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Density-dependent fecundity in Schistosoma mansoni infections in man
532 TRANSACTIONS OF THE ROYAL SOCIETY OF TROPICAL MEDICINE
Density-dependent
fecundity
AND HYGIENE (1985) 79, 532-534
in SchistusorHa man
mansoni
infections
in
G. MEDLEY AND R. M. ANDERSON Dept. of Pure and Applied Biology, imperial College, London SW7 2BB Abstract
An analysis is presented df autopsy data collected by Cheever in 1968 on the association between Sc~i~~s~ foci worm burdens and faecai egg counts. Signet negative re~tionships are found between the number of ~slworm pair/g faecesand both the number of worm pairs and the total number of worms. A general, non-linear, statistical routine is shown to be the most practicai method of fitting non-linear models to the observed density-dependent fecundity relationship. The conceptual and practical implications of density-dependence are discussedin the context of epidemiological field studies and chemotherapy programmes. Introduction
The per capita fecundity of helminth parasitesoften appeari to be-inversely reiated to the density of worms within a host (SARLES,1929; KRUPP, 1961; ANDERSON, 1978; K~YMER, -1982): This observaiion is of epidemiological significance in the context of understanding the impact of control measures on parasite transmission, and with respect to the use of faecal egg counts as indirect measuresof worm burden (AND% SON& MAY. 19851.In the caseof intestinal nematode infections oi man ihe evidence for density-dependent fecundity is substantial (CROLL et al., 1982; SINNAH, 1982: HALL. 1982: ANDERSON MAY. 1982: MARTIN
et al.; 1983;‘THEI&HLIANG et al., 1984; ANDERSON & SCHAD, 1985). However, the situation is more confused with respect to schistosomeinfections due to the limited data available for analvsis. The most extensive information is that providid by Cheever’s autopsy studies, but various authors have published conflicting reports on the patterns revealed in the data (CHEEVER, 1968; ANDERSON& MAY, 1982). We report the results of the re-analysis of Cheever’s data for evidence of any systematic decreasein eggs per gram of faecesper worm pair (EPG) with increase in to& worm burden (TW) or number of worm pairs PW. Re-analysis
Of the 103 casesdescribed (CHEEVER,1968), only 65 were used in the analyses. 15 were excluded for lack of egg counts, 10 cases for non-quantitative scoresof egg counts, sevenfor absenceof worm pairs, three for the use of antimonials before investigation and three for failure to examine the intestinal mucosa. In order to normalize the highly-skewed distributions of the EPG, TW and WP, the natural logarithms of the variate; were used doughout the a&lysis. No significant deviation from normality was observed in the transformed data. Evidence for the existence of a relationship between two normal variates can be established by examination of the correlation coefficient of the bivariate normal distribution. Pearson’s correlation coefficient was calcuiated between In (EPG) and In (TW) (1:= -0.256, PtO.05) and between In (EPG) and In (WP) (r = -0.277, P
provides clear evidence of a statistically significant negative association between per capita fecundity and worm burden c’both TW and WPJ. Analvsis of variance showed-that sex and race had no sig&icant effect on EPG, TW and WP; linear regression gave the same result for age. In order to make quantitative use of the relationship demonstrated above, the parameters of several models were estimated from the data. Since no discernible non-linearity was present in the logarithmtransformed data, a simple linear function was fitted to the data to capture the over-all pattern of densitydependent fecundity; where In (EPG) = In (a,) + b, . In (TW) In (EPG) = In (az>+ b2 . In (WP)
(1)
A hidden assumption in such models is that sampling error is multiplicative in the untransformed data. Evidence from studies of intestinal nematode burdens and faecai egg counts (whose logarithmic plots of EPG variances and means yield linear relationships) suggests that this assumption is probably correct for Cheever’s data (CROLL et al., 1982). The ~~sfo~ed data was homoscedastic(variance in EPG equal for all values of TW and WP). Parameter estimation for equation (1) depends on the sources of variation influencing the observations (KENDALL & STUART, 1973; KENDALL, 1951, 1952). It was not possible to distinguish between natural variation and errors of measurement within Cheever’s data since replicated counts (of either EPG, TW or WP) for each casewere not recorded. Thus, variation was assumedto derived from one, combined, additive source. The assumedunderlying model for the transformed data of equation (1) is a power function of the form, EPG=al.TWb’;
EPG=az.WPhZ.
(2)
Other models, such as the exponential, could also be employed. Parameter values for the power and exponential models were obtained with a general, non-linear, least squares routine. The results of fitting three linear models (RICKER, 1973; KUHRY & MARCUS, 1977), and the two non-linear models are
G.
MEDLEY
AND
R.
recorded in Table I. There are considerable differences between the parameter estimates due, in the main, to the different assumptions made when applying the various models. The non-linear technique is considered the most reliable and robust method of parameter estimation given the uncertainty concerning the sourcesand magnitudes of the error terms in equations (1) and (2), and controversy about the Table
I-Parameter
estimates
for equations
M.
statistical techniques used in fitting the linear models (JOLICOEUR,
(1) and (2) derived
95% C.L.
Major Axis
106.9
-0.6960
Standard Major Axis
244.5
-1.815
17.81
Non-Linear Power
45.48
-0.4417
-0.1922
-0.0052
23.12
-0.0188
14.61 31.64
50.50
-0.0007
b 95% C.L. -0.6528 -1.516
101.0
22.97 48.36 23.76 15.09 32.43
-0.7065
-0.2454 -0.4244
35.67
-0.1708
-0.8856 -1.065
15.00
-0.0483
-0.2887
28.99 61.97 Non-Linear Exponential
a2 95% CL.
-0.7244
-0.2332 -0.4181
1975; SOKAL & ROHLF,
methods
-0.1500
-0.9095 -1.095
Linear Regression
by various
bt
aI
1975; RICKER,
1981). The power function provides the best simple empirical description of the observed pattern (Fig. 1). These results have two important implications. First, the estimation of the worm burden by faecal egg counts is made difficult by the inherent variability in EPG combined with the density-dependent factors.
95% C.L.
Method
533
ANDERSON
-0.0664
-0.2980 -0.5201
-0.1694
-0.0141 -0.0499
-0.0023
1
+ 30
+
25
-
0
0.2
0.4
0.8 1 1.4 1.2 ousands) Total Worm Burden (TW) Fig. 1. Density-dependent relationship between egg production (eggs per gram of faecesper worm pair) and total worm burden. The data are taken in blocks of five and the mean plotted to make the trend clearer; they are untransformed and therefore heteroscedastic(greater variance in EPG at lower worm burdens) and highly skewed. The line is the best-fit, non-linear, power model.
534
DENSITY-DEPENDENT
FECUNDITY
The accuracy of estimation would be greatly improved if replicate egg counts were recorded for each individual patient. However, it is interesting to note that above burdens of 100 worms, EPG remains approximately constant (Fig. 1,; the rate of depression of per capita fecundity with changes in worm burden is greatest at low to moderate parasite loads. Similar trends have been recorded for intestinal nematode infections of man (ANDERSON & SCHAD, 1985; CROLL et al., 1982). Second, chemotherapeutic treatment increases the per capita fecundity of the surviving worms within a patient, as a consequence of reduced parasite density. This has two effects: faecal egg counts tend to underestimate treatment efficacy (with respect to reduction in worm load), and increased egg production offsets the antipathological benefit of treatment (pathology being related to egg production). The practical epidemiological significance of density-dependent fecundity is best illustrated by a simple quantitative example. If a patient with 100 worms is treated with a drug of 90% efficacy, then, on the basis of the power relationship (Fig. I), net EPG (EPG/ worm pair times the number of worm pairs) will change from 602 to 117. That is, a 90% reduction in worm burden causes an 80% reduction in egg output. Treatment is clearly beneficial as a means of reducing egg production, but not quite as successful as suggested by reference to the efficacy of the drug in killing adult worms. These factors should be borne in mind when designing community-based chemotherapy programmes, and when interpreting their impact. References Anderson, R. M. (1978). The regulation of host population growth by parasite species. Parasitology, 76, 119-157. Anderson, R. M. & May, R. M. (1982). Population dynamics of human helrninth infections: control by chemotherapy. Nature, 297, 557-563. Anderson, R. M. & May, R. M. (1985). Helminth infections of humans: mathematical models, population dynamics and control. Advances in Parasitology, 24, l-101. Anderson, R. M. & Schad, G. A. (1985). Hookworm burdens and faecal egg counts: an analysis of the biological basis of variation. Transactions of the Royal Society of Tropical Medicine and Hygiene (in press). Cheever, A. W. (1968). A quantitative post-mortem study of schistosomiasis mansoni in man. American Journal of Tropical Medicine and Hygiene, 17, 38-64.
IN
SCHISTOSOMIASIS
MANSONI
IN
MAN
Croll, N. A., Anderson, R. M., Gyorkos, T. W. & Ghadirian, E. (1982). The population biology and control of Ascaris lumbricoides in a rural community in Iran. Transactions of the Royal Socieg of Tropical Medicine and Hygiene, 76, 187-197.
Hall, A. (1982). Intestinal helmimhs of man: the interpretation of egg counts. Parasitology, 85, 605-613. Jolicoeur, I’. (1975). Linear regressionsin fishery research: some comments. Journal of the FisheriesResearch Board of Canada, 32, 1491-1494. Kendall, M. G. (1951). Regression, structure and functional relationship. I. Biometrika, 38, 11-25. Kendall? M. G. (1952). Regression, structure and functional relationship. II. Biometrika, 39, 96-108. Kendall,, M. G. & Stuart, A. (1973). The Advanced Theory of Statsstics. Vol. 2 (3rd edit.). London: Griffin. Keymer, A. (1982). Density-dependent mechanisms in the regulation of intestinal hehninth populations. Parasitologv, 84, 573-587.
Krupp, I. M. (1961). Effects of crowding and of superinfection on habitat selection and egg production in Ancylostoma caninum. 3oumal of Parasitology,
47., 957-961.
Kuhry, B. & Marcus, L. F. (1977). Bivariate hnear models in biometry. Systematic Zoology, 26, 201-209. Martin, J., Keymer, A., Isherwood, R. J. & Wainwright, S. M. (1983). The prevalence and intensity of Ascaris lumbricoides infection in Moslem children in northern Bangladesh. Transactions of the Royal Society of Tropical Medicine and Hygiene, 77, 702-706.
Ricker, W. E. (1973). Linear regression in fishery research. Journal of the Fisheries Research Board of Canada, 30,
409-434 .__ ._ Ricker, W. E. (1975). A note concerning Professor Jolicoeur’s comments. ‘fournal of the FisheriesResearch Board of Canada, 32, 1494-1498.. Sarles, M. I’. (1929). The effect of age and size of infestation of the egg production of the dog hookworm, Ancylostoma caninum. American Joumal of Hygiene, 10, 658-666. Sinnah B. (1982). Daily egg production of Ascuris lumbricoides: the distribution of eggs in the faeces and variability of egg counts. Parasitology, 84, 167-175. Sokal, R. R. & Rohlf, F. J. (1981). Biometry. The Principles and Practice of Statistics in Biological Research (2nd edit.). San Francisco: W. H. Freeman & Co. Thein-Hliang, Than-Saw, Htay-Htay-Aye, Myint-Lwin St Thein-Maung-Myint (1984). Epidemiology and transmission dynamics of Ascaris lumbricoides in Okpo village, rural Burma. Transactions of the Royal Society of Tropical Medicine and Hygiene, 78, 497-504.
Accepted for publication
1st March,
1985.
Density-dependent
fecundity
AND HYGIENE (1985) 79, 532-534
in SchistusorHa man
mansoni
infections
in
G. MEDLEY AND R. M. ANDERSON Dept. of Pure and Applied Biology, imperial College, London SW7 2BB Abstract
An analysis is presented df autopsy data collected by Cheever in 1968 on the association between Sc~i~~s~ foci worm burdens and faecai egg counts. Signet negative re~tionships are found between the number of ~slworm pair/g faecesand both the number of worm pairs and the total number of worms. A general, non-linear, statistical routine is shown to be the most practicai method of fitting non-linear models to the observed density-dependent fecundity relationship. The conceptual and practical implications of density-dependence are discussedin the context of epidemiological field studies and chemotherapy programmes. Introduction
The per capita fecundity of helminth parasitesoften appeari to be-inversely reiated to the density of worms within a host (SARLES,1929; KRUPP, 1961; ANDERSON, 1978; K~YMER, -1982): This observaiion is of epidemiological significance in the context of understanding the impact of control measures on parasite transmission, and with respect to the use of faecal egg counts as indirect measuresof worm burden (AND% SON& MAY. 19851.In the caseof intestinal nematode infections oi man ihe evidence for density-dependent fecundity is substantial (CROLL et al., 1982; SINNAH, 1982: HALL. 1982: ANDERSON MAY. 1982: MARTIN
et al.; 1983;‘THEI&HLIANG et al., 1984; ANDERSON & SCHAD, 1985). However, the situation is more confused with respect to schistosomeinfections due to the limited data available for analvsis. The most extensive information is that providid by Cheever’s autopsy studies, but various authors have published conflicting reports on the patterns revealed in the data (CHEEVER, 1968; ANDERSON& MAY, 1982). We report the results of the re-analysis of Cheever’s data for evidence of any systematic decreasein eggs per gram of faecesper worm pair (EPG) with increase in to& worm burden (TW) or number of worm pairs PW. Re-analysis
Of the 103 casesdescribed (CHEEVER,1968), only 65 were used in the analyses. 15 were excluded for lack of egg counts, 10 cases for non-quantitative scoresof egg counts, sevenfor absenceof worm pairs, three for the use of antimonials before investigation and three for failure to examine the intestinal mucosa. In order to normalize the highly-skewed distributions of the EPG, TW and WP, the natural logarithms of the variate; were used doughout the a&lysis. No significant deviation from normality was observed in the transformed data. Evidence for the existence of a relationship between two normal variates can be established by examination of the correlation coefficient of the bivariate normal distribution. Pearson’s correlation coefficient was calcuiated between In (EPG) and In (TW) (1:= -0.256, PtO.05) and between In (EPG) and In (WP) (r = -0.277, P
provides clear evidence of a statistically significant negative association between per capita fecundity and worm burden c’both TW and WPJ. Analvsis of variance showed-that sex and race had no sig&icant effect on EPG, TW and WP; linear regression gave the same result for age. In order to make quantitative use of the relationship demonstrated above, the parameters of several models were estimated from the data. Since no discernible non-linearity was present in the logarithmtransformed data, a simple linear function was fitted to the data to capture the over-all pattern of densitydependent fecundity; where In (EPG) = In (a,) + b, . In (TW) In (EPG) = In (az>+ b2 . In (WP)
(1)
A hidden assumption in such models is that sampling error is multiplicative in the untransformed data. Evidence from studies of intestinal nematode burdens and faecai egg counts (whose logarithmic plots of EPG variances and means yield linear relationships) suggests that this assumption is probably correct for Cheever’s data (CROLL et al., 1982). The ~~sfo~ed data was homoscedastic(variance in EPG equal for all values of TW and WP). Parameter estimation for equation (1) depends on the sources of variation influencing the observations (KENDALL & STUART, 1973; KENDALL, 1951, 1952). It was not possible to distinguish between natural variation and errors of measurement within Cheever’s data since replicated counts (of either EPG, TW or WP) for each casewere not recorded. Thus, variation was assumedto derived from one, combined, additive source. The assumedunderlying model for the transformed data of equation (1) is a power function of the form, EPG=al.TWb’;
EPG=az.WPhZ.
(2)
Other models, such as the exponential, could also be employed. Parameter values for the power and exponential models were obtained with a general, non-linear, least squares routine. The results of fitting three linear models (RICKER, 1973; KUHRY & MARCUS, 1977), and the two non-linear models are
G.
MEDLEY
AND
R.
recorded in Table I. There are considerable differences between the parameter estimates due, in the main, to the different assumptions made when applying the various models. The non-linear technique is considered the most reliable and robust method of parameter estimation given the uncertainty concerning the sourcesand magnitudes of the error terms in equations (1) and (2), and controversy about the Table
I-Parameter
estimates
for equations
M.
statistical techniques used in fitting the linear models (JOLICOEUR,
(1) and (2) derived
95% C.L.
Major Axis
106.9
-0.6960
Standard Major Axis
244.5
-1.815
17.81
Non-Linear Power
45.48
-0.4417
-0.1922
-0.0052
23.12
-0.0188
14.61 31.64
50.50
-0.0007
b 95% C.L. -0.6528 -1.516
101.0
22.97 48.36 23.76 15.09 32.43
-0.7065
-0.2454 -0.4244
35.67
-0.1708
-0.8856 -1.065
15.00
-0.0483
-0.2887
28.99 61.97 Non-Linear Exponential
a2 95% CL.
-0.7244
-0.2332 -0.4181
1975; SOKAL & ROHLF,
methods
-0.1500
-0.9095 -1.095
Linear Regression
by various
bt
aI
1975; RICKER,
1981). The power function provides the best simple empirical description of the observed pattern (Fig. 1). These results have two important implications. First, the estimation of the worm burden by faecal egg counts is made difficult by the inherent variability in EPG combined with the density-dependent factors.
95% C.L.
Method
533
ANDERSON
-0.0664
-0.2980 -0.5201
-0.1694
-0.0141 -0.0499
-0.0023
1
+ 30
+
25
-
0
0.2
0.4
0.8 1 1.4 1.2 ousands) Total Worm Burden (TW) Fig. 1. Density-dependent relationship between egg production (eggs per gram of faecesper worm pair) and total worm burden. The data are taken in blocks of five and the mean plotted to make the trend clearer; they are untransformed and therefore heteroscedastic(greater variance in EPG at lower worm burdens) and highly skewed. The line is the best-fit, non-linear, power model.
534
DENSITY-DEPENDENT
FECUNDITY
The accuracy of estimation would be greatly improved if replicate egg counts were recorded for each individual patient. However, it is interesting to note that above burdens of 100 worms, EPG remains approximately constant (Fig. 1,; the rate of depression of per capita fecundity with changes in worm burden is greatest at low to moderate parasite loads. Similar trends have been recorded for intestinal nematode infections of man (ANDERSON & SCHAD, 1985; CROLL et al., 1982). Second, chemotherapeutic treatment increases the per capita fecundity of the surviving worms within a patient, as a consequence of reduced parasite density. This has two effects: faecal egg counts tend to underestimate treatment efficacy (with respect to reduction in worm load), and increased egg production offsets the antipathological benefit of treatment (pathology being related to egg production). The practical epidemiological significance of density-dependent fecundity is best illustrated by a simple quantitative example. If a patient with 100 worms is treated with a drug of 90% efficacy, then, on the basis of the power relationship (Fig. I), net EPG (EPG/ worm pair times the number of worm pairs) will change from 602 to 117. That is, a 90% reduction in worm burden causes an 80% reduction in egg output. Treatment is clearly beneficial as a means of reducing egg production, but not quite as successful as suggested by reference to the efficacy of the drug in killing adult worms. These factors should be borne in mind when designing community-based chemotherapy programmes, and when interpreting their impact. References Anderson, R. M. (1978). The regulation of host population growth by parasite species. Parasitology, 76, 119-157. Anderson, R. M. & May, R. M. (1982). Population dynamics of human helrninth infections: control by chemotherapy. Nature, 297, 557-563. Anderson, R. M. & May, R. M. (1985). Helminth infections of humans: mathematical models, population dynamics and control. Advances in Parasitology, 24, l-101. Anderson, R. M. & Schad, G. A. (1985). Hookworm burdens and faecal egg counts: an analysis of the biological basis of variation. Transactions of the Royal Society of Tropical Medicine and Hygiene (in press). Cheever, A. W. (1968). A quantitative post-mortem study of schistosomiasis mansoni in man. American Journal of Tropical Medicine and Hygiene, 17, 38-64.
IN
SCHISTOSOMIASIS
MANSONI
IN
MAN
Croll, N. A., Anderson, R. M., Gyorkos, T. W. & Ghadirian, E. (1982). The population biology and control of Ascaris lumbricoides in a rural community in Iran. Transactions of the Royal Socieg of Tropical Medicine and Hygiene, 76, 187-197.
Hall, A. (1982). Intestinal helmimhs of man: the interpretation of egg counts. Parasitology, 85, 605-613. Jolicoeur, I’. (1975). Linear regressionsin fishery research: some comments. Journal of the FisheriesResearch Board of Canada, 32, 1491-1494. Kendall, M. G. (1951). Regression, structure and functional relationship. I. Biometrika, 38, 11-25. Kendall? M. G. (1952). Regression, structure and functional relationship. II. Biometrika, 39, 96-108. Kendall,, M. G. & Stuart, A. (1973). The Advanced Theory of Statsstics. Vol. 2 (3rd edit.). London: Griffin. Keymer, A. (1982). Density-dependent mechanisms in the regulation of intestinal hehninth populations. Parasitologv, 84, 573-587.
Krupp, I. M. (1961). Effects of crowding and of superinfection on habitat selection and egg production in Ancylostoma caninum. 3oumal of Parasitology,
47., 957-961.
Kuhry, B. & Marcus, L. F. (1977). Bivariate hnear models in biometry. Systematic Zoology, 26, 201-209. Martin, J., Keymer, A., Isherwood, R. J. & Wainwright, S. M. (1983). The prevalence and intensity of Ascaris lumbricoides infection in Moslem children in northern Bangladesh. Transactions of the Royal Society of Tropical Medicine and Hygiene, 77, 702-706.
Ricker, W. E. (1973). Linear regression in fishery research. Journal of the Fisheries Research Board of Canada, 30,
409-434 .__ ._ Ricker, W. E. (1975). A note concerning Professor Jolicoeur’s comments. ‘fournal of the FisheriesResearch Board of Canada, 32, 1494-1498.. Sarles, M. I’. (1929). The effect of age and size of infestation of the egg production of the dog hookworm, Ancylostoma caninum. American Joumal of Hygiene, 10, 658-666. Sinnah B. (1982). Daily egg production of Ascuris lumbricoides: the distribution of eggs in the faeces and variability of egg counts. Parasitology, 84, 167-175. Sokal, R. R. & Rohlf, F. J. (1981). Biometry. The Principles and Practice of Statistics in Biological Research (2nd edit.). San Francisco: W. H. Freeman & Co. Thein-Hliang, Than-Saw, Htay-Htay-Aye, Myint-Lwin St Thein-Maung-Myint (1984). Epidemiology and transmission dynamics of Ascaris lumbricoides in Okpo village, rural Burma. Transactions of the Royal Society of Tropical Medicine and Hygiene, 78, 497-504.
Accepted for publication
1st March,
1985.