Bulletin of Mathematical Biology

, Volume 74, Issue 10, pp 2474–2487 | Cite as

A Vaccination Model for a Multi-City System

Original Article

Abstract

A modelling approach is used for studying the effects of population vaccination on the epidemic dynamics of a set of n cities interconnected by a complex transportation network. The model is based on a sophisticated mover-stayer formulation of inter-city population migration, upon which is included the classical SIS dynamics of disease transmission which operates within each city. Our analysis studies the stability properties of the Disease-Free Equilibrium (DFE) of the full n-city system in terms of the reproductive number R 0. Should vaccination reduce R 0 below unity, the disease will be eradicated in all n-cities. We determine the precise conditions for which this occurs, and show that disease eradication by vaccination depend on the transportation structure of the migration network in a very direct manner. Several concrete examples are presented and discussed, and some counter-intuitive results found.

Keywords

Vaccination Network model Epidemic model Reproduction number 

Notes

Acknowledgements

We are grateful for funding support from the EU FP7 Epiwork grant, the Israel Science Foundation, and the Israel Ministry of Health.

References

  1. Arino, J. (2009). Diseases in metapopulations. In: Modeling and dynamics of infectious diseases. Series in contemporary applied mathematics (Vol. 11, 65–123). Google Scholar
  2. Arino, J., & van den Driessche, P. (2003). A multi-city epidemic model. Math. Popul. Stud., 10, 175–193. MathSciNetCrossRefMATHGoogle Scholar
  3. Arino, J., & van den Driessche, P. (2004). The basic reproduction number in a multi-city compartmental epidemic model. Lect. Notes Control Inf. Sci., 294, 100. Google Scholar
  4. Arino, J., & van den Driessche, P. (2006). Metapopulations epidemic models. A survey. Fields Inst. Commun., 48, 1–12. Google Scholar
  5. Arino, J., Ducrot, A., & Zongo, P. (2012). A metapopulation model for malaria with transmission-blocking partial immunity in hosts. J. Math. Biol., 64, 423–448. MathSciNetCrossRefMATHGoogle Scholar
  6. Berman, A., & Plemmons, R. J. (1979). Nonnegative matrices in the mathematical sciences. San Diego: Academic Press. MATHGoogle Scholar
  7. Brauer, F. (2008). Epidemic models with heterogeneous mixing and treatment. Bull. Math. Biol., 70, 1869–1885. MathSciNetCrossRefMATHGoogle Scholar
  8. Brockmann, D., et al. (2006). The scaling laws of human travel. Nature, 439, 462–465. CrossRefGoogle Scholar
  9. Diekmann, O., & Heesterbeek, J. A. P. (2000). Mathematical epidemiology of infectious diseases: model building, analysis and interpretation. New York: Wiley. Google Scholar
  10. Diekmann, O., et al. (1990). On the definition and the computation of the basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations. J. Math. Biol., 28, 365. MathSciNetCrossRefMATHGoogle Scholar
  11. Eames, K. T., & Keeling, M. J. (2002). Modeling dynamic and network heterogeneities in the spread of sexually transmitted diseases. Proc. Natl. Acad. Sci. USA, 99(20), 13330–13335. CrossRefGoogle Scholar
  12. Eubank, S., et al. (2004). Modelling disease outbreaks in realistic urban social networks. Nature, 429, 180–184. CrossRefGoogle Scholar
  13. Fine, P. E. (1993). Herd immunity: history, theory, practice. Epidemiol. Rev., 15, 265–302. Google Scholar
  14. Hadeler, K. P., & Castillo-Chavez, C. (1995). A core group model for disease transmission. Math. Biosci., 128, 41–55. CrossRefMATHGoogle Scholar
  15. Hadeler, K. P., & van den Driessche, P. (1997). Backward bifurcation in epidemic control. Math. Biosci., 146, 15–35. MathSciNetCrossRefMATHGoogle Scholar
  16. Hethcote, H. W., & Yorke, J. A. (1984). Lecture notes in biomathematics: Vol. 56. Gonorrhea transmission dynamics and control. Berlin: Springer. MATHGoogle Scholar
  17. Keeling, M. J., & Eames, K. T. (2005). Networks and epidemic model. J. R. Soc. Interface, 2, 295–307. CrossRefGoogle Scholar
  18. Kribs-Zaleta, C., & Martcheva, M. (2002). Vaccination strategies and backward bifurcation in an age-since-infection structured model. Math. Biosci., 177, 317–332. MathSciNetCrossRefGoogle Scholar
  19. Kribs-Zaleta, C. M., & Velasco-Hernandez, J. X. (2000). A simple vaccination model with multiple endemic states. Math. Biosci., 164, 183–201. CrossRefMATHGoogle Scholar
  20. Lloyd, A. L., & May, R. M. (2001). How viruses spread among computers and people. Science, 292, 1316–1317. CrossRefGoogle Scholar
  21. McCluskey, C.C., et al. (2003). Global results for an epidemic model with vaccination that exhibits backward bifurcation. SIAM J. Appl. Math., 64(1), 260–276. MathSciNetCrossRefMATHGoogle Scholar
  22. Pastor-Satorras, R., & Vespignani, A. (2001). Epidemic spreading in scale-free networks. Phys. Rev. Lett., 86, 3200–3203. CrossRefGoogle Scholar
  23. Pastor-Satorras, R., & Vespignani, A. (2001). Epidemic dynamics and endemic states in complex networks. Phys. Rev. E, Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics, 63, 066117. CrossRefGoogle Scholar
  24. Pastor-Satorras, R., & Vespignani, A. (2002). Immunization of complex networks. Phys. Rev. E, Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics, 65, 036104. CrossRefGoogle Scholar
  25. Riley, S. (2007). Large-scale spatial-transmission models of infectious diseases. Science, 316, 1298–1301. CrossRefGoogle Scholar
  26. Ruan, S., et al. (2006). The effect of global travel on the spread of Sars. Math. Biosci. Eng., 3, 205–218. MathSciNetCrossRefMATHGoogle Scholar
  27. Sattenspiel, L., & Dietz, K. (1995). A structured epidemic model incorporating geographic mobility among regions. Math. Biosci., 128, 71–91. CrossRefMATHGoogle Scholar
  28. van den Driessche, P., & Watmough, J. (2002). Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math. Biosci., 180, 29–48. MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Society for Mathematical Biology 2012

Authors and Affiliations

  1. 1.Department of PhysicsBar Ilan UniversityRamat GanIsrael
  2. 2.BioMathematics Unit, Department of Zoology, Faculty of Life SciencesTel Aviv UniversityTel AvivIsrael

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