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

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 analyse 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 systems 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 attractors reconstruction. We identify different dynamical behaviors of the system varying two biologically meaningful parameters: r1, representing the uncontrolled proliferation rate of cancer cells, and k1, denoting the immune systems 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 construct the explicit series solution 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.