Journal of Mathematical Biology

, Volume 67, Issue 6, pp 1507–1532

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


DOI: 10.1007/s00285-012-0600-3

Cite this article as:
Chan, B.S. & Yu, P. J. Math. Biol. (2013) 67: 1507. doi:10.1007/s00285-012-0600-3


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.


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

Mathematics Subject Classification (2000)

34C23 37G10 62P10 

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

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

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