Journal of Mathematical Biology

, Volume 55, Issue 1, pp 1–19 | Cite as

Using dimension reduction to improve outbreak predictability of multistrain diseases

Article

Abstract

Multistrain diseases have multiple distinct coexisting serotypes (strains). For some diseases, such as dengue fever, the serotypes interact by antibody-dependent enhancement (ADE), in which infection with a single serotype is asymptomatic, but contact with a second serotype leads to higher viral load and greater infectivity. We present and analyze a dynamic compartmental model for multiple serotypes exhibiting ADE. Using center manifold techniques, we show how the dynamics rapidly collapses to a lower dimensional system. Using the constructed reduced model, we can explain previously observed synchrony between certain classes of primary and secondary infectives (Schwartz et al. in Phys Rev E 72:066201, 2005). Additionally, we show numerically that the center manifold equations apply even to noisy systems. Both deterministic and stochastic versions of the model enable prediction of asymptomatic individuals that are difficult to track during an epidemic. We also show how this technique may be applicable to other multistrain disease models, such as those with cross-immunity.

Keywords

Center manifold analysis Epidemic models Multistrain disease Dengue 

Mathematics Subject Classification

92D30 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Andreasen V., Lin J. and Levin S.A. (1997). J. Math. Bio. 35: 825–842 MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Billings, L., Schwartz, I.B., Shaw, L.B., McCrary, M., Barke, D.S., Cummings, D.A.T.: J. Theor. Biol. (to appear) (2007)Google Scholar
  3. 3.
    Carr J. (1981). Applications of centre manifold theory. Springer, New York MATHGoogle Scholar
  4. 4.
    Castillo-Chavez C., Hethcote H.W., Andreasen V., Levin S.A. and Liu W.M. (1989). J. Math. Biol.  27: 233–258 MATHMathSciNetGoogle Scholar
  5. 5.
    Cummings D.A.T., Schwartz I.B., Billings L., Shaw L.B. and Burke D.S. (2005). Proc. Natl. Acad. Sci. USA 102: 15259–15264 CrossRefGoogle Scholar
  6. 6.
    Dawes J. and Gog J. (2002). J. Math. Biol. 45: 471–510 MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Endy T.P., Nisalak A., Chunsuttiwat S., Libraty D.H., Green S., Rothman A.L., Vaughn D.W. and Ennis F.E. (2002). Am. J. Epidemiol. 156: 52–59 CrossRefGoogle Scholar
  8. 8.
    Esteva L. and Vargas C. (2003). J. Math. Biol. 46: 31–47 MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Feng Z. and Velasco-Hernandez J.X. (1997). J. Math. Bio. 35: 523–544 MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Ferguson N., Anderson R. and Gupta S. (1999a). Proc. Natl. Acad. Sci. USA 96: 790–794 CrossRefGoogle Scholar
  11. 11.
    Ferguson N.M., Donnelly C.A. and Anderson R.M. (1999b). Proc. R. Soc. Lond. Ser. B 354: 757–768   CrossRefGoogle Scholar
  12. 12.
    Gupta S., Trenholme K., Anderson R.M. and Day K.P. (1994). Science 263: 961–963 CrossRefGoogle Scholar
  13. 13.
    Nisalak A., Endy T.P., Nimmannitya S., Kalayanarooj S., Thisayakorn U., Scott R.M., Burke D.S., Hoke C.H., Innis B.L. and Vaughn D.W. (2003). Am. J. Trop. Med. Hyg. 68: 191–202 Google Scholar
  14. 14.
    Sangkawibha N., Rojanasuphot S., Ahandrik S., Viriyapongse S., Jatanasen S., Salitul V., Phanthumachinda B. and Halstead S.B. (1984). Am. J. Epidemiol. 120: 653–669 Google Scholar
  15. 15.
    Schwartz I.B., Shaw L.B., Cummings D.A.T., Billings L., McCrary M. and Burke D.S. (2005). Phys. Rev. E 72: 066201 CrossRefGoogle Scholar
  16. 16.
    Vaughn, D.W., Green, S., Kalayanarooj, S., Innis, B.L., et al.: J. Infect. Dis. 181, 2 (2000)Google Scholar
  17. 17.
    Webster R.G. and Hulse D.J. (2004). Rev. Sci. Tech. Off. Int. Epiz. 23: 453–465 Google Scholar
  18. 18.
    Yang Z.-Y., Werner H.C., Kong W.P. and Leung K. (2005). PNAS 102: 797–801 CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  • Leah B. Shaw
    • 1
  • Lora Billings
    • 2
  • Ira B. Schwartz
    • 1
  1. 1.Naval Research Laboratory, Plasma PhysicsDivisionNonlinear Systems Dynamics SectionWashingtonUSA
  2. 2.Department of Mathematical SciencesMontclair State UniversityUpper MontclairUSA

Personalised recommendations