^{1}

^{*}

^{1}

^{1}

^{1}

BSC, WJE, and NJG conceived the study. RJP obtained and processed the travel data. BSC implemented and analysed the model. All authors contributed to writing the manuscript.

The authors have declared that no competing interests exist.

The recent emergence of hypervirulent subtypes of avian influenza has underlined the potentially devastating effects of pandemic influenza. Were such a virus to acquire the ability to spread efficiently between humans, control would almost certainly be hampered by limited vaccine supplies unless global spread could be substantially delayed. Moreover, the large increases that have occurred in international air travel might be expected to lead to more rapid global dissemination than in previous pandemics.

To evaluate the potential of local control measures and travel restrictions to impede global dissemination, we developed stochastic models of the international spread of influenza based on extensions of coupled epidemic transmission models. These models have been shown to be capable of accurately forecasting local and global spread of epidemic and pandemic influenza. We show that under most scenarios restrictions on air travel are likely to be of surprisingly little value in delaying epidemics, unless almost all travel ceases very soon after epidemics are detected.

Interventions to reduce local transmission of influenza are likely to be more effective at reducing the rate of global spread and less vulnerable to implementation delays than air travel restrictions. Nevertheless, under the most plausible scenarios, achievable delays are small compared with the time needed to accumulate substantial vaccine stocks.

Air travel might contribute to the spread of influenza in future pandemics. However, this modelling study concluded that restrictions on air travel would not provide effective control.

Most people who get influenza (flu) recover quickly, although it can cause serious illness and death, most often in the elderly. Sometimes a new type of flu virus appears that is much more likely to kill. This happened, for example, in 1918, when a worldwide flu pandemic killed between 20 million and 100 million people. Recently, there have been concerns about a flu virus that affects birds, and often kills them. At present the virus does not pass easily from birds to humans, and it does not seem to pass from one human to another. However, the fear is that this virus might change and that human-to-human infection could then be possible. Should all this happen, the changed virus would be a major threat to human health. With current technology, it would take several months to produce enough vaccine for even a small proportion of the world's population. By that time, it would probably be too late; the virus would already have spread to most parts of the world. It is therefore important for health authorities to consider all the methods that might control the spread of the virus. With the increase in international travel that has taken place, the virus could spread more quickly than in previous worldwide pandemics. Restrictions on international travel might, therefore, be considered necessary, particularly travel by air.

It is important to estimate how useful restrictions on air travel might be in controlling the spread of a flu virus. Travel restrictions are usually unpopular and could themselves be harmful, and, if they are not effective, resources could be wasted on enforcing them.

This research involved mathematical modelling. In other words, complex calculations were done using information that is already available about how flu viruses spread, particularly information recorded during a worldwide flu outbreak in 1968–1969. Using this information, virtual experiments were carried out by simulating worldwide outbreaks on a computer. The researchers looked at how the virus might spread from one city to another and how travel restrictions might reduce the rate of spread. Their calculations allowed for such factors as the time of the year, the number of air passengers who might travel between the cities, and the fact that some people are more resistant to infection than others. From the use of their mathematical model, the researchers concluded that restrictions on air travel would achieve very little. This is probably because, compared with some other viruses, the flu virus is transmitted from one person to another very quickly and affects many people. Once a major outbreak was under way, banning flights from affected cities would be effective at significantly delaying worldwide spread only if almost all travel between cities could be stopped almost as soon as an outbreak was detected in each city. It would be more effective to take other measures that would control the spread of the virus locally. These measures could include use of vaccines and antiviral drugs if they were available and effective against the virus.

Please access these Web sites via the online version of this summary at

• Fact sheets are available about various aspects of flu from the Web site of the World Health Organization, which takes a global overview of the impact of the infection

Many health Web sites aimed at patients provide basic information about flu.

• US National Institute of Allergy and Infectious Diseases page about flu

• National Institute of Allergy and Infectious Diseases fact sheet about cold and flu symptoms

• US Centers for Disease Control and Prevention page about flu

• The Journal of the American Medical Association's patient page about influenza

• Page on flu from BBC Health

• Information about pandemic influenza from The Health Protection Agency

The scale of threat posed by hypervirulent avian influenza subtypes [

Border controls and World Health Organization travel advisories formed central and sometimes controversial components of the control efforts during the severe acute respiratory syndrome (SARS) epidemic [_{t}) to below one, making sustained transmission impossible. This happened during the SARS epidemic, where isolation, quarantine, and behaviour change were able to bring about control [

To evaluate the potential of travel restriction and local control measures to impede global dissemination we developed a stochastic (i.e., probabilistic) model of the international spread of influenza based on extensions of coupled deterministic epidemic transmission models [

We used a metapopulation model that consists of a set of coupled dynamic epidemic transmission models (

Previous work has used deterministic approximations to study the evolution of this system [

Coupling between cities was estimated using data from the International Air Transport Association for 2002 that gives the number of seats on flights between 105 cities, including the 100 with the highest number of international scheduled passengers and all 52 used in the 1968/9 data. City sizes were taken from the United Nations urban agglomeration data (available at:

To select from variants of the basic model, we compared deterministic and stochastic model fits to data from the 1968/9 pandemic by choosing parameters to minimize the sum of squared deviations (SSQ) between times of observed and predicted peaks (this contrasts with previous work that aimed to forecast the pandemic, and therefore based parameter estimates only on data from the first affected city [

We evaluated models with sine wave, square wave, and no seasonal variation in transmission parameters for cities outside the tropics. In the sine wave formulation, the peak transmissibility occurred on the shortest day in each hemisphere, while in the square wave formulation, peak transmissibility lasted 6 mo, also centred on the winter solstice. We also considered model formulations where the transmission parameter in the tropics was taken as the maximum, minimum, and mean over 1 y of that outside the tropics. For some parameter values, the model predicted no epidemic peaks in some cities for which an epidemic peak was in fact recorded. When fitting the models we penalized these regions of parameter space by arbitrarily assigning a deviation between model and data of 500 d.

We used the stochastic model to consider the effects of (i) reducing local transmission (this simulates the effects of isolation, behaviour changes, antiviral use, or other measures that may reduce the average number of secondary cases produced by one primary case); and (ii) restricting travel to and from affected cities. We assessed the ability of these measures to delay epidemics in individual cities. We considered only major epidemics, which we defined as those peaking with at least one case per 10,000 people per day. We assumed that measures were introduced only after the first 100 symptomatic cases in each city except the originating city, for which 1,000 cases were required, although we also evaluated the sensitivity of the results to these assumptions.

We considered a number of other scenarios to assess the sensitivity of the results to the most important unknowns: patterns of seasonal variation in influenza transmission; variation in transmissibility between tropical and temperate regions; the proportion of individuals initially susceptible to the virus; the basic reproduction number, _{0} (defined as the mean number of secondary cases in a local and susceptible population caused by the introduction of one primary case); the distribution of the infectious period; the city in which the pandemic begins; and the date on which the virus first begins to spread.

Amongst the model variants considered, the best fit to data from the 1968/9 pandemic was achieved when transmissibility varied sinusoidally in temperate regions and was constant and equal to the north/south maximum in the tropics. We used this model to estimate key parameters using 1968/9 data, and to evaluate the impact of interventions. Models without seasonal forcing terms gave poor fits to data and could not account for the large differences in epidemic timing between cities in the north and south temperate regions. Models in which transmissibility in the tropics was set to the north/south mean also performed surprisingly poorly, with best-fit SSQs approximately three times greater than those obtained when transmissibility in the tropics was set to the north/south maximum (

(A and B) SSQ/_{0} (_{0,max}) is held constant at 3.0 in (A), and the seasonal minimum _{0} (_{0,min}) is held constant at 1.6 (the value giving the minimum SSQ/n for model 2) in (B). In both cases, the fraction of the population initially susceptible was fixed at 0.6.

(C and D) SSQ/_{0,min} held constant at 1.2 in (D); broken lines indicate regions of the parameter space with constant effective reproduction numbers, _{max}_{max}_{0,max}

An exploration of the parameter space for the best-fit model showed that, assuming 60% of the population to be initially susceptible (the approximate value estimated previously [_{0} values (_{0,max}) ranging from about 2.5 to 3.5 gave the best fits to data, while minimum _{0} values (_{0,min}) between about 0.5 and 1.5 had the most support (_{0} value and the fraction initially susceptible could not be identified simultaneously: A high value of one implied a low value of the other (_{max} (equal to the product of the two and giving the average number of secondary cases produced by one primary case in an actual population, accounting for immunity) was well defined, with only a narrow range of values between about 1.5 and 2.2 supported by the data. This result is consistent with other estimates from influenza pandemics [_{0,max} value of 3 and an _{0,min} of 1.2, assumed 60% of the population to be initially susceptible, and used a model in which the _{0} value varied sinusoidally and peaked in midwinter, and in which the pandemic originated in Hong Kong on 1 June.

The model showed good agreement with data from the 1968/9 pandemic, with observed epidemic peaks almost always occurring at times when the model predicted a very high probability of influenza activity (

(A) Predicted combined incidence using baseline model assumptions (bold lines show mean incidence).

(B) Observed and predicted times in individual cities. Peak times from individual simulation runs and mean peak times with 1968/9 data are shown as blue and white dots, respectively. Mean peak times that would have occurred with 2002 travel patterns are shown as yellow dots. Predictions are based on 100 simulation runs. Influenza activity was defined as at least one new symptomatic case per 100,000 people in a given week.

Despite large variation in the timing of predicted epidemic peaks in individual cities between simulation runs, the overall course of the pandemic was quite predictable (

When we used the model to evaluate interventions using contemporary air travel and demographic data, we found that travel restrictions to and from affected cities would slow epidemic spread, but unless almost all air travel from affected cities (i.e., greater than 99%) was suspended, the potential for delaying the pandemic was limited (_{t} was reduced to slightly above one were these sufficient to delay epidemics until the next influenza season. These findings were not highly sensitive to assumptions about initial susceptibility and transmissibility (

Maps show the extent of epidemic spread and average impact on aviation network (taken from 100 simulation runs) 2, 5, 8, and 11 mo after the first cases on 1 June. The intervention is made after 100 cases in each city (or 1,000 cases for Hong Kong, the city of origin). Blue lines represent flights, with darker blues representing greater mean weekly passenger numbers (after accounting for interventions to suspend travel and averaging over all simulation runs). Flights are not shown when travel restrictions have been imposed by the given time in more than 95% of simulation runs. Area of circles is proportional to city population size, and shading indicates the probability of each city having experienced a major epidemic (greater than one case per 10,000 people per day) by the given time.

Median Delays in Epidemic Peak for Diverse Interventions and Percentage of Cities Experiencing Major Epidemics

Effect of reducing travel (A, C, E, and G) and transmission (B, D, F, and H) on the timing of major outbreaks (≥ 1 case per 10,000 per day) in 105 cities using contemporary transport and demographic data and baseline parameters. Transmission reductions are imposed in each city after one case in the given city (A and B); 100 cases (C and D); 1,000 cases (E and F); and 10,000 cases (G and H). Interventions in the originating city (Hong Kong) occur after 1,000 cases (A–F) or 10,000 cases (G and H). Lines and shaded regions show means of 100 simulation runs and ± standard deviation, respectively. In all cases, _{0,max} = 3. and the proportion of the population initially susceptible is 0.6.

Impact of interventions under different assumptions about how the probability of being infectious and the degree of infectiousness varies with time since infection (A–I) and with latitude (J–L). Interventions, key, and other details are as in

(A–C) Variable infectiousness (baseline daily progression and recovery probabilities, but degree of infectiousness declines sharply after day 1 since infection). Mean serial interval = 2.6 d; mean infectious period = 3.0 d; mean latent period = 1.9 d.

(D–F) Reduced latent period (constant infectiousness): Mean serial interval = 3.8 d; mean infectious period = 3.4 d; mean latent period = 1.2 d.

(G–I) Extended infectious period (constant infectiousness): Mean serial interval = 8.6 d; mean infectious period = 3.9 d; mean latent period = 5 d.

(J–L) Baseline parameters (constant infectiousness), but transmission in the tropics set to the mean of that in the temperate region.

(A, D, G, and J) Proportion of secondary transmission that occurs 0–20 d after infection. In all cases, no transmission occurs on day 0 and _{0,max} = 3.

Decreasing the number initially susceptible (while holding _{max} constant) has two opposing effects (_{t} (the epidemic peaks when _{t} = 1). Conversely, between-city dynamics are slowed because there are fewer infectious people to spread the disease. Which effect dominates varies between cities; those affected at the start of the pandemic tend to experience peak activity earlier when there are fewer initial susceptibles; for the rest it usually occurs later. Although travel restriction always reduces the rate of spread between cities, under most scenarios so many people become infected that even near-total restriction has remarkably little effect. However, for a given _{max}, the smaller the number of susceptibles the greater the impact of this intervention. For example, when 90% of the population are initially immune, the most extreme travel restrictions can be quite effective in preventing international spread. Conversely, reducing transmission has the greatest effect on impeding international spread when (for a given _{max}) more people are susceptible. The large delays and reductions in the number of affected cities result from two effects acting in the same direction: The reduced _{t} slows the epidemic within each city (delaying epidemic peaks), and the reduced total number of cases reduces the rate of spread between cities. Larger reductions in transmission led, in extreme cases, to smaller delays in epidemic peaks (_{t} was reduced to below one, causing the epidemic decline to begin immediately; the peak therefore occurred at the time of the intervention, earlier than it would have done with a less effective intervention. Under such circumstances the time of the epidemic peak is not a good measure for fully evaluating local control measures.

Previous influenza modelling work has used both square and sine wave seasonal forcing terms [

Median Delays in Timing of Epidemic Peaks with Different Dates and Locations for the Start of the Pandemic

The course of infection with a future pandemic influenza virus might differ in important ways from our baseline assumptions, and could be quite unlike typical interpandemic influenza. We therefore assessed the robustness of our conclusions to the assumed latent and infectious periods. We found that assuming a greater degree of infectiousness early in the course of infection (reducing the serial interval from 4.2 to 2.6 d, as suggested by recent analysis of household influenza transmission data [_{max} was reduced from about 1.8 to 1.5. Conclusions were also robust to moderate variation in the distribution of the latent period (

The relative ineffectiveness of travel restrictions for controlling pandemic influenza is a consequence of the rapid initial rate of growth of the epidemic in each city and the large number of people infected. For example, with a serial interval of 3 d, ignoring depletion of susceptibles, an _{t} of two would cause a 128-fold increase in new cases within 21 d (128 = 2^{21/3}). This means that if travel from the first affected city was restricted to 1/128 of its former value on (and after) day 1, there would be approximately the same number of influenza cases leaving the city on day 21 +

Hufnagel et al. [

Large and important uncertainties abound in influenza epidemiology: We do not know whether or not a significant proportion of transmission occurs before the onset of symptoms or whether subclinical infections are an important source of transmission, and we know very little about the determinants of seasonality [_{t} to close to one. Elsewhere it has been shown that airport entry screening would be unlikely to detect more than 10% of passengers latently infected with influenza when boarding [

The results also raise interesting questions about the importance of seasonality in influenza transmission. The evidence for strong seasonal effects in temperate regions found here with 1968/9 data is supported by a recent analysis of interpandemic influenza [

Recent models of pandemic influenza have accounted for household and social contact patterns [_{0}, assuming nonhomogeneous local mixing patterns would result in a somewhat reduced attack rate and rate of spread within each city, causing a slight decrease in the rate of global spread. For this reason, estimates of _{max} based on fitting models that assume homogeneous local mixing to pandemic data may underestimate the true value.

A new pandemic strain might not show the same pattern of seasonality as in 1968/9 and could potentially have greater transmissibility than strains seen previously. Both SARS and smallpox transmission can be greatly amplified by nosocomial spread [

(65 KB DOC)

We thank Emilia Vynnycky and Maria Zambon (Health Protection Agency) for helpful discussions; the International Air Transport Association for supplying data; and Professor Neil Ferguson (Imperial College) for helpful suggestions that led to significant improvements in the manuscript.

interquartile range

severe acute respiratory syndrome

sum of squared deviations