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Modeling Infection Transmission— The Pursuit of Complexities That Matter

Koopman, James S.


From the Center for the Study of Complex Systems and Department of Epidemiology, University of Michigan, Ann Arbor, Michigan.

Address correspondence to: Jim Koopman, Dept. of Epidemiology SPH-1, 109 Observatory Street, Ann Arbor, MI 48109;

Support for the work providing a basis for this commentary comes from EPA grant CR 827427–01–0 and NIAID grant F002732.

The assessment of risk and benefit in public health often assumes that a population is merely the sum of its individuals. It assumes that population outcomes are not affected by interactions among individuals. This assumption implies that there is no feedback through which outcomes in one individual affect outcomes in others. In the case of transmissible infections this assumption seems likely to invalidate the conclusions of many analyses based on it. Moreover, since infection levels can often be more successfully reduced by changing contact patterns than by changing individual risk factors, assuming away contact pattern effects blinds us to the most effective interventions.

Epidemiology is rapidly improving its capacity to model the complexities of transmission of infectious diseases. New approaches are adding flexibility and power to transmission system analyses. 1 But the challenges are formidable. One major problem is how to model the contact patterns between susceptible and infectious individuals. Another is how to model the genetic diversity of the infectious agent. Both of these add new levels of complexity to the analysis.

In this issue of the journal, Hughes et al. 2 model the potential effects of human papilloma virus (HPV) vaccines on the prevention of cervical cancer. Their initial steps to address this question are hopefully the beginning of a successful journey. Some of the key issues are discussed here.

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Modeling the Diversity of the Infectious Agent

Most infection transmission system models treat the infectious agent as a static and homogeneous entity. In real life, common infectious agents often demonstrate great antigenic and pathogenic diversity and they evolve rapidly. Agents can reach a high level of sustained endemic circulation by developing antigenic diversity, thus escaping immune responses. The immune system responds by increasing its own complexity. The diversity of our human leukocyte antigen (HLA) system most likely arose from such a process. Antigenic diversity and the diverse human responses to pathogens are a fascinating adaptive system process. 3 The evolutionarily stable solution is for the agent and host each to develop a high level of diversity in adapting to the other.

“The models of Hughes et al. make unrealistic assumptions. The very nature of models requires that they make unrealistic assumptions.”

Hughes et al. 2 have made some extreme simplifying assumptions in addressing agent diversity. They assumed that the circulation of diverse HPV agents is not influenced by cross-reactions and interference among infectious agents. They did this by first using a model of the circulation of a homogeneous population of HPV viruses. They then modeled the effects of infection with diverse HPV types under the assumption that all HPV variants have the same transmission dynamics. This approach requires the assumption that the dynamics of all strains taken together is merely the sum of the dynamics of each strain taken separately. Of course all HPV variants do not have the same transmission dynamics, and there may well be immune cross-reactions or interference among HPV strains.

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Modeling Contact Patterns Among Persons

Hughes et al. 2 used a simple mixing structure to describe contact patterns among individuals. They assumed that contacts are events with no duration in time, and that populations mix completely and thoroughly in the instant after a contact. These assumptions are intrinsic to compartmental models assuming a single mixing site. A further assumption intrinsic to the deterministic differential equation form of the compartmental models used by Hughes et al. is that every subgroup in the population behaves as if it were an infinite size, with no stochastic events such as the die-out of infection in some local part of a population.

To their credit, Hughes et al. 2 tested the robustness of their conclusions to certain aspects of contact patterns, namely the assumptions regarding patterns of mixing. They validated this aspect of their model for the control program issues they address by demonstrating that changes in this class of model assumptions do not change their control program conclusions.

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Validating Model Use

The models of Hughes et al. 2 make unrealistic assumptions. The very nature of models requires that they make unrealistic assumptions. Models do not have to be perfect to provide insights and to identify new research directions. They need only be valid for the specific purposes to which they are applied.

Hughes et al. 2 seek to understand how HPV vaccines that have been directed to HPV variants and to various population subgroups could affect the spread and control of cervical cancer. Their work advances our understanding of the HPV transmission system. But the validity of their models for informing the control program conclusions they reach needs further assessment. Their conclusions might be reversed when some of the simplifying assumptions they make are relaxed and further model realism is added. Just making a model more realistic does not necessarily enhance model validity for a specific purpose. Model validity is supported, however, by finding that further model realism will not change conclusions based on model analyses. Sensitivity analyses must explore the vulnerability of the conclusions to adjustments in the assumptions. There are no hard and fast rules to guide such work. It relies on the intuitions of the investigators. Different researchers may have different intuitions. This argues for full availability of the details (and software) involved in any given model, so that in the spirit of scientific inquiry, others might then pursue their own intuitions.

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Assessing the Conclusion About Targeting High-Risk Groups

Hughes et al. 2 conclude that there is little advantage in directing HPV vaccines toward high-risk groups. This is supported by the following intuition. When infection rates are high in low-risk groups, those groups could be sustaining transmission on their own. Eliminating the introduction of infection from high-risk groups (eg, by targeted immunization) will in that case not have the dramatic indirect effects that would occur if the low-risk group could not sustain circulation. Still, this conclusion could be sensitive to assumptions intrinsic to the model form used by Hughes et al., which enables low-risk groups to sustain transmission in the model when they do not do so in the real world. Assumptions that might have this effect are that mixing is instantaneous and thorough in the population, and that the population size is effectively infinite. These assumptions do not allow the model of Hughes et al. to simulate the following realistic scenario.

In a population where contacts are frequently local (in either geographic or social space), infection might disappear by chance in a local population. This would happen more frequently in a population of low-risk individuals than high-risk individuals. In that case, some high-risk individuals might introduce infection into infection-free low-risk populations, and thus set off new chains of transmission. This might amplify the effect of interventions on the high-risk group.

We have demonstrated that such effects from local mixing can be very strong and that the general type of model used by Hughes et al. 2 is incapable of capturing these effects. 4 Therefore, the sensitivity of their conclusion to assumptions intrinsic to the model form they used should be assessed. This can be done by elaborating the model from a deterministic compartmental form to its stochastic analog, and then adding geographic and social structure in a way that adds local contacts to the purely global contacts in the original model.

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Assessing the Need to Retain Pap Smear Programs

Hughes et al. 2 conclude that Pap smears will continue to be cost effective even after an immunization program. Is this conclusion sensitive to the limitations described above? For geographically related stochastic effects to reverse this conclusion, these effects would probably have to enhance the effect of vaccine in reducing population levels of infection in the high-risk group as well as the low-risk group. Whether vaccines would have greater effects on high-risk groups in stochastic models with realistic contact structures than in the deterministic models used by Hughes et al. is unclear. Finding that vaccine effects on high-risk groups are not greater in stochastic models would help validate the use of deterministic models to assess this issue of the continuing need for Pap smears. However, lacking a clear scenario as to how vaccine effects in the high-risk group would be greater in a stochastic model, there may be less motivation to carry out a stochastic analysis to assess the validity of this conclusion than in the case of focusing interventions on the high-risk group.

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Promoting a Science of Infection Transmission System Analysis

The value of infection transmission system models such as those presented by Hughes et al. 2 goes beyond specific policy questions. Each effort to model a transmission system brings new insights about that transmission system. These insights may be about potential limitations of the model, or unanticipated effects, or simply a clearer picture of the forces in motion. All such benefits contribute to the science of modeling infection transmission systems. But the key to advancing such a science is to promote communication about models in a fashion that makes it easier to build on the work of others.

In a flourishing science of infection transmission system analysis, the publication of a new model would ideally stimulate other scientists to consider what assumptions are most vulnerable in the model. Publication would stimulate new questions that the model could address, and suggest possible sources of data that might resolve questions raised by the model analysis. Towards these goals, it would help to have software that allowed modelers easily to reproduce a published model analysis and then to explore the robustness of the assumptions of that model.

Such model flexibility would be particularly valuable in suggesting new data that might help resolve key questions raised in the model analysis. For example, one classic approach is to use two models expressing alternative theories to define the key observations that would be consistent with one model but not the other. Another strategy is to use the models to simulate data, and then use the simulated data to assess the power and validity of alternative study designs. For either purpose, it helps if the models can generate the type of data one would collect in a study.

In the case of the Hughes et al. 2 model, this would involve further elaboration to a stochastic compartmental model form that permits individual event histories to be followed separately. Such models, if elaborated to encompass the various types of data that might be useful, could help define the data needed to resolve the issue of whether high-risk individuals are setting off chains of infection in populations of lower-risk individuals. Models in which the high-risk group plays this role could be compared with models in which it does not, under various conditions of person, place and time patterns of infection, risk factor associations, and strain diversity patterns by risk group. This would help identify populations for study and the data to be collected from these populations that could best demonstrate whether models with or without a key introduction of infection role for high-risk groups are closer to reality.

Funded by a grant from that National Institute of Allergies and Infectious Diseases (NIAID), Biomedware Inc. (Ann Arbor, MI) is developing software that will facilitate model communication and model transition to new model forms. Its Model Transition Sensitivity Analysis (MTSA) software is being designed to permit scientists to transit between model forms in a way that can relax unrealistic assumptions and enable the elaboration of model details in new dimensions not possible in the preceding model form. The initial parameter values of the new model form can be set so that behavior of both model forms is identical. This provides a test of code validity that is essential for a science so dependent upon computer code. The transitions currently being incorporated into this software have been recently discussed. 1

The intent of this MTSA software development is to provide open-source software that can be used by scientists taking diverse approaches to the modeling of infection transmission systems. If a scientist has a new model form, and if transitions to this new model form can be established, the common basis for communication between infection transmission system scientists will be expanded. Although there should never be any limitations on the model forms (or the computer implementation of those forms) that scientists can publish, publishing models in a form that facilitates communication and further analysis of one’s work will always be beneficial.

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About the Author

JIM KOOPMAN set his sights on epidemiology in 1963, but pursued a medical degree and a residency in pediatrics before launching his epidemiology career. Although always interested in theory, he spent the first 16 years of his career on fieldwork, mostly with public health responsibility and mostly in developing countries. For the last 16 years he has been developing new theory and exploring new sources of data that can help epidemiology analyze the system that circulates infection in populations, assess transmission risks, and design effective control efforts. He loves to escape to Michigan’s Upper Peninsula to do this.

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1. Koopman JS, Jacquez G, Chick SE. New data and tools for integrating discrete and continuous modeling strategies. In: Weinstein M, Hermalin A, Stoto MA, eds. Population Health and Aging: Strengthening the Dialogue Between Epidemiology and Demography. New York: New York Academy of Sciences, 2001; 268–294.
2. Hughes JP, Garnett GP, Koutsky L. The theoretical impact of a human papilloma virus vaccine. Epidemiology 2002; 13: 631–639.
3. Holland JH. Hidden Order: How Adaptation Builds Complexity. Reading, Massachusetts: Addison-Wesley, 1995.
4. Koopman JS, Chick SE, Simon CP, Riolo CS, Jacquez G. Stochastic effects on endemic infection levels of disseminating versus local contacts. Math Biosci 2002; in press.
© 2002 Lippincott Williams & Wilkins, Inc.