چكيده به لاتين
In this dissertation, the qualitative behavior of the model arising from the biophysics properties of the brain neocortical neurons is predicted by a mathematical method. This nonlinear model is complicated due to its nonhomogeneity. As a result, the investigation of dynamical behavior and analytical solutions of the model is difficult. Thus, we employed step Homotopy analysis method to solve the nonlinear system. The Homotopy analysis method is a strong approach to solve nonlinear problems, specifically in dynamical systems. In this dissertation, for the first time, the analytical solutions of the human and mammalian neocortical neurons model are obtained and the epileptic patient electroencephalography is simulated. The behavior of the membrane potential and the gating variable versus time are illustrated. Furthermore, we consider the discretized classical Susceptible-Infected-Recovered (SIR) forced epidemic model to investigate the consequences of the introduction of different transmission rates and the effect of a constant vaccination strategy, providing new numerical and topological insights into the complex dynamics of recurrent diseases. Starting with a constant contact (or transmission) rate, the computation of the spectrum of Lyapunov exponents allows us to identify different chaotic regimes. Studying the evolution of the dynamical variables, a family of unimodal-type iterated maps with a striking biological meaning is detected among those dynamical regimes of the densities of the susceptibles. Using the theory of symbolic dynamics, these iterated maps are characterized based on the computation of an important numerical invariant, the topological entropy. The introduction of a degree (or amplitude) of seasonality, ɛ, is responsible for inducing complexity into the population dynamics. The resulting dynamical behaviors are studied using some of the previous tools for particular values of the strength of the seasonality forcing, ɛ. Finally, we carry out a study of the discrete SIR epidemic model under a planned constant vaccination strategy.
