THE JOURNAL OF INFECTIOUS DISEASES. VOL. 139. NO.5. l\/AY 1979 © 1979 by The University of Chicago. 0022·1899179/3905-0018$00.75
CORRESPONDENCE Models of Transmission of Infectious Agents SIR-In the editorial on tautologies in epidemic models [1], Stille and Gersten imply that stratification of populations into multiple and more homogeneous mixing groups makes models of disease transmission based on the mass-action law correspond only slightly better to reality than single-group models. Further, they imply that the group-structured models are "tautologous" because the process of stratification would have to be carried to absurd extremes before the models become realistic and because the parameters of these models (the average contact rates) are "unique to a population, setting, and circumstance." The authors then make a laudable call for new modeling approaches whose parameters can be tested by means independent of the models themselves. They imply that these approaches should not be based on the mass-action law. I believe that in trying to adhere to Popper's admonition to devise refutable theories [2], Stille and Gersten have confused tautologous models with irrefutable, and therefore untestable, theories; consequently, they underestimate the epidemiological usefulness of contact group-structured models. All mathematical models are tautologous when they are viewed as purely mathematical phenomena. The mathematical relationships between variables established by the model determine the relationships between variables and the outcomes obtained imply the original mathematical relationships. This is what tautology means. When we try to use mathematical models to abstract characteristics of interest in the real world, the tautologous nature of the models does not make them useless for projecting the behavior of the real world or for deriving information about the concepts implicit in the model. Models are useless for predicting behavior of the real world when the assumptions of the model or the parameters are in error. A model is useless in assessing the appropriateness of the assumptions inherent in it when in any situation that may be encountered in the real world, the parameters of the model can be manipulated to make the model
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fit that situation. When treated as a theory about the real world, this kind of model cannot be tested; therefore, it is irrefutable and can tell us nothing about the ways in which it does or does not correspond to reality. The basic assumption of the Reed-Frost and other mass-action epidemic models is that agents causing infection originate in one infectious individual and are transmitted directly or indirectly, with a uniform probability, to a susceptible individual. Most models also contain assumptions about the natural history of infection and susceptibility to infection. The fact that these models are based on the physical reality of transmissibility is one of their strengths. The problems of fitting data with the models should not cause us to abandon the physical reality of transmission in our modeling attempts, but rather to reject the assumption of uniform probability of transmission or assumptions about the natural history of infection. The structuring of populations into contact groups is a first attempt to go beyond the assumption of uniformity. Paying close attention to Popper's definition of good theory, we see that the difficulties in fitting contact group-structured models to some data could indicate the potential power of these models. These difficulties are perceived because the predictions made by the contact group-structured models are multiple and specific. These predictions include age and mixing group-specific epidemic curves. Consequently, there are many areas in which inconsistencies may be sought. The failure to fit such specific data to models of influenza or other infectious diseases whose natural, history and agent are well understood should not indicate that transmissibility via mass-action probabilities should be abandoned in our models. The weight of evidence is too strong for this. Rather, the failure to fit the data with the models should indicate either that our assumptions about the natural history of disease implicit in the model are in error or that we have failed to define the population groups with enough differences between transmission probabilities to affect significantly the dynamic behavior of the model.
617
Correspondence
Statements about measurable variables such as excretion rates, environmental survivability, or proliferation of agents, and about infective dose in relation to measurable indicators of host resistance could be usefully integrated into many models. I only wish to emphasize that the failure of disease transmission models to advance more dramatically our understanding of the behavior of infectious diseases in populations can be attributed neither to lack of creativity in devising empirically applicable mathematical statements, nor to the tautologous nature of the statements derived. Instead, progress is held up because epidemiologists do not tend to use conceptual models to generate testable hypotheses in cases where these models might be expected to conform reasonably closely to reality. There has not been a real effort on the part of epidemiologists to define the contact groups that make practically significant differences in infection rates. This effort should not be discouraged because tautology in models is confused with irrefutable theory nor because some approaches to the fitting of data to group-structured models have involved the manipulation of so many parameters that, when used as theory, their assumptions are irrefutable. Instead, epidemiologists should be encouraged to gather the data for age and mixing group-specific epidemic curves and for the natural history of infection that will enable them, through the use of the models, to develop theories about transmission of disease-producing agents. JAMES
S.
KOOPMAN
Department of Epidemiology The University of Michigan School of Public Health Ann Arbor, Michigan References 1. Stille, W. T., Gersten, J. C. Tautology in epidemic models. J. Infect. Dis. 138:99-101, 1978. 2. Popper, K. R. Conjectures and refutations: the growth of scientific knowledge. Basic Books, New York, 1965. 3. Elvcback, L. R., Fox, J. P., Ackerman, E., Langworthy, A., Boyd, M. M., Gatewood, L. An influenza simulation model for immunization studies. Am. J. Epidemiol. 103: 152-165,1976. 4. Fox, J. P., Elveback. L. R. Herd immunity: basic con-
Downloaded from http://jid.oxfordjournals.org/ at New York University on September 2, 2015
Assumptions about the natural history of disease should be testable independently of the model. When one begins to use models, however, it is discouraging to see what little work epidemiologists have done to measure duration of infection or relative excretion rates with or without symptoms. The search for appropriate definitions of contact groups is slowed when one can make the model fit the observed data by altering the parameters of the natural history of infection. For some uses of this kind of model, precise definition of contact groups and precise fitting of specific epidemiologic data may not be necessary. Contact group-structured models of transmission have been of considerable use in clarifying concepts of secondary attack rates [3] and herd immunity [4]. They provided a way to evaluate, at least qualitatively, the advisability of immunizing one population group, with intentions of helping another [3,5], and they have shown how our value system may affect those decisions [5]. Many public health problems, on the other hand, could be better attacked if appropriate group divisions were known. With diarrheal disease and many other infectious problems of children in developing countries, we believe that the infant and toddler groups at high risk are not the key population groups in the maintenance of the high endemic levels of the agents. If we can focus on the key groups, we will greatly increase the efficiency of our public health efforts. Developing countries are now debating the use of health service delivery systems that would emphasize prevention alternatively at a family or community level. The separate approaches involve huge sums of money, and models that would help us evaluate which approaches are most efficient would be useful. Contact group-structured models are a natural approach to this kind of problem. In the case of infectious diarrhea, contact group-structured models might indicate where it would be more efficient to focus efforts to interrupt transmission in groups such as schools [6] or working adults [7], rather than orienting interventions toward the classic highrisk infant groups in a family setting. I do not mean to discourage the use of "new empirical statements," as called for by Stille and Gersten "that could be tested in regard to the multiple types of contact information" [1].
CORRESPONDENCE Models of Transmission of Infectious Agents SIR-In the editorial on tautologies in epidemic models [1], Stille and Gersten imply that stratification of populations into multiple and more homogeneous mixing groups makes models of disease transmission based on the mass-action law correspond only slightly better to reality than single-group models. Further, they imply that the group-structured models are "tautologous" because the process of stratification would have to be carried to absurd extremes before the models become realistic and because the parameters of these models (the average contact rates) are "unique to a population, setting, and circumstance." The authors then make a laudable call for new modeling approaches whose parameters can be tested by means independent of the models themselves. They imply that these approaches should not be based on the mass-action law. I believe that in trying to adhere to Popper's admonition to devise refutable theories [2], Stille and Gersten have confused tautologous models with irrefutable, and therefore untestable, theories; consequently, they underestimate the epidemiological usefulness of contact group-structured models. All mathematical models are tautologous when they are viewed as purely mathematical phenomena. The mathematical relationships between variables established by the model determine the relationships between variables and the outcomes obtained imply the original mathematical relationships. This is what tautology means. When we try to use mathematical models to abstract characteristics of interest in the real world, the tautologous nature of the models does not make them useless for projecting the behavior of the real world or for deriving information about the concepts implicit in the model. Models are useless for predicting behavior of the real world when the assumptions of the model or the parameters are in error. A model is useless in assessing the appropriateness of the assumptions inherent in it when in any situation that may be encountered in the real world, the parameters of the model can be manipulated to make the model
616
Downloaded from http://jid.oxfordjournals.org/ at New York University on September 2, 2015
fit that situation. When treated as a theory about the real world, this kind of model cannot be tested; therefore, it is irrefutable and can tell us nothing about the ways in which it does or does not correspond to reality. The basic assumption of the Reed-Frost and other mass-action epidemic models is that agents causing infection originate in one infectious individual and are transmitted directly or indirectly, with a uniform probability, to a susceptible individual. Most models also contain assumptions about the natural history of infection and susceptibility to infection. The fact that these models are based on the physical reality of transmissibility is one of their strengths. The problems of fitting data with the models should not cause us to abandon the physical reality of transmission in our modeling attempts, but rather to reject the assumption of uniform probability of transmission or assumptions about the natural history of infection. The structuring of populations into contact groups is a first attempt to go beyond the assumption of uniformity. Paying close attention to Popper's definition of good theory, we see that the difficulties in fitting contact group-structured models to some data could indicate the potential power of these models. These difficulties are perceived because the predictions made by the contact group-structured models are multiple and specific. These predictions include age and mixing group-specific epidemic curves. Consequently, there are many areas in which inconsistencies may be sought. The failure to fit such specific data to models of influenza or other infectious diseases whose natural, history and agent are well understood should not indicate that transmissibility via mass-action probabilities should be abandoned in our models. The weight of evidence is too strong for this. Rather, the failure to fit the data with the models should indicate either that our assumptions about the natural history of disease implicit in the model are in error or that we have failed to define the population groups with enough differences between transmission probabilities to affect significantly the dynamic behavior of the model.
617
Correspondence
Statements about measurable variables such as excretion rates, environmental survivability, or proliferation of agents, and about infective dose in relation to measurable indicators of host resistance could be usefully integrated into many models. I only wish to emphasize that the failure of disease transmission models to advance more dramatically our understanding of the behavior of infectious diseases in populations can be attributed neither to lack of creativity in devising empirically applicable mathematical statements, nor to the tautologous nature of the statements derived. Instead, progress is held up because epidemiologists do not tend to use conceptual models to generate testable hypotheses in cases where these models might be expected to conform reasonably closely to reality. There has not been a real effort on the part of epidemiologists to define the contact groups that make practically significant differences in infection rates. This effort should not be discouraged because tautology in models is confused with irrefutable theory nor because some approaches to the fitting of data to group-structured models have involved the manipulation of so many parameters that, when used as theory, their assumptions are irrefutable. Instead, epidemiologists should be encouraged to gather the data for age and mixing group-specific epidemic curves and for the natural history of infection that will enable them, through the use of the models, to develop theories about transmission of disease-producing agents. JAMES
S.
KOOPMAN
Department of Epidemiology The University of Michigan School of Public Health Ann Arbor, Michigan References 1. Stille, W. T., Gersten, J. C. Tautology in epidemic models. J. Infect. Dis. 138:99-101, 1978. 2. Popper, K. R. Conjectures and refutations: the growth of scientific knowledge. Basic Books, New York, 1965. 3. Elvcback, L. R., Fox, J. P., Ackerman, E., Langworthy, A., Boyd, M. M., Gatewood, L. An influenza simulation model for immunization studies. Am. J. Epidemiol. 103: 152-165,1976. 4. Fox, J. P., Elveback. L. R. Herd immunity: basic con-
Downloaded from http://jid.oxfordjournals.org/ at New York University on September 2, 2015
Assumptions about the natural history of disease should be testable independently of the model. When one begins to use models, however, it is discouraging to see what little work epidemiologists have done to measure duration of infection or relative excretion rates with or without symptoms. The search for appropriate definitions of contact groups is slowed when one can make the model fit the observed data by altering the parameters of the natural history of infection. For some uses of this kind of model, precise definition of contact groups and precise fitting of specific epidemiologic data may not be necessary. Contact group-structured models of transmission have been of considerable use in clarifying concepts of secondary attack rates [3] and herd immunity [4]. They provided a way to evaluate, at least qualitatively, the advisability of immunizing one population group, with intentions of helping another [3,5], and they have shown how our value system may affect those decisions [5]. Many public health problems, on the other hand, could be better attacked if appropriate group divisions were known. With diarrheal disease and many other infectious problems of children in developing countries, we believe that the infant and toddler groups at high risk are not the key population groups in the maintenance of the high endemic levels of the agents. If we can focus on the key groups, we will greatly increase the efficiency of our public health efforts. Developing countries are now debating the use of health service delivery systems that would emphasize prevention alternatively at a family or community level. The separate approaches involve huge sums of money, and models that would help us evaluate which approaches are most efficient would be useful. Contact group-structured models are a natural approach to this kind of problem. In the case of infectious diarrhea, contact group-structured models might indicate where it would be more efficient to focus efforts to interrupt transmission in groups such as schools [6] or working adults [7], rather than orienting interventions toward the classic highrisk infant groups in a family setting. I do not mean to discourage the use of "new empirical statements," as called for by Stille and Gersten "that could be tested in regard to the multiple types of contact information" [1].
