Journal of Mathematical Biology

, Volume 67, Issue 6–7, pp 1507–1532 | Cite as

Bifurcation, stability, and cluster formation of multi-strain infection models

Article

Abstract

Clustering behaviours have been found in numerous multi-strain transmission models. Numerical solutions of these models have shown that steady-states, periodic, or even chaotic motions can be self-organized into clusters. Such clustering behaviours are not a priori expected. It has been proposed that the cross-protection from multiple strains of pathogens is responsible for the clustering phenomenon. In this paper, we show that the steady-state clusterings in existing models can be analytically predicted. The clusterings occur via semi-simple double zero bifurcation from the quotient networks of the models and the patterns which follow can be predicted through the stability analysis of the bifurcation. We calculate the stability criteria for the clustering patterns and show that some patterns are inherently unstable. Finally, the biological implications of these results are discussed.

Keywords

Multi-strain infection model Semi-simple double zero bifurcation Clustering Stability 

Mathematics Subject Classification (2000)

34C23 37G10 62P10 

References

  1. Buono P, Golubitsky M (2001) Models of central pattern generators for quadruped locomotion: I. primary gaits. J Math Biol 42(4):291–326MathSciNetCrossRefMATHGoogle Scholar
  2. Buono P, Palacios A (2004) A mathematical model of motorneuron dynamics in the heartbeat of the leech. Phys D Nonlinear Phenom 188(3):292–313MathSciNetCrossRefMATHGoogle Scholar
  3. Calvez V, Korobeinikov A, Maini P (2005) Cluster formation for multi-strain infections with cross-immunity. J Theor Biol 233(1):75–83MathSciNetCrossRefGoogle Scholar
  4. Chan BS, Yu P (2012) Synchrony-breaking Hopf bifurcation in a model of antigenic variation. Int J Bifurc Chaos (in press)Google Scholar
  5. Coico R, Sunshine G (2009) Immunology: a short course. Blackwell, New JerseyGoogle Scholar
  6. Dawes J, Gog J (2002) The onset of oscillatory dynamics in models of multiple disease strains. J Math Biol 45(6):471–510MathSciNetCrossRefMATHGoogle Scholar
  7. Ferguson N, Andreasen V (2002) The influence of different forms of cross-protective immunity on the population dynamics of antigenically diverse pathogens. IMA Vol Math Appl 126:157–170MathSciNetCrossRefGoogle Scholar
  8. Ferguson N, Galvani A (2003) The impact of antigenic variation on pathogen population structure, fitness and dynamics. Antigenic Variation, pp 403–433Google Scholar
  9. Gjini E, Haydon D, Barry J, Cobbold C (2010) Critical interplay between parasite differentiation, host immunity, and antigenic variation in Trypanosome infections. Am Nat 176(4):424–439CrossRefGoogle Scholar
  10. Gog J, Grenfell B (2002) Dynamics and selection of many-strain pathogens. Proc Natl Acad Sci USA 99(26):17–209CrossRefGoogle Scholar
  11. Gog J, Swinton J (2002) A status-based approach to multiple strain dynamics. J Math Biol 44(2):169–184MathSciNetCrossRefMATHGoogle Scholar
  12. Golubitsky M, Stewart I (2006) Nonlinear dynamics of networks: the groupoid formalism. Bull Am Math Soc 43(3):305MathSciNetCrossRefMATHGoogle Scholar
  13. Golubitsky M, Stewart I, Buono P, Collins J (1999) Symmetry in locomotor central pattern generators and animal gaits. Nature 401(6754):693–695CrossRefGoogle Scholar
  14. Golubitsky M, Stewart I, Török A (2005) Patterns of synchrony in coupled cell networks with multiple arrows. SIAM J Appl Dynam Sys 4(1):78–100CrossRefMATHGoogle Scholar
  15. Guo H, Li M, Shuai Z (2008) A graph-theoretic approach to the method of global Lyapunov functions. Proc Am Math Soc 136(8):2793–2802MathSciNetCrossRefMATHGoogle Scholar
  16. Gupta S, Galvani A (1999) The effects of host heterogeneity on pathogen population structure. Philos Trans R Soc Lond Ser B Biol Sci 354(1384):711CrossRefGoogle Scholar
  17. Gupta S, Maiden M, Feavers I, Nee S, May R, Anderson R (1996) The maintenance of strain structure in populations of recombining infectious agents. Nat Med 2(4):437–442CrossRefGoogle Scholar
  18. Gupta S, Ferguson N, Anderson R (1998) Chaos, persistence, and evolution of strain structure in antigenically diverse infectious agents. Science 280(5365):912CrossRefGoogle Scholar
  19. Iooss G, Joseph D (1990) Elementary stability and bifurcation theory. Springer, BerlinCrossRefMATHGoogle Scholar
  20. Nowak M, May R, Phillips R, Rowland-Jones S, Lalloo D, McAdam S, Klenerman P, Köppe B, Sigmund K, Bangham C et al (1995a) Antigenic oscillations and shifting immunodominance in HIV-1 infections. Nature 375(6532):606–611CrossRefGoogle Scholar
  21. Nowak M, May R, Sigmund K (1995b) Immune responses against multiple epitopes. J Theor Biol 175(3):325–353CrossRefGoogle Scholar
  22. Omori R, Adams B, Sasaki A (2010) Coexistence conditions for strains of influenza with immune cross-reaction. J Theor Biol 262(1):48–57MathSciNetCrossRefGoogle Scholar
  23. Recker M, Gupta S (2005) A model for pathogen population structure with cross-protection depending on the extent of overlap in antigenic variant repertoires. J Theor Biol 232(3):363–373MathSciNetCrossRefGoogle Scholar
  24. Recker M, Nee S, Bull P, Kinyanjui S, Marsh K, Newbold C, Gupta S (2004) Transient cross-reactive immune responses can orchestrate antigenic variation in malaria. Nature 429(6991):555–558CrossRefGoogle Scholar
  25. Recker M, Blyuss K, Simmons C, Hien T, Wills B, Farrar J, Gupta S (2009) Immunological serotype interactions and their effect on the epidemiological pattern of dengue. Proc R Soc B Biol Sci 276(1667):2541CrossRefGoogle Scholar
  26. Stewart I, Golubitsky M, Pivato M (2003) Symmetry groupoids and patterns of synchrony in coupled cell networks. SIAM J Appl Dynam Sys 2(4):609–646MathSciNetCrossRefMATHGoogle Scholar
  27. Wiggins S (2003) Introduction to applied nonlinear dynamical systems and chaos. Springer, BerlinMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Department of Applied MathematicsThe University of Western OntarioLondonCanada

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