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Optimal homotopy analysis of a chaotic HIV-1 model incorporating AIDS-related cancer cells

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Abstract

The studies of nonlinear models in epidemiology have generated a deep interest in gaining insight into the mechanisms that underlie AIDS-related cancers, providing us with a better understanding of cancer immunity and viral oncogenesis. In this article, we analyze an HIV-1 model incorporating the relations between three dynamical variables: cancer cells, healthy CD4 + T lymphocytes, and infected CD4 + T lymphocytes. Recent theoretical investigations indicate that these cells interactions lead to different dynamical outcomes, for instance to periodic or chaotic behavior. Firstly, we analytically prove the boundedness of the trajectories in the system’s attractor. The complexity of the coupling between the dynamical variables is quantified using observability indices. Our calculations reveal that the highest observable variable is the population of cancer cells, thus indicating that these cells could be monitored in future experiments in order to obtain time series for attractor’s reconstruction. We identify different dynamical behaviors of the system varying two biologically meaningful parameters: r 1, representing the uncontrolled proliferation rate of cancer cells, and k 1, denoting the immune system’s killing rate of cancer cells. The maximum Lyapunov exponent is computed to identify the chaotic regimes. Considering very recent developments in the literature related to the homotopy analysis method (HAM), we calculate the explicit series solutions of the cancer model and focus our analysis on the dynamical variable with the highest observability index. An optimal homotopy analysis approach is used to improve the computational efficiency of HAM by means of appropriate values for the convergence control parameter, which greatly accelerate the convergence of the series solution. The approximated analytical solutions are used to compute density plots, which allow us to discuss additional dynamical features of the model.

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Acknowledgments

This work was partially funded by FCT/Portugal through UID/MAT/04459/2013. JS has been partially funded by the CERCA Programme of the Generalitat de Catalunya. The research leading to these results has received funding from “la Caixa” Foundation.

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Authors and Affiliations

  1. Instituto Superior de Engenharia de Lisboa - ISEL, Department of Mathematics, Rua Conselheiro Emídio Navarro 1, 1949-014, Lisboa, Portugal

    Jorge Duarte & Cristina Januário

  2. Center for Mathematical Analysis, Geometry and Dynamical Systems, Mathematics Department, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais 1, 1049-001, Lisboa, Portugal

    Jorge Duarte & Nuno Martins

  3. Department of Mathematics, University of Évora, Rua Romão Ramalho, 59, 7000-671, Évora, Portugal

    C. Correia Ramos

  4. ESTS - Technology Superior School of Setubal, Department of Mathematics, Campus do IPS, Rua Vale de Chaves, Estefanilha, 2914-761, Setubal, Portugal

    Carla Rodrigues

  5. Centre de Recerca Matemàtica, Campus de Bellaterra, Edifici C, 08193, Bellaterra, Barcelona, Spain

    Josep Sardanyés

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  1. Jorge Duarte
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  2. Cristina Januário
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  4. C. Correia Ramos
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Correspondence to Jorge Duarte or Josep Sardanyés.

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Duarte, J., Januário, C., Martins, N. et al. Optimal homotopy analysis of a chaotic HIV-1 model incorporating AIDS-related cancer cells. Numer Algor 77, 261–288 (2018). https://doi.org/10.1007/s11075-017-0314-0

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  • DOI: https://doi.org/10.1007/s11075-017-0314-0

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