Mathematical Methods in Medical and Biological Sciences

1st Edition - November 5, 2024
Editors: Harendra Singh, Hari M Srivastava
Language: English
Paperback ISBN:
9 7 8 - 0 - 4 4 3 - 2 8 8 1 4 - 2
eBook ISBN:
9 7 8 - 0 - 4 4 3 - 2 8 8 1 5 - 9

Mathematical Methods in Medical and Biological Sciences presents mathematical methods for computational models arising in the medical and biological sciences. The book presents… Read more

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Mathematical Methods in Medical and Biological Sciences presents mathematical methods for computational models arising in the medical and biological sciences. The book presents several real-life medical and biological models, such as infectious and non-infectious diseases that can be modeled mathematically to accomplish　profound research in virtual environments when the cost of laboratory expenses is relatively high. It focuses on mathematical techniques that provide global solutions for models arising in medical and biological sciences by considering their long-term benefits.

In addition, the book provides leading-edge developments and insights for a range of applications, including epidemiological modeling of pandemic dynamics, viral infection developments, cancer developments, blood oxygen dynamics, HIV infection spread, reaction-diffusion models, polio infection spread, and chaos modeling with fractional order derivatives.

Title of Book
Cover image
Title page
Table of Contents
Copyright
Contributors
Chapter 1: Mathematical tools and techniques in the applied scientific disciplines
1.1. Introduction, motivation, and definitions
1.2. The Fox–Wright-type hypergeometric function and associated families of higher transcendental functions
1.3. Fractional-calculus operators with Eα,β(φ;z,s,κ) as the kernel
1.4. Fractional-order modeling and analysis of initial-value problems
1.5. General hybrid-type families of fractional-order kinetic equations
1.6. Corollaries and consequences of Theorems 1.1, 1.2, and 1.3
1.7. Miscellaneous applications developed in recent years
1.8. Concluding remarks and observations
Chapter 2: Fractional and fractal extensions of epidemiological models
2.1. Introduction
2.2. Non-integer calculus
2.2.1. Fractional operators
2.2.2. Fractal operator
2.3. Epidemiological models
2.3.1. Susceptible-infected (SI) model
2.3.2. Susceptible-infected-susceptible (SIS) model
2.3.3. Susceptible-infected-recovered (SIR) model
2.3.4. Susceptible-infected-exposed-recovered (SEIR) model
2.4. Conclusion
Chapter 3: Mathematical analysis of the COVID-19 pandemic dynamics in the emergence of resistant strains of SARS-CoV-2 in a vaccinated and non-vaccinated population: a real-world scenario in most countries
3.1. Introduction
3.2. Description of the model
3.3. Theoretical analysis
3.3.1. Disease-free equilibrium
3.3.2. Control reproduction number
3.3.3. Endemic equilibria
3.4. Estimation of parameters
3.5. Results
3.5.1. Simulations of the model with action of the bivalent vaccine
3.5.2. Evaluating the impact of vaccination rate (VR) in a population with the bivalent booster
3.5.3. Varying the mutation rate (MR) of the emergence of a new variant
3.5.4. Evaluating whether the use of face masks can further decrease the infection burden
3.5.5. Will the monovalent vaccine behave better compared with the bivalent vaccine?
3.5.6. Behavior of the control reproduction number of both strains in a bivalent booster population
3.5.7. Behavior of the control reproduction number considering the new variant in a slow booster vaccination rate
3.5.8. Sensitivity analysis
3.6. Discussion
Chapter 4: A fractional study on the vaccination effect to control the COVID-19 epidemic
4.1. Introduction
4.2. Preliminaries concepts
4.3. Fractional Caputo COVID-19 model
4.3.1. Positivity of solution
4.3.2. Equilibrium points
4.3.3. Stability results of DFE
4.3.4. Numerical solution of Caputo model
4.4. Fractional Atangana–Baleanu COVID-19 model
4.4.1. Existence and uniqueness
Hyers–Ulam stability
4.4.2. Numerical solution of AB model
4.5. Sensitivity analysis
4.6. Numerical simulation
4.7. Conclusion
Chapter 5: An epidemiological model of the monkeypox virus and its quarantine effects
5.1. Introduction
5.2. Nomenclature
5.3. Mathematical assumption and formation of the mathematical model
5.4. Derivation of the basic reproduction number
5.5. Stability analysis for the monkeypox virus
5.5.1. Global stability analysis for monkeypox-free equilibrium
5.5.2. Global stability for monkeypox endemic equilibrium
5.6. Discussion of the results
5.7. Conclusions and future scope
Chapter 6: Dynamically consistent nonstandard discretization methods for some mathematical models of infectious diseases
6.1. Introduction
6.2. Preliminaries and concepts
6.3. Mathematical models and their qualitative dynamical properties
6.3.1. An SEI model of an infectious disease with immigration
6.3.2. An SVIR model of disease transmission with immigration
6.3.3. An epidemic model of computer viruses with vaccination and a generalized non-linear incidence rate
6.4. Construct of dynamically consistent NSFD methods
6.4.1. NSFD methods for the model (6.3)
6.4.2. NSFD methods for the model (6.6)
6.4.3. NSFD methods for the model (6.10)
6.5. Numerical experiments
6.5.1. Numerical simulation of the model (6.3)
6.5.2. Numerical simulation of the model (6.6)
6.5.3. Numerical simulation of the model (6.10)
6.6. Concluding remarks and conclusions
Chapter 7: A comparative study of different disease-incidence functions in the mathematical modeling of infectious diseases
7.1. Introduction
7.2. General mathematical model
7.3. Boundedness and positivity of model
7.4. Model 1: mathematical model with bilinear incidence function
(i) Basic reproduction number
(ii) Sensitivity analysis
7.5. Model 2: mathematical model with Holling type-II incidence function
(i) Basic reproduction number
(ii) Sensitivity analysis
7.6. Model 3: mathematical model with the Beddington–DeAngelis incidence function
(i) Basic reproduction number
(ii) Sensitivity analysis
7.7. Model 4: mathematical model with the Crowley–Martin incidence function
(i) Basic reproduction number
(ii) Sensitivity analysis
7.8. Numerical simulation and conclusions
Chapter 8: Numerical investigation of HIV infection of CD4+ T-cells via fractional Vieta–Lucas wavelets
8.1. Introduction
8.2. Preliminaries
8.3. Fractional-order Vieta–Lucas wavelets
8.3.1. Fractional-order shifted Vieta–Lucas polynomials
8.3.2. Fractional-order Vieta–Lucas wavelets
8.3.3. Function approximation
8.3.4. Matrix transformations
8.3.5. Matrix representation of FSVLPs and FVLWs
8.3.6. Convergence and error analysis
8.4. Numerical scheme
8.5. Numerical simulations
8.6. Conclusion
Chapter 9: A numerical study of the fractional SIR epidemic model of an infectious disease via the reproducing kernel Hilbert space method
9.1. Introduction
9.2. Foundational elements
9.3. Methodology
9.4. Convergence analysis
9.5. Numerical results
9.6. Conclusion
Chapter 10: The numerical study of the cancer model in biological science
10.1. Introduction
10.2. Essential definitions
10.3. Mathematical model
10.4. Cancer model in the generalized Caputo sense
10.4.1. Adaptive predictor-corrector numerical technique
10.4.2. Implementation of the adaptive P-C technique on proposed model
10.5. Cancer model with fractional conformable derivatives
10.5.1. Numerical technique
10.5.2. Implementation of the Adams–Moulton method on fractional conformable and β-conformable cancer models
10.6. Numerical simulation
10.7. Conclusion
Chapter 11: Mathematical analysis of the non-linear dynamics of bone mineralization
11.1. Introduction
11.2. Mathematical model
11.3. Qualitative analysis
11.3.1. Boundedness of the solution
11.3.2. Existence and uniqueness of the solution of a system of bone mineralization
11.3.3. Stability analysis
11.4. Numerical simulation and sensitivity analysis
11.5. Conclusion
Chapter 12: Mathematical analysis and computational study of a novel Covid-19 model in the presence of a nonsingular, non-local kernel
12.1. Introduction
12.2. Preliminaries
12.3. Mathematical model
12.4. Global asymptotic stability
12.5. Positive solutions
12.6. Numerical method for the Caputo–Fabrizio fractional derivative
12.7. AB fractional derivative
12.8. Application to the covid model with CF and AB fractional derivatives
12.9. Numerical simulation
12.10. Conclusions
Competing interests
Chapter 13: Parametric analysis of two reaction–diffusion models leading to pattern formation
13.1. Introduction
13.2. Bifurcation analysis
13.2.1. Fisher-KPP equation
13.2.2. FitzHugh–Nagumo equation
13.3. Analytical solution using the G′/G-expansion method
Key steps for implementing the G′/G-expansion method
13.3.1. Solution of Fisher-KPP equation
13.3.2. Solution of FitzHugh–Nagumo equation
13.4. Conclusions
Appendix 13.A. Maple code for analytical solution of the FitzHugh–Nagumo equation
Chapter 14: Existence, uniqueness, and stability analysis results for the SIR epidemic model with a fractional operator
14.1. Introduction
14.2. Fractional operators and some results
14.3. Main results
14.4. Reproduction number and stability analysis in the Ulam–Hyers sense
14.5. Numerical scheme and simulations
14.6. Conclusion
Conflict of interest
Chapter 15: Analysis of the chaotic behavior of a fractional biological system: chaos control via a novel numerical method
15.1. Introduction
15.2. Important theorems
15.3. Mathematical model formulation
15.4. Boundedness, existence, and uniqueness of solutions of the model
15.5. Existence and stability of the points of equilibrium of the system (15.5)
15.5.1. Existence of points of equilibrium
15.5.2. Stability analysis
15.6. Sliding-mode controller
15.7. Numerical simulation
15.8. Conclusion
Index

Harendra Singh

Dr. Harendra Singh is an Assistant Professor in the Department of Mathematics at Post-Graduate College Ghazipur, Uttar Pradesh, India. He teaches post-graduate mathematics courses including Real and Complex Analysis, Functional Analysis, Abstract Algebra, and Measure Theory. His research areas of interest include Mathematical Modelling, Fractional Differential Equations, Integral Equations, Calculus of Variations, and Analytical and Numerical Methods. He is the co-Editor with Dr. Srivastava of Special Functions in Fractional Calculus and Engineering, Taylor and Francis/CRC Press.

Affiliations and expertise

Assistant Professor, Department of Mathematics, Post-Graduate College, Ghazipur, India

Hari M Srivastava

Dr. Hari M. Srivastava is Professor Emeritus in the Department of Mathematics and Statistics at the University of Victoria, British Columbia, Canada. He earned his Ph.D. degree in 1965 while he was a full-time member of the teaching faculty at the Jai Narain Vyas University of Jodhpur, India. Dr. Srivastava has held (and continues to hold) numerous Visiting, Honorary and Chair Professorships at many universities and research institutes in diﬀerent parts of the world. Having received several D.Sc. (honoris causa) degrees as well as honorary memberships and fellowships of many scientiﬁc academies and scientiﬁc societies around the world, he is also actively associated editorially with numerous international scientiﬁc research journals as an Honorary or Advisory Editor or as an Editorial Board Member. He has also edited many Special Issues of scientiﬁc research journals as the Lead or Joint Guest Editor, including the MDPI journal Axioms, Mathematics, and Symmetry, the Elsevier journals, Journal of Computational and Applied Mathematics, Applied Mathematics and Computation, Chaos, Solitons & Fractals, Alexandria Engineering Journal, and Journal of King Saud University – Science, the Wiley journal, Mathematical Methods in the Applied Sciences, the Springer journals, Advances in Diﬀerence Equations, Journal of Inequalities and Applications, Fixed Point Theory and Applications, and Boundary Value Problems, the American Institute of Physics journal, Chaos: An Interdisciplinary Journal of Nonlinear Science, and the American Institute of Mathematical Sciences journal, AIMS Mathematics, among many others. Dr. Srivastava has been a Clarivate Analytics (Web of Science) Highly-Cited Researcher since 2015. Dr. Srivastava’s research interests include several areas of Pure and Applied Mathematical Sciences, such as Real and Complex Analysis, Fractional Calculus and Its Applications, Integral Equations and Transforms, Higher Transcendental Functions and Their Applications, q-Series and q-Polynomials, Analytic Number Theory, Analytic and Geometric Inequalities, Probability and Statistics, and Inventory Modeling and Optimization. He has published 36 books, monographs, and edited volumes, 36 book (and encyclopedia) chapters, 48 papers in international conference proceedings, and more than 1450 peer-reviewed international scientiﬁc research journal articles, as well as Forewords and Prefaces to many books and journals.

Affiliations and expertise

Professor Emeritus, Department of Mathematics and Statistics, University of Victoria, British Columbia, Canada

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Harendra Singh

Hari M Srivastava