In This Issue
Engineering and Vaccine Production for an Influenza Pandemic
September 1, 2006 Volume 36 Issue 3

Planning for Pandemics of Infectious Diseases

Wednesday, December 3, 2008

Author: Roy M. Anderson

Health care providers and public health officials can use proven mathematical models to plan responses to a pandemic.

In the past decade, major changes have been made in the way governments and international agencies plan for the management and control of epidemics. These changes have been brought about by the growing realization that mathematical models can provide accurate descriptions and analytical tools for predicting and interpreting the typical course of an epidemic of a given aetiological agent and the consequences of various combinations of control options (Anderson and May, 1991). Just as physicists and engineers have done for much of the past century, health care providers and public health officials are beginning to use well defined mathematical templates to help them understand natural phenomena.

The mathematical theory of epidemics has a long history, dating back at least to the beginning of the twentieth century—and perhaps much earlier, as illustrated by the work of Bernouilli published in 1761 on the mortality caused by smallpox epidemics. Until relatively recently, however, most of this research has been published in specialized technical journals and has not been on the everyday reading lists of physicians and public health workers—let alone policy makers.

A number of recent events have led to changes in this pattern. First, there is a growing appreciation among scientists of the power of mathematics as a tool for dissecting the causal factors that underlie complex dynamic behaviors in biological systems, ranging from ecosystems to the human immune system. Second, and perhaps more important, mathematical models have recently been used successfully to inform responses to epidemics that were covered heavily in the media: (1) an epidemic of foot and mouth disease, a viral disease in cattle and sheep, in Great Britain in 2001; and (2) the 2003 epidemic of SARS, a lethal respiratory tract virus that spread out of China to many other countries at alarming speed.

Thankfully, although both epidemics caused suffering and mortality and much anxiety, they were brought under control relatively quickly. In both cases, mathematical models, constructed and analyzed as events unfolded, provided insight and guidance to veterinary and public health authorities on optimal control policies and the expected course of the epidemics under different intervention scenarios (Ferguson et al., 2001; Lipsitch et al., 2003; Riley et al., 2003). Mathematical models also provided estimates of key epidemiological parameters: typical incubation periods (the average time from infection to the appearance of symptoms); the duration of infectiousness; and the basic reproductive number, R0, the average number of secondary infections generated by one primary infection in a susceptible population.

We are now facing a serious threat of a highly pathogenic strain of avian influenza, H5N1, mutating or reassorting with a human strain to permit sustained transmission within human communities. This threat has stimulated a great deal of research on mathematical models of an influenza A pandemic and the best ways to control its spread and limit consequent morbidity and mortality before the event occurs. Both government departments of health (e.g., in the United Kingdom and the United States) and the World Health Organisation have encouraged this research and are actively participating in the development of quantitative tools to assist in planning for a pandemic.

In the past, such planning would have been based on the consensus opinion of expert committees, consisting largely of specialists in medicine, infectious diseases, biology, and public health. Today, however, calculation and simulation are replacing opinion and consensus—although not yet in all countries. This article includes a brief review of recent work on the development of a mathematical model for an influenza pandemic and offers conclusions based on this research to date.

Simple and Complex Models
Simple mathematical models, which permit a degree of analytical exploration, can improve the general understanding of determinants in the course of an epidemic and the impact of intervention. Even the simplest models are nonlinear and include analyses of once-simple heterogeneities, which are integral to disease spread (Anderson and May, 1991). However, because only a small number of parameters can be estimated from observed trends, the advantages of simple models are limited.

At the other end of the spectrum, complex individual-based stochastic frameworks can simulate the behavior and disease state of all individuals in a defined population, and, given sufficient computational power, can explore the efficacy of various interventions. However, because of the large number of parameters involved, there is always a degree of uncertainty about the validity of simulated patterns.

Ideally, different approaches with varying degrees of complexity should be adopted. After comparing the results, attention should then be focused on differences in model predictions that could influence policy making.

Both simple and complex formulations require a subset of key parameters, such as incubation and infectious periods and the basic R0. Complex models require much more information, such as the details of demography, the movement of people, mixing patterns (at household, local community, and larger scales), and the spatial distribution of the population in a defined area. Ideally, international travel patterns would also be mirrored in these models.

The simplest models view the human population as consisting of individuals classified by disease state—such as susceptible, X; infected but not yet infectious (i.e., incubating), H; infectious, Y; and recovered, Z. The key equations follow:


Here, ? is the transmission coefficient, 1/? represents life expectancy, 1/(?+γ) is the average incubation period, 1/(?+α+γ) is the average duration of infectiousness, and a is the disease-induced death rate. Given the very short time scale of influenza epidemics, which are typically over in a given country within six months, births are not accounted for. In some circumstances, an epidemic may last for two influenza seasons (a season usually lasts from October to February in the northern hemisphere).

Ferguson et al. (2001) have described more complex models, in which the fine details of age structure and spatial distribution of population density are accounted for, as is the capability of embedding distributions for incubation and infectious periods, mixing patterns, and spatial kernels for the probability that someone in a given location will move a certain distance in a defined period of time. The day of the week, spatial location (i.e., city or remote rural area), and time of year are important parameters in these probability distributions.

Measurements of movements and mixing patterns can be made in a variety of ways, including using population-based anonymized samples of tracking data based on mobile phone use, census and survey records of travel patterns, and questionnaires. Work to date suggests that the probability distributions of movement are not scale-free except for a middle section of distance movement between local and long distance. The vast majority of people move mostly locally. A few, so-called “super-travelers,” often contribute most to the early stages of epidemic spread. This was well illustrated by the SARS epidemic in March 2003 (Hollingsworth et al., 2006).

If, as Ferguson et al. (2006) suggest, reasonable parameterization is possible for complex models, they would have the great advantage of including a great amount of detail for possible control options. For influenza A, the options are focused on: increasing “social distance” (i.e., restricting travel or encouraging people to stay at home and not attend social, entertainment, or sporting events); therapeutic or prophylactic treatment with anti-influenza drugs (such as oseltamivir); contact tracing and treatment; and vaccination, either before or after the start of an epidemic. Complex, individual-based, stochastic simulation structures can mirror the logistics of drug and vaccine delivery, as well as delays in the diagnosis, treatment, and isolation of infectious patients.

Despite the differences in the level of detail in simple and complex models, their predictions of overall patterns of disease spread through time are surprisingly similar. Figure 1a (see PDF version for figures) shows outcomes for the simple equations defined above (eqn. 1). Figure 1b (see PDF version) shows the outcome from the individual-based stochastic model for the United Kingdom and United States. Note the similar shape and timing of the overall epidemic for a typical influenza A virus.

This simple comparison shows that much can be gained by using both simple and complex models. Where their predictions differ, we can take those opportunities to further our understanding of the key determinants of observed patterns.

Control of an Influenza Pandemic
The rapid control of the SARS epidemic in 2003 left some with the impression that “we have done it once, and we can do it again for the next emerging pathogen.” However, a sense of complacency is misplaced for one particular reason—the typical course of the SARS infection in the infected patient. Detailed clinical virological studies showed that peak viraemia (i.e., peak infectiousness to susceptible contacts) in nasal or faecal secretions and excretions occurred many days after the onset of clinical symptoms of disease, often as long as 10 days after the end of the incubation period. Therefore, if a patient was isolated a few days after the onset of illness, the duration of the infectious period was significantly decreased. In other words, contact tracing and isolation were very effective public health control measures for SARS.

In the case of influenza A, the clinical pattern of infection is very different. First, the average incubation period is very short—on the order of two days—roughly half the incubation period for SARS. Thus, the generation time for influenza (the time from infection to transmission to a susceptible contact) is only three to five days. An epidemic of influenza A, therefore, can develop very quickly in dense, highly mixed populations, with the vast majority of cases occurring within the first 200 days. More important, peak viraemia, hence peak infectiousness, is synchronous with the onset of clinical symptoms (e.g., see equ 2). Thus, contact tracing and isolation will be much less effective because a good part of the infectious period occurs before the onset of clinical symptoms. Even with the rapid diagnosis and isolation of contacts, these simple public health measures are very unlikely to have a substantial impact on the course of the epidemic.

Simple and complex models both suggest that the best option for suppressing an epidemic significantly is via prevaccination of the population with a vaccine that has some antigenic similarity to the emerging strain. The development of such a vaccine will require making a guess about (1) the antigenic composition arising from mutations in the H5N1 virus that would make it transmissible in human communities or (2) the reassortment of two viral genomes, one human and one the H5N1 bird virus.

Some countries have chosen to go this route, with the key antigens related to the H5 haemaglutinin and the N1 neuraminidase of the bird virus. If the guess about the vaccine antigens is close to what actually emerges, this is clearly a good option. If not—then the vaccine will not be effective. Thus, this approach is a high-risk, costly strategy—but one that could work well.

To prevent an epidemic, the fraction of the population that must be immunized, p, is shown in the following simple expression:

(eqn 2)

Here, p represents vaccine efficacy (unity represents perfect protection and zero represents no protection). R0, the basic reproductive number, typically has a value between 1 and 2 for new strains of influenza A in human communities. With perfect efficacy and an R0 value of 1.6, 37.5 percent of the population would have to be pre-immunized to prevent an epidemic. The percentage would be lower for the total population if vaccination were targeted to 5 to 15 year olds (where most transmission typically occurs in respiratory tract infectious-disease epidemics).

If vaccine development is delayed until after the emergence of a new strain, the vaccine will, of course, be precisely targeted on the correct antigens. But the formulation and scale-up of production will take time, perhaps more than the 200 days of a typical epidemic.

The conclusions of the very detailed studies done by Ferguson et al. (2006) exploring a wide range of options other than vaccination alone or in combination can be summed up very simply. Unless delivery systems and logistics of the supply of antiviral drugs are extremely efficient, it will be very difficult to control the spread of the epidemic. Prophylactic drug treatment can inhibit infection, and, if the infection still occurs, it can limit infectiousness by lowering viraemia. Even after the onset of clinical symptoms, it can reduce morbidity and mortality. However, the timing of drug delivery is key. Optimally, it must be given one or two days after the onset of symptoms.

Ferguson et al. show that a combination of (1) treatment of sick patients; (2) contact tracing; and (3) prophylactic treatment of household contacts plus restrictions on travel (i.e., increasing social distance) can very significantly reduce the scale of a potential epidemic. This is encouraging and gives us a target to aim for. However, these options must be delivered optimally in terms of very rapid tracing (the first day after a sick patient is seen in a primary or secondary care setting because that patient will probably come in one or two days after the onset of symptoms); very rapid delivery of treatment; and excellent compliance with treatment regimens and “stay at home” requests. To be successful, simulations suggest that logistics and delivery must be efficient and uniform across districts, counties, and states.

Both simple and complex models suggest that controlling the spread of a highly pathogenic influenza A strain will be very difficult. Although the emergence of such a virus is all but certain in the coming decades, the timing and antigenic nature of the virus are very uncertain. Restrictions on international travel and closing borders will have little effect on the rate of spread unless more than 99.5 percent of entries are stopped (Hollingsworth et al., 2006). In practice, this seems impossible to achieve. Although some time might be gained by these restrictions, the gains would be measured in weeks, not months.

Substantial reductions in morbidity and mortality are possible via drug treatment and increasing social distance, but only if the availability, logistics, and delivery of the drug are extremely efficient and compliance with treatment regimens is strictly observed. Prevaccination, by far the best option, will require significant expenditures and good guesswork about the nature of the novel strain that will emerge.

The key message of analyses based on models to date is that logistics and delivery must be carefully planned and efficiently carried out. Once an epidemic starts, health authorities must be able to get drugs and care to sick individuals quickly—in all areas of the country. Rehearsals are essential—as is uniform performance across county, district, or state boundaries.

The worst option is to delegate authority to the lower tiers of local governments, which are likely to adopt diverse responses. The authority to deliver drugs and care can be delegated, but, as simulation studies show, the fine details of contingency plans must be decided at the national level, communicated to everyone, and applied uniformly. The present need is for national contingency planning based on extensive simulations and calculations rather than on qualitative opinions.

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About the Author:Roy M. Anderson is professor of epidemiology, Department of Infectious Disease Epidemiology, Faculty of Medicine, Imperial College, University of London, United Kingdom.