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Title: Mathematical Modeling of Some Infectious Diseases with Integer and Non-Integer Order Derivatives
Authors: , Saif Ullah
Keywords: Mathematics
Issue Date: 2019
Publisher: University of Peshawar, Peshawar.
Abstract: Mathematical models play an important role to understand the spread, per sistence and prevention mechanism of infectious diseases. In this thesis, we present some mathematical models and their analysis on the dynamics of Tuberculosis (TB) and Hepatitis B virus (HBV). Firstly, we develop these models with classical integer-order derivative and present a detailed qualita tive analysis including, existence and stability of the equilibria, sensitivity of the model parameters and the existence of the bifurcation phenomena. The threshold quantity also called the basic reproduction numberR0 is presented for each model that shows the disease persistence or elimination for their par ticular cases. Further, we develop some suitable optimal control strategies which would be useful for public health department and other health agen cies, in order to reduce and eradicate TB and HBV from the community. The reported TB infected cases in Khyber Pakhtunkhwa province of Pakistan, for the period 2002-2017 are used to parameterize the proposed TB model and an excellent agreement is shown with the field data. The models are solved numerically using Runge-Kutta order four (RK4) method and numerous nu merical simulations carried out to illustrate the disease dynamics and some of the theoretical results. Mathematical models with fractional differential equations (FDEs) are more realistic and provide comparatively better fit to the real data instead of integer order models. Moreover, FDEs possess the memory effect which plays an essential role in the spreading of a disease. Therefore, the second main mathematical findings of this thesis is that we extend the proposed models using fractional order derivatives considering three different fractional xvi operators namely; Caputo, Caputo-Fabrizio and Atangana-Baleanu-Caputo operators. The proposed fractional models are analyzed rigorously and solved numerically using fractional Adams-Bashforth scheme. The graphical results reveal that the models with fractional derivatives give useful and biologically more feasible consequences.
Gov't Doc #: 18963
Appears in Collections:PhD Thesis of All Public / Private Sector Universities / DAIs.

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