^{1}

^{*}

^{2}

^{2}

^{2}

^{3}

^{2}

Conceived and designed the experiments: JE DG GB. Performed the experiments: DG FY RM. Analyzed the data: JE DG GB DW. Wrote the paper: JE DG GB DW. Other: Built and tested the model: FY RM DG.

The authors have declared that no competing interests exist.

Planning for a possible influenza pandemic is an extremely high priority, as social and economic effects of an unmitigated pandemic would be devastating. Mathematical models can be used to explore different scenarios and provide insight into potential costs, benefits, and effectiveness of prevention and control strategies under consideration.

A stochastic, equation-based epidemic model is used to study global transmission of pandemic flu, including the effects of travel restrictions and vaccination. Economic costs of intervention are also considered. The distribution of First Passage Times (FPT) to the United States and the numbers of infected persons in metropolitan areas worldwide are studied assuming various times and locations of the initial outbreak. International air travel restrictions alone provide a small delay in FPT to the U.S. When other containment measures are applied at the source in conjunction with travel restrictions, delays could be much longer. If in addition, control measures are instituted worldwide, there is a significant reduction in cases worldwide and specifically in the U.S. However, if travel restrictions are not combined with other measures, local epidemic severity may increase, because restriction-induced delays can push local outbreaks into high epidemic season. The

International air travel restrictions may provide a small but important delay in the spread of a pandemic, especially if other disease control measures are implemented during the afforded time. However, if other measures are not instituted, delays may worsen regional epidemics by pushing the outbreak into high epidemic season. This important interaction between policy and seasonality is only evident with a global-scale model. Since the benefit of travel restrictions can be substantial while their costs are minimal, dismissal of travel restrictions as an aid in dealing with a global pandemic seems premature.

Planning for a possible influenza pandemic is obviously an extremely high priority for the U.S. government. Less obvious, perhaps, is the fact that, in the well-connected world of the 21^{st} century, no country is isolated from the potential spread of infection. Therefore, there is a pressing need to study the global spread of flu to understand the impact of the global epidemic on U.S. preparedness.

Rvachev and Longini

Here, we argue that international air travel restrictions sometimes could be useful to slow the progression of pandemic flu and sometimes could be harmful. While travel restrictions alone do little to directly ameliorate the pandemic, they can buy time to develop and deliver vaccine and institute a range of powerful nonpharmaceutical interventions (e.g., social distancing, public education, staging of medical equipment), all of which could sharply reduce cases. Of course, travel restrictions can directly decrease the influx of new infected persons into an area. More importantly, the restrictions reduce the probability of an infected individual leaving the area in which an outbreak is developing. Consequently, travel restrictions are one among a range of strategies that could be used to address a global pandemic. In a recent paper, Brownstein,

From a public health perspective, it becomes clear that the

To estimate this delay, we model the distribution of first passage times (FPT) for infected persons to the United States. We define the start of the epidemic as the day on which the first 100 individuals are exposed in a single city, and we define the FPT as the number of days from the epidemic start until the first infected individual crosses the United States border. Seemingly small increases in FPT can translate into significant delays in peak incidence times and values. In all pandemic plans, local social interaction restrictions are recommended. Once these are implemented, the course of the epidemic will be altered. In this model, both simultaneous, global restrictions and sequential, city by city restrictions were tested. The impact of the travel restrictions on both the mean FPT to the United States and the full course of the epidemic are considered, as well as the costs of the intervention.

The initial version of our model was based on (and calibrated to) the global influenza model of Rvachev and Longini

The model consists of a set of stochastic difference equations describing the disease dynamics within each city and air travel by individuals from one city to another. Time is measured in discrete units of 1 day. The population of each city is divided into mutually exclusive nonsusceptible (NS), susceptible (S), exposed (E), infectious (I), and recovered (R) classes. We do not estimate deaths, although readers can easily compute them by multiplying our infection levels by any assumed case fatality rate. The exposed period is assumed to coincide with the viral incubation period, and the infectious period is assumed to coincide with the symptomatic period. Infectious persons are assumed not to travel. Within each city, individuals are assumed to be well-mixed. Parameter values are the same for all cities. The parameter values used in the model are given in

The model includes 155 major cities around the world, including the cities with the 100 busiest airports, the 100 largest cities worldwide, and the 52 cities in the Rvachev-Longini model. The 155 cities modeled include 34 major U.S. cities. The population and transportation data have been updated to include year 2000–2004 values. Population data are taken from the U.S. Census Bureau, the United Nations Department of Economic and Social Affairs, the Instituto Brasileiro de Geografia e Estatística, and several other sources

Travel data are taken from OAG statistics on flight schedules provided by L. Amaral

Natural history parameters for the H5N1 influenza virus align with those used previously _{0} has been chosen to be 1.7. We have also studied a range of R_{0} values but specifically focused on the values of 2.0 and 1.4, which in combination with 1.7, correspond to the world pandemics of 1918, 1957, and 1968. Seasonality was implemented based on the assumptions that cities within the tropics have peak viral transmission year round, while in cities outside the tropics, transmissibility varies sinusoidally, with peak transmission occurring on January 1 in the northern hemisphere and on July 2 in the southern hemisphere. To avoid abrupt pattern changes at the boundaries of the tropics, we modeled a smooth latitudinal variation of the amplitude by implementing a corresponding sine wave. In _{0}. Full details are provided in

The severity and the speed of an epidemic both increase as the value of R_{0} increases. Results are shown for an epidemic starting in Hong Kong on July 1. The actual values of R_{0} are modified by seasonal and geographical factors. (A) Worldwide daily number of infected individuals. (B) Worldwide cumulative number of influenza cases. (C) U.S. daily number of infected individuals. (D) U.S. cumulative number of influenza cases. (green: R_{0} = 1.4, blue: R_{0} = 1.7, red: R_{0} = 2.0)

Deterministic models can be effective in describing the mean behavior of a stochastic epidemic, particularly when the number of persons in each disease class is large enough to model the system behavior in terms of population proportions, rather than numbers of individuals. However, in the early stages of an outbreak in a city, when very few exposed or infectious individuals are present, individual actions are important and random factors may easily affect the course of the outbreak. In our model, the realized number of newly infected persons each day within each city is drawn from a Poisson distribution with the mean calculated from the numbers of susceptible and infectious persons in that city and the local, seasonally-adjusted, infectious contact rate. The numbers of individuals in a particular disease state traveling from one city to each of the other cities directly connected to it on a particular day are drawn from a multinomial distribution based on the average daily numbers of travelers from that city and the proportion of the city's population in that disease state on that day.

Travel restrictions are implemented in the model as a reduction in the probability of travel between cities that occurs after a threshold cumulative number of infectious influenza cases has been reached. Sequential restrictions are applied to travel to and from a city that has crossed the threshold of 1,000 cumulative infectious cases. Note that because travel restrictions reduce the travel both into and out of a city, those cities directly connected to a restricted city are also affected by the restrictions, even if they have not yet reached the intervention threshold. We have also considered simultaneous, worldwide interventions, in which travel restrictions are applied to all cities after 1,000 infectious cases have occurred in the initially exposed city. Obviously, one could assume thresholds proportional to city sizes and many other variations.

Vaccination is implemented as a transfer of a percentage of the susceptible population to the permanently nonsusceptible population, and can be implemented as an initial vaccination at time zero (i.e., prevaccination), or as an ongoing, daily vaccination of the population during the epidemic. Daily vaccinations are implemented either simultaneously or sequentially, similar to the imposition of travel restrictions. To more closely parallel travel restrictions, vaccination is implemented at the same time in those cities directly connected to cities which have crossed the intervention threshold. Note, however, that we use the term “vaccination” broadly, to denote simply the product of the number of vaccine courses administered and the effectiveness per course, so that the nonsusceptible population consists of those who have been effectively removed from the susceptible population before becoming infected. Our baseline value for vaccination is that 0.1% of the susceptible population is vaccinated daily.

We have run a number of scenarios varying the origin of the infection (Hong Kong, London, Sydney), the origination date (January 1, July 1), the level of travel restriction (90%, 95%, 99%), the vaccination strategy (sequential, simultaneous), the initial vaccination level (0%, 10%, 20%), the daily vaccination rate (0.05%, 0.1%, 1%), and the severity of flu transmission (R_{0} = 1.4, 1.7, 2.0). Note that the value of R_{0} that we routinely report in this paper corresponds to a baseline value that is further modified by seasonal variations and latitude. The actual value of R_{0} depends on the location and the season, and thus may be lower than the baseline value. In this paper we illustrate our point by using the most commonly published parameters and scenarios _{0} = 1.7 as in our main analysis without asymptomatic individuals, we observed no significant qualitative differences in results. We recognize that a more detailed exploration of parameter space may yield more information, and such an analysis may be part of future research.

The model was implemented in AnyLogic™, a Java-based modeling platform developed by XJ Technologies Company Ltd. (

Consistent with previous work

High levels of international travel restrictions are necessary to reduce the total number of infected individuals worldwide. There is little difference in effect between sequential, city-by-city implementation of travel restrictions and simultaneous, worldwide implementation. Results are shown for an epidemic with R_{0} = 1.7 starting in Hong Kong on July 1. (blue: sequential travel restrictions; red: simultaneous travel restrictions, mean values shown, error bars = 95% confidence intervals)

Location and Time of InitialCases | Travel Restrictions Implemented | Total Metropolitan Cases Worldwide after 6 Months | Total Metropolitan Cases Worldwide after 12 Months | Total Metropolitan Cases Worldwide at End of Epidemic | |||

mean | sd | mean | sd | mean | sd | ||

Hong Kong - Jan 1 | no | 193,609,206 | 4,345,032 | 293,636,107 | 3,096,894 | 358,390,361 | 1,342,560 |

yes | 81,531,156 | 9,783,597 | 331,162,274 | 3,836,716 | 391,746,313 | 2,736,224 | |

Hong Kong - July 1 | no | 323,819,238 | 4,071,117 | 414,093,710 | 255,211 | 414,198,937 | 244,499 |

yes | 132,230,576 | 9,451,456 | 409,718,662 | 1,974,674 | 415,947,262 | 2,462,781 | |

London - Jan 1 | no | 216,643,706 | 2,791,062 | 275,413,403 | 2,270,138 | 347,348,752 | 2,986,580 |

yes | 118,523,844 | 10,690,524 | 321,370,868 | 5,570,406 | 385,633,413 | 3,058,182 | |

London - July 1 | no | 22,673,116 | 57,638,959 | 81,867,867 | 164,641,526 | 82,021,371 | 164,941,514 |

yes | 7,134,433 | 19,098,146 | 61,749,309 | 141,663,297 | 67,074,165 | 149,098,629 | |

Sydney - Jan 1 | no | 80,356,144 | 25,615,355 | 335,303,211 | 10,261,001 | 373,149,982 | 2,987,185 |

yes | 33,068,217 | 18,255,000 | 327,274,492 | 10,724,921 | 406,597,417 | 5,940,327 | |

Sydney - July 1 | no | 298,429,077 | 6,434,137 | 417,607,112 | 400,989 | 417,718,338 | 416,499 |

yes | 94,823,730 | 13,494,412 | 406,339,496 | 2,846,810 | 412,396,914 | 3,138,013 |

The end of the epidemic is determined when there are no further cases worldwide.

These data represent means and standard deviations for all 100 runs, including the runs in which the disease did not develop a pandemic state and did not reach the U.S.

In _{0}. In _{0} = 1.7. When travel restrictions are imposed, the FPT increases by two to three weeks when the outbreak originates in Hong Kong (from 18 days to 31 days) or Sydney (from 7 days to 27 days). There is no delay in FPT when the outbreak originates in London. These delays are larger for smaller values of R_{0}; for example, for an R_{0} = 1.4 the delay in FPT from Hong Kong to the U.S. due to travel restrictions increases to 20–23 days (data not shown).

Location and Time of Initial Cases | No Intervention | 95% Travel Restriction Only | 0.1% Daily Vaccination Only | Both Travel and Vaccination | ||||

mean | sd | mean | sd | mean | sd | mean | sd | |

Hong Kong - Jan 1 | 17.58 | 7.23 | 31.12 | 12.44 | 17.75 | 5.02 | 30.40 | 13.17 |

Hong Kong - July 1 | 17.86 | 6.17 | 31.33 | 14.42 | 18.94 | 7.07 | 30.06 | 14.19 |

London - Jan 1 | 5.50 | 3.94 | 5.50 | 3.88 | 5.34 | 4.47 | 5.91 | 4.44 |

London - July 1 | 16.26 | 32.97 | 16.15 | 33.20 | 23.86 | 40.35 | 25.95 | 44.95 |

Sydney - Jan 1 | 34.91 | 17.80 | 62.03 | 33.50 | 32.96 | 15.85 | 69.07 | 33.79 |

Sydney - July 1 | 14.63 | 6.23 | 21.32 | 14.60 | 14.20 | 6.16 | 23.10 | 14.09 |

_{0} = 1.7.

These data represent means and standard deviations for all 100 runs, including the runs in which the disease did not develop a pandemic state and did not reach the U.S.

Vaccination alone, even at low rates, reduces the total number of cases worldwide and in the United States (data not shown). As expected, vaccination reduces the effective R_{0}, which leads to the reduction of the total number of cases, and also increases the duration of the epidemics. The FPT, however, is little affected by the vaccination-only intervention (see

The speed and severity of an epidemic can be reduced by implementation of travel restriction and vaccination policies. Implementing both travel restrictions and vaccination can have a greater effect than implementing either policy alone. Results are shown for an epidemic with R_{0} = 1.7 starting in Hong Kong on July 1. (A) Worldwide daily number of infected individuals. (B) Worldwide cumulative number of influenza cases. (C) U.S. daily number of infected individuals. (D) U.S. cumulative number of influenza cases. (red: no intervention, blue: sequential 95% restriction of international travel, green: daily vaccination of 0.1% of susceptible population, orange: both travel restriction and vaccination)

Location and Time of Initial Cases | Intervention | Total Metropolitan U.S. Cases 6 Months after the Start of the Epidemic | Total Metropolitan U.S. Cases 12 Months after the Start of the Epidemic | Total Metropolitan U.S. Cases at the End of the Epidemic | |||

mean | sd | mean | sd | mean | sd | ||

Hong Kong - Jan 1 | N | 18,245,753 | 1,657,562 | 62,118,714 | 2,517,623 | 82,833,403 | 318,722 |

TO | 2,951,395 | 1,765,465 | 90,173,754 | 1,667,679 | 96,429,042 | 1,107,950 | |

VO | 6,017,992 | 820,342 | 13,231,241 | 238,007 | 16,386,410 | 494,161 | |

TV | 812,576 | 700,550 | 8,006,939 | 1,861,928 | 17,910,022 | 2,491,967 | |

Hong Kong - July 1 | N | 83,701,712 | 1,004,370 | 102,368,352 | 76,848 | 102,368,456 | 76,846 |

TO | 18,913,221 | 1,474,799 | 102,418,028 | 409,462 | 102,418,055 | 409,465 | |

VO | 32,642,187 | 992,303 | 72,958,924 | 216,288 | 73,008,133 | 247,663 | |

TV | 3,942,933 | 837,907 | 56,928,367 | 2,087,690 | 56,928,594 | 2,087,572 | |

London - Jan 1 | N | 30,099,814 | 1,256,785 | 41,865,074 | 1,378,565 | 76,508,738 | 1,527,186 |

TO | 13,591,127 | 2,772,253 | 77,390,536 | 4,173,168 | 92,464,670 | 994,422 | |

VO | 12,660,235 | 987,928 | 14,420,437 | 737,109 | 14,806,721 | 621,159 | |

TV | 4,344,538 | 1,311,592 | 8,600,742 | 968,822 | 12,602,370 | 524,534 | |

London - July 1 | N | 5,277,589 | 16,653,990 | 20,382,433 | 40,984,845 | 20,382,469 | 40,984,918 |

TO | 1,030,904 | 5,350,070 | 15,484,489 | 35,542,037 | 16,277,110 | 36,239,497 | |

VO | 1,482,819 | 5,799,144 | 12,336,496 | 27,433,567 | 12,336,532 | 27,433,647 | |

TV | 757,059 | 2,828,242 | 10,225,818 | 22,137,451 | 10,231,246 | 22,140,760 | |

Sydney - Jan 1 | N | 4,032,772 | 4,944,879 | 79,002,872 | 4,418,787 | 84,376,383 | 731,748 |

TO | 1,646,404 | 2,792,800 | 82,209,426 | 6,160,091 | 101,034,551 | 3,632,634 | |

VO | 1,163,532 | 1,546,781 | 21,035,146 | 3,261,573 | 33,057,868 | 5,607,727 | |

TV | 248,120 | 622,647 | 6,055,356 | 2,372,328 | 26,395,864 | 7,881,035 | |

Sydney - July 1 | N | 82,454,611 | 2,075,752 | 102,519,057 | 105,857 | 102,519,138 | 105,861 |

TO | 17,977,729 | 1,887,605 | 101,981,636 | 556,065 | 101,981,640 | 556,065 | |

VO | 34,503,408 | 1,997,174 | 74,304,850 | 315,807 | 74,305,013 | 315,807 | |

TV | 4,954,351 | 1,428,691 | 58,365,274 | 2,605,232 | 58,365,422 | 2,605,185 |

_{0} = 1.7.

N: no intervention, TO: 95% travel restriction only, VO: 0.1% vaccination only, TV: both 95% travel restriction and 0.1% vaccination

The end of the epidemic is determined when there are no further cases worldwide.

These data represent means and standard deviations for all 100 runs, including the runs in which the disease did not develop a pandemic state and did not reach the U.S.

An important result of the model is that the delay of viral introduction caused by travel restrictions may interact with seasonality to cause a larger initial epidemic peak or total number of infected individuals in a region such as the United States. This can happen when the restrictions push the local epidemic outbreak into a period of higher seasonal transmission of the virus, causing it to spread more rapidly through the local population. By the same token, depending on the timing of the initial outbreak in the world, delays caused by travel restrictions can shift introduction of the virus to a period of lower transmissibility, making the local outbreak less severe. Thus travel restrictions alone may have either a positive or a negative local effect.

The timing of an outbreak can greatly influence the effects of international travel restrictions on the severity of the epidemic in a region such as the United States. Results are shown for epidemics with R_{0} = 1.7 beginning in Hong Kong on either January 1 or July 1. For an epidemic beginning in January, the initial epidemic wave in the United States is suppressed, although without other interventions, the second epidemic wave would be more severe. It is thus important to implement additional measures during the time gained. For an epidemic beginning in July, the delay in the epidemic is much smaller, but the overall severity is reduced. (A) U.S. daily number of infected individuals. (B) U.S. cumulative number of influenza cases. (red: January 1 epidemic start in Hong Kong with no intervention, blue: January 1 start in Hong Kong with sequential 95% restriction of international travel, green: July 1 epidemic start in Hong Kong with no intervention, orange: July 1 start in Hong Kong with sequential 95% restriction of international travel)

A number of factors can modify the results of the simulations. For example, with a higher value of R_{0}, a global epidemic would be more “synchronized” (individual cities' peaks would be more clustered) and the delay would be less pronounced. This might be the case in the early stages of an epidemic, when the public is not yet aware of basic contact reduction measures that reduce the effective reproduction number of the virus. However, with the implementation of such measures, the disease transmission rate could be reduced and the delays would become more pronounced. In _{0}. For example, if the value of R_{0} is lowered from 1.7 to 1.4 by either self-isolation or other means of reducing contact rates, the delay due to travel restrictions will be increased to 20–23 days, giving public health officials more time to prepare for the upcoming epidemic. The value of R_{0} can be crudely calculated from the early stage of the epidemic growth curve from an equation such as

Screenshots showing that with higher values of R_{0}, individual cities' epidemic peaks are more clustered in time and the number of infected persons is much higher. (A) Time series diagram for major metropolitan areas for an uncontrolled influenza epidemic with R_{0} = 1.4. (B) Time series diagram for an epidemic with R_{0} = 2.0.

If R_{0} is so estimated during the early stages of the epidemic, then it would be possible to predict the amount of time available to increase public health preparedness before the epidemic strikes the United States.

In deciding whether to adopt a policy such as the imposition of international air travel restrictions, one must compare benefits to costs. The foregoing analysis suggests very strongly that restrictions on international air passenger travel can be of substantial benefit to the U.S. It is not widely appreciated that the associated cost is minimal. To economists, a

First, of the 155 international cities included in the analysis, 34 are U.S. cities with major airports. These account for the activity of airlines classified in the Bureau of Transportation Statistics as Major carriers. Ranked below them in size are the National, Large Regional, Medium Regional, Small Certified, and Commuter carriers

With all of this information as background, we estimate the cost of closing the Major airlines as if that were the equivalent of shutting down the entire system, and we will cost it as a simultaneous shutdown, although as discussed above, we model a sequential shutdown. Under these worst case assumptions, the estimated cost falls between $93 billion and $100 billion

Labor deserves a separate discussion. First, the impact on labor depends on whether the economy is operating at full employment. If not (as in the U.S. at present), many workers (managers, executives, baggage handlers, agents, mechanics, etc.) would find alternative employment (i.e., there would be some factor mobility). For conservatism's sake, let us assume no such mobility. Roughly 60% of the airline industry remains non-unionized. These individuals received no severance pay after the layoffs of 9/11 and would likely be treated similarly in a pandemic flu shutdown. Severance packages are unlikely; hence, labor costs will not likely weigh heavily on the calculation of costs. We are not condoning this treatment of workers, merely reporting the likely GNP impact. Indeed, a more generous labor policy is altogether feasible. The Senate Joint Economic Committee estimates that “a government funded severance package that covered 100 percent of wages and benefits would cost roughly $500 million per month.”

In summary, considering substitution possibilities, and even including labor compensation, it is extremely difficult to drive the cost of air travel restrictions beyond 1% of the U.S. GNP

We have presented a study of the impact of a number of interventions on the mean first passage time of a pandemic virus to the United States and on the total number of cases both worldwide and in the U.S. We have shown that although international travel restrictions alone will not contain a pandemic, they can buy time in which to take important steps. Our results suggest that the delay can be significant (about 2 to 3 weeks). Although this is not enough time to develop and produce large quantities of a vaccine, from a public health perspective, a delay of even 1 or 2 weeks can be a big help in preparing for vaccination, developing public awareness, instituting social distancing, organizing vaccination centers, and preparing other means of disease containment. One should also note that the effect of the travel restrictions is not limited to delaying the initial disease introduction. Restrictions also help to limit continuous reintroduction of the disease to the United States, and thus allow development of more efficient local containment measures.

The impact of travel restrictions on the total number of cases in an epidemic is roughly comparable to vaccination of a substantial portion of the population. However, because of the delay in FPT, the number of cases at intermediate time points such as 6 or 12 months following the initial outbreak of the epidemic worldwide can be substantially reduced when travel restrictions are used.

A number of factors could modify the effects of the delay caused by travel restrictions. Seasonality is one of them. Seemingly counterintuitively, due to the interaction with the global seasonality of influenza, travel restrictions alone may lead to a higher number of total cases in a given region than would an unmitigated epidemic. This occurs because the increased FPT may delay the regional introduction of the virus until the influenza season. For example, an outbreak in Hong Kong occurring in January would lead to a slow epidemic start in the United States in the spring, when the seasonal transmission rate is low. As the seasonal transmission rate increases around September, one would expect to see a large epidemic outbreak. Any delay of the epidemic introduction in the United States would only push the disease into a season with a higher effective reproduction number. Conversely, when an epidemic starts in Hong Kong in July and becomes visible in the United States around October it peaks around February. Any delay in introduction will push the epidemic out of the high transmissibility season and thus reduce the total number of cases. The actual seasonality pattern might vary slightly depending on how seasonality is introduced; however, the general relationship between seasonality and the delay will hold and is worth considering when planning prevention measures.

Because of seasonality and travel patterns, first passage times differ greatly depending on the location of the outbreak. For instance, for January outbreaks, the mean FPT to the U.S. is 18 days from Hong Kong, 6 days from London, and 35 days from Sydney. January is outside influenza season in Sydney, so the outbreak requires more time to reach a level such that a number of travelers would carry the infection to other cities. The short FPT to the United States from London reflects the heavy volume of air travel between London and the United States.

In the study we used a conservative operational estimate for the imposition of intervention policies of 1,000 total infectious cases. One could use 500 total cases or even just a single case; however, the use of a single case as a signal of an epidemic start might be dramatically misleading because of the stochasticity at the small size of the infectious population. As a modeling assumption, the choice of a single fixed threshold was based on parsimony. It seems likely that different countries would have different thresholds. However, lacking detailed data to support city-specific estimates, we chose not to add further model complexity based on undocumented hypotheses. The base case value of 1000 infectious cases as the threshold for implementation of travel restrictions was meant as a conservative bound; that is, we chose a relatively high number to ensure that the analysis was not biased in favor of travel restrictions. Obviously, they look better the earlier they are implemented. This threshold should not be confused with the onset of local containment measures. Presumably these could begin earlier, and after the first cases are reported anywhere, vigilance will likely increase everywhere. That is, the country response thresholds should fall as the disease spreads. This makes our use of the constant threshold more conservative.

Our simulations show (see

In the simulations presented here, we have used a single-leg travel matrix, primarily to be consistent with other authors

This mathematical model is focused on the description of the disease spread across the continents and has a number of limitations. The model is based on the largest metropolitan areas. It does not include the heterogeneous populations around these cities and in rural areas. Other types of heterogeneity, such as population age structure or social networks and the consequent differences in transmission probability are not considered. Further, the model does not include ground transportation.

Our mathematical model does allow one to evaluate the impact of travel restrictions combined with other types of interventions, such as quarantine, self-isolation, wearing masks, closing schools, etc. We have presented a number of scenarios illustrative of the interactions between location, seasonal timing, travel restrictions, and vaccination. Our future work is focused on more complex scenarios involving other disease characteristics and other factors effectively reducing disease transmission beyond vaccination-type strategies.

Economically, a 1-year total ban on international and major U.S. domestic air passenger travel is estimated to cost the United States less than 1% of GNP. Because our model predicts that regionally implemented sequential travel restrictions may be just as effective as simultaneous global restrictions, we expect the direct economic impact would be even smaller. Given that the benefits of air travel restrictions can clearly be substantial, while the costs are clearly minimal, their dismissal is premature; the approach deserves serious consideration as an adjunct to other direct disease control measures.

Model Equations and Initial Conditions

(0.31 MB DOC)

Parameters and Values for the Model that Do Not Vary over Time

(0.09 MB DOC)

Modifications to the Travel Matrix to Account for Multiple Legs of Travel

(0.16 MB DOC)

A user can select one of three visualization screens: a world map view, time series plots, or numeric tables for each of the cities. Before running the model, one can choose to produce stochastic or deterministic runs and choose the types of intervention. Each spot on the map corresponds to a metropolitan area. Clicking on a spot will display the city name and a snapshot of the city disease status. Arrows link each infected city with its initial source of infection.

(0.45 MB TIF)

We thank Moshe Feder and Clifford Winston for helpful discussions and comments during the preparation of this manuscript; Steven Naron for useful suggestions during model building; and Eric Solano for his help and insight during model testing.