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**Frequency dependence and viral diversity imply chaos in an HIV model.**
*(English)*
Zbl 1126.34032

New mathematical models are suggested in which cytotoxic T-lymphocytes (CTLs) proliferation and elimination of infected cells by CTLs depend on a frequency characterized by the viral diversity. These are based on the refinement of the immune model of infectious disease developed by M. A. Nowak and C. R. Bangham [Science 272, 74-79 (1996)] which does not assume interaction between different types of CTLs.

The authors investigate the stability of equilibria for the one-virus model and proceed with numerical simulations for the two-virus model. It turns out that the frequency dependence caused by the random search with viral diversity leads to continuous changes of dominant infected cells and corresponding specific immune cells which, in turn, yield complex behavior. In particular, it has been demonstrated that the frequency dependence may lead to the loss of stability of the interior equilibrium of the one-virus model. On the other hand, numerical simulations suggest that only a stable limit cycle exists when the interior equilibrium is unstable. For the two-viral model, numerical experiments indicate the existence of strange attractors, and the frequency dependence due to viral diversity causes chaotic behavior in the system. Consequently, both mathematical models suggest that viral diversity along with the frequency dependent proliferation of CTLs and elimination of the infected cells may induce the collapse of the immune system.

The authors investigate the stability of equilibria for the one-virus model and proceed with numerical simulations for the two-virus model. It turns out that the frequency dependence caused by the random search with viral diversity leads to continuous changes of dominant infected cells and corresponding specific immune cells which, in turn, yield complex behavior. In particular, it has been demonstrated that the frequency dependence may lead to the loss of stability of the interior equilibrium of the one-virus model. On the other hand, numerical simulations suggest that only a stable limit cycle exists when the interior equilibrium is unstable. For the two-viral model, numerical experiments indicate the existence of strange attractors, and the frequency dependence due to viral diversity causes chaotic behavior in the system. Consequently, both mathematical models suggest that viral diversity along with the frequency dependent proliferation of CTLs and elimination of the infected cells may induce the collapse of the immune system.

Reviewer: Svitlana P. Rogovchenko (Famagusta)

### MSC:

34C60 | Qualitative investigation and simulation of ordinary differential equation models |

92C60 | Medical epidemiology |

92D30 | Epidemiology |

34C25 | Periodic solutions to ordinary differential equations |

34C28 | Complex behavior and chaotic systems of ordinary differential equations |

34D45 | Attractors of solutions to ordinary differential equations |

37D45 | Strange attractors, chaotic dynamics of systems with hyperbolic behavior |

### Keywords:

epidemiology; immune model; viral diversity; chaotic behavior; strange attractors; stability; limit cycles### Software:

MATCONT
Full Text:
DOI

### References:

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