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Mathematical Modeling of Some Infectious Diseases With Integer and Non-Integer Order Derivatives

Thesis Info

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Author

Saif Ullah

Program

PhD

Institute

University of Peshawar

City

Peshawar

Province

KPK

Country

Pakistan

Thesis Completing Year

2019

Thesis Completion Status

Completed

Subject

Mathemaics

Language

English

Link

http://prr.hec.gov.pk/jspui/bitstream/123456789/11733/1/Saif%20Ullah%20maths%202019%20uop%20peshwr%20prr.pdf

Added

2021-02-17 19:49:13

Modified

2024-03-24 20:25:49

ARI ID

1676726620680

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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.
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100. Al-‘Adiyat/The Chargers

I/We begin by the Blessed Name of Allah

The Immensely Merciful to all, The Infinitely Compassionate to everyone.

100:01
a. By the war-horses, charging, snorting, rushing to the battlefield,

100:02
a. striking sparks with their hooves,

100:03
a. and charging by the dawn,

100:04
a. raising a trail of dust,

100:05
a. and storming into the midst of the enemy troops together.

100:06
a. Indeed, the human being has always been ungrateful and grudging to his Rabb - The Lord,

100:07
a. and truly he is a witness to it,

100:08
a. and he is truly very excessive as well as aggressive in his passion for wealth.

100:09
a. But does he not realize what will happen to him when the contents of the graves are thrown out -

100:10
a. and that which is within the hearts will be made known,

100:11
a. at that Time, their Rabb - The Lord will be Fully Aware of them?

لأزمة الأخلاقية في المجتمع الباكستاني المظاهر- العوامل- المعالجة

Moral values are seen as the basis of human civilization. Absence of moral values and responsibilities results in the justification of every evil in the society, as it is the case being observed in the present-day societies in many parts of the world. A nation, whose collective morals are high, is capable to lead other nations, irrespective of caste, creed and religious affiliations. If a nation, Muslim or non-Muslim, ignores the high moral values, it cannot avoid its decadence and destruction. Due to this utmost importance of morality for humanity, Islām regards morality as one of the integral parts of the Divine Revelation. Islām aims to create a sense of moral responsibility in its adherents, so that, they may show a complete picture of an ideal society, and enjoy their freedom to carry out the best possible moral deeds. The author of this paper, chose to study the present moral crisis in the Pakistani society and tried to determine the causes, which has brought about this moral crisis and also presents its cure in the light of the Qur’ān and Sunnah. The study focuses on the following aspects: Definitions of moral values & society, Prevalent social evils in our society, Causes of crimes and social evils, Remedies to root out unethical practices and evils from the society, Conclusion and recommendations.

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The thesis proposes area classification after construction of datasets with pre-processing for Hyperion Hyperspectral, Operational Land Imager (OLI) and Advanced Land Imager (ALI) orthoimages. The techniques perform comparative analysis of Hyperion Hyperspectral, OLI and ALI orthoimages in terms of high Signal to Noise (SNR), spectral band configuration, technical superiority, improved system design and high radiometric resolution. The thesis further proposes criteria for selection of parameters like gamma parameter, penalty parameter, pyramid parameter and classification probability threshold to achieve higher classification accuracies of Hyperion Hyperspectral, OLI and ALI Satellite orthoimages by using Support Vector Machine (SVM), Spectral Angle Mapper (SAM) and Spectral Information Divergence (SID) classifiers. After performing the comparison, the thesis selects SVM as the most appropriate classifier in terms of overall accuracy, individual classes and the best orthoimagery i.e. Hyperion, OLI and ALI respectively. The thesis also presents application of classification accuracy assessment on Hyperion Hyperspectral, OLI and ALI orthoimages by using different classifiers i.e. SVM, SAM and SID. The thesis also proposes high accuracy based seasonal change detection analysis technique on Hyperion Hyperspectral, ALI and different datasets of OLI by using change detection matrix and difference maps. As a result of these contributions, 1x Journal and 6 x Conference papers duly peer reviewed have been published. The pre-processing of 242 bands of hyperspectral data results in 136 calibrated bands. Quick Atmospheric Correction (QUAC) are applied to Hyperion Hyperspectral and Fast Line-of-sight Atmospheric Analysis of Hyper cubes (FLAASH) are applied to OLI and ALI imagery respectively for atmospheric correction. Principal Component Analysis (PCA) is used for dimensional reduction of the hyperspectral data. PCA reveals that 99.94% of the hyperspectral data are contained in the first 15 Principal Components (PCs). Distinct spectral profiles are identified for all classes which are highly beneficial for feature identification and classification of images. Novel parameters are selected for high accuracy area classification in hyperspectral, OLI and ALI imagery via SVM, SAM and SID classification techniques. High accuracy based post classification change detection analysis is used on Hyperion Hyperspectral, ALI and different OLI datasets to produce difference maps which provide information not only about change of category but also type of change i.e. “from-to” of category of classes. Change detection matrix is also used which shows an overall decrease and increase of corresponding spatial extension of classes whereas diagonal elements of the change detection matrix show the unchanged pixels for the individual classes. The post classification technique is selected because of its ability for accurate change detection analysis of imagery of different sensors and its advantages over pre-classification methods that it compensates for variation in atmospheric correction and in conditions where the change is limited due to small rate of change. The results show that Hyperion hyperspectral and Landsat-8 OLI data achieved higher accuracies in mapping applications and high accuracy based post classification seasonal change detection analysis on different datasets on OLI results extraction of accurate change detection information as compared to previous Landsat satellite series.