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Computational and Applied Mathematics

, Volume 36, Issue 1, pp 415–431 | Cite as

Dynamics and bifurcations in a simple quasispecies model of tumorigenesis

  • Vanessa Castillo
  • J. Tomás LázaroEmail author
  • Josep SardanyésEmail author
Article

Abstract

Cancer is a complex disease and thus is complicated to model. However, simple models that describe the main processes involved in tumoral dynamics, e.g., competition and mutation, can give us clues about cancer behavior, at least qualitatively, also allowing us to make predictions. Here, we analyze a simplified quasispecies mathematical model given by differential equations describing the time behavior of tumor cell populations with different levels of genomic instability. We find the equilibrium points, also characterizing their stability and bifurcations focusing on replication and mutation rates. We identify a transcritical bifurcation at increasing mutation rates of the tumor cells. Such a bifurcation involves a scenario with dominance of healthy cells and impairment of tumor populations. Finally, we characterize the transient times for this scenario, showing that a slight increase beyond the critical mutation rate may be enough to have a fast response towards the desired state (i.e., low tumor populations) by applying directed mutagenic therapies.

Keywords

Applied mathematics Bifurcations Cancer Complex systems Quasispecies dynamics 

Mathematics Subject Classification

92Bxx Mathematical biology in general 65Lxx ODEs 58Kxx catastrophe theory 37G35 Attractors and their bifurcations 70K50 bifurcations and instability 

Notes

Acknowledgments

We want to thank Ricard V. Solé for helpful comments and suggestions. JTL has been partially supported by the Spanish MICIN/FEDER grant MTM2012-31714, by the Generalitat de Catalunya grant number 2014SGR-504, and by grant 14-41-00044 of RSF at the Lobachevsky University of Nizhny Novgorod. JS has been funded by the Fundación Botín.

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Copyright information

© SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2015

Authors and Affiliations

  1. 1.Department of Applied MathematicsUniversitat Politècnica de CatalunyaBarcelonaSpain
  2. 2.ICREA-Complex Systems Lab, Department of Experimental and Health SciencesUniversitat Pompeu Fabra, Parc de Recerca Biomèdica de Barcelona (PRBB)BarcelonaSpain
  3. 3.Institut de Biologia Evolutiva (CSIC-Universitat Pompeu Fabra)BarcelonaSpain

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