A STOCHASTIC EQUATION-BASED MODEL OF THE VALUE OF INTERNATIONAL AIR-TRAVEL RESTRICTIONS FOR CONTROLLING PANDEMIC FLU

Transcription

1 Proceedngs of the 2007 Wnter Smulaton Conference S. G. Henderson, B. Bller, M.-H. Hseh, J. Shortle, J. D. Tew, and R. R. Barton, eds. A STOCHASTIC EQUATION-BASED MODEL OF THE VALUE OF INTERNATIONAL AIR-TRAVEL RESTRICTIONS FOR CONTROLLING PANDEMIC FLU D. Mchael Goedecke Georgy V. Bobashev Feng Yu Statstcs and Epdemology Dvson RTI Internatonal 3040 Cornwalls Road Research Trangle Park, NC 27709, U. S. A. ABSTRACT Internatonal ar travel can be an mportant contrbutng factor to the global spread of nfectous dseases, as evdenced by the outbreak of Severe Acute Respratory Syndrome n Restrctons on ar travel may therefore be one response to attempt to control a wdespread epdemc of a dsease such as nfluenza. We present results from a stochastc, equaton-based, global epdemc model whch suggest that ar travel restrctons often provde only a slght delay n the epdemc. Ths delay may gve valuable tme n whch to mplement other dsease control strateges; however, f other strateges are not mplemented, the use of travel restrctons alone may lead to a more severe epdemc than f they had not been mposed. Our results also ndcate that the partcular network of ctes chosen for modelng can have a great nfluence on the model results. 1 INTRODUCTION The outbreak of Severe Acute Respratory Syndrome (SARS) n 2003 hghlghted the role that nternatonal ar travel can play n the rapd spread of contagous dseases throughout the world. No country s solated from the threat of a pandemc or a large regonal epdemc. A global model of the spread of a dsease can therefore provde an mportant complementary vew to those gven by more focused country-, regon-, or cty-specfc models. Such a global epdemc model could allow estmaton of the tme avalable to prepare for a local or regonal outbreak of a dsease gven an outbreak n another part of the world, and of the lkely effects of potental nterventon and dsease contanment strateges. Early global nfluenza epdemc models were based on determnstc systems of coupled, cty-level compartmental models (Baroyan, Mronov, and Rvachev 1981; Rvachev and Longn 1985). More recent models have updated populaton and travel data, used larger networks of ctes, added seasonalty, nterventons, or stochastcty, or focused on other dseases (Gras, Ells, and Glass 2003; Gras et al. 2004; Cooper et al. 2006; Colzza et al. 2006a; Colzza et al. 2006b; Colzza et al. 2007; Hufnagel, Brockmann, and Gesel 2004; Epsten et al. 2007). Equaton-based models often assume homogeneous mxng wthn the populaton and mass-acton knetcs, and thus provde a mean-feld vew of the dynamcs of the system. In contrast, agent-based models, whch track ndvduals rather than groups, have been developed to capture fner levels of behavoral detal and ther effects on the macrolevel behavor of the model (Ferguson et al. 2005, Ferguson et al. 2006, Longn et al. 2005, Germann et al. 2006, Eubank et al. 2004). However, the ncreased complexty and level of detal of agent-based models means that they are often much slower and more computatonally expensve than equaton-based models. The choce of usng an equaton-based model or an agent-based model for a gven system often depends on ths trade-off between the desred level of detal and the computatonal resources avalable. In ths paper, we present a stochastc, equaton-based model for the spread of pandemc nfluenza, and some results regardng the potental effectveness of ar travel restrctons as a dsease control strategy. In Secton 2, we gve a bref overvew of the detals of the model. Results are presented n Secton 3, and conclusons and possble future work are descrbed n Secton 4. 2 MODEL DETAILS We created a stochastc, equaton-based model utlzng a global network of ctes connected by ar travel. Travel and populaton data are from several publcly avalable sources (Brnkhoff 2005; ESRI 2005; Helders 2005; IBGE 2006; Mongabay.com 2004; Populaton Dvson of the Department of Economc and Socal Affars of the Unted /07/$ IEEE 1538

2 Natons Secretarat, World Populaton Prospects 2004; Populaton Dvson, U.S. Census Bureau 2004; Gumerà et al. 2005). Natural hstory parameters for the H5N1 nfluenza vrus algn wth those publshed prevously (Ferguson et al. 2005, Ferguson et al. 2006, Longn et al. 2005, Germann et al. 2006). Wthn each cty s a homogeneously mxed, SEIRtype epdemologcal model, wth the cty s populaton dvded among four possble dsease states: Susceptble (S), Exposed (E), Infectous (I), and Recovered (R). Members of the Susceptble group are those who have not yet been nfected, but who are susceptble to nfecton gven contact wth a member of the Infectous group. Those n the Exposed group have been nfected but the dsease s n a latent dsease state, n whch they show no symptoms and are not yet able to nfect others. Those n the Infectous group are symptomatc and able to nfect others. Recovered are no longer nfectous, and have acqured mmunty to further nfecton. Members of the Susceptble, Exposed, and Recovered groups can travel between ctes. Members of the Infectous group are assumed not to travel. Thus, the dsease s spread wthn ctes by contact between Susceptble and Infectous, whle an outbreak n a prevously unnfected cty s caused by Exposed travelers who become Infectous whle n that cty. Stochastcty n the model allows for characterzaton of varablty of the possble outcomes for a gven scenaro, rather than descrbng only the mean behavor of the system. There are two sources of stochastcty n ths model: the daly number of nfectous contacts between Susceptble and Infectous, and the daly numbers of travelers between ctes. The number of newly Exposed wthn a cty on a partcular day s the expected daly number of nfectous contacts between susceptble and nfectous members of that cty s populaton. In the determnstc case, wthn cty at the start of day t + 1, ths number s gven by S () t E ( 0, t +1) = λ ( t) I () t, (1) T () t where λ s the nfectous contact rate and T s the total populaton of cty. We ncorporate dsease seasonalty and geographc effects nto the model by allowng the nfectous contact rate to vary both wth tme and locaton. Thus λ () t = λ( t, lattude ). Assumng that random contacts between pars of ndvduals are ndependent events and that the total number of contacts that occurs n the tme nterval from t to t + Δt does not depend on ether the prevous number of contacts or on t, we can make equaton (1) stochastc by nstead choosng the value of E ( 0, t + 1) from a Posson dstrbuton wth ts mean value equal λ t S t I t T t. to () () () () The determnstc number of Susceptble travelers from cty to cty j on day t s gven by S ( t) ptj, where pt j s the probablty of travelng from cty to cty j. Wth n ctes n the model, the net number of travelers nto and out of cty s then gven by Ω n [ j j j ] [ S () t ] = S () t pt () t S () t pt () t j= 1 Because a traveler from one cty could travel to any one of multple destnatons, the stochastc number of daly travelers to each destnaton s drawn from a multnomal dstrbuton. The numbers of Exposed or Recovered travelers are calculated smlarly. To prevent small fractons of Exposed travelers from prematurely ntatng outbreaks n prevously unnfected ctes, f the net number of Exposed travelers to a cty s below a threshold value, t s set equal to zero. Scenaros nvolvng the mposton of travel restrctons are mplemented by reducng the probablty of all travel nto or out of a gven cty by a pre-set percentage when the cumulatve dsease ncdence n that cty or n another cty to whch t s drectly connected reaches a threshold value. Travel restrctons are thus mplemented on a cty-by-cty bass as the epdemc approaches, rather than beng mplemented smultaneously worldwde. Once restrctons are mplemented n a gven cty, they reman n effect for the duraton of the model run. For the results presented ths paper, the threshold value for mposng restrctons was set to 1000 cases, and the travel reducton level was set to 95%. 3 MODEL RESULTS For ths paper, we ran several model scenaros varyng three factors: epdemc orgnaton date (January 1 vs. July 1), mposton of travel restrctons (no restrcton vs. 95% travel restrcton), and cty network (largest populaton vs. largest ar travel). In each scenaro, the orgn of nfecton was Hong Kong, the number of ctes n the network was 155, and the basc reproducton number of the vrus, R 0, was 1.7. Each scenaro was run 50 tmes. Results reported here are mean results unless stated otherwse. The maps n Fgures 1 and 2 clearly show the dfference n the populaton-based and the travel-volume-based networks. The network based on largest populaton szes s much more geographcally dverse, provdng sgnfcantly more coverage of Afrca and South Amerca. The network based on largest ar travel volumes s sgnfcantly more concentrated n the Unted States and Western Europe

3 Los Angeles daly new nfectous - populaton-based network Fgure 1: Map of the network of 155 ctes by largest cty populaton that was used n the model. 1.00E E E E E E E E E E+04 Jan - no restrcton Jan - 95% travel restrcton July - no restrcton July - 95% travel restrcton days snce January 1 Fgure 3: Epdemc curves for Los Angeles, usng the largest-populaton based network. Daly number of newly Infectous. The dark blue and pnk lnes represent an epdemc orgnatng n Hong Kong on January 1, wth and wthout 95% travel restrctons, respectvely. The red and lght blue lnes represent an epdemc orgnatng n Hong Kong on July 1, wth and wthout 95% travel restrctons, respectvely. Los Angeles cumulatve new nfectous - populaton-based network Fgure 2: Map of the network of 155 ctes by largest ar travel volume that was used n the model. Fgures 3 and 4 present the mean epdemc curves for Los Angeles, usng the largest-populaton based network, to demonstrate that mposng travel restrctons can have a sgnfcant effect at the local level on the tme course of an epdemc, and that those effects can be ether postve or negatve, based on the tmng of the ntal outbreak. For an outbreak startng n Hong Kong n January, f there are no nterventons, the ncdence peaks n Los Angeles n May, past the usual flu season. Wth travel restrctons mposed, the peak s pushed to md-june and ts sze s greatly reduced. For an outbreak orgnatng n Hong Kong n July, however, the epdemc peaks n Los Angeles n late November, and s much more severe. In ths case, travel restrctons delay the peak to late December, at the heght of the flu season, and thus make the local epdemc worse than f the restrctons had not been mposed. From Fgure 4, t can be seen that the fnal cumulatve number of cases n Los Angeles s not sgnfcantly changed by the mposton of travel restrctons, n the case of an ntal outbreak n ether January or July. 3.50E E E E E E E+05 Jan - no restrcton Jan - 95% travel restrcton July - no restrcton July - 95% travel restrcton days snce January 1 Fgure 4: Epdemc curves for Los Angeles, usng the largest-populaton based network. Cumulatve number of newly Infectous. The dark blue and pnk lnes represent an epdemc orgnatng n Hong Kong on January 1, wth and wthout 95% travel restrctons, respectvely. The red and lght blue lnes represent an epdemc orgnatng n Hong Kong on July 1, wth and wthout 95% travel restrctons, respectvely. Fgures 5 and 6 show very dfferent results for Los Angeles when the epdemc s modeled on the network based on largest ar travel volumes. For both January and July outbreaks n Hong Kong, the peak ncdence n Los Angeles s earler than n the case of the populaton-based network, and the delays nduced by the travel restrctons are much shorter a few days, rather than approxmately a month. However, the epdemc duraton n Los Angeles s 1540

4 also shortened n the case of travel restrctons, wth the net effect that there s a sgnfcant reducton n the cumulatve number of cases. 1.00E E E E E E E E E E+04 Los Angeles daly new nfectous - volume-based network Jan - no restrcton Jan - 95% travel restrcton July - no restrcton July - 95% travel restrcton days snce January 1 Fgure 5: Epdemc curves for Los Angeles, usng the largest-travel-volume based network. Daly number of newly Infectous. The dark blue and pnk lnes represent an epdemc orgnatng n Hong Kong on January 1, wth and wthout 95% travel restrctons, respectvely. The red and lght blue lnes represent an epdemc orgnatng n Hong Kong on July 1, wth and wthout 95% travel restrctons, respectvely. 4 CONCLUSIONS We have presented a model of the potental effects of usng nternatonal ar travel restrctons as a method to control an epdemc of a contagous dsease such as H5N1 nfluenza. Our results ndcate that such restrctons can delay the spread of the dsease, but that delays may be short. If no other control measures than travel restrctons are mplemented, then local epdemcs may be ether more or less severe, as they are delayed ether nto or out of the local peak epdemc season. The choce of the underlyng network of ctes to be modeled s also extremely mportant. The tme course and the cumulatve severty of the epdemcs were sgnfcantly dfferent for our populaton-based and travel-based networks. Combnng a number of such crtera as well as crtera based on network characterstcs such as degree or betweenness, may be most approprate. Ths model does not currently nclude rural populatons, ground transportaton, or detaled heterogenety wthn the ctes. Each of these may have a large mpact on model results, and are the subjects of contnung work. ACKNOWLEDGMENTS Ths work was supported n part by the Plot Studes of Modelng of Infectous Dsease Agents Study (MIDAS) cooperatve agreement from NIGMS, (1U01 GM070698). 3.50E E E E E E E+05 Los Angeles cumulatve new nfectous - volume-based network Jan - no restrcton Jan - 95% travel restrcton July - no restrcton July - 95% travel restrcton days snce January 1 Fgure 6: Epdemc curves for Los Angeles, usng the largest-travel-volume based network. Cumulatve number of newly Infectous. The dark blue and pnk lnes represent an epdemc orgnatng n Hong Kong on January 1, wth and wthout 95% travel restrctons, respectvely. The red and lght blue lnes represent an epdemc orgnatng n Hong Kong on July 1, wth and wthout 95% travel restrctons, respectvely. REFERENCES Baroyan, O. V., G. A. Mronov, and L. A. Rvachev An algorthm modelng global epdemcs of mutant orgn. Programmng and Computer Software 6(5): Englsh translaton from Programmrovane 5: (n Russan). Brnkhoff, T Mato Grosso Cty Populaton. Avalable va < Brazl-MatoGrosso.html> Colzza, V., A. Barrat, M. Barthélemy, and A. Vespgnan. 2006a. The role of the arlne transportaton network n the predcton and predctablty of global epdemcs. Proceedngs of the Natonal Academy of Scences of the U. S. A. 103(7): Colzza, V., A. Barrat, M. Barthélemy, and A. Vespgnan. 2006b. The modelng of global epdemcs: Stochastc dynamcs and predctablty. Bulletn of Mathematcal Bology 68: Colzza, V., A. Barrat, M. Barthelemy, A. J. Valleron, and A. Vespgnan Modelng the worldwde spread of pandemc nfluenza: Baselne case and contanment nterventons. PLoS Medcne 4(1):e13. Cooper, B. S., R. J. Ptman, W. J. Edmunds, and N. J. Gay Delayng the nternatonal spread of pandemc nfluenza. PLoS Medcne 3(6):e

5 Epsten, J. M., D. M. Goedecke, F. Yu, R. J. Morrs, D. K. Wagener, and G. V. Bobashev Controllng Pandemc Flu: The Value of Internatonal Ar Travel Restrctons. PLoS ONE 2(5):e401. ESRI ArcGIS 9 World, Europe, Canada, and Mexco: 1996, 1998, Wnter 1993/1994. [Computer software and data fles , 2000, , , , , , 2000, 2003]. Redlands, CA: ESRI. Eubank, S., H. Guclu, V. S. A. Kumar, M. V. Marathe, A. Srnvasan, Z. Toroczka, and N. Wang Modelng dsease outbreaks n realstc urban socal networks. Nature 429: Ferguson, N. M., D. A. T. Cummngs, S. Cauchemez, C. Fraser, S. Rley, A. Meeya, S. Iamsrthaworn, and D. S. Burke Strateges for contanng an emergng nfluenza pandemc n Southeast Asa. Nature 437: Ferguson, N. M., D. A. T. Cummngs, C. Fraser, J. C. Cajka, P. C. Cooley, and D. S. Burke Strateges for mtgatng an nfluenza pandemc. Nature 442: Germann, T. C., K. Kadau, I. M. Longn Jr, C. A. Macken Mtgaton strateges for pandemc nfluenza n the Unted States. Proceedngs of the Natonal Academy of Scences of the U. S. A. 103(15): Gras, R. F., J. H. Ells, and G. E. Glass Assessng the mpact of arlne travel on the geographc spread of pandemc nfluenza. European Journal of Epdemology. 18: Gras, R. F., J. H. Ells, A. Kress, and G. E. Glass Modelng the spread of annual nfluenza epdemcs n the U.S.: The potental role of ar travel. Health Care Management Scence 7: Gumerà, R., S. Mossa, A. Turtsch, L. A. N. Amaral The worldwde ar transportaton network: Anomalous centralty, communty structure, and ctes global roles. Proceedngs of the Natonal Academy of Scences of the U. S. A. 102(22): Helders, S World Gazetteer. Avalable va < [accessed Aprl 20, 2006]. Hufnagel, L., D. Brockmann, and T. Gesel Forecast and control of epdemcs n a globalzed world. Proceedngs of the Natonal Academy of Scences of the U. S. A. 101(42): Insttuto Braslero de Geografa e Estatístca (IBGE) Avalable va < [accessed Aprl 20, 2006]. Longn, Jr., I. M., A. Nzam, S. Xu, K. Ungchusak, W. Hanshaoworakul, D. A. T. Cummngs, and M. E. Halloran Contanng pandemc nfluenza at the source. Scence 309: Mongabay.com World Populaton Fgures. Avalable va < [accessed Aprl 24, 2006]. Populaton Dvson of the Department of Economc and Socal Affars of the Unted Natons Secretarat, World Populaton Prospects World Urbanzaton Prospects: The 2003 Revson Populaton Database. Avalable va < unup> [accessed Aprl, 2006]. Populaton Dvson, U.S. Census Bureau Table 1. Annual Estmates of the Populaton of Metropoltan and Mcropoltan Statstcal Areas: Aprl 1, 2000 to July 1, 2004 (CBSA-EST ). Avalable va < w/estmates/estmates%20pages_fnal. html> [accessed Aprl, 2006]. Rvachev, L. A., I. M. Longn, Jr A mathematcal model for the global spread of nfluenza. Mathematcal Boscences 75:3 22. AUTHOR BIOGRAPHIES D. MICHAEL GOEDECKE s a Research Statstcan at RTI Internatonal. He receved hs PhD n Bomathematcs from North Carolna State Unversty. Hs research nterests are focused on the mathematcal modelng of bologcal systems at ether the cellular or populaton levels. Hs emal address s <mgoedecke@rt.org>. GEORGIY V. BOBASHEV s a Senor Research Statstcan at RTI nternatonal. He has receved hs PhD n Bomathematcs from North Carolna State Unversty. Dr. Bobashev have been drectng and managng statstcal and mathematcal projects that employ both determnstc and stochastc modelng. Hs research nterests are n modelng and analyss of complex systems wth focus on healthrelated problems. Hs e-mal address s <bobashev@rt.org>. FENG YU s a Research Statstcan at RTI Internatonal. She receved her PhD n Chemcal Engneerng from the Unversty of Tennessee, Knoxvlle, and her MS n Statstcs from the Unversty of Nebraska-Lncoln. Feng has been nvolved n numerous statstcal and modelng projects. Her research nterests nclude mathematcal modelng n engneerng and epdemology and statstcal analyss of complex survey data. Her emal address s <fyu@rt.org>. 1542