Mathematical analysis and numerical simulation for fractal-fractional cancer model

The mathematical oncology has received a lot of interest in recent years since it helps illuminate pathways and provides valuable quantitative predictions, which will shape more effective and focused future therapies. We discuss a new fractal-fractional-order model of the interaction among tumor cells, healthy host cells and immune cells. The subject of this work appears to show the relevance and ramifications of the fractal-fractional order cancer mathematical model. We use fractal-fractional derivatives in the Caputo senses to increase the accuracy of the cancer and give a mathematical analysis of the proposed model. First, we obtain a general requirement for the existence and uniqueness of exact solutions via Perov's fixed point theorem. The numerical approaches used in this paper are based on the Grünwald-Letnikov nonstandard finite difference method due to its usefulness to discretize the derivative of the fractal-fractional order. Then, two types of stabilities, Lyapunov's and Ulam-Hyers' stabilities, are established for the Incommensurate fractional-order and the Incommensurate fractal-fractional, respectively. The numerical results of this study are compatible with the theoretical analysis. Our approaches generalize some published ones because we employ the fractal-fractional derivative in the Caputo sense, which is more suitable for considering biological phenomena due to the significant memory impact of these processes. Aside from that, our findings are new in that we use Perov's fixed point result to demonstrate the existence and uniqueness of the solutions. The way of expressing the Ulam-Hyers' stabilities by utilizing the matrices that converge to zero is also novel in this area

Comparative Study for Multi-Strain Tuberculosis (TB) Model of Fractional Order

In this paper, we introduce the multi−strain TB model of fractional-order derivatives, which incorporates three strains: drug−sensitive, emerging multi−drug resistant(MDR) and extensively drug−resistant(XDR ). Numerical simulations for this extended fractional order model is the main aim of this work, where the adopted model is described by a system of non-linear ordinary differential equations and the fractional derivative is defined in the sense of the Grünwald−Letnikov definition. Two numerical methods are presented for this model, the standard finite difference method (SFDM) and the nonstandard finite difference method (NSFDM). Numerical comparisons between SFDM and NSFDM are presented. It is concluded that the proposed NSFDM preserves the positivity of the solutions, and it is numerically stable in large regions than SFDM

Numerical Study for Time Delay Multistrain Tuberculosis Model of Fractional Order

A novel mathematical fractional model of multistrain tuberculosis with time delay memory is presented. The proposed model is governed by a system of fractional delay differential equations, where the fractional derivative is defined in the sense of the Grünwald–Letinkov definition. Modified parameters are introduced to account for the fractional order. The stability of the equilibrium points is investigated for any time delay. Nonstandard finite deference method is proposed to solve the resulting system of fractional-order delay differential equations. Numerical simulations show that nonstandard finite difference method can be applied to solve such fractional delay differential equations simply and effectively