Thesis (M.Sc.) - Cairo University - Faculty of Science - Department of Mathematics

In this thesis, numerical studies for the multi-strain Tuberculosis (TB) model, that incor- porates three strains: i.e., drug{u2212}sensitive, emerging multi{u2212}drug resistant (MDR) and ex- tensively drug{u2212}resistant (XDR), which given in [5], are introduced. The adopted model is described by a system of nonlinear ordinary di erential equations(ODEs). Special class of numerical methods, known as nonstandard nite di erence method (NSFDM) is intro- duced. The obtained results of using NSFDM are compared with other known numerical methods such as implicit Euler method and fourth-order Runge-Kutta (RK4) method. Also, the fractional order multi-strain TB model (FOTBM) as a novel model is presented. The fractional derivative is de ned in the sense of Grünwald-Letinkov de nition. Two numerical methods are presented to study this model; the standard nite di erence method (SFDM) and NSFDM. The stability of equilibrium points is studied. As an extension of FOTBM, the variable-order fractional multi-strain TB model (VOFTBM) is presented. The variable-order fractional derivative is de ned in the sense of Grünwald-Letinkov de nition. Two numeri- cal methods are presented for this model, SFDM and NSFDM. The stability of equilibrium points is studied. At the end of thesis, the optimal control for multi-strain TB model is presented. The optimal control problem of TB model is formulated and studied theoretically using the Pontryagin maximum principle. Di erent optimal control strategies are proposed to minimize the cost of interventions. The resultant system of ODEs is numerically studied using foreword and backward Euler method

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