Methods of Mathematical Modelling: Infectious Diseases

By Hari M Srivastava and Dumitru Baleanu

Methods of Mathematical Modeling: Infectious Diseases presents computational methods related to biological systems and their numerical treatment via mathematical tools and techniques. Edited by renowned experts in the field, Dr. Hari Mohan Srivastava, Dr. Dumitru Baleanu, and Dr. Harendra Singh, the book examines advanced numerical methods to provide global solutions for biological models. These results are important for medical professionals, biomedical engineers, mathematicians, scientists and researchers working on biological models with real-life applications. The authors deal with methods as well as applications, including stability analysis of biological models, bifurcation scenarios, chaotic dynamics, and non-linear differential equations arising in biology.

The book focuses primarily on infectious disease modeling and computational modeling of other real-world medical issues, including COVID-19, smoking,  cancer and diabetes. The book provides the solution of these models so as to provide actual remedies.

• Includes mathematical modeling for a variety of infectious diseases and disease processes, including SIR/SIRA,  COVID-19, cancer, smoking and diabetes
• Offers a complete and foundational understanding of modeling algorithms and techniques such as stability analysis, bifurcation scenarios, chaotic dynamics, and non-linear differential equations
• Provides readers with datasets for applied learning of the various algorithms and modeling techniques

• Language: English
• Publisher: Academic Press
• Release date: Jun 10, 2022
• ISBN: 9780323999472

Book preview

9780323999472_FC

Methods of Mathematical Modeling

Infectious Disease

First Edition

Harendra Singh

Department of Mathematics, Post-Graduate College, Ghazipur, Uttar Pradesh, India

Hari Mohan Srivastava

University of Victoria, Victoria, BC, Canada

Dumitru Baleanu

Cankaya University, Ankara, Turkey

Table of Contents

Cover image

Title page

Copyright

Contributors

1: Epidemic theory: Studying the effective and basic reproduction numbers, epidemic thresholds and techniques for the analysis of infectious diseases with particular emphasis on tuberculosis

Abstract

Acknowledgment

1: Introduction

2: Basic and effective reproduction numbers

3: Significance and limitations of basic reproduction number

4: Herd immunity

5: Different methods to determine R0

6: Computation of R0 by the next-generation matrix method

7: Computation of BRN of a mathematical model on multidrug-resistant tuberculosis (MDR-TB)

8: Literature review on BRN of TB

9: Other measures to study an epidemic

10: Reproduction number using epidemic curve and serial interval distribution

11: Conclusions

References

2: Numerical methods applied to a class of SEIR epidemic models described by the Caputo derivative

Abstract

1: Introduction

2: On fractional operators in fractional calculus

3: Description of the model

4: Existence of the equilibrium points

5: Numerical schemes and applications

6: Stability analysis

7: Simulations and illustrations

8: Conclusion

References

Further reading

3: Mathematical model and interpretation of crowding effects on SARS-CoV-2 using Atangana-Baleanu fractional operator

Abstract

1: Introduction

2: Spread of new SARS-CoV-2 variant in India

3: Model for crowding effects on COVID-19

4: Conclusions

Availability of data and materials

Competing interests

Authors’ contributions

References

4: Analysis for modified fractional epidemiological model for computer viruses

Abstract

1: Introduction

2: Model description

3: Preliminaries

4: Outline of method

5: Stability analysis

6: Numerical discussion

7: Conclusions

References

5: Analysis of e-cigarette smoking model by a novel technique

Abstract

1: Introduction

2: Fundamental definitions

3: Analysis with the exponential-decay kernel

4: Numerical simulations

5: Conclusions

References

6: Stability analysis of an unhealthy diet model with the effect of antiangiogenesis treatment

Abstract

1: Introduction

2: Description of the antiangiogenic model

3: Invariant region

4: Existing equilibrium points of the system

5: Stability analysis

6: Global stability analysis at healthy equilibrium point E1

7: Diet-antiangiogenic drug model with delay

8: Positivity and boundedness of the solution

9: Existence of equilibrium points

10: Conclusion

References

7: Analysis of the spread of infectious diseases with the effects of consciousness programs by media using three fractional operators

Abstract

1: Introduction

2: Preliminaries

3: Solution for FKDV equation

4: Results and discussion

5: Conclusion

References

8: Modeling and analysis of computer virus fractional order model

Abstract

1: Introduction

2: Basic concept of fractional order

3: Mathematical model formulations

4: Atangana-Baleanu Caputo sense

5: Result and discussion

6: Conclusion

References

9: Stochastic analysis and disease transmission

Abstract

Acknowledgment

1: Introduction

2: Deterministic epidemic models

3: DTMC epidemic models

4: CTMC epidemic models formulation

5: SDE epidemic models formulation

References

Further reading

10: Analysis of the Adomian decomposition method to estimate the COVID-19 pandemic

Abstract

Funding

Conflicts of interests

1: Introduction

2: Methodology

3: Theory and calculations

4: Results and discussion

5: Conclusion

References

11: Study of a COVID-19 mathematical model

Abstract

1: Introduction

2: Fundamental results

3: Feasibility of solution and stability analysis

4: Qualitative analysis

5: Series solution for model (2)

6: Numerical solution for Eq. (3)

7: Computational of the numerical solution of the COVID-19 model for model (2)

8: Graphical results and discussion for model (3)

9: Concluding remarks

References

Index

Copyright

Academic Press is an imprint of Elsevier

125 London Wall, London EC2Y 5AS, United Kingdom

525 B Street, Suite 1650, San Diego, CA 92101, United States

50 Hampshire Street, 5th Floor, Cambridge, MA 02139, United States

The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, United Kingdom

Copyright © 2022 Elsevier Inc. All rights reserved.

No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Details on how to seek permission, further information about the Publisher’s permissions policies and our arrangements with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our website: www.elsevier.com/permissions.

This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be noted herein).

Notices

Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary.

Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility.

To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein.

Library of Congress Cataloging-in-Publication Data

A catalog record for this book is available from the Library of Congress

British Library Cataloguing-in-Publication Data

A catalogue record for this book is available from the British Library

ISBN 978-0-323-99888-8

For information on all Academic Press publications visit our website at https://www.elsevier.com/books-and-journals

Image 1

Publisher: Mara Conner

Acquisitions Editor: Chris Katsaropoulos

Editorial Project Manager: Mariana L. Kuhl

Production Project Manager: Omer Mukthar

Cover Designer: Greg Harris

Typeset by STRAIVE, India

Contributors

Garima Agarwal     Department of Mathematics and Statistics, Manipal University Jaipur, Rajasthan, India

Ritu Agarwal     Department of Mathematics, Malaviya National Institute of Technology, Jaipur, Rajasthan, India

Aqeel Ahmad     Department of Mathematics and Statistics, University of Lahore, Lahore, Pakistan

M.O. Ahmad     Department of Mathematics and Statistics, University of Lahore, Lahore, Pakistan

Ali Akgül     Art and Science Faculty, Department of Mathematics, Siirt University, Siirt, Turkey

Esra Karatas Akgül     Art and Science Faculty, Department of Mathematics, Siirt University, Siirt, Turkey

Muhammad Arfan     Department of Mathematics, University of Malakand, Khyber Pakhtunkhwa, Pakistan

Archana Singh Bhadauria     Deen Dayal Upadhyaya Gorakhpur University, Gorakhpur, India

Ankita Chandola     Amity Institute of Applied Sciences, Amity University Uttar Pradesh, Noida, India

Tikam Chand Dakal     Genome and Computational Biology Lab, Department of Biotechnology, Mohanlal Sukhadia University, Udaipur, Rajasthan, India

Anusmita Das     Department of Mathematics, Gauhati University, Assam, India

Kaushik Dehingia     Department of Mathematics, Sonari College, Sonari, Assam, India

Hom Nath Dhungana     School of Mathematical Science, University of Technology Sydney, Ultimo, NSW, Australia

Anwarud Din     Department of Mathematics, Sun Yat-sen University, Guangzhou, Peoples Republic of China

Eiman     Department of Mathematics, University of Malakand, Khyber Pakhtunkhwa, Pakistan

Muhammad Farman     Department of Mathematics and Statistics, University of Lahore, Lahore, Pakistan

Rozi Gul     Department of Mathematics, University of Malakand, Khyber Pakhtunkhwa, Pakistan

Tariq Hussain     Department of Mathematics, University of Malakand, Khyber Pakhtunkhwa, Pakistan

Jogendra Kumar     School of Physical Sciences DIT University, Dehradun, India

Yongjin Li     Department of Mathematics, Sun Yat-sen University, Guangzhou, Peoples Republic of China

Khalid Mahmood     Department of Mathematics, University of Malakand, Khyber Pakhtunkhwa, Pakistan

Athira M. Menon     Genome and Computational Biology Lab, Department of Biotechnology, Mohanlal Sukhadia University, Udaipur, Rajasthan, India

Vishnu Narayan Mishra     Department of Mathematics, Indira Gandhi National Tribal University, Madhya Pradesh, India

Man Mohan     Department of Mathematics and Statistics, Manipal University Jaipur, Rajasthan, India

Rupakshi Mishra Pandey     Amity Institute of Applied Sciences, Amity University Uttar Pradesh, Noida, India

Sunil Dutt Purohit     Department of HEAS (Mathematics), Rajasthan Technical University, Kota, Rajasthan, India

Muhammad Umer Saleem     Department of Mathematics, University of Education, Lahore, Pakistan

Ndolane Sene     Department of Mathematics, Cheikh Anta Diop University, Dakar Fann, Senegal

Kamal Shah     Department of Mathematics, University of Malakand, Khyber Pakhtunkhwa, Pakistan

Amit Sharma     Department of Ophthalmology, University Clinic Bonn, Bonn, Germany

C.S. Singh     Department of Mathematics, Uttar Pradesh Textile Technology Institute, Kanpur, India

Harendra Singh     Department of Mathematics, Post-Graduate College, Ghazipur, Uttar Pradesh, India

Mayank Srivastava     Department of Mathematics, Post-Graduate College, Ghazipur, Uttar Pradesh, India

Hayat Ullah     Department of Mathematics, University of Malakand, Khyber Pakhtunkhwa, Pakistan

P. Veeresha     Centre for Mathematical Needs, Department of Mathematics, CHRIST (Deemed to Be University), Bengaluru, India

1: Epidemic theory: Studying the effective and basic reproduction numbers, epidemic thresholds and techniques for the analysis of infectious diseases with particular emphasis on tuberculosis

Archana Singh Bhadauriaa; Hom Nath Dhunganab    a Deen Dayal Upadhyaya Gorakhpur University, Gorakhpur, India

b School of Mathematical Science, University of Technology Sydney, Ultimo, NSW, Australia

Abstract

Basic reproduction number (BRN) plays a significant role in studying communicable diseases in the starting phase of an epidemic. However, when the disease spreads in the population, the effective reproduction number replaces it taking the primary role in the study of disease dynamics. This book deals with the significance of basic and effective reproduction numbers in the study of communicable diseases, their threshold behavior, and different approaches to determine them in a mathematical model. Starting with the history of mathematical epidemiology and determination of BRN using deterministic and stochastic approaches, this chapter presents a brief introduction of all the terms required to be aware of while studying BRN such as herd immunity, epidemic curve, index cases, primary cases, generation time, and serial interval. This chapter also covers the scope of exception reporting, clusters, and study of BRN using epidemic curve and serial interval distribution. The next-generation matrix (NGM) method to determine BRN is discussed in detail. In addition, a mathematical model on multidrug-resistant tuberculosis (MDR-TB) has been formulated and its BRN is determined by the NGM approach. A brief review of BRN for tuberculosis (TB) disease is also given in the chapter to emphasize the role of studying BRN in its control.

Keywords

Mathematical epidemiology; Communicable diseases; Basic reproduction number; Effective reproduction number; Tuberculosis

Acknowledgment

The first author thankfully acknowledges UGC-BSR Start-Up research Grant (sanction No. F.30–466/2019(BSR)) for financial support in the preparation of the chapter.

1: Introduction

The baseline level of any disease prevailing in a region is the expected level of disease. A sudden increase in the number of expected levels of disease in that population is known as an epidemic. Furthermore, an epidemic restricted to a geographic area is termed as an outbreak. Investigation becomes imperative for single case of some rare diseases such as plague, rabies, etc., whereas some diseases need investigation when the number of cases exceeds the expected or baseline number of cases. Apart from epidemics, some diseases become endemic in some populations and cause many disease-related deaths, particularly in developing countries with inadequate health-care systems. Diseases such as malaria, typhus, schistosomiasis, cholera, Japanese encephalitis, and sleeping sickness are endemic in many parts. When an epidemic becomes worldwide, it is known as a pandemic, for example, SARS, MERS, and the recent COVID-19.

Epidemiologists play a significant role in understanding the causes of disease, predicting the course of the disease, and guiding the health-care system in controlling it by comparing different possible approaches to control the spread of disease. The first model in mathematical epidemiology had been given by Daniel Bernoulli (1700–82) on inoculation against smallpox. Bernoulli, through his mathematical model, determined the efficacy of variolation against disease. A brief outline of his work was published in 1760 [1] and a detailed study in 1766 [2]. However, his approach was generalized by Dietz and Heesterbeek in 2002 [3].

In 1906, Hamer [4] suggested that the transmission of infection depends on both the number of susceptible and infective individuals. He proposed mass action law for determining the rate of new infections. Since that time, this idea of Hamer has been used as a basic assumption in compartmental models. The foundation of the compartmental model in mathematical epidemiology was laid by public health physicians Sir R.A. Ross, W.H. Hamer, A.G. McKendrick, and W.O. Kermack between 1900 and 1935.

Dr. Ross was awarded the second Nobel Prize in Medicine in 1902 for his work on the transmission of malaria between mosquitoes and humans. Ross gave a simple compartmental model using differential equations in 1911 [4], including mosquitoes and humans. He showed that spread of malaria could be controlled by reducing the mosquito population below a critical level. It was believed earlier that malaria could not be eliminated as long as mosquitoes are present in a population. He introduced the concept of basic reproduction number in the field of mathematical epidemiology. This study of Ross was supported by field trials, which led to a significant success in malaria control. Later on, Kermack and McKendrick provided the basic compartmental models in 1927 [5], 1932 [6], and 1933 [7] to describe the transmission of communicable diseases. A special case of the Kermack-McKendrick epidemic model is given by

si1_esi2_e

where N is the total population size, and S and I are susceptible and infected populations, respectively, β is the transmission coefficient of infection, and γ is the recovery rate. Since then, various epidemic models have been framed and studied based on the compartment models given by Kermack and McKendrick. Numerous research papers have been published to date related to the dynamics of infectious diseases based on these compartment models using the theory of ordinary and fractional differential equations. Some of the reported literatures [8–16] provide a good insight into the dynamics of the recent COVID-19 pandemic using fractional differential equations.

Basic reproduction number (BRN), however, is unnamed but appeared as a threshold quantity in the work of Ross, and Kermack and McKendrick, which is now almost universally denoted by R0. But, neither Ross nor Kermack and McKendrick identified this threshold quantity or named it. MacDonald [17] was the first person to identify and name the threshold quantity explicitly as BRN in his work on malaria.

Although Kermack-McKendrick models are yet used as a basis of epidemic modeling, there are serious shortcomings in the Kermack-McKendrick model. The first shortcoming lies in a description of the beginning of a disease outbreak. The simple Kermack-McKendrick compartmental epidemic model assumes the large size of the compartment such that the mixing of the population is homogeneous. However, at the beginning of a disease outbreak, the population of the infected individuals is not very large. Therefore, the transmission of infection is a stochastic event that depends on the pattern of contact between individuals of the population. The process is known as a Galton-Watson process first given in Ref. [18]. The first description of this process in an epidemic study was mentioned by Metz in 1978 [19]. In the stochastic process, the beginning of the disease outbreak has been assumed to be a network of contact of individuals. This process has been defined by a graph in which vertices are represented by individuals of the population, and contacts between individuals are represented by the edges. The number of edges of a graph at a vertex is termed as the degree of that vertex. Let the degree distribution of a graph is {pk}, where pk is the fraction of vertices having degree k. In addition, it has been assumed that the initial number of infectives in a population is so small that a decrease in the number of susceptibles may be neglected. The number of contacts made by the infected population