Public Defense of a Ph.D Research Scholar at Department of Mathematics

Mr. Saif Ullah, a Ph.D. Research Scholar at the Department of Mathematics, University of Peshawar has submitted his thesis entitled "Mathematical Modeling of some Infectious Diseases with Integer and Non-Integer Order Derivatives"  to the University of Peshawar, in partial fulfillment of the requirement for the award of degree of  Doctor of Philosophy (Ph. D) in the discipline of Mathematics.

All those having interest in the said research work, are cordially invited to attend the occasion. The participants would be allowed to raise relevant questions after the presentation by the scholar for further judgment and evaluation of the examiners.

Abstract:

Mathematical models play an important role to understand the spread, persistence 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 qualitative 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 number R0 is presented for each model that shows the disease persistence or elimination for their particular cases. Further, in order to reduce and eradicate the infection of TB and HBV from the community, we develop some suitable optimal control strategies. 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 numerical 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 _t 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, in this research, we extend the proposed models using fractional order derivatives considering three different fractional 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.