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). 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.
Model Fitting and Comparison by Sum of Squared Deviations
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 R0
) ranging from about 2.5 to 3.5 gave the best fits to data, while minimum R0
) between about 0.5 and 1.5 had the most support (C). The maximum R0
value and the fraction initially susceptible could not be identified simultaneously: A high value of one implied a low value of the other (D). However, the initial maximum effective
reproduction number, Rmax
(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
]. We therefore took as our baseline scenario an R0,max
value of 3 and an R0,min
of 1.2, assumed 60% of the population to be initially susceptible, and used a model in which the R0
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 (). 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.
Predicted and Observed Times of Influenza Activity and Epidemic Peaks in 1968/9
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 (A), 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 (B). 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 (– and ). Even when 99.9% of air traffic was suspended, most cities had a low probability of ultimately escaping the pandemic (), 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 (). Interventions that reduced transmission could typically lead to more pronounced delays ( and and ), although only when Rt 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 ().
Time Course of a Pandemic with and without an Intervention to Suspend 99.9% of Air Travel from Affected Cities
Impact of Interventions: Sensitivity to Disease Assumptions
Median Delays in Epidemic Peak for Diverse Interventions and Percentage of Cities Experiencing Major Epidemics
Impact of Interventions and Implementation Delays on Rate and Extent of Spread
Decreasing the number initially susceptible (while holding Rmax constant) has two opposing effects (). 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 Rt (the epidemic peaks when Rt = 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 Rmax, 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 Rmax) 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 Rt 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 (). This happened only when Rt 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
]. 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 (), 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 (). 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 (A and B).
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 [11
]) did not substantially alter the conclusions about the value of the interventions (A–C) compared with the baseline scenario (C and D), although if this assumption was used when fitting the model to the 1968/9 data the estimated value of Rmax
was reduced from about 1.8 to 1.5. Conclusions were also robust to moderate variation in the distribution of the latent period (D–F). If, however, the virus behaved more like the SARS coronavirus, with extended latent and infectious periods (G–I), 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 (J–L) 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.