Abstract
We examine the properties of a recently proposed model for antigenic variation in malaria which incorporates multiple epitopes and both long-lasting and transient immune responses. We show that in the case of a vanishing decay rate for the long-lasting immune response, the system exhibits the so-called “bifurcations without parameters” due to the existence of a hypersurface of equilibria in the phase space. When the decay rate of the long-lasting immune response is different from zero, the hypersurface of equilibria degenerates, and a multitude of other steady states are born, many of which are related by a permutation symmetry of the system. The robustness of the fully symmetric state of the system was investigated by means of numerical computation of transverse Lyapunov exponents. The results of this exercise indicate that for a vanishing decay of long-lasting immune response, the fully symmetric state is not robust in the substantial part of the parameter space, and instead all variants develop their own temporal dynamics contributing to the overall time evolution. At the same time, if the decay rate of the long-lasting immune response is increased, the fully symmetric state can become robust provided the growth rate of the long-lasting immune response is rapid.
Similar content being viewed by others
References
Ashwin P, Field M (1999) Heteroclinic networks in coupled cell systems. Arch Rat Mech Anal 148: 107–43
Dias APS, Stewart I (2003) Secondary nifurcations in systems with all-to-all coupling. Proc R Soc Lond A 459: 1969–986
Elmhirst T (2004) S N -equivariant symmetry-breaking bifurcations. Int J Bifurcat Chaos 14: 1017–036
Fiedler B, Liebscher S, Alexander JC (2000) Generic Hopf bifurcation from lines of equilibria without parameters. J Differ Equ 167: 16–5
Golubitsky M, Schaeffer D (1985) Singularities and groups in bifurcation theory. Springer, New York
Gupta S (2005) Parasite immune escape: new views into host–parasite interactions. Curr Opin Microbiol 8: 428–33
Pecora LM, Carroll TL (1990) Synchronization in chaotic systems. Phys Rev Lett 64: 821–24
Postlethwaite C, Dawes JHP (2005) Regular and irregular cycling near a heteroclinic network. Nonlinearity 18: 1477–509
Recker M, New S, Bull PC, Linyanjui S, Marsh K, Newbold C, Gupta S (2004) Transient cross-reactive immune responses can orchestrate antigenic variation in malaria. Nature 429: 555–58
Recker M, Gupta S (2006) Conflicting immune responses can prolong the length of infection in Plasmodium falciparum malaria. Bull Math Biol 68: 821–35
Rokni Lamooki GR, Townley S, Osinga HM (2005) Bifurcations and limit dynamics in adaptive control systems. Int J Bifurcat Chaos 15: 1641–664
Shampine LF, Reichelt MW (1997) The MATLAB ODE Suite. SIAM J Sci Comp 18: 1–2
Author information
Authors and Affiliations
Corresponding author
Additional information
This work was partially supported by the ATRJVVO grant from the James Martin 21st Century School, University of Oxford.
Rights and permissions
About this article
Cite this article
Blyuss, K.B., Gupta, S. Stability and bifurcations in a model of antigenic variation in malaria. J. Math. Biol. 58, 923–937 (2009). https://doi.org/10.1007/s00285-008-0204-0
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00285-008-0204-0