Journal of Mathematical Biology

, Volume 70, Issue 6, pp 1411–1456

# Global analysis for spread of infectious diseases via transportation networks

• Yukihiko Nakata
• Gergely Röst
Article

## Abstract

We formulate an epidemic model for the spread of an infectious disease along with population dispersal over an arbitrary number of distinct regions. Structuring the population by the time elapsed since the start of travel, we describe the infectious disease dynamics during transportation as well as in the regions. As a result, we obtain a system of delay differential equations. We define the basic reproduction number $${\mathcal {R}}_0$$ as the spectral radius of a next generation matrix. For multi-regional systems with strongly connected transportation networks, we prove that if $${\mathcal {R}}_0\le 1$$ then the disease will be eradicated from each region, while if $${\mathcal {R}}_0> 1$$ there is a globally asymptotically stable equilibrium, which is endemic in every region. If the transportation network is not strongly connected, then the model analysis shows that numerous endemic patterns can exist by admitting a globally asymptotically stable equilibrium, which may be disease free in some regions while endemic in other regions. We provide a procedure to detect the disease free and the endemic regions according to the network topology and local reproduction numbers. The main ingredients of the mathematical proofs are the inductive applications of the theory of asymptotically autonomous semiflows and cooperative dynamical systems. We visualise stability boundaries of equilibria in a parameter plane to illustrate the influence of the transportation network on the disease dynamics. For a system consisting of two regions, we find that due to spatial heterogeneity characterised by different local reproduction numbers, $${\mathcal {R}}_0$$ may depend non-monotonically on the dispersal rates, thus travel restrictions are not always beneficial.

## Keywords

Epidemic models Transportation networks Global dynamics  Delay differential systems

34K05 92D30

## Notes

### Acknowledgments

The authors are grateful for Professor Eduardo Liz for his kind hospitality at the Universidade de Vigo, October, 2010, where YN and GR started this project. The authors are grateful for Professor Teresa Faria for her kind hospitality at the University of Lisbon, July 2012, where the authors had stimulating discussions about multipatch models. YN and GR were supported by European Research Council StG Nr. 259559. YN was also supported by Spanish Ministry of Science and Innovation (MICINN), MTM2010-18318 and by the European Union and the State of Hungary, co-financed by the European Social Fund in the framework of TÁMOP-4.2.4. A/2-11-1-2012-0001 ‘National Excellence Program’. Since April 2014 YN was also supported by JSPS Fellowship, No. 268448. GR was supported by European Union and the European Social Fund through project FuturICT.hu (Grant No. TÁMOP-4.2.2.C-11/1/KONV-2012-0013), and Hungarian Scientific Research Fund OTKA K75517. The authors are particularly grateful for the three referees, whose comments significantly improved the manuscript.

## References

1. Alirol E, Getaz L, Stoll B, Chappuis F, Loutan L (2011) Urbanisation and infectious diseases in a globalised world. Lancet Infect Dis 11(2):131–141
2. Arino J (2009) Diseases in metapopulations. Modeling and dynamics of infectious diseases. In: Series in contemporary applied mathematics, vol 11. World Scientific Publishing, Singapore, pp 65–123Google Scholar
3. Arino J, Brauer F, van den Driessche P, Watmough J, Wu J (2006) Simple models for containment of a pandemic. J R Soci Interface 3(8):453–457
4. Arino J, Davis J, Hartley D, Jordan R, Miller J, van den Driessche P (2005) A multi-species epidemic model with spatial migration. Math Med Biol 22(2):129–142
5. Arino J, van den Driessche P (2003) A multi-city epidemic model. Math Popul Stud 10(3):175–193
6. Arino J, Jordan R, van den Driessche P (2007) Quarantine in a multi-species epidemic model with spatial dynamics. Math Biosci 206(1):46–60
7. Bajardi P, Barrat A, Natale F, Savini L, Colizza V (2011) Dynamical patterns of cattle trade movements. PLoS One 6(5):e19869
8. Bajardi P, Poletto C, Ramasco JJ, Tizzoni M, Colizza V, Vespignani A (2011) Human mobility networks, travel restrictions, and the global spread of 2009 H1N1 pandemic. PLoS One 6(1):e16591
9. Balcan D, Vespignani A (2011) Phase transitions in contagion processes mediated by recurrent mobility patterns. Nat Phys 7:581–586
10. Bell DM (2004) World Health Organization Working Group on Prevention of International and Community Transmission of SARS.: Public health interventions and SARS spread, 2003. Emerg Infect Dis. http://wwwnc.cdc.gov/eid/article/10/11/04-0729.htm
11. Berman A, Plemmons RJ (1994) Nonnegative matrices in the mathematical sciences. Classics in Applied Mathematics. SIAM, PhiladelphiaGoogle Scholar
12. Brauer F, van den Driessche P (2001) Models for transmission of disease with immigration of infectives. Math Biosci 171:143–154
13. Colizza V, Barrat A, Barthélemy M, Vespignani A (2006) The role of the airline transportation network in the prediction and predictability of global epidemics. Proc Natl Acad Sci USA 103(7):2015–2020
14. Colizza V, Vespignani A (2007) Invasion threshold in heterogeneous metapopulation networks. Phys Rev Lett 99(14):148701
15. Colizza V, Vespignani A (2008) Epidemic modeling in metapopulation systems with heterogeneous coupling pattern: theory and simulations. J Theor Biol 251(3):450–467
16. Cui J, Takeuchi Y, Saito Y (2006) Spreading disease with transport-related infection. J Theor Biol 239(3):376–390
17. Diekmann O, van Gils SA, Walther HO (1995) Delay equations: functional-, complex-, and nonlinear analysis. In: Applied mathematical sciences vol 110. Springer, New YorkGoogle Scholar
18. Diekmann O, Heesterbeek JAP, Metz JAJ (1990) On the definition and the computation of the basic reproduction ratio $$\cal {R}_{0}$$ in models for infectious diseases in heterogeneous populations. J Math Biol 28(4):365–382
19. European Centre for Disease Prevention and Control (2009a) Risk assessment guidelines for diseases transmitted on aircraft. ECDC Technical Report. http://ecdc.europa.eu/en/publications/Publications/0906_TER_Risk_Assessment_Guidelines_for_Infectious_Diseases_Transmitted_on_Aircraft.pdf
20. European Centre for Disease Prevention and Control (2009b) Risk assessment guidelines for diseases transmitted on aircraft. Part 2: operational guidelines for assisting in the evaluation of risk for transmission by disease. ECDC Technical Report. http://ecdc.europa.eu/en/publications/Publications/1012_GUI_RAGIDA_2.pdf
21. Epstein JM, Goedecke DM, Yu F, Morris RJ, Wagener DK, Bobashev GV (2007) Controlling pandemic flu: the value of international air travel restrictions. PLoS One 2(5):e401
22. Faria T (2011) Global asymptotic behaviour for a Nicholson model with patch structure and multiple delays. Nonlinear Anal Theory Methods Appl 74(18):7033–7046
23. Fiedler M (1986) Special matrices and their applications in numerical mathematics. Martinus Nijhoff Publishers, The Hague
24. Gao D, Ruan S (2012) A multipatch malaria model with logistic growth populations. SIAM J Appl Math 72(3):819–841
25. Győri I (1992) Stability in a class of integrodifferential systems. Recent trends in differential equations. World Scientific Publishing, Singapore, pp 269–284Google Scholar
26. Hale JK, Verduyn Lunel SM (1993) Introduction to functional-differential equations. In: Applied mathematical sciences, vol 99. Springer, New YorkGoogle Scholar
27. Hofbauer J, So JWH (2000) Diagonal dominance and harmless off-diagonal delays. Proc Am Math Soc 128(9):2675–2682Google Scholar
28. Hollingsworth TD, Ferguson NM, Anderson RM (2006) Will travel restrictions control the international spread of pandemic influenza? Nat Med 12(5):497–499
29. Hsieh YH, van den Driessche P, Wang L (2007) Impact of travel between patches for spatial spread of disease. Bull Math Biol 69(4):1355–1375
30. Khan K, Arino J, Hu W, Raposo P, Sears J, Calderon F, Heidebrecht C, Macdonald M, Liauw J, Chan A, Gardam M (2009) Spread of a novel influenza A (H1N1) virus via global airline transportation. N Engl J Med 361(2):212–214
31. Knipl DH, Röst G, Wu J (2013) Epidemic spread and variation of peak times in connected regions due to travel-related infections–dynamics of an antigravity-type delay differential model. SIAM J Appl Dyn Syst 12(4):1722–1762
32. Li J, Zou X (2010) Dynamics of an epidemic model with non-local infections for diseases with latency over a patchy environment. J Math Biol 60(5):645–686
33. Liu J, Wu J, Zhou Y (2008) Modeling disease spread via transport-related infection by a delay differential equation. Rocky Mt J Math 38(5):1525–1540
34. Liu X, Takeuchi Y (2006) Spread of disease with transport-related infection and entry screening. J Theor Biol 242(2):517–528
35. Mangili A, Gendreau MA (2005) Transmission of infectious diseases during commercial air travel. Lancet 365(9463):989–996
36. Meloni S, Perra N, Arenas A, Gómez S, Moreno Y, Vespignani A (2011) Modeling human mobility responses to the large-scale spreading of infectious diseases. Sci Rep 1:62
37. Nakata Y (2011) On the global stability of a delayed epidemic model with transport-related infection. Nonlinear Anal Real World Appl 12(6):3028–3034
38. Nichol KL, Tummers K, Hoyer-Leitzel A, Marsh J, Moynihan M, McKelvey S (2010) Modeling seasonal influenza outbreak in a closed college campus: impact of pre-season vaccination, in-season vaccination and holidays/breaks. PLoS One 5(3):e9548
39. Olsen SJ, Chang HL, Cheung TYY, Tang AFY, Fisk TL, Ooi SPL, Kuo HW, Jiang DDS, Chen KT, Lando J, Hsu KH, Chen TJ, Dowell SF (2003) Transmission of the severe acute respiratory syndrome on aircraft. N Engl J Med 349(25):2416–2422
40. Poletto C, Tizzoni M, Colizza V (2012) Heterogeneous length of stay of hosts’ movements and spatial epidemic spread. Sci Rep 2:476
41. Silverman D, Gendreau M (2009) Medical issues associated with commercial flights. Lancet 373(9680):2067–2077
42. Smith H, Waltman P (1995) The theory of the chemostat: dynamics of microbial competition. In: Cambridge studies in mathematical biology, vol 13. Cambridge University Press, CambridgeGoogle Scholar
43. Smith HL (1995) Monotone dynamical systems: an introduction to the theory of competitive and cooperative systems. In: Mathematical surveys and monographs, vol 41. American Mathematical SocietyGoogle Scholar
44. Smith HL (2011) An introduction to delay differential equations with applications to the life sciences. In: Texts in applied mathematics, vol 57. Springer, New YorkGoogle Scholar
45. Takeuchi Y, Liu X, Cui J (2007) Global dynamics of SIS models with transport-related infection. J Math Anal Appl 329(2):1460–1471
46. Thieme HR (1992) Convergence results and a Poincaré–Bendixson trichotomy for asymptotically autonomous differential equations. J Math Biol 30(7):755–763
47. van den Driessche P, Watmough J (2002) Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math Biosci 180(1–2):29–48
48. Wang W (2007) Epidemic models with population dispersal. In: Mathematics for life science and medicine, biological and medical physics, biomedical engineering. Springer, New YorkGoogle Scholar
49. Wang W, Zhao X (2004) An epidemic model in a patchy environment. Math Biosci 190(1):97–112
50. Wang W, Zhao X (2005) An age-structured epidemic model in a patchy environment. SIAM J Appl Math 65(5):1597–1614
51. WHO (2003) Severe acute respiratory syndrome (SARS): status of the outbreak and lessons for the immediate future. Geneva. http://www.who.int/csr/media/sars_wha.pdf
52. Zhou Y, Khan K, Feng Z, Wu J (2008) Projection of tuberculosis incidence with increasing immigration trends. J Theor Biol 254(2):215–228
53. Zhao XQ, Jing ZJ (1996) Global asymptotic behavior in some cooperative systems of functional differential equations. Can Appl Math Q 4(4):421–444