitrary, since there is no proved relationship between the numbers of indicator bacteria and the transmission of waterborne illness, and urge that bacteriological tests be abandoned in favor of sanitary inspections or purely aesthetic criteria. In view of the scarcity of recreational waters in many urban areas, this criticism must be taken seriously. On the other hand, authorities are quick to respond that waterborne disease is not a myth, but a reality and that present bacteriological standards have been important in preventing outbreaks and improving the general epidemiological picture, which in turn reduces the probability of outbreaks. Arbitrary relaxation or abandonment of bacteriological standards, they charge, would amount to experimentation with human health on a grandiose scale. A difficulty in setting rational bacteriological standards for bathing beaches is the lack not only of pertinent bacteriological and epidemiological data, but also and primarily of a conceptual framework which connects the public health risk of bathing in polluted waters with the available data. Not until 1972 was an attempt made to define mathematically the risk of contracting a waterborne disease as the function of 165
two probabilities, that of ingesting a certain dose of an infectious agent and t h a t of contracting the disease as a result of such a dose (Mechalas et al.4). The model described by these authors, however, fails to incorporate several important elements. The model presented here was developed independently and deals with the mathematical components in a different manner. Since the concept on which both models are based has not yet received wide attention, I shall present the approach in its entirety, referring to the report by Mechalas et al. 4 whenever it is appropriate to do so. BASIC ASSUMPTIONS AND DERIVATION OF THE MODEL
Infektious dose The risk of infection depends on the dose received and on the probability that one single microorganism or infectious particle causes infection. The latter parameter is determined by the type and viability of the microorganism and by the effectiveness of immunological and other defense mechanisms. Under favorable conditions one microorganism could cause infection; and for the most feared, most virulent, but best controlled pathogens, the immunity in the population may be low or practically non-existent. For a mathematical representation of the infectious process involving particles which act independently of one another 5, the Poisson distribution appears most appropriate. It is characterized by a single parameter, p, which at a value o f 0.2 or lower approximately represents the probability of infection by a single microorganism. The probability Q of acquiring the disease by ingesting k bacteria becomes Q1 = 1 - e -pk
(1)
Mechalas et al. 4, in plotting data from the literature on probability of infection vs. dose received, favored a lognormal distribution, but they did not consider the Poisson distribution. If eqn. (1) is plotted on a log-probability grid, an almost straight line at prbbabilities of infection of 10% and less suggests that some data sets m a y be compatible with either model. At higher probabilities, however, an upward curvature of the model represented in eqn. (1) invariably leads to a higher probability, of infection than would be predicted from a lognormal model. Unfortunately there are few accurate estimates of infectious doses of organisms which are of concern in water sanitation. For Salmonella typhosa, estimates vary widely. Earlier data suggest p = 0.07 (ref. 6) and p = 0.015 (ref. 7), corresponding to 50% probabilities of infection (IDso) from doses of 10 and 45 bacteria, respectively. Steiniger 8, 9 assumes S. typhosa to be a highly virulent organism but agrees with McCullough and Eisele 1°-13 that the salmonellae which are commoner in polluted waters than S. typhosa, particularly S. paratyphi B, cause infection only in doses of several thousand organisms. Data from volunteer experiments with S. typhosa 14' 15 suggest a higher infectious dose also for this organism. The result of ID2~ = 105 converts to p = 3 x 10 -6 and an IDso of 2.3 x 105 using eqn (1) and to an IDso of 107 using the lognormal model 4. In these experiments, 105 bacteria were administered in a glass of milk. If large volumes of liquid are ingested, dilution and neutralization 166
of the gastric juice by ingested fluid will reduce bacterial die-off and therefore will decrease the apparent infectious dose.
Volume of water containing infectious dose To determine the probability of contracting a disease from water of a given bacteriological quality, it is necessary to estimate the volume of water likely to be ingested by a bather. Mechalas et al. 4 specify a volume of 10 ml, whereas Steiniger s' 9 cites a consensus o f physicians involved in physical education that 50 ml is a good estimate, probably during active swimming in high-quality freshwater. The calculations below are based on the latter figure but are easily adjusted. While the number of bacteria taken up during continuous sampling is related to the arithmetic rather than to the geometric mean s, the sample ingested by a bather, although it may be received in several portions, is better represented by a single sample taken from a lognormal distribution. Distribution of pathoyens and indicator bacteria in the environment The distribution of bacteria in the environment is generally lognormal with respect to both time and space 16' 17. The biological processes of growth and decay and the physical process of dispersion favor such a distribution. Also, the concentrations of chemical species are quite often distributed in this fashion is. A lognormal distribution of bacterial numbers was readily apparent in all data sets used in connection with this study (Fig. 1), and there is no reason to believe that pathogens, although present in much lower numbers than the indicator bacteria, are distributed in any other way. Of the parameters/z (geometric mean or log mean) and s (standard deviation; antilog s = error factor) of the lognormal distribution, the importance of # is generally recognized, although its calculation seems to present difficulties when the data set contains results with the notation "less t h a n " and when the true value o f / t lies near, or even below, the detection limit of the bacteriological technique. In these cases, the cumulative frequency distribution of the experimental value should be plotted or fitted to obtain estimates of both ~ and s. (This method could conceivably be applied to the characterization of populations of bacterial pathogens in water using presence/ absence texts at two o r more different detection limits, a modification of the technique of most probable numbers.) While the /z values of indicator bacterial densities have become standards which determine the acceptability of recreational water quality, the meaning and significance of s is little understood or appreciated. It will be shown below that the value of s, which indicates the degree of dispersion or of homogeneity of the pathogenic and indicator bacterial populations, has great influence on the risk of infection. Legal specification o f a maximum permissible s often takes the form of a percentile clause (x percent of the samples not to exceed a count of y bacteria per 100 ml). While this approach is perfectly sound, the data presented below show that the numerical values in these clauses are often wholly unrelated to the actual distribution of pathogenic or indicator bacteria in nature. 167
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Fig. 1. Mean fecal and total coliform counts for 58 bathing beaches in New York State. T h e log mean,/~, is plotted against the standard deviation in base 10 logs, s. The symbols, in the order shown at the bottom of the figure, indicate increasing relative risk, by order of magnitude and on t h e basis that fecal coliforms are a valid indicator of waterborne enteric pathogens. Several very low-risk classes are not differentiated (blank circles). The symbols in the total coliform plot are those assigned to the respective beaches in the fecal coliform plot. The different distribution of the shaded symbols is therefor~ due to widely varying fecal coliform/total coliform ratios. See text for data points marked with arrows.
P h y s i c a l processes o f dispersion s h o u l d affect b o t h p a t h o g e n i c a n d i n d i c a t o r b a c t e r i a i n the same m a n n e r . W i t h s o m e specifications o u t l i n e d below, w e shall a s s u m e t h a t s values for the two p o p u l a t i o n s are the same.
Factors affecting the ratio of pathogens to indicator bacteria T h e use o f i n d i c a t o r bacteria in the asssessment o f r e c r e a t i o n a l w a t e r q u a l i t y is j u s t i f i e d b y the relative scarcity o f pathogens. In p o t a b l e waters one s t r i v e s f o r c o m p l e t e b l o c k a g e o f the r o u t e o f t r a n s m i s s i o n o f i n t e s t i n a l pathogens. R e c r e a t i o n a l activities, however, c a n n o t be restricted to waters free o f i n d i c a t o r b a c t e r i a , a n d q u a n t i t a t i v e c o n s i d e r a t i o n s o f the ratio o f p a t h o g e n i c a n d i n d i c a t o r o r g a n i s m s a r e in o r d e r . T h i s r a t i o d e p e n d s on the n u m b e r o f excreters in the p o p u l a t i o n a n d o n t h e n u m b e r o f p a t h o g e n s excreted by them. A u t h o r i t i e s express confidence t h a t t h e n u m b e r o f e x c r e t e r s for certain diseases within a given w a t e r s h e d can be k n o w n w i t h a 168
high degree of accuracy. This applies particularly to typhoid fever and other conditions for which constant surveillance and reporting are mandatory in many areas. In contrast, the ratio of pathogens to indicator bacteria excreted by carrier persons is known only for a few cases of typhoid fever reported by Thomson 19'2°. According to this author, the ratio of S. typhosa to Escherichia coli in the feces decreases with time. From acute infection to the last phases of the carrier state it can vary over at least six orders of magnitude, with an average of 1/250 in a selection of 12 cases. Better estimates of this ratio are necessary for other situations and diseases and could be easily obtained if the presence/absence tests in routine surveillance of known excreters were replaced by appropriate quantitative tests.
Factors affecting the ratio of pathoyens and indicator bacteria in small polluting populations In large populations the ratio of pathogens to indicator bacteria is rather stable and predictable once the number of excreters and the average number of pathogenic organisms excreted by them are approximately known. In particular, the ratio will not be affected significantly by normal population mobility. The risk arising from small polluting populations, however, is significantly different depending on whether several, one, or no excreter is present. In fact, a beach polluted by a small population which includes a single excreter may be unsafe by any of the commonly accepted bacteriological standards. Conversely, in the absence of an excreter in the polluting population the beach is safe even when it is heavily polluted. Mathematically this situation should be dealt with in two different ways, depending on the intended application of the model. For scientific-epidemiological studies one may utilize data from a larger population (county, state); and assuming equal and random distribution of the excreters, one may use a Poisson distribution to predict the likelihood or fraction of time that 0, 1, 2, etc., excreters are present in the polluting population, multiplying this probability by the overall probability of infection. For public health policy making, however, this approach is not feasible because even the presence of a single excreter in a small polluting population might cause an outbreak involving all persons exposed to the pollution. Such an outbreak not only significantly alters the epidemiological situation upon which bathing beach standards are (or should be) based but in itself constitutes an event which the public health community is called upon to prevent. For setting public health standards, therefore, the number of excreters of a disease in any given population should be assumed equal to the expected number plus one. It will be shown below that this does not change the basis for computing standards for beaches contaminated by large metropolitan populations, but it forces the adoption of more stringent standards for beaches affected by small populations, and in this manner it establishes a logical link to the long-recognized need for zero contamination (in terms of indicator bacteria) in swimming pools and drinking water systems.
169
Die-off Among other factors affecting the ratio of pathogenic and indicator bacteria are the differences in die-off and other forms of elimination, e.g., sedimentation. Die-off rates of S. typhosa and E. coli appear to be of the same general order of magnitude ~5. In cases where they are not, a function of time o f travel from the source of pollution may have to be introduced into the model.
Calculation In principle, the probabilities of ingesting certain doses of pathogenic microorganisms are calculated from the /~ values and s of the population of indicator bacteria and the pathogen/indicator ratio, corrected for small population effects as required. Each of these probabilities is multipled by the probability of contracting the disease from such a dose, and the results are summed to give the probability of infection from ingesting a given volume of the water (50 ml). By using a Wang 720C calculator with 12-digit a.ccuracy and a Wang library program for the calculation o f the normal probability integral, the probabilities Q i = Qx(k-ll2)--Qx(k+ll2) o f encountering between k - 1 / 2 and k + 1/2 organisms were calculated for k = 1, 2, 3... from the arguments x(k+l/2)=[log(k+l/2)--lt]/s. The probabilities of acquiring the disease were calculated by multiplying each Qi by the term 1 - e -pk and by summing the products obtained in this manner for all values of k. Whenever k reached a value which let the term 1 - e -pk assume a value of 0.95 or higher, the probability of ingesting a greater number of organisms k was added as such, assuming that the probability of such a dose causing disease is unity. This step terminated the calculation. For small values of p, the calculations become lengthy because k must be increased by 1 many times before 1 - e -pk reaches 0.95. To avoid this, k was replaced by the term i" t, where t is a scaling factor and i assumes the values .1, 2, 3... An error is introduced insofar as i = 1 at t = 1 represents 0 . 5 < i . t < 1 . 5 , whereas i = 1 at t = 10 covers the range 5 < i. t < 15. Five or less organisms represent a dose with a low but definitive risk of infection, whereas less than one organism does not. Trial runs showed, however, that the error is so small that it does not affect the results to any great extent. The principal risk is associated with the probability of ingesting large doses. Table 1 shows the results for p = 0.2, representing a highly virulent agent causing infection in approximately one out of five cases upon ingestion of a single organism (p = 0.2). For other values of p, the values of the columns can be scaled accordingly. This is shown in an example represented by a column marked " p = 0.0002". The s is given in two forms, in base-10 logarithms and in multiples of # which are exceeded in 20% of the samples in a population (80% percentile value, Qo.a, divided by/~). Table 1 illustrates the fact that the public health risk increases not only with increasing/~ but also significantly with increasing s. A change in s from 0.2 to 0.3 (in base-10 logarithms) corresponds to a 10-fold change in the log mean which, either directly or converted to units of indicator organisms, is the numerical standard, now representing a defined public health risk. Introduction of the other elements into the model is a relatively straightfor170
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A PROBABILISTIC MODEL OF BATHING BEACH SAFETY
G. W O L F G A N G F U H S Division of Laboratories and Research, New York State Department of Health, Albany, N.Y. 12201
(U.S.A.) (Received February 8th, 1975)
ABSTRACT
An improved mathematical model for bathing beach safety is proposed. It is derived by joining the probability of infection from a given dose (Poisson distribution and the probability of acquiring such a dose (lognormal distribution). Even in the absence of better clinical and epidemiological data, the model permits an assessment of relative risk from certain hazards and the design of more meaningful bacteriological standards for individual beaches.
INTRODUCTION
Bacteriological standards for bathing beaches have for some time been the subject of controversy 1-3. Most critics contend that the standards are overly restrictive, citing the extremely favorable epidemiological record of beaches considered of marginal bacteriological quality and even of areas used by bathers despite poor bacteriological quality. Other critics point out that bacteriological standards are arbitrary, since there is no proved relationship between the numbers of indicator bacteria and the transmission of waterborne illness, and urge that bacteriological tests be abandoned in favor of sanitary inspections or purely aesthetic criteria. In view of the scarcity of recreational waters in many urban areas, this criticism must be taken seriously. On the other hand, authorities are quick to respond that waterborne disease is not a myth, but a reality and that present bacteriological standards have been important in preventing outbreaks and improving the general epidemiological picture, which in turn reduces the probability of outbreaks. Arbitrary relaxation or abandonment of bacteriological standards, they charge, would amount to experimentation with human health on a grandiose scale. A difficulty in setting rational bacteriological standards for bathing beaches is the lack not only of pertinent bacteriological and epidemiological data, but also and primarily of a conceptual framework which connects the public health risk of bathing in polluted waters with the available data. Not until 1972 was an attempt made to define mathematically the risk of contracting a waterborne disease as the function of 165
two probabilities, that of ingesting a certain dose of an infectious agent and t h a t of contracting the disease as a result of such a dose (Mechalas et al.4). The model described by these authors, however, fails to incorporate several important elements. The model presented here was developed independently and deals with the mathematical components in a different manner. Since the concept on which both models are based has not yet received wide attention, I shall present the approach in its entirety, referring to the report by Mechalas et al. 4 whenever it is appropriate to do so. BASIC ASSUMPTIONS AND DERIVATION OF THE MODEL
Infektious dose The risk of infection depends on the dose received and on the probability that one single microorganism or infectious particle causes infection. The latter parameter is determined by the type and viability of the microorganism and by the effectiveness of immunological and other defense mechanisms. Under favorable conditions one microorganism could cause infection; and for the most feared, most virulent, but best controlled pathogens, the immunity in the population may be low or practically non-existent. For a mathematical representation of the infectious process involving particles which act independently of one another 5, the Poisson distribution appears most appropriate. It is characterized by a single parameter, p, which at a value o f 0.2 or lower approximately represents the probability of infection by a single microorganism. The probability Q of acquiring the disease by ingesting k bacteria becomes Q1 = 1 - e -pk
(1)
Mechalas et al. 4, in plotting data from the literature on probability of infection vs. dose received, favored a lognormal distribution, but they did not consider the Poisson distribution. If eqn. (1) is plotted on a log-probability grid, an almost straight line at prbbabilities of infection of 10% and less suggests that some data sets m a y be compatible with either model. At higher probabilities, however, an upward curvature of the model represented in eqn. (1) invariably leads to a higher probability, of infection than would be predicted from a lognormal model. Unfortunately there are few accurate estimates of infectious doses of organisms which are of concern in water sanitation. For Salmonella typhosa, estimates vary widely. Earlier data suggest p = 0.07 (ref. 6) and p = 0.015 (ref. 7), corresponding to 50% probabilities of infection (IDso) from doses of 10 and 45 bacteria, respectively. Steiniger 8, 9 assumes S. typhosa to be a highly virulent organism but agrees with McCullough and Eisele 1°-13 that the salmonellae which are commoner in polluted waters than S. typhosa, particularly S. paratyphi B, cause infection only in doses of several thousand organisms. Data from volunteer experiments with S. typhosa 14' 15 suggest a higher infectious dose also for this organism. The result of ID2~ = 105 converts to p = 3 x 10 -6 and an IDso of 2.3 x 105 using eqn (1) and to an IDso of 107 using the lognormal model 4. In these experiments, 105 bacteria were administered in a glass of milk. If large volumes of liquid are ingested, dilution and neutralization 166
of the gastric juice by ingested fluid will reduce bacterial die-off and therefore will decrease the apparent infectious dose.
Volume of water containing infectious dose To determine the probability of contracting a disease from water of a given bacteriological quality, it is necessary to estimate the volume of water likely to be ingested by a bather. Mechalas et al. 4 specify a volume of 10 ml, whereas Steiniger s' 9 cites a consensus o f physicians involved in physical education that 50 ml is a good estimate, probably during active swimming in high-quality freshwater. The calculations below are based on the latter figure but are easily adjusted. While the number of bacteria taken up during continuous sampling is related to the arithmetic rather than to the geometric mean s, the sample ingested by a bather, although it may be received in several portions, is better represented by a single sample taken from a lognormal distribution. Distribution of pathoyens and indicator bacteria in the environment The distribution of bacteria in the environment is generally lognormal with respect to both time and space 16' 17. The biological processes of growth and decay and the physical process of dispersion favor such a distribution. Also, the concentrations of chemical species are quite often distributed in this fashion is. A lognormal distribution of bacterial numbers was readily apparent in all data sets used in connection with this study (Fig. 1), and there is no reason to believe that pathogens, although present in much lower numbers than the indicator bacteria, are distributed in any other way. Of the parameters/z (geometric mean or log mean) and s (standard deviation; antilog s = error factor) of the lognormal distribution, the importance of # is generally recognized, although its calculation seems to present difficulties when the data set contains results with the notation "less t h a n " and when the true value o f / t lies near, or even below, the detection limit of the bacteriological technique. In these cases, the cumulative frequency distribution of the experimental value should be plotted or fitted to obtain estimates of both ~ and s. (This method could conceivably be applied to the characterization of populations of bacterial pathogens in water using presence/ absence texts at two o r more different detection limits, a modification of the technique of most probable numbers.) While the /z values of indicator bacterial densities have become standards which determine the acceptability of recreational water quality, the meaning and significance of s is little understood or appreciated. It will be shown below that the value of s, which indicates the degree of dispersion or of homogeneity of the pathogenic and indicator bacterial populations, has great influence on the risk of infection. Legal specification o f a maximum permissible s often takes the form of a percentile clause (x percent of the samples not to exceed a count of y bacteria per 100 ml). While this approach is perfectly sound, the data presented below show that the numerical values in these clauses are often wholly unrelated to the actual distribution of pathogenic or indicator bacteria in nature. 167
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Fig. 1. Mean fecal and total coliform counts for 58 bathing beaches in New York State. T h e log mean,/~, is plotted against the standard deviation in base 10 logs, s. The symbols, in the order shown at the bottom of the figure, indicate increasing relative risk, by order of magnitude and on t h e basis that fecal coliforms are a valid indicator of waterborne enteric pathogens. Several very low-risk classes are not differentiated (blank circles). The symbols in the total coliform plot are those assigned to the respective beaches in the fecal coliform plot. The different distribution of the shaded symbols is therefor~ due to widely varying fecal coliform/total coliform ratios. See text for data points marked with arrows.
P h y s i c a l processes o f dispersion s h o u l d affect b o t h p a t h o g e n i c a n d i n d i c a t o r b a c t e r i a i n the same m a n n e r . W i t h s o m e specifications o u t l i n e d below, w e shall a s s u m e t h a t s values for the two p o p u l a t i o n s are the same.
Factors affecting the ratio of pathogens to indicator bacteria T h e use o f i n d i c a t o r bacteria in the asssessment o f r e c r e a t i o n a l w a t e r q u a l i t y is j u s t i f i e d b y the relative scarcity o f pathogens. In p o t a b l e waters one s t r i v e s f o r c o m p l e t e b l o c k a g e o f the r o u t e o f t r a n s m i s s i o n o f i n t e s t i n a l pathogens. R e c r e a t i o n a l activities, however, c a n n o t be restricted to waters free o f i n d i c a t o r b a c t e r i a , a n d q u a n t i t a t i v e c o n s i d e r a t i o n s o f the ratio o f p a t h o g e n i c a n d i n d i c a t o r o r g a n i s m s a r e in o r d e r . T h i s r a t i o d e p e n d s on the n u m b e r o f excreters in the p o p u l a t i o n a n d o n t h e n u m b e r o f p a t h o g e n s excreted by them. A u t h o r i t i e s express confidence t h a t t h e n u m b e r o f e x c r e t e r s for certain diseases within a given w a t e r s h e d can be k n o w n w i t h a 168
high degree of accuracy. This applies particularly to typhoid fever and other conditions for which constant surveillance and reporting are mandatory in many areas. In contrast, the ratio of pathogens to indicator bacteria excreted by carrier persons is known only for a few cases of typhoid fever reported by Thomson 19'2°. According to this author, the ratio of S. typhosa to Escherichia coli in the feces decreases with time. From acute infection to the last phases of the carrier state it can vary over at least six orders of magnitude, with an average of 1/250 in a selection of 12 cases. Better estimates of this ratio are necessary for other situations and diseases and could be easily obtained if the presence/absence tests in routine surveillance of known excreters were replaced by appropriate quantitative tests.
Factors affecting the ratio of pathoyens and indicator bacteria in small polluting populations In large populations the ratio of pathogens to indicator bacteria is rather stable and predictable once the number of excreters and the average number of pathogenic organisms excreted by them are approximately known. In particular, the ratio will not be affected significantly by normal population mobility. The risk arising from small polluting populations, however, is significantly different depending on whether several, one, or no excreter is present. In fact, a beach polluted by a small population which includes a single excreter may be unsafe by any of the commonly accepted bacteriological standards. Conversely, in the absence of an excreter in the polluting population the beach is safe even when it is heavily polluted. Mathematically this situation should be dealt with in two different ways, depending on the intended application of the model. For scientific-epidemiological studies one may utilize data from a larger population (county, state); and assuming equal and random distribution of the excreters, one may use a Poisson distribution to predict the likelihood or fraction of time that 0, 1, 2, etc., excreters are present in the polluting population, multiplying this probability by the overall probability of infection. For public health policy making, however, this approach is not feasible because even the presence of a single excreter in a small polluting population might cause an outbreak involving all persons exposed to the pollution. Such an outbreak not only significantly alters the epidemiological situation upon which bathing beach standards are (or should be) based but in itself constitutes an event which the public health community is called upon to prevent. For setting public health standards, therefore, the number of excreters of a disease in any given population should be assumed equal to the expected number plus one. It will be shown below that this does not change the basis for computing standards for beaches contaminated by large metropolitan populations, but it forces the adoption of more stringent standards for beaches affected by small populations, and in this manner it establishes a logical link to the long-recognized need for zero contamination (in terms of indicator bacteria) in swimming pools and drinking water systems.
169
Die-off Among other factors affecting the ratio of pathogenic and indicator bacteria are the differences in die-off and other forms of elimination, e.g., sedimentation. Die-off rates of S. typhosa and E. coli appear to be of the same general order of magnitude ~5. In cases where they are not, a function of time o f travel from the source of pollution may have to be introduced into the model.
Calculation In principle, the probabilities of ingesting certain doses of pathogenic microorganisms are calculated from the /~ values and s of the population of indicator bacteria and the pathogen/indicator ratio, corrected for small population effects as required. Each of these probabilities is multipled by the probability of contracting the disease from such a dose, and the results are summed to give the probability of infection from ingesting a given volume of the water (50 ml). By using a Wang 720C calculator with 12-digit a.ccuracy and a Wang library program for the calculation o f the normal probability integral, the probabilities Q i = Qx(k-ll2)--Qx(k+ll2) o f encountering between k - 1 / 2 and k + 1/2 organisms were calculated for k = 1, 2, 3... from the arguments x(k+l/2)=[log(k+l/2)--lt]/s. The probabilities of acquiring the disease were calculated by multiplying each Qi by the term 1 - e -pk and by summing the products obtained in this manner for all values of k. Whenever k reached a value which let the term 1 - e -pk assume a value of 0.95 or higher, the probability of ingesting a greater number of organisms k was added as such, assuming that the probability of such a dose causing disease is unity. This step terminated the calculation. For small values of p, the calculations become lengthy because k must be increased by 1 many times before 1 - e -pk reaches 0.95. To avoid this, k was replaced by the term i" t, where t is a scaling factor and i assumes the values .1, 2, 3... An error is introduced insofar as i = 1 at t = 1 represents 0 . 5 < i . t < 1 . 5 , whereas i = 1 at t = 10 covers the range 5 < i. t < 15. Five or less organisms represent a dose with a low but definitive risk of infection, whereas less than one organism does not. Trial runs showed, however, that the error is so small that it does not affect the results to any great extent. The principal risk is associated with the probability of ingesting large doses. Table 1 shows the results for p = 0.2, representing a highly virulent agent causing infection in approximately one out of five cases upon ingestion of a single organism (p = 0.2). For other values of p, the values of the columns can be scaled accordingly. This is shown in an example represented by a column marked " p = 0.0002". The s is given in two forms, in base-10 logarithms and in multiples of # which are exceeded in 20% of the samples in a population (80% percentile value, Qo.a, divided by/~). Table 1 illustrates the fact that the public health risk increases not only with increasing/~ but also significantly with increasing s. A change in s from 0.2 to 0.3 (in base-10 logarithms) corresponds to a 10-fold change in the log mean which, either directly or converted to units of indicator organisms, is the numerical standard, now representing a defined public health risk. Introduction of the other elements into the model is a relatively straightfor170
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