Preface
Recently fractional calculus has gained much attention of the scientists due to its application in the fields of science and engineering like fluid dynamics, bio engineering, heat transform Fuzzy analysis. Modelling of dynamical problems with fractional order differential equations are the base of different existing systems, solving these fraction order differential systems are challenging. It is worth mentioning that various aspects of fractional order (singular/non-singular kernels) modelling that may include deterministic or uncertain (viz. fuzzy or interval or stochastic) scenarios are also important to understand the behaviour of the physical systems. As such, the aim of this book will be to include computation and modelling for obtaining exact and/or numerical solutions for fractional order systems.
Author
Prof. Dr. Muhammad Aslam Noor
Department of Mathematics, COMSATS University, Islamabad, 45550, Pakistan.
Dr. Idris Ahmed
Department of Mathematics, Sule Lamido University, KafinHausa, 741103, JigawaState, Nigeria.
Dr. Muhammad Jamilu Ibrahim
Department of Mathematics, Sule Lamido University, 741103, Kafin Hausa, Jigawa State, Nigeria.
Dr. Mujahid Abdullahi
Department of Mathematics, Sule Lamido University, 741103, Kafin Hausa, Jigawa State, Nigeria.
Dr. Abbas Umar Saje
Department of Mathematics, Sule Lamido University, 741103, Kafin Hausa, Jigawa State, Nigeria.
Dr. Mohsan Raza
Department of Mathematics, Government College University, Faisalabad, 38000, Pakistan
Dr. Muhamamd Ahsan Binyamin
Department of Mathematics, Government College University, Faisalabad, 38000, Pakistan
Dr. Muhammad Uzair Awan
Department of Mathematics, Government College University, Faisalabad, 38000, Pakistan
Dr. Khadija Tul Kubra
Department of Mathematics, Government College University, Faisalabad, 38000, Pakistan
Rooh Ali
Department of Mathematics, Government College University, Faisalabad, 38000, Pakistan
Muhammad Zakria Javed
Department of Mathematics, Government College University, Faisalabad, 38000, Pakistan
Sehrish Rafique
Department of Mathematics, Government College University, Faisalabad, 38000, Pakistan
Samra Gulshan
Department of Mathematics, Government College University, Faisalabad, 38000, Pakistan
Editor
Dr. Muhammad Imran
Government College University,
Faisalabad, Pakistan
drmimranchaudhry@gcuf.edu.pk
Prof. Dr. Saima Akran
Government College Women University,
Faisalabad, Pakistan
saimaakram@gcwuf.edu.pk
Dr. Madeeha Tahir
Government College Women University,
Faisalabad, Pakistan
madeehatahir@gcwuf.edu.pk
Mr. Muhammad Abdul Basit
Government College University,
Faisalabad, Pakistan
mabdulbasit50581@gcuf.edu.pk
Pages: 1-169
Publication Date: 18 December 2023
ISBN: 978-627-7623-03-6
1 A Mathematical Analysis of a Caputo Fractional-order Cholera Model
and its Sensitivity Analysis 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Preliminaries Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Formulation of the Caputo Fractional-order Cholera Model . . . . . . . . 5
1.3.1 Theoretical Analysis of the Caputo fractional-order Cholera
Disease Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.3.2 Positivity and boundedness of solution.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.4 Sensitivity analysis in relation to R0 . . . . . . . . . . . . . . . . . . . . . 11
1.5 Numerical simulations and discussions . . . . . . . . . . . . . . . . . . . . 13
1.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2 Univalence Criteria for Integral Operators Defined by Rabotnov Fractional Exponential Functions 25
2.1 Introduction and preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.2 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3 Newly Discovered Inclusions through Generalized Mittag-Leffler Functions in Double Fractional Integrals: Novel Discoveries 39
3.1 Introduction and Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . 40
3.2 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.2.1 Generalized fractional Hermite-Hadamar’s type inclusions . . 50
3.2.2 Generalized fractional Pachpatte type inclusions . . . . . . . 62
3.3 Fractional Hermite-Hadamard-Fejer type inclusions . . . . . . . . . . . . 68
3.4 Numerical examples with graphical analysis . . . . . . . . . . . . . . . . . 71
3.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4 Analysis of Monkey Pox Transmission Dynamics in Society with
Control Strategies Under Caputo-Fabrizio Fractal-Fractional Derivative 83
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.2 Model Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
4.3 Parameter estimation and model fitting . . . . . . . . . . . . . . . . . . . 90
4.4 Qualitative analysis of the model . . . . . . . . . . . . . . . . . . . . . . . 93
4.4.1 Positivity and Boundedness of the Model Solution . . . . . . 93
4.4.2 The Equilibrium State of the system . . . . . . . . . . . . . . 97
4.4.3 Basic Reproduction Number . . . . . . . . . . . . . . . . . . . 98
4.4.4 Stability of Monkey Pox Free Equilibrium (MFE) . . . . . . . 101
4.5 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
4.5.1 Existence and Uniqueness . . . . . . . . . . . . . . . . . . . . 106
4.5.2 Hyers-Ulam Stability Analysis . . . . . . . . . . . . . . . . . . 109
4.6 Numerical scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
4.7 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
4.7.1 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
4.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
5 Modeling and Analysis of Corruption Dynamics in Society under FractalFractional Derivative in Caputo Sense with Power-Law 131
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
5.2 Mathematical Model and Formulation . . . . . . . . . . . . . . . . . . . . 134
5.3 Positivity and Invariant region of the model solution . . . . . . . . . . . . 138
5.4 Qualitative analysis of the model . . . . . . . . . . . . . . . . . . . . . . . 139
5.4.1 The CFE Point . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
5.4.2 Corruption Transmission Generation Number . . . . . . . . . 140
5.4.3 The CPE Point . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
5.5 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
5.5.1 Existence and uniqueness under Caputo (power law) case . . 148
5.5.2 Hyers-Ulam Stability . . . . . . . . . . . . . . . . . . . . . . . 150
5.6 Numerical scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
5.7 Results and Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
5.7.1 Discussion/Recommendations . . . . . . . . . . . . . . . . . . 159
5.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
