Bulletin of Mathematical Biology

, Volume 78, Issue 2, pp 185–209 | Cite as

Modeling Importations and Exportations of Infectious Diseases via Travelers

  • Luis Fernandez Lopez
  • Marcos Amaku
  • Francisco Antonio Bezerra Coutinho
  • Mikkel Quam
  • Marcelo Nascimento Burattini
  • Claudio José Struchiner
  • Annelies Wilder-Smith
  • Eduardo MassadEmail author
Original Article


This paper is an attempt to estimate the risk of infection importation and exportation by travelers. Two countries are considered: one disease-free country and one visited or source country with a running endemic or epidemic infectious disease. Two models are considered. In the first model (disease importation), susceptible individuals travel from their disease-free home country to the endemic country and come back after some weeks. The risk of infection spreading in their home country is then estimated supposing the visitors are submitted to the same force of infection as the local population but do not contribute to it. In the second model (disease exportation), it is calculated the probability that an individual from the endemic (or epidemic) country travels to a disease-free country in the condition of latent infected and eventually introduces the infection there. The input of both models is the force of infection at the visited/source country, assumed known. The models are deterministic, but a preliminary stochastic formulation is presented as an appendix. The models are exemplified with two distinct real situations: the risk of dengue importation from Thailand to Europe and the risk of Ebola exportation from Liberia to the USA.


Infectious disease importation Infectious disease exportation Travelers Modeling Risk 



This study received partial funding from LIM01-HCFMUSP, HSP/ UNIFESP, CNPq, FAPESP, Ministry of Health (Fundo Nacional de Saúde, grant 27835/2012), and was also partially funded by DengueTools (22) under the Health theme of the Seventh Framework Programme of the European Community (Grant Agreement Number 282589), DengueTools: innovative tools and strategies for the surveillance and control of dengue. Open image in new window Open image in new window (Wilder-Smith et al. 2012).

Conflict of interest

The authors declare no conflict of interest.


  1. Amaku M, Azevedo F, Burattini MN, Coutinho FAB, Lopez LF, Massad E (2014) Interpretations and pitfalls in modeling vector-transmitted infections. Epidemiol Infect 143:1803–1815. doi: 10.1017/S0950268814002660 CrossRefGoogle Scholar
  2. Anderson RM, May RM (1991) Infectious diseases of humans: dynamics and control. Oxford University Press, OxfordGoogle Scholar
  3. Bossak BH, Welford MR (2009) Did medieval trade activity and a viral etiology control the spatial extent and seasonal distribution of Black Death mortality? Med Hypotheses 72:749–752CrossRefGoogle Scholar
  4. Caley P, Becker NG, Philp DJ (2007) The waiting time for inter-country spread of pandemic influenza. PLoS One 2:e143CrossRefGoogle Scholar
  5. Cox DR, Miller HD (1965) The theory of stochastic processes. Chapman and Hall, LondonzbMATHGoogle Scholar
  6. Gomes MFC, Piontti AP, Rossi L, Chao D, Longini I, Halloran ME, Vespignani A (2014) Assessing the international spreading risk associated with the 2014 West African Ebola outbreak. PLoS Current Outbreaks 2014 Sep2. Edition 1. doi: 10.1371/currents.outbreaks.cd818f63d40e24aef769dda7df9e0da5
  7. Jones KE, Patel NG, Levy MA et al (2008) Global trends in emerging infectious diseases. Nature 451:990–993CrossRefGoogle Scholar
  8. Khan K, Arino J, Hu W et al (2009) Spread of a novel influenza A (H1N1) virus via global airline transportation. N Engl J Med 361:212–214CrossRefGoogle Scholar
  9. Khan K, Memish ZA, Chabbra A et al (2010) Global public health implications of a mass gathering in Mecca, Saudi Arabia during the midst of an influenza pandemic. J Travel Med 17:75–81CrossRefGoogle Scholar
  10. Leder K, Torresi J, Brownstein JS et al (2013) Travel-associated illness trends and clusters, 2000–2010. Emerg Infect Dis. 19:1049–1073CrossRefGoogle Scholar
  11. Leder K, Wilson ME, Freedman DO, Torresi JA (2008) comparative analysis of methodological approaches used for estimating risk in travel medicine. J Travel Med 15:263–272CrossRefGoogle Scholar
  12. Lewnard JA, Ndeffo Mbah ML, Alfaro-Murillo JA, Altice FL, Bawo L, Nyenswah TG, Galvani A (2014) Dynamics and control of Ebola virus transmission in Montserrado, Liberia: a mathematical modeling analysis. Lancet Infect Dis 14:1189–1195CrossRefGoogle Scholar
  13. Massad E, Burattini MN, Coutinho FAB, Lopez LF (2007) The 1918 influenza a epidemic in the city of Sao Paulo. Braz Med Hypotheses 68:442–445CrossRefGoogle Scholar
  14. Massad E, Burattini MN, Coutinho FAB, Struchiner CJ (2010) The risk of acquiring the new influenza A(H1N1) for Brazilian travelers to Chile, Argentina and the USA. Mem Inst Oswaldo Cruz 105:179–183CrossRefGoogle Scholar
  15. Massad E, Burattini MN, Lopez LF, Coutinho FAB (2005a) Forecasting versus projection models in epidemiology: the case of the SARS epidemics. Med Hypotheses 65:17–22CrossRefGoogle Scholar
  16. Massad E, Coutinho FAB, Burattini MN, Lopez LF (2004) The Eyam plague revisited: did the village isolation change transmission from fleas to pulmonary? Med Hypotheses 63:911–915CrossRefGoogle Scholar
  17. Massad E, Coutinho FAB, Burattini MN, Lopez LF, Struchiner CJ (2005b) Yellow fever vaccination: how much is enough. Vaccine 23(30):3908–3914CrossRefGoogle Scholar
  18. Massad E, Coutinho FAB, Yang HM, De Carvalho HB, Mesquita F, Burattini MN (1994) The basic reproduction ratio of HIV among intravenous-drug-users. Math Biosci 123:227–247CrossRefzbMATHGoogle Scholar
  19. Massad E, Wilder-Smith A (2009) Risk estimates of dengue in travelers to dengue endemic areas using mathematical models. J Travel Med 16:191–193CrossRefGoogle Scholar
  20. Massad E, Rocklov J, Wilder-Smith A (2013) Dengue infections in non-immune travelers to Thailand. Epidemiol Infect 141:412–417CrossRefGoogle Scholar
  21. Massad E, Wilder-Smith A, Ximenes R et al (2014) Risk of symptomatic dengue for foreign visitors to the 2014 FIFA World Cup in Brazil. Mem Inst Oswaldo Cruz 109(3):394–397CrossRefGoogle Scholar
  22. Pandey A, Atkins KE, Medlock J, Wenzel N, Townsend JP, Childs JE, Nyenswah TG, Ndeffo-Mbah ML, Galvani AP (2014) Strategies for containing Ebola in West Africa. Science 346:991. doi: 10.1126/science.1260612 CrossRefGoogle Scholar
  23. Quam M, Massad E, Wilder-Smith A (2014) Effects of India’s new polio policy on travelers. Lancet 383:1632CrossRefGoogle Scholar
  24. Stannard DE (1993) Disease, human migration, and history. In: Kipple KE (ed) The Cambridge world history of human disease. Cambridge University Press, Cambridge, pp 35–44CrossRefGoogle Scholar
  25. Tatem AJ, Huang Z, Das A, Qi Q, Roth J, Qiu Y (2012) Air travel and vector-borne disease movement. Parasitology 139:1816–1830CrossRefGoogle Scholar
  26. UNWTO World Tourism Barometer. Accessed Aug 2014
  27. Wilder-Smith A (2006) The severe acute respiratory syndrome: impact on travel and tourism. Travel Med Infect Dis 4:53–60CrossRefGoogle Scholar
  28. Wilder-Smith A, Freedman DO (2003) Confronting the new challenge in travel medicine: SARS. J Travel Med 10:257–258CrossRefGoogle Scholar
  29. Wilder-Smith A, Renhorn KE, Tissera H, Abu Bakar S, Alphey L, Kittayapong P, Lindsay S, Logan J, Hatz C, Reiter P, Rocklöv J, Byass P, Louis VR, Tozan Y, Massad E, Tenorio A, Lagneau C, L’Ambert G, Brooks D, Wegerdt J, Gubler D (2012) DengueTools: innovative tools and strategies for the surveillance and control of dengue. Glob Health Action 5. doi: 10.3402/gha.v5i0.17273
  30. Wilder-Smith A, Tambyah PA (2007) The spread of poliomyelitis via international travel. Trop Med Int Health 12:1137–1138CrossRefGoogle Scholar

Copyright information

© Society for Mathematical Biology 2016

Authors and Affiliations

  • Luis Fernandez Lopez
    • 1
    • 2
  • Marcos Amaku
    • 1
  • Francisco Antonio Bezerra Coutinho
    • 1
  • Mikkel Quam
    • 3
  • Marcelo Nascimento Burattini
    • 1
    • 4
  • Claudio José Struchiner
    • 5
  • Annelies Wilder-Smith
    • 6
    • 7
  • Eduardo Massad
    • 1
    • 8
    Email author
  1. 1.School of MedicineUniversity of São PauloSão PauloBrazil
  2. 2.CIARAFlorida International UniversityMiamiUSA
  3. 3.Epidemiology and Global HealthUmeå UniversityUmeåSweden
  4. 4.Hospital São PauloEscola Paulista de MedicinaSão PauloBrazil
  5. 5.PROCCFIOCRUZRio de JaneiroBrazil
  6. 6.Lee Kong Chian School of MedicineNanyangSingapore
  7. 7.Technological UniversitySingaporeSingapore
  8. 8.London School of Hygiene and Tropical MedicineLondonUK

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