Despite major advances in healthcare over the past century, the successful treatment of cancer has remained a significant challenge, and cancers are the second leading cause of death worldwide behind cardiovascular disease. Early detection and survival are important issues to control cancer. The development of quantitative methods and computer technology has facilitated the formation of new models in medical and biological sciences. The application of mathematical modelling in solving many real-world problems in medicine and biology has yielded fruitful results. In spite of advancements in instrumentations technology and biomedical equipment, it is not always possible to perform experiments in medicine and biology for various reasons. Thus, mathematical modelling and simulation are viewed as viable alternatives in such situations, and are discussed in this book. The conventional diagnostic techniques of cancer are not always effective as they rely on the physical and morphological appearance of the tumour. Early stage prediction and diagnosis is very difficult with conventional techniques. It is well known that cancers are involved in genome level changes. As of now, the prognosis of various types of cancer depends upon findings related to the data generated through different experiments. Several machine learning techniques exist in analysing the data of expressed genes; however, the recent results related with deep learning algorithms are more accurate and accommodative, as they are effective in selecting and classifying informative genes. This book explores the probabilistic computational deep learning model for cancer classification and prediction.

Preface xiii List of Figures xv List of Tables xxi List of Contributors xxiii List of Abbreviations xxvii 1 Modeling of Smoking Transmission Dynamics using Caputo⁰́₃Fabrizio Type Fractional Derivative 1 1.1 Introduction 2 1.2 Basic Concept of Fractional Operators 4 1.3 Model Formulation 5 1.4 Caputo⁰́₃Fabrizio Fractional Order Derivative 6 1.4.1 Stability Analysis of Model by Using Fixed-Point Theory 8 1.5 Numerical Results and Discussion 17 1.6 Conclusion 20 References 20 2 Hybrid Feature Selection Techniques Utilizing Soft Computing Methods for Classifying Microarray Cancer Data 23 2.1 Introduction 24 2.2 Proposed Framework 27 2.2.1 Genes Extraction by ICA 27 2.2.2 Genetic Bee Colony (GBC) Algorithm 28 2.3 Used Classifier 28 2.3.1 Naïve Bayes Classifier (NBC) 28 2.3.2 Support Vector Machine (SVM) Classifier 29 2.4 Experimental Setups 29 2.5 Experimental Result 31 2.6 Conclusion 35 References 35 3 Finite Element Technique to Explicate Calcium Diffusion in Alzheimer⁰́₉s Disease 41 3.1 Introduction 42 3.2 Literature Survey 43 3.3 Mathematical Formulations 44 3.3.1 Calcium Buffering 44 3.3.2 Voltage Gated Calcium Channel (VGCC) 45 3.3.3 Endoplasmic Reticulum (ER) 46 3.4 The Finite Element Technique 48 3.4.1 Approximated Geometry of the Cell 49 3.4.2 Physiological Boundary Conditions 50 3.4.3 Meshing of the domain 51 3.5 Results and Discussion 53 3.5.1 For Hippocampal Neuron 53 3.5.2 For Basal Forebrain Neuron 55 3.6 Conclusion 57 References 58 4 Comparative Analysis of Computational Methods used in Protein⁰́₃Protein Interaction (PPI) Studies 63 4.1 Introduction 64 4.1.1 Protein 64 4.1.2 Protein⁰́₃Protein Interaction 64 4.1.2.1 Protein⁰́₃protein interfacial characteristics 65 4.1.2.1.1 Size and shape 65 4.1.2.1.2 Complementarity between surfaces 65 4.1.2.1.3 Residue interface propensities 66 4.1.2.1.4 Hydrophobicity including Hydrogen bonding 66 4.1.2.1.5 Segmentation and secondary structure 66 4.1.2.1.6 Conformational changes on complex formation 67 4.1.2.2 PPI types 67 4.1.2.2.1 Homo oligomeric and Hetero oligomeric 67 4.1.2.2.2 Obligate and non obligate complexes 67 4.1.2.2.3 Transient and permanent complexes 68 4.1.2.2.4 Disordered to ordered complexes 69 4.1.2.3 PPI methods classification 69 4.2 In Silico Methods 69 4.2.1 Sequence Based Approaches 70 4.2.1.1 Ortholog based sequence approach 71 4.2.1.2 Domain pairs-based sequence approach 71 4.2.1.3 Statistical sequence-based approaches 72 4.2.1.3.1 Mirror tree method 72 4.2.1.3.2 PIPE 72 4.2.1.3.3 Co-evolutionary divergence 73 4.2.1.4 Machine learning sequence-based approaches 74 4.2.1.4.1 Auto-covariance 74 4.2.1.4.2 Pairwise similarity 74 4.2.1.4.3 Amino acid composition 75 4.2.1.4.4 Amino acid triad 75 4.2.1.4.5 UNISPPI 76 4.2.1.4.6 ETB viterbi 76 4.2.2 Structure Based Approaches 77 4.2.2.1 Template structure-based approaches 77 4.2.2.1.1 PRISM 77 4.2.2.1.2 PREPPI 78 4.2.2.2 Statistical structure-based approach 78 4.2.2.2.1 PID matrix score 78 4.2.2.2.2 Pre SPI 78 4.2.2.2.3 Domain cohesion and coupling 79 4.2.2.2.4 MEGADOCK 79 4.2.2.2.5 MetaApproach 80 4.2.2.3 Machine learning structure-based approaches 81 4.2.2.3.1 Random forest 81 4.2.2.3.2 Struct2Net 81 4.2.3 Gene Neighbourhood 82 4.2.4 Gene Fusion 82 4.2.5 In Silico Two-hybrid (I2h) 82 4.2.6 Phylogenetic Tree 83 4.2.7 Phylogenetic Profile 83 4.2.8 Gene Expression 84 4.3 PPI Networks and Databases 85 4.3.1 Creation of the PPI Networks 85 4.3.1.1 Choice of databases and data selection 85 4.3.1.2 Visualising PPI network 85 4.3.2 Different Databases 86 4.3.2.1 Interaction database 86 4.3.2.2 Metamining databases 87 4.3.2.3 Predictive interaction databases 88 4.3.2.4 Pathway database 89 4.3.2.5 Unifying database 90 4.4 Softwares Available for PPI 91 4.5 Conclusion 95 References 95 5 Optimization of COVID-19 Risk Factors Using Fuzzy Logic Inference System 101 5.1 Introduction 102 5.2 Methodology 103 5.2.1 Proposed Mamdani Fuzzy Control System 103 5.2.2 Fuzzy Controller Design 104 5.2.3 Parameters Identification 104 5.2.4 Fuzzification 104 5.2.5 Fuzzy Inference Rule Base 108 5.2.6 Rule Evaluation By Fuzzy Inference Engine 109 5.2.7 Defuzzification 109 5.3 Results 110 5.4 Discussion 113 5.5 Conclusion and Future Work 115 References 115 6 Dynamical Analysis of the Fractional-Order Mathematical Model of Hashimoto⁰́₉s Thyroiditis 119 6.1 Introduction 120 6.2 Preliminaries 121 6.3 Formulation of Fractional-Order Model of Hashimoto⁰́₉s Thyroiditis 122 6.4 Stability Analysis 123 6.5 Construction of a Numerical Solution Scheme 127 6.6 Numerical Segment 128 6.7 Conclusions 133 References 133 7 Heated Laminar Vertical Jet of Pseudoplastic Fluids-Against Gravity 137 7.1 Introduction 137 7.2 Basic Equations 142 7.3 Results and Discussions 147 7.4 Graphical Presentation 148 7.5 Conclusion 150 References 151 8 Analytical Solutions For Hydromagnetic Flow of Chemically Reacting Williamson Fluid Over a Vertical Cone and Wedge with Heat Source/Sink 155 8.1 Introduction 156 8.2 Formulation of the Problem 157 8.3 Solution of the Problem 159 8.4 Results and Discussion 160 8.5 Conclusion 164 References 164 9 Aboodh Transform Homotopy Perturbation Method for Solving Newell-Whitehead-Segel Equation 167 9.1 Introduction 167 9.2 Basic Definition of Aboodh Transform 168 9.2.1 Some Properies of Aboodh Transform 169 9.3 Idea of Aboodh Transform Homotopy Perturbation Method 169 9.4 Some Illustrations 170 9.5 Conclusion 176 References 177 10 Transmission and Control of Droplet Infection from Exotic to Native Population: A Mathematical Model 179 10.1 Introduction 180 10.2 Basic Assumptions and Formulation of the Model 181 10.3 Disease Free Equilibrium Point and Basic Reproduction Number 184 10.4 Stability Around the Disease Free Equilibrium Point 185 10.4.1 Linear Stability Around the Disease Free Equilibrium Point E0 186 10.4.2 Non Linear Stability Around the Disease Free Equilibrium Point E0 187 10.5 Existence of Endemic Equilibrium Point 190 10.6 Stability Around the Endemic Equilibrium Point 191 10.6.1 Linear Stability Around the Endemic Equilibrium Point E1 191 10.6.2 Non Linear Stability Around the Endemic Equilibrium Point E1 192 10.7 Sensitivity Analysis of Basic Reproduction Number and State Variables 196 10.8 Optimal Control Problem 196 10.9 Numerical Simulation 200 10.10 Conclusions 202 References 206 11 A Fractional Calculus Model to Depict the Calcium Diffusion for Neurodegenerative Disease 209 11.1 Introduction 210 11.2 Mathematical Preliminaries 211 11.3 Mathematical Modelling of Advection Diffusion Equation 212 11.3.1 A Fractional Calculus Model of Advection Diffusion Equation 214 11.4 Results and Discussion 216 11.5 Conclusion 221 References 222 12 Approximate Analytic Solution for Tumour Growth and Human Head Heat Distribution Singular Boundary Value Model by High-Resolution Order-Preserving Fuzzy Transform 227 12.1 Introduction 228 12.2 Preliminaries of Fuzzy Transform 230 12.3 Exponential Basis Approximated Fuzzy Components 231 12.4 Order Preserving Fuzzy Component Scheme 234 12.5 Tumor Growth and Oxygen Diffusion in a Spherical Cell Model 236 12.6 Heat Distribution in the Human Head 237 12.7 Numerical Simulations and Performance Evaluation 239 12.8 Conclusion and Remarks 245 References 245 13 Analysis of One-Dimensional Groundwater Recharge by Spreading using Hybrid Differential Transform and Finite Difference Method 247 13.1 Introduction 248 13.2 Research Gap 249 13.3 Mathematical Formulation 250 13.4 Methodology 252 13.5 Hybrid Differential Transform and Finite Difference Method 253 13.6 Solution 253 13.7 Results and Discussion 255 13.8 Conclusions 257 13.9 Utilities of Research 257 References 258 14 Numerical Solution of Physiological Thermoregulatory Disturbances in Cold Environment 261 14.1 Introduction 262 14.2 Material and Methods 263 14.3 Result 266 14.4 Discussion 267 References 267 15 Mathematical Modelling of Transient Heat Conduction in Biological System by Finite Element Method and Coding in MATLAB 271 15.1 Introduction 271 15.2 General Procedure of Finite Element Method 272 15.3 Process of Finite Element Method 273 15.3.1 Definition of the Problem and its Domain: One Dimensional Thermal Equation of Biological System 274 15.4 Steps Involved in Finite Element Process 275 15.5 Model-2: One Dimensional Quadratic Interpolation Model 277 15.5.1 Assumption of a Suitable form of Variation in T for Quadratic Element 277 15.6 Assembly of Elements 280 15.7 Matrix Form of Element Equation 280 15.8 Algorithm and Computer Program to Solve Heat Equation using FEM in Matlab: 281 15.9 Result and Discussion 287 References 289 Index 293 About the Editors 295.

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