AMBHICAN JOUBNAL OF EPIDEMIOLOGY
Vol. 110, No. 1
Copyright © 1979 by The Johns Hopkins University School of Hygiene and Public Health All rights reserved
Printed in USA.
ANTIBODY AGAINST HEPATITIS A IN SEVEN EUROPEAN COUNTRIES II. STATISTICAL ANALYSIS OF CROSS-SECTIONAL SURVEYS D. SCHENZLE,1 K. DIETZ1 AND G. G. FROSNER1 Schenzle, D. (Instltut fur Med. Blometrie, D-7400 Tubingen, West Germany), K. Dletz and G. G. Frosner. Antibody against hepatitis A in seven European countries. II. Statistical analysis of cross-sectional surveys. Am J Epidemiol 110:70-76, 1979. The age-specific prevalence of hepatitis A antibodies has been analyzed using a catalytic epidemic model for populations in seven European countries: West Germany, Norway, Greece, Switzerland, Holland, France and Sweden. The results Indicate a significant decline In the force of infection in recent decades. However, there are substantial differences between the countries, especially between the Scandinavian countries and Greece. The incidence of hepatitis A in Norway and Sweden has declined since 1930, while a downward trend In incidence In Greece may have started only recently. cross-sectional studies; epldemlologlc methods; hepatitis A; prevalence studies; statistics
Frdsner et al. in part I of this series (1) infection. This corresponds to the sohave determined the prevalence of anti- called simple catalytic model of infection body to hepatitis A virus (anti-HAV) in (2). In other countries, however, the dissamples of sera from seven European tributions are distinctly s-shaped, indicountries. These cross-sectional data, cating that the force of infection is agewhich represent only a "spot survey" of and/or time-dependent. the corresponding populations, nevertheUsually age-dependency of exposure is less contain information on the endemic considered to be responsible for deviasituation of hepatitis A in the past. The tions from the exponential form, but in observed age distributions of anti-HAV the present study this could hardly exprevalence were found to vary greatly be- plain the differences between countries. tween the countries (see figure 1 in Therefore, in this paper, we will describe Frosner et al. (1)). In Greece, for example, a catalytic model with a time-dependent the form of the distribution is nearly of an force of infection. It will be shown that exponential type, and this can be under- the data can be consistently interpreted stood easily assuming a constant force of if we assume that in recent decades the exposure to hepatitis A has declined at different rates in the various countries. Received for publication September 29, 1978, and in final form November 20, 1978. Abbreviation: anti-HAV, antibody to hepatitis A virus. 1 Institut fur Medizinische Biometrie, University of Tubingen, Tubingen, West Germany. 1 Max von Pettenkofer-Institut fur Hygiene und Mikrobiologie, University of Munich, Munich, West Germany. Reprint requests to Dr. G. Frosner, Max von Pettenkofer-Institut der Universitat Munchen, Pettenkoferstr. 9a, D-8000 Munchen 2, West Germany.
THE MODEL
The derivations and results will rely on the following set of assumptions: 1) Members of a population are susceptible to infection after disappearance of maternal antibody and remain so, until they become infected. 70
71
HEPATITIS A ANTIBODY IN EUROPEAN COUNTRIES. U.
2) Mortality due to infection is negligible. 3) Evidence of exposure to infection is definite during life. 4) Populations are homogeneous with respect to susceptibility and exposure to infection. 5) We assume to have samples of individuals who have spent their entire life in the same population. According to the catalytic model, at any time tc, the fraction u(tc) of susceptibles in a subpopulation of age tc — t0 > 0 will be found by solving the differential equation du(t) = - kit)uit), dt
populations with age i, collected at time
N,itc)'
where S ( « c ) denotes the number of susceptibles in a sample of Niitc) persons of age i at time tc. Although the following analysis does not require that the collection dates be the same for all sampled age groups, we will neglect slight differences and put tc = 0 corresponding to the year 1976. We write u,(0) = u,, S,(0) = S, andN,iO) = N,. We will try to fit the observed frequencies i2| with the following form of time(1) dependency for the force of infection:
with the initial condition
kit) = X. - Lit),
u(t0) = 1.
with
Here kit) denotes the force of infection, which we assume to be a function only of time and not of age. Integrating equation 1, the solution becomes
uitc) = exp - J \(t)dt\.
(5)
(6)
Lit) = kj{l + exp[-a(* - 6)]},
the well-known symmetric logistic function. Thus, kit) approaches the value ka for t —* - oo, whereas it goes to zero for t —> + oo. The parameter a determines the slope of kit) at t = 0. At t = 0, the time of (2) collection of the data, we have
The mean age at attack for individuals born at time t0 will be given by
Lit = 0) = XJ[1 + exp(afl)].
(7)
Therefore, choosing large positive values for a and 6, we can simulate the limiting case kit) = const, for t < 0. T(t0) = ( it - to)kit)uit)dt. (3) The symmetric logistic function in equation 6 has been chosen on purely heuristic grounds. It is a simple threeIf kit) = const. = Xo, we obtain parameter function which we believe is u(tc) = e-Vr - 'o> suited to describe the overall decline of the force of infection. If the data are not and inconsistent with the model, the choice of (4) Lit) will be justified. Using more complio cated functions for kit) could only improve In this case, there is no longer a depen- the fit. dency on the date of birth t0. Integrating in equation 2 from t = -i to t = 0, we obtain
T =f f .
u, = {[1 + exp(a0)]/[l + expiaid + i))]}k*/a (8) The data we have at our disposal are estimates udtc) from samples out of sub- as the probability that an individual of APPLICATION OF THE MODEL
72
SCHENZLE, DIETZ AND FROSNER
age i at time tc = 0 will be susceptible. It is now a straightforward procedure, using maximum likelihood, to deduce estimates for the parameters \m, a and 6 from the ungrouped raw data. If, among those sampled at age i, there are (N, - S,) who have had an infection (i.e, who have antibodies) and S, who have not, then the likelihood function is
L = [I I =
-
«f) cv
Vol. 110, No. 1
Copyright © 1979 by The Johns Hopkins University School of Hygiene and Public Health All rights reserved
Printed in USA.
ANTIBODY AGAINST HEPATITIS A IN SEVEN EUROPEAN COUNTRIES II. STATISTICAL ANALYSIS OF CROSS-SECTIONAL SURVEYS D. SCHENZLE,1 K. DIETZ1 AND G. G. FROSNER1 Schenzle, D. (Instltut fur Med. Blometrie, D-7400 Tubingen, West Germany), K. Dletz and G. G. Frosner. Antibody against hepatitis A in seven European countries. II. Statistical analysis of cross-sectional surveys. Am J Epidemiol 110:70-76, 1979. The age-specific prevalence of hepatitis A antibodies has been analyzed using a catalytic epidemic model for populations in seven European countries: West Germany, Norway, Greece, Switzerland, Holland, France and Sweden. The results Indicate a significant decline In the force of infection in recent decades. However, there are substantial differences between the countries, especially between the Scandinavian countries and Greece. The incidence of hepatitis A in Norway and Sweden has declined since 1930, while a downward trend In incidence In Greece may have started only recently. cross-sectional studies; epldemlologlc methods; hepatitis A; prevalence studies; statistics
Frdsner et al. in part I of this series (1) infection. This corresponds to the sohave determined the prevalence of anti- called simple catalytic model of infection body to hepatitis A virus (anti-HAV) in (2). In other countries, however, the dissamples of sera from seven European tributions are distinctly s-shaped, indicountries. These cross-sectional data, cating that the force of infection is agewhich represent only a "spot survey" of and/or time-dependent. the corresponding populations, nevertheUsually age-dependency of exposure is less contain information on the endemic considered to be responsible for deviasituation of hepatitis A in the past. The tions from the exponential form, but in observed age distributions of anti-HAV the present study this could hardly exprevalence were found to vary greatly be- plain the differences between countries. tween the countries (see figure 1 in Therefore, in this paper, we will describe Frosner et al. (1)). In Greece, for example, a catalytic model with a time-dependent the form of the distribution is nearly of an force of infection. It will be shown that exponential type, and this can be under- the data can be consistently interpreted stood easily assuming a constant force of if we assume that in recent decades the exposure to hepatitis A has declined at different rates in the various countries. Received for publication September 29, 1978, and in final form November 20, 1978. Abbreviation: anti-HAV, antibody to hepatitis A virus. 1 Institut fur Medizinische Biometrie, University of Tubingen, Tubingen, West Germany. 1 Max von Pettenkofer-Institut fur Hygiene und Mikrobiologie, University of Munich, Munich, West Germany. Reprint requests to Dr. G. Frosner, Max von Pettenkofer-Institut der Universitat Munchen, Pettenkoferstr. 9a, D-8000 Munchen 2, West Germany.
THE MODEL
The derivations and results will rely on the following set of assumptions: 1) Members of a population are susceptible to infection after disappearance of maternal antibody and remain so, until they become infected. 70
71
HEPATITIS A ANTIBODY IN EUROPEAN COUNTRIES. U.
2) Mortality due to infection is negligible. 3) Evidence of exposure to infection is definite during life. 4) Populations are homogeneous with respect to susceptibility and exposure to infection. 5) We assume to have samples of individuals who have spent their entire life in the same population. According to the catalytic model, at any time tc, the fraction u(tc) of susceptibles in a subpopulation of age tc — t0 > 0 will be found by solving the differential equation du(t) = - kit)uit), dt
populations with age i, collected at time
N,itc)'
where S ( « c ) denotes the number of susceptibles in a sample of Niitc) persons of age i at time tc. Although the following analysis does not require that the collection dates be the same for all sampled age groups, we will neglect slight differences and put tc = 0 corresponding to the year 1976. We write u,(0) = u,, S,(0) = S, andN,iO) = N,. We will try to fit the observed frequencies i2| with the following form of time(1) dependency for the force of infection:
with the initial condition
kit) = X. - Lit),
u(t0) = 1.
with
Here kit) denotes the force of infection, which we assume to be a function only of time and not of age. Integrating equation 1, the solution becomes
uitc) = exp - J \(t)dt\.
(5)
(6)
Lit) = kj{l + exp[-a(* - 6)]},
the well-known symmetric logistic function. Thus, kit) approaches the value ka for t —* - oo, whereas it goes to zero for t —> + oo. The parameter a determines the slope of kit) at t = 0. At t = 0, the time of (2) collection of the data, we have
The mean age at attack for individuals born at time t0 will be given by
Lit = 0) = XJ[1 + exp(afl)].
(7)
Therefore, choosing large positive values for a and 6, we can simulate the limiting case kit) = const, for t < 0. T(t0) = ( it - to)kit)uit)dt. (3) The symmetric logistic function in equation 6 has been chosen on purely heuristic grounds. It is a simple threeIf kit) = const. = Xo, we obtain parameter function which we believe is u(tc) = e-Vr - 'o> suited to describe the overall decline of the force of infection. If the data are not and inconsistent with the model, the choice of (4) Lit) will be justified. Using more complio cated functions for kit) could only improve In this case, there is no longer a depen- the fit. dency on the date of birth t0. Integrating in equation 2 from t = -i to t = 0, we obtain
T =f f .
u, = {[1 + exp(a0)]/[l + expiaid + i))]}k*/a (8) The data we have at our disposal are estimates udtc) from samples out of sub- as the probability that an individual of APPLICATION OF THE MODEL
72
SCHENZLE, DIETZ AND FROSNER
age i at time tc = 0 will be susceptible. It is now a straightforward procedure, using maximum likelihood, to deduce estimates for the parameters \m, a and 6 from the ungrouped raw data. If, among those sampled at age i, there are (N, - S,) who have had an infection (i.e, who have antibodies) and S, who have not, then the likelihood function is
L = [I I =
-
«f) cv
