Simple Models of Influenza Progression within a Heterogeneous Population
In the May-June, 2007, issue of Operations Research, Professor Richard C. Larson looks at the role that operations research can place in one of the most pressing issues of our time: handling a possible influenza pandemic. In his abstract, he outlines his goals:
The focus of this ‘OR framing paper’ is to introduce the OR community to the need for new mathematical modeling of an influenza pandemic and its control. By reviewing relevant history and literature, one key concern that emerges relates to how a population’s heterogeneity may affect disease progression. Another is to explore within a modeling framework ‘social distancing’ as a disease progression control method, where social distancing refers to steps aimed at reducing the frequency and intensity of daily human to human contacts. To depict social contact behavior of a heterogeneous population susceptible to infection, a non-homogeneous probabilistic mixing model is developed. Partitioning the population of susceptibles into subgroups, based on frequency of daily human contacts and infection propensities, a stylistic difference equation model is then developed depicting the day-to-day evolution of the disease. This simple model is then used to develop a preliminary set of results. Two key findings are (1) early exponential growth of the disease may be dominated by susceptibles with high human contact frequencies and may not be indicative of the general population’s susceptibility to the disease; and (2) social distancing may be an effective non-medical way to limit and perhaps even eradicate the disease. Much more decision-focused research needs to be done before any of these preliminary findings may be used in practice.
In the paper, Prof. Larson provides a number of simple, yet plausible, difference equations and uses them to model influenza spread in and environment with a population that is heterogeneous in the amount of social interaction made. In his conclusions, Prof. Larson describes his view of the rationale and importance of this research:
No one knows how or when the next pandemic influenza will emerge and what its intrinsic properties will be. If history can be a guide, the next influenza will have ‘emergent properties,’ meaning that it will mutate during the course of the epidemic and its intrinsic properties will evolve accordingly. Any mathematical model of the disease and its control is bound to be incorrect. But we are not seeking multi-decimal numerical accuracy but rather insights on how to limit the spread of the disease. We firmly believe that fresh eyes from the OR community can play a significant role in this quest.
You can read the 
full paper here
full paper here along with its
online companion
.
The editors of this journal have invited three individuals and groups to comment on Prof. Larson’s paper.
Atul Bhandari, Andrew Schaefer, and Derek Cummings. Atul Bhandari is a Post-Doctoral Associate at the Department of Industrial Engineering, University of Pittsburgh. He does research in the fields of Medical Decision Making (Therapeutic Optimization) and Public Health (Evaluation of Pandemic Flu Strategies). Atul obtained his PhD in Operations Research at the Tepper School of Business, Carnegie Mellon University in 2006.
Andrew Schaefer is an Associate Professor of Industrial Engineering and Wellington C. Carl Fellow at the University of Pittsburgh. He received his PhD in Industrial and Systems Engineering from Georgia Tech. His research interests are in therapeutic optimization, stochastic programming and integer programming
Derek Cummings is a Visiting Assistant Professor at the Department of Epidemiology, University of Pittsburgh Graduate School of Public Health. His research focuses on the dynamics of infectious disease transmission including influenza, and dengue transmission. Derek received his doctoral training in Environmental Engineering at the Johns Hopkins School of Engineering in 2004. He received an M.H.S. in International Health from the Johns Hopkins Bloomberg School of Public Health in 2004 as well.
In their commentary, Drs. Bhandari, Schaefer, and Cummings note some of the literature related to Prof. Larson’s work:
"Larson has done a commendable job in reviewing relevant history and literature and, most importantly, pointing out the limitations of some existing epidemiological models, such as assuming a homogeneous population. We are familiar with the papers by Ferguson et al. (2005, 2006) as well as the inner workings of the mathematical simulation used by these authors to study the spread of pandemic flu in Southeast Asia, the UK, and the USA. We are involved in the further use and development of this mathematical simulation and base our comments on our experience with this model. First, we discuss how
the models of Ferguson et al. address most of the issues raised by Larson, while at the same time noting some of the difficulties in obtaining the relevant data. We also comment on the multiple, possibly conflicting, objectives involved in containment/mitigation of pandemic flu. Finally, we would like to comment on the increase in computational burden as the models of flu progression are made increasingly heterogeneous."They note the large amount of time required by detailed, individual-agent based models, which limits the role such models can play:
"Finally, we comment on the fact that making influenza models increasingly realistic while simultaneously accounting for the entire population of the region under consideration increases computational burden. For example the simulation model by Ferguson et al. required 2-5 hours on 16 CPUs of the NCSA SGI ALtix 3700 system and used 55GB of RAM for each realization for the US. Obtaining statistically significant results required 5 to 20 realizations which increases total computation time significantly. Overall, approximately 20,000 CPU hours were used to generate the US results, and approximately 8,000 CPU hours to generate the Great Britain results. While such overhead is acceptable to obtain offline results, such models can not be used to support dynamic real-time decision making during a flu pandemic. These models can be used to develop pre-pandemic strategies and for simulated-training of authorities and enforcers. Hence, we are of the view that models considering large populations should make acceptable tradeoffs by modeling certain heterogeneous aspects of the population while making certain other homogeneous assumptions.
"They, too, close with a comment on the role OR plays in influenza modeling:
"We are of the opinion that various OR techniques such as stochastic programming, queuing theory, graph and network theory, etc. would be of immense value in providing decision support for the control/mitigation of pandemic flu. At the same time, any proposed strategy should be evaluated via detailed simulation if policy makers are to be convinced of the value of operations research.
"Stephen Chick. Stephen Chick is a Professor of Technology and Operations Management at INSEAD. He received his MS and PhD from the University of California at Berkeley in Industrial Engineering and Operations Research and his BS in Mathematics from Stanford University. His research brings together simulation and statistical decision making tools to help improve process design and public health decisions. His work on epidemic modeling and simulation and vaccine and intervention trial analysis has been funded by the US Environmental Protection Agency and the Centers for Disease Control. Most recently he working on integrating operations management and health care concepts to find cost-effective ways of managing the influenza vaccine supply chain, and cost-benefit analysis for the control of infectious diseases, such as vCJD.
Prof. Chick also believes there are a number of opportunities for OR to play in influenza modeling, and he too supplements Prof. Larson’s discussion of the literature:
"“Simple Models of Influenza Progression within a Heterogeneous Population” targets the important goal of engaging the operations research (OR) community to work on challenging and important problems in infectious disease management and control. Valuable points Larson makes include (1) the importance of developing models with a decision-context in mind, (2) the examination of system-level control alternatives such as public health interventions as well as medical interventions, including state-dependent control decisions, (3) a discussion of behavioral issues and disease control that may depend upon the progression of an outbreak. That said, I would like to challenge some statements in the paper with respect to the modeling of heterogeneity in populations, and with respect to the so-called “basic reproductive ratio” R0, a concept that is fundamental to epidemiology and to the control of infectious diseases. This commentary first reviews some existing literature on heterogeneous population models and R0 that supplements the discussion in Larson’s paper. With this supplemented view of epidemic models, some different high-level perspectives emerge about ways that OR modelers can contribute to the field of infectious disease control. There are a number of exciting opportunities.
"Prof. Chick gives a number of areas where OR expertise would be valuable. These include
"- Complexity. How complex must a model be in order to embed the important system-level features?
- Uncertainty. Can richer models be created that better handle uncertainty?
- Input/Output perspectives. How can measures like disease incidence, an output measure, be used to create the inputs that are normally used in OR models?
- Broader health delivery view. How can issues like supply chains, predictive planning and other OR applications be integrated into pandemic research?
- Implementation. How can these models best be solved, and what are the effects of the approximations used?
- Model selection. With the choices of discrete-time versus continuous time, stochastic or differential equations, Markovian or generalized Markovian and the many other choices of modeling paradigm, there is a need for attention to the choice of model for these problems.
"Prof Chick concludes with a call for more OR on these problems:
"Health care in general, and public health or even influenza control in specific, can benefit greatly from operations research thinking. In the other direction, what is particularly exciting is the richness of existing models and analytical tools from mathematical biology, epidemiology and biostatistics that is available to inform individuals with an OR background. It takes time to learn about these results, but it is time well spent – the payoff is greater sophistication in the OR
tool kit and the opportunity to contribute to resolving pressing problems."Steven Hinrichs. Dr. Steven Hinrichs is the Director of the Nebraska Public Health Laboratory (NPHL) and Director of the University of Nebraska Center for Biosecurity. He is the Stokes-Shackleford Professor of Pathology and Microbiology at the University of Nebraska Medical Center (UNMC) in Omaha and principal investigator of awards from the Department of Defense for projects related to Chemical and Biological terrorism preparedness. One of the programs within the Public Health Lab has received an award from the Association of Public Health Laboratories (APHL), the Centers for Disease Control and Prevention (CDC) for the development of an outreach program to extend training and expertise in the early recognition of biological agents at the community level. As part of this program his laboratory has developed several advanced technologies for the identification of infectious agents and electronic reporting mechanisms. The Center for Biosecurity is responsible for coordinating a broad range of activities across the University system. Dr. Hinrichs’ medical background includes board certification in anatomic and clinical pathology after completion of a residency at the University of California, Davis Medical Center. He obtained research training in molecular biology at the NIH and maintains an active research program in molecular pathogenesis related to infectious disease. One component of the work focuses on antibiotic resistance mechanisms in level A or select organisms. Dr. Hinrichs has published over 100 papers in basic science and medical journals.
In his commentary, Prof. Hinrichs notes that Prof. Larson’s models, while simple, offer significant policy insight:
"The need for situational awareness has become the goal not only for military operations but also public health officials charged with monitoring and controlling the outbreak of disease. In this case, situational awareness refers to the ability to gather information and create knowledge about what is happening on the front lines and whether response measures are impacting the challenge or serving to reduce the risks. The paper by Richard Larson significantly adds to the effort to obtain a better understanding of the impact of social factors on the spread of an infectious disease such as influenza and also introduces important topics that influence the accuracy of models to predict the event.
My interest in “Simple Models of Influenza Progression within a Heterogeneous Population” is two-fold. First, as many authors have described, the impact of pandemic influenza on our society and its economic engines could be immense. As Larson correctly points out, the challenge is so great that we must take advantage of the expertise from all available sources. The contribution by Larson, an expert in systems engineering and operational research illustrates the value of bringing the entire scientific community to bear on a challenge of this magnitude. Secondly, the primary focus of Larson’s work is characterizing the impact of a heterogeneous population on the spread of influenza. The observations that are made lend additional credence to the use of social distancing as an approach for reducing the impact of pandemic influenza. No doubt others will
critique the mathematical models and representations as overly simplistic, but any progress made towards recognizing the important factors contributing to a model is valuable and an important contribution."Prof. Hinrichs notes the importance of the social network aspects of Dr. Larson’s work throughout the entire course of a pandemic, and the role OR plays in policies during that time:
"The final observation from the studies of Dr. Larson relate to a well described phenomenon of influenza outbreaks, and that is they occur in waves, or a successive pattern of waxing and waning of new cases, analogous to the aftershocks of a major earthquake. This emphasizes the need to maintain appropriate controls or continue the explanation for the need for social distancing until the peak risk has passed. The model predicts that if control measures are relaxed too early, that another wave of influenza is likely to hit. But what is the ability of the government to influence social behavior over an extended period of time? Can other social institutions be brought to bear and do we have general agreement on their utility, such as religious and academic organizations? Taken together, these observations and questions arising from the paper by Larson, illustrate the value of broadening the research base of public health so that the perspective and insights from a
variety of academic fields can be used for the benefit of the world."
Prof. Larson provides a detailed response to the commentary. In that response, he states:
Before we get into details, we discuss a few general ideas. As the paper states, all mathematical models of reality are wrong, it is just that some offer better insights for decision making than others. Our purpose is to ignite the OR community to start to create better decision-consequential models for severe infectious diseases such as pandemic influenza. The very existence of the paper as published followed by this on-line forum suggests that the first steps have been successfully negotiated. In modeling there is always the tradeoff between analytical simplicity, often with accompanying theoretical elegance, and operational detail as one can achieve with simulations. Pandemic flu is no exception. One can use the S-I-R approach with coupled differential equations with constant coefficients to generate often-elegant models. But these models have many problems in reality as they are deterministic and usually suggest that the physical dynamics of disease progression remains unchanged during the course of the epidemic. I agree with Steven Chick that such models play a role in epidemiology analogous to that played by the M/M/1 queue model in waiting line systems; they provide initial insights but are simplistic. At the other end are agent-based simulations that may give the impression of accuracy with up to 300 million agents, one for every man, woman and child in the United States. Hours and hours of super computer time are required to run such models, with other untold hours required to set up runs and then to “diagnose results.” This logic that “size is everything” reminds me of a colleague recently who said, “Dick, this Systems Dynamics model is perfect for my research problem because it has 420 flow variables.” In modeling, size is not everything, and sometimes less is more. The trick, an art as much as a science, is to find the right balance: the least modeling complexity to offer valuable decision insights that may be available from data and known process dynamics.
His response then continues with a discussion of R0:
Regarding the key parameter of so many epidemiology models, R0, I have several comments. It is clear that R0 is an important parameter in many perhaps most such models. But the hyper strong reaction against those who criticize R0, suggests that it has become a type a sacred cow of epidemiological modeling. It is not healthy in a profession for certain long-held assumptions to be placed above criticism. Many perhaps most major breakthroughs in science over the years involved successful attacks on the sacred cows of the time, and these attacks too were usually met with massive criticism. I stand by my comments on R0 in the paper. There are at least five reasons why I think R0 is a flawed concept…
He then provides a detailed response to each comment. The
full response is available here
.
What role should the OR community play in influenza pandemic research and policy? What limits OR from having more of a role to play? What are important directions for our field to take? Your thoughts and comments on the paper, commentary, and area of study are welcome.
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Posted by Mark Temple on 07/10/2007 at 12:00 pm