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- Homotopy analysis of explicit solutions in a chronic hepatitis C virus modelPublication . Duarte, Jorge; Januário, Cristina; Martins, NunoMathematical analysis of nonlinear models in epidemiology has generated a deep interest in gaining insights into the mechanisms that underlie hepatitis C virus (HCV) infections. In this article, we provide a study of a chronic HCV infection model with immune response, incorporating the effect of dendritic cells (DC) and cytotoxic T lymphocytes (CTL). Considering very recent developments in the literature related to the Homotopy Analysis Method (HAM), we calculate the explicit series solutions of the HCV model, focusing our analysis on a particular set of dynamical variables. An optimal homotopy analysis approach is used to improve the computational efficiency of HAM by means of appropriate values for a convergence control parameter, which greatly accelerates the convergence of the series solutions. The approximated analytical solutions, with the variation of a parameter representing the expansion rate of CTL, are used to compute density plots, which allow us to discuss additional dynamical features of the model.
- On the analytical solutions of the Hindmarsh-Rose neuronal modelPublication . Duarte, Jorge; Januário, Cristina; Martins, NunoIn this article we analytically solve the Hindmarsh-Rose model (Proc R Soc Lond B221:87-102, 1984) by means of a technique developed for strongly nonlinear problems-the step homotopy analysis method. This analytical algorithm, based on a modification of the standard homotopy analysis method, allows us to obtain a one-parameter family of explicit series solutions for the studied neuronal model. The Hindmarsh-Rose system represents a paradigmatic example of models developed to qualitatively reproduce the electrical activity of cell membranes. By using the homotopy solutions, we investigate the dynamical effect of two chosen biologically meaningful bifurcation parameters: the injected current I and the parameter r, representing the ratio of time scales between spiking (fast dynamics) and resting (slow dynamics). The auxiliary parameter involved in the analytical method provides us with an elegant way to ensure convergent series solutions of the neuronal model. Our analytical results are found to be in excellent agreement with the numerical simulations.