Introduction
The scale of threat posed by hypervirulent avian influenza subtypes [1,2], and the
memory of the 20–100 million who died in the 1918 pandemic [3,4], warrant consideration
of large-scale, concerted, and potentially highly disruptive control measures [5].
Were such a virus to acquire the ability to spread efficiently between humans, control
would almost certainly be hampered by limited vaccine supplies [6]. Interventions
able to substantially impede global spread, by providing time for vaccine stocks to
accumulate, could have profound public health benefits.
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 [7,8], and travel restriction is thought likely
to occur during an influenza pandemic (although enforcement is currently considered
by the World Health Organization to be impractical in most countries) [9]. In the
absence of sufficient vaccine stocks, other control measures such as the use of antiviral
agents could also be used [10,11]. Ideally, such measures would reduce the average
number of secondary cases caused by each primary case (the effective reproduction
number, R
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 [12]. The much shorter serial interval for influenza makes the chances
for early epidemic termination much lower [13]. The main value of interventions is
more likely to be in reducing the incidence and slowing the rate of spread of the
virus.
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 [14–19]. This class of models has been shown to be capable of
accurately forecasting local and global spread of epidemic and pandemic influenza
[14–19] and accounting for the global distribution of other pathogens [20,21], but
has not previously been used to assess the impact of travel restrictions or other
control options for pandemic influenza.
Methods
We used a metapopulation model that consists of a set of coupled dynamic epidemic
transmission models (Figure 1). Each component model represents one city and tracks
the progression of individuals through four classes: susceptible to infection (S);
exposed to the virus but not yet infectious (E); infectious (I); and recovered and
no longer susceptible (R). We assumed that infectiousness coincides with disease onset
and that infectious cases do not travel.
Previous work has used deterministic approximations to study the evolution of this
system [14–19]. With this approach the first case in each city (except the originating
city) was assumed to occur only when the average number of incubating cases arriving
from other cities exceeded one, an approximation that will artificially slow the rate
of spread between cities. Our stochastic model, which has a similar underlying structure
to its deterministic counterpart, avoids this distortion and uses probabilistic transitions
to capture the inherent uncertainty in the course of the epidemic. This is more appropriate
than the deterministic approaches previously adopted, because chance effects dominate
in the early stages of the epidemic in each city and in the seeding of each city's
epidemic. As well as providing greater realism, this approach allowed us to quantify
the uncertainty in model predictions due to demographic stochasticity. Thus, rather
than assuming that each city's epidemic starts at a determined time, we assumed that
the initiation of an epidemic depends on the timing of a sequence of chance events:
a person incubating the virus must board a plane; that person must infect others in
the destination city; some of those others must cause further transmission, and so
on. Each time the model is run, even with identical starting conditions and parameter
values, a different answer is obtained. This stochastic model was used to estimate
key parameters using data from the 1968–1969 (1968/9) influenza pandemic and to evaluate
the impact of interventions using contemporary demographic and transport data. Mathematical
details of the model are provided in Protocol S1.
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: http://unstats.un.org/unsd/demographic/sconcerns/densurb/urban.aspx).
When fitting models to 1968/9 data, air transport data, sizes of urban agglomerations,
and influenza data were taken from a previous study [14,15].
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 [14]). A major strength
of our approach is that the observed time of an epidemic peak should be unaffected
even by large between-country variation in influenza reporting rates. The predicted
epidemic peak for a given city was defined as the day on which the highest number
of infected people first developed symptoms. When a city had more than one observed
peak in the 1968/9 data, only the first peak was used. We used the statistic SSQ(m)/(n
− 2m) to compare model fits [22], where m is the number of fitted parameters (between
one and four), and n the number of fitted data points (the number of cities with observed
and predicted epidemic peak times). This formula selects models that are parsimonious
and fit the data well. The full stochastic model was then used to estimate parameters
for the best-fitting model by choosing parameters that minimised the mean SSQ from
ten simulation runs for each combination of parameter values.
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.
Interventions
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,
R
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.
Results
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 (Figure 2A and 2B). Less surprisingly, models that assumed
all cities were equally connected by air travel (but with the same total volume of
air traffic) also performed poorly, with best-fit SSQs about twice as large as those
from the models that used the air travel data. Previous work with the deterministic
version of the model has assumed a square wave variation in transmissibility, assigning
transmission outside the influenza season to be one-tenth of the value during the
season [14]. We found the fit to data under this assumption to be substantially poorer
compared with models in which maximum and minimum seasonal transmission parameters
were both estimated.
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 [14]), maximum R
0 values (R
0,max) ranging from about 2.5 to 3.5 gave the best fits to data, while minimum R
0 values (R
0,min) between about 0.5 and 1.5 had the most support (Figure 2C). The maximum R
0 value and the fraction initially susceptible could not be identified simultaneously:
A high value of one implied a low value of the other (Figure 2D). However, the initial
maximum effective reproduction number, R
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
[14,23]. We therefore took as our baseline scenario an R
0,max value of 3 and an R
0,min of 1.2, assumed 60% of the population to be initially susceptible, and used
a model in which the R
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 (Figure 3). Observed and predicted times of epidemic
peaks differed, on average, by 31 d. There were, however, some anomalies: the first
epidemic peaks occurred much later than might have been expected in London and Tokyo,
and somewhat earlier than predicted in Manila and Madras.
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
(Figure 3A), although there was markedly more between-run variability in the tropics
and the south than in the north. The roughly ten-fold increase in air traffic since
1968 causes epidemics in most cities to peak between 1 and 2 mo earlier than they
would have done in 1968 (in some southern hemisphere cities the epidemic peaks 1 y
earlier) and substantially reduces variation between simulation runs (Figure 3B).
The model reproduced another interesting aspect of influenza epidemiology: the tendency
for peak periods of influenza activity in the tropics to shift with latitude, so that
in the northern tropics they are closer to countries north of the tropics, while the
southern tropics tend to be more closely aligned with countries south of the tropics
[24]. This occurs despite the fact that the model has no explicit assumptions about
seasonality for cities in the tropics; the behaviour arises only as a result of the
strength of transport connections between different regions. It is also notable that
the pandemic starts early enough to allow some probability of influenza activity in
the south during the end of the flu season in 1968. Despite this, predicted epidemic
peaks (the weeks with the greatest number of reported cases in each location) still
occur in 1969 in the south.
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
(Figures 4–6 and Table 1). Even when 99.9% of air traffic was suspended, most cities
had a low probability of ultimately escaping the pandemic (Figure 4), and delays large
enough to be of clinical significance (6 mo or more) were common only if interventions
were made after the first few cases (Figure 5). Interventions that reduced transmission
could typically lead to more pronounced delays (Figures 5 and 6 and Table 1), although
only when R
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 (Table 1).
Decreasing the number initially susceptible (while holding R
max constant) has two opposing effects (Table 1). First, within cities the time between
seeding with influenza cases and the epidemic peak decreases. This is because the
initial epidemic growth rate is unaffected, but each new case represents a greater
proportional reduction in the susceptibles and causes a greater reduction in R
t (the epidemic peaks when R
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 R
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 R
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
R
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 (Table 1). This happened
only when R
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 [14,25]. We found that the outcomes of interventions were not highly sensitive
to the precise assumptions made. The delays in the timing of epidemic peaks depended
only to a limited extent on the city in which the pandemic started and to a somewhat
greater extent on the date of release (Table 2), with larger delays more likely when
the first cases occurred towards the end of the influenza season in the place of origin.
Results were, however, highly sensitive to the timing of the intervention (Figure
5). Large delays in the timing of epidemic peaks and the prevention of epidemics in
a large number of locations could be achieved with the most extreme interventions,
but only when they were made sufficiently early. However, making the interventions
after fewer than 1,000 cases in the place of origin had minimal additional benefit
in slowing pandemic spread. Similarly, preemptive travel restrictions had no advantage
over interventions made after one case in affected cities (Figure 5A and 5B).
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 [11,26])
did not substantially alter the conclusions about the value of the interventions (Figure
6A–6C) compared with the baseline scenario (Figure 5C and 5D), although if this assumption
was used when fitting the model to the 1968/9 data the estimated value of R
max was reduced from about 1.8 to 1.5. Conclusions were also robust to moderate variation
in the distribution of the latent period (Figure 6D–6F). If, however, the virus behaved
more like the SARS coronavirus, with extended latent and infectious periods (Figure
6G–6I), a greatly delayed rate of global spread could be expected, giving more chance
of delaying epidemics until the next influenza season. In this case, smaller reductions
in travel and transmission can achieve clinically significant delays (6 mo or more)
in epidemic take-off in many cities. Assuming reduced transmission in the tropics
(Figure 6J–6L) also led to a substantial reduction in the rate of global dissemination.
Under this scenario much smaller reductions in transmission would be sufficient to
greatly reduce the chance of a pandemic; this happens because the lower transmission
in the tropics (where the virus is assumed to originate) means that a further transmission
reduction of just 21% would be sufficient to make sustained spread impossible in this
region.
Discussion
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 R
t of two would cause a 128-fold increase in new cases within 21 d (128 = 221/3). 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 + t as there would have been on day t had there been
no intervention; even such an extreme intervention would therefore buy only about
3 wk. The highly connected nature of the air travel network prevents such minor delays
between pairs of cities combining into substantial delays over the whole network.
Hufnagel et al. [20] used a related model to study the global spread of SARS. Although
this model differed in important respects from the one used here (the implicit assumptions
that air travel frequency varies with neither infection state nor country would not
be tenable in the context of pandemic influenza), the conclusion that “remarkable
success [in SARS epidemic control] is guaranteed if the largest cities are isolated
in response to an outbreak” might, at first sight, be thought to apply equally to
influenza. In fact, pandemic influenza is expected to have a much shorter serial interval
than SARS, and delays in international spread that could be achieved by restricting
almost all travel would be far more modest. Even if 99.9% of all travel could be stopped,
epidemics in most cities would be delayed by no more than 4 mo. Moreover, the conclusion
that a policy of isolating only the largest cities would guarantee success implicitly
assumes that closing major airports would cause infected individuals who would have
travelled through them to abandon their journeys rather than seek alternative routes,
and that disease spread by routes other than air travel can be ignored without substantially
altering the conclusions. This seems rather implausible, and for these reasons we
think that the conclusions of Hufnagel and colleagues, while of undoubted theoretical
interest, would be misleading if taken too literally.
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 [24,27,28]. In evaluating
the potential to delay the spread of influenza by restricting travel and reducing
transmission, we have systematically adopted optimistic assumptions, chosen to give
the interventions the greatest chance of success. Thus we have assumed that seasonal
effects are important (delaying the rate of spread outside the influenza season),
and that asymptomatic cases do not contribute to transmission (minimizing the numbers
capable of spreading the virus, and maximising the chance of detecting them); we have
ignored travel that is not by air and not between major airports; and we have ignored
the possibility of transmission during flights themselves. Despite these optimistic
assumptions we found that even large and widely enforced travel restrictions would
usually delay epidemic peaks by only a few days; to have a major impact, restrictions
would have to be almost total and almost instantaneous. Only if a pandemic strain
were considerably less transmissible, or had a considerably longer serial interval
than influenza strains seen in the past, or if very few people were initially susceptible,
would such measures be likely to have an important impact on the rate of pandemic
spread. Local control measures able to reduce influenza transmission were found to
have greater potential for reducing the rate of global spread (they could also substantially
reduce the total number of cases, although an evaluation of this benefit is beyond
the scope of this paper). Under most plausible scenarios, however, delays would still
fall far short of those required to produce large quantities of vaccine unless they
were implemented early and able to reduce R
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 [29]. The results in this paper show that such an intervention would
have a negligible impact on the course of a pandemic once it was underway.
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
[30]. However, it is not clear how important such seasonal effects have been in previous
pandemics, nor is it clear why a much better model fit should be obtained when transmission
in the tropics is assumed to be the maximum (rather than the mean) of that in temperate
regions. Indeed, a fuller understanding of the determinants of seasonal effects and
their variation with latitude remains one of the outstanding problems of influenza
epidemiology [24,27,28].
Recent models of pandemic influenza have accounted for household and social contact
patterns [10,11]. While such details are needed for evaluating the possibility of
containment at source, they would not be expected to affect the broad conclusions
presented here. However, for a given R
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 R
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
[31,32]; a similar amplification effect could occur with an unusually virulent influenza
virus that led to many hospitalisations. In these more pessimistic scenarios, even
more heroic efforts would be required to have any chance of significantly delaying
the virus's spread by restricting travel. The results here suggest that resources
might be better directed at reducing transmission locally and at attempting to control
outbreaks during the earliest stages of sustained human-to-human spread, when movement
restrictions are likely to be a more valuable containment measure [10,11]
Supporting Information
Protocol S1
Detailed Description of the Model
(65 KB DOC)
Click here for additional data file.
Editors' Summary
Background.
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.
Why Was This Study Done?
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.
What Did the Researchers Do and Find?
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.
Additional Information
Please access these Web sites via the online version of this summary at http://dx.doi.org/10.1371/journal.pmed.0030212.
• 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