Swedish Institute for Infectious Disease Control, Solna, Sweden
Department of Medical Epidemiology and Biostatistics, Karolinska Institute, Solna, Sweden
Department of Sociology, Stockholm University, Stockholm, Sweden
Theoretical Biological Physics, Department of Physics, Royal Institute of Technology, Stockholm, Sweden
Abstract
Background
Much research in epidemiology has been focused on evaluating conventional methods of control strategies in the event of an epidemic or pandemic. Travel restrictions are often suggested as an efficient way to reduce the spread of a contagious disease that threatens public health, but few papers have studied in depth the effects of travel restrictions. In this study, we investigated what effect different levels of travel restrictions might have on the speed and geographical spread of an outbreak of a disease similar to severe acute respiratory syndrome (SARS).
Methods
We used a stochastic simulation model incorporating survey data of travel patterns between municipalities in Sweden collected over 3 years. We tested scenarios of travel restrictions in which travel over distances >50 km and 20 km would be banned, taking into account different levels of compliance.
Results
We found that a ban on journeys >50 km would drastically reduce the speed and geographical spread of outbreaks, even when compliance is < 100%. The result was found to be robust for different rates of intermunicipality transmission intensities.
Conclusion
This study supports travel restrictions as an effective way to mitigate the effect of a future disease outbreak.
Background
Knowledge of the speed at which a contagious disease travels between geographical regions is vital for making decisions about the most effective intervention strategies. The actual routes a disease will take are strongly determined by how individuals travel within and between regions
Several authors have responded to the call, resulting in nowclassic papers. Rvachev and Longini
Our study applied a version of the Hufnagel model to Sweden in order to predict the effect that travel restrictions might have on the geographical spread of an outbreak. Instead of using only the aviation network, which connects only some 30 towns in Sweden, we used survey data on all intermunicipal travel, including all forms of travel.
Sweden is, by European standards, a large country, with a small population. Just over 9 million people share 450000 square kilometers. The population is, however, largely urbanized, and in that respect similar to other industrialized nations with large areas.
Eubank et al
The choice of a stochastic modeling approach
Methods
For this study, we used data from a random survey carried out by Statistics Sweden from 1999 to 2001, inclusive
As it turned out, roughly 1% of the data was significantly erroneous and was consequently removed*. From the remaining set, we estimated a travel intensity matrix with each element corresponding to the oneway travel intensity between two municipalities. The number of nonzero elements was 11611 (to be compared with the size of the matrix: 83521). The matrix elements stood in direct correspondence with the underlying data, weighted for time and population. Even though the matrix gives a good picture of the traveling pattern in Sweden, we must treat any travel intensity between two specific communities with care. This is especially true for small communities with only a single or very few journeys made between them.
A total of nine scenarios, with 1000 realizations each, was simulated to study the effects of three levels of travel restrictions as a control measure, for three different levels of the global intercommunity infectiousness parameter,
The intermunicipal travel network
The intermunicipal travel network. The intermunicipal travel network with travel intensities indicated by color lines. The scale is logarithmic in trips per day. SIM shows the complete dataset. In SIM50 and SIM20, all journeys > 50 km and 20 km, respectively, have been removed. The lines are drawn between the population centers of each municipality, so in many cases the trips are shorter than the lines representing them.
We also considered the case if the travel restrictions were not obeyed wholly by the public. Perhaps 5% might not heed the restrictions, resulting in a small but nonzero intensity for trips longer than the set restrictions. Full 1000run simulations were made at varying levels of distance restrictions and compliance, resulting in a mesh surface of the incidence.
We used a simplified version of the model suggested by Hufnagel et al
▪ S: susceptible
▪ L: latent, meaning infected but not infectious
▪ I: infectious
▪ R: recovered and/or immune.
The rate at which individuals move from one category to the next is governed by the intensity parameters:
As the process is assumed to be Markovian, as in Hufnagel's model, the time between two events,
Δ
where Q is the total intensity, the sum of all independent transmission rates:
These are the equations that govern the simulations and give us the continuous time setting. The component
We would have liked to calibrate our model in a similar way, but as we have no outbreak data for Sweden, we needed to see whether changes in
Although this is not mentioned in the original work by Hufnagel et al, the expression above means that everybody, regardless of where they live, is equally prone to travel outside their home, the uptake area of the airport or, in our case, the municipality. This is a heavy assumption indeed, as it depends on the function of the municipality varying. The municipality may be a suburb or selfsufficient community, just as airports may be transit hubs or terminals. One of the strengths of Hufnagel's model is that it seems to be forgiving towards many simplifications, this one included, with the correct choice of
After the initial conditions were set up, including a single infected person in Stockholm, the simulation ran as follows. First, we moved forward in time with a random step
Results
The results for all nine scenarios were plotted geographically and colorcoded according to the mean incidence (Figure
Epidemic spread for different restrictions and values of
Epidemic spread for different restrictions and values of
A scenario with no restrictions resulted in an outbreak in which a majority of the municipalities became affected regardless of
Table
Main results
SIM
SIM50
SIM20
Results
Mean
95% SI
Mean
95% SI
Mean
95% SI
Total number of infected
320 555
301 587
339 243
154 517
145 664
163 678
64 307
60 326
68 293
Percentage of population
3.6
3.4
3.8
1.7
1.6
1.8
0.72
0.67
0.76
Intermunicipal infections (n)
0.3
0.3
0.3
0.3
0.3
0.3
0.2
0.2
0.2
Incidence after 60 days (n)
77 184
72 760
81 784
37 065
34 941
39 321
15 240
14 307
16 190
Percentage of population
0.9
0.8
0.9
0.4
0.4
0.4
0.17
0,16
0.18
Afflicted municipalities (n)
262.1
258.5
265.4
47.2
46.6
47.8
34.0
33.6
34.5
Mean incidence in municipalities (n)
267.1
251.4
283.1
128.3
120.5
136.0
52.7
49.5
56.1
Mean influence distance (km)
1 222

245.1

153.8

Travel intensity matrix
Value
Value
Value
Total travel intensity (millions/day)
4.2

2.9

1.5

Intermunicipal oneway routes (n)
11 611

1 386

797

Summary
Value
Value
Value
Extinction runs (n)
262

268

305

Mean time for extinction (days)
3.48
2.84
4.14
3.48
2.78
4.25
3.61
2.85
4.46
Mean number of afflicted municipalities before extinction (n)
1.33
1.26
1.41
1.29
1.21
1.36
1.27
1.22
1.34
Total number of realizations
1 000

1 000

1 000

The table shows the main results along with miscellaneous information about the simulation.
Figures refer to simulated values at the end of the run, 60 days. The mean, where applicable, was taken over the set of runs that ran their course through the full 60 days.
The extinction runs hence did not affect the means but their numbers are of course interesting in their own right.
The 95% simulation intervals (SI) were calculated by bootstrapping 10 000 samples.
By incidence, we mean the number of infectious people.
Intermunicipal infections is the percentage of the total number of infected that caught the disease via intermunicipal infection.
There are 289 municipalities in Sweden and the population is approximately 8.9 million.
Municipalities of key interest
SIM
SIM50
SIM20
Municipality
Mean
95% SI
Mean
95% SI
Mean
95% SI
Stockholm
18 563
17470
19645
13 231
12437
14066
6029
5653
6412
Göteborg
730.7
654.4
813.9






Malmö
338.6
295.4
390.2






Huddinge
3473
3277
3668
2607
2453
2761
1218
1136
1298
UpplandsBro
573.7
537.0
610.7
362.2
337.7
388.1
84.1
76.2
92.4
Norrtälje
939.1
882.0
998.2
214.6
197.9
232.3
37.4
33.5
41.8
Södertälje
1133
1060
1205
638.2
593.5
685.1
60.7
51.4
72.3
Västerås
864.4
798.9
934.1
27.0
23.1
31.3
2.9
1.9
4.0
Eskilstuna
692.4
639.8
748.8
60.7
53.4
68.9
26.0
22.1
30.5
Umeå
118.2
98.1
144.6






Luleå
237.4
201.9
278.4






Örebro
557.0
507.7
611.1
0.3
0.1
0.4



Jönköping
227.6
206.4
250.9






Linköping
528.5
479.5
582.6
1.7
1.3
2.3



Helsingborg
143.0
128.9
158.3






Borås
140.3
127.6
154.5






Gävle
559.2
517.5
601.1
21.9
18.7
25.5
1.8
1.3
2.4
Ljungby
29.7
26.7
33.0






Hofors
72.9
66.8
79.2
2.9
2.1
3.9



Örkelljunga
4.9
3,8
5,0






A selection of municipalities with the mean incidence and bootstrapped 95 % simulation intervals with extinction runs filtered out. This set and its ordering is the same for the individual rows in the table, which explains the zerovalued lower interval bounds and other discrepancies.
After Stockholm, Göteborg and Malmö are the largest cities in Sweden. The single most traveled route is that between Stockholm and neighboring Huddinge, traveled by approximately 37 000 people daily, each way. The decline in incidence closely follows that in Stockholm.
UpplandsBro is representative of an outer suburb to Stockholm. Södertälje and Norrtälje are nearby towns but are not considered suburbs. Västerås and Eskilstuna are more distant, but have a fair number of commuters. Örebro through Luleå are larger towns at some distance from Stockholm with no notable commuter traffic. Finally, the last four are small towns in southern Sweden.
SI, simulation intervals
The reason for the decrease in incidence is of course the limited transmission paths available to the disease. The disease, after having spread from one municipality to another will constantly be transmitted back into the originating municipality, provided that there is a flow of travelers in the opposite direction in the travel intensity matrix. Travel restrictions limit both spread to other municipalities and reintroduction. For comparison, if traffic is removed altogether, the mean incidence in Stockholm will be 917.
The process outlined above is also responsible for the decreased number of extinction runs. For a regular continuous time branching process where the number new cases is completely independent between individuals, one would expect the probability of extinction to be I/R_{0 }= 37%. and this was confirmed in the course of testing our model.
It is also clear how travel restrictions confer increasing protection on cities that are further from the capital, the focal point of the infection. The major cities of Göteborg (Gothenburg) and Malmö would be protected even though traffic into these cities is heavy. In fact, the farthest the disease woudl ever make it in SIM50 is Ljungby, 1471 km from Stockholm and still some 200 km from Malmö. For SIM20, the farthest city is Uddevalla, 441 km away and a suburb of Göteborg. The mean reach of the epidemic in those cases is only 276 and 34 km, respectively.
An objection to the applicability of this model is that in all probability, complete enforcement of the restrictions may not be achievable or even desirable, as in the case of highpriority professionals with crucial functions in society during a crisis situation. Incidence does indeed climb the more restrictions are ignored, but not to such an extent as to render the travel restrictions dubious as a means of disease control (Figure
Epidemic spread for different restrictions and compliance
Epidemic spread for different restrictions and compliance. Geographical distribution of the incidence after 60 days shown for SIM50 and SIM20 for different levels of compliance. The left plot shows the unrestricted case with Hufnagels original
Total incidence for varying compliance and restrictions
Total incidence for varying compliance and restrictions. A surface plot showing incidence after 60 days with the parameters of compliance and distance restrictions on the data axes. 1000 realizations were made for each point. The surface has its highest values at high set distance limit and low compliance. Its low values are found at opposite corner.
Discussion
Our results show clearly that traveling restrictions would have a significant beneficial effect, reducing both the geographical spread and the total and local incidence. This holds true for all three levels of intercommunity infectiousness,
The same reasoning supports generalization of the results to other countries or regions. The survey travel data give a fairly accurate picture of travel patterns in Sweden and mirror many western countries. Some countries, e.g. the USA, are more dependent on motor vehicles for commuter traffic, and infectiousness would therefore be anticipated to be lower. Such effects are, again, included in calibration of
In light of the fact that intermunicipal travel heavily influences incidence even at a local level, we may justifiably be concerned about boundary conditions. We treated Sweden as an isolated country, but quite obviously, the incidence will be underestimated for areas with frequent traffic across the borders. This includes in particular the Öresund region around Malmö, and to a lesser extent, international airports and the small towns bordering Norway and Finland.
We would like to point out that, as in most epidemiologic simulations, individuals are not explicitly represented in the model. This is also true for individuals who are traveling. In reality, people who travel run an increased risk of contracting the disease. This is correctly modeled, as individuals who are traveling are included in the travel influx into the municipalities. The influx in turn affects the probability of additional infections at any given time. Of course, it is highly probable that this would be the travelers themselves, as, almost without exception, they return to the origin of their journey.
Even though there is presently no treatment or vaccine for SARS, results show that limited quarantine as suggested here drastically decreases the risk of transmission, and this may well turn out to be the most expedient form of intervention. In many countries, Sweden included, limiting freedom of travel is unconstitutional and must take the form of general recommendations. Additionally, certain professions of crucial importance to society during a crisis situation must be exempt from travel restrictions. The study shows that even if a substantial fraction of the population breaks the restrictions, this strategy is still viable. For other types of disease for which preventive treatment (pandemic flu) or vaccine (smallpox) are available, our results show that longdistance travelers are an important group for targeted control measures.
It is worth noting on the travel intensity matrix, where the elements directly reflect the underlying survey data, that there are several proposed alternatives, using smoothing techniques or datagenerating simulation. Completing the matrix in such a way would correctly introduce many connections that are missing from our data, but a substantial number would be falsely represented, and could endanger the validity of the model due to unforeseen stochastic mechanisms. The methods all have inherent imprecisions and flaws, which unfortunately in this context would be difficult to estimate. The choice of one in preference over another would certainly be contended. Our scheme of direct extrapolation from the raw data is certainly no better but does have the benefit of transparency and reasonable control over errors. As is explained in further detail in the appendix, this means that certain connections between municipalities that are used in reality, however infrequently, are missing, while on the other hand some will be heavily overestimated. This is especially true for certain unusual municipalities. The routes between the more populated communities and other heavy connections are much better estimated, as crude statistical analysis will indicate. Also close to the true value is the travel intensity as a whole, as well as the summed influx and outflux of any municipality.
Conclusion
Our methods show that restricting travel between municipalities in such a way that travel above a certain distance is banned, would indeed have a beneficial effect on the speed of transmission of a highly contagious disease, geographically and in absolute numbers. This conclusion is true for a range of plausible values of the intermunicipal infectiousness. Even in scenarios of compliance as low as 70%, travel restrictions are effective. Thus, the effectiveness of travel restrictions as a means of mitigating a future epidemic is supported. The model and results are robust and there is no reason to believe that the results are not generally applicable to any country or region.
Competing interests
The author(s) have received financial support from the organizations mentioned in acknowledgements.
Authors' contributions
MC performed all coding and simulations, carried out analyses and is the main author of the manuscript. FL conceived the project and design and initiated the work. He participated in analyses and drafting of the paper. FL approved the final version of the paper.
Notes
* The erroneous records were longdistance journeys, mostly between individual communities in an unreasonably short time. Had they not been removed, their influence would have been significant, accelerating the spread across the country. The correct data were irretrievable but the effect of their absence was deemed within the margin of error for longdistance journeys.
^{† }Some authors refer to this as the "attack rate" although this is not the commonest definition.
Supplementary material
Appendix A
This appendix describes the travel survey data from Statistics Sweden in greater detail.
Sample animations. The animations in wmf format provided with this manuscript show different single realizations or possible scenario of an unrestricted epidemic where one initial infected person is located in Stockholm. They are provided for interest only and should not be seen as contributing results, which require the combined analysis of many more realizations. The first file, Stockholm.wmf, shows a common scenario of spread throughout the country from Stockholm.
Click here for file
Sample animations. The animations in wmf format provided with this manuscript show different single realizations or possible scenario of an unrestricted epidemic where one initial infected person is located in Stockholm. They are provided for interest only and should not be seen as contributing results, which require the combined analysis of many more realizations. Uppsala.wmf clearly demonstrates the benefits of stochastic modeling in that the epicenter is spontaneously translocated to Uppsala, north of Stockholm, and the epidemic is hence delayed. This is quite a plausible event that would not be captured in deterministic models.
Click here for file
Sample animations. The animations in wmf format provided with this manuscript show different single realizations or possible scenario of an unrestricted epidemic where one initial infected person is located in Stockholm. They are provided for interest only and should not be seen as contributing results, which require the combined analysis of many more realizations. Dies_out.wmf shows the common event of the epidemic not catching on but rather dying out within a few days. All these realizations have been aborted after 60 days.
Click here for file
Sample animations. The animations in wmf format provided with this manuscript show different single realizations or possible scenario of an unrestricted epidemic where one initial infected person is located in Stockholm. They are provided for interest only and should not be seen as contributing results, which require the combined analysis of many more realizations. Biggy.wmf is similar to the first but was not aborted until the epidemic has peaked and died out due to depletion of susceptible individuals.
Click here for file
Acknowledgements
This study was supported by The Swedish Institute for Infectious Diseases Control, Swedish Council for Working Life and Social Research, and the European Union Research NEST Project (DYSONET 012911). The authors would like to express their gratitude to Tom Britton and Åke Svensson, Department of Mathematics, Stockhholm University, Monica Nordvik, Department of Sociology, Stockholm University and Alden Klovdahl, Social Sciences, Australian National University and the members of SGEM, Stockholm Group of Epidemic Modeling
Prepublication history
The prepublication history for this paper can be accessed here: