Mathematical Models on Population Biology and Epidemiology
Harkaran Singh, PhD
Joydip Dhar, PhD

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Mathematical Models on Population Biology and Epidemiology

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In today’s era, the spread of diseases can occur extremely rapidly as a large populations migrate from one part to another of the world with the readily available transportation facilities. In this century, mankind faces even more challenging environment- and health-related problems than ever before. This new book, Mathematical Models on Population Biology and Epidemiology, explores the use of mathematical applications in biology for the prediction and control infectious diseases.

The volume focuses on studies on prey-predator models to predict and control the spread of infectious diseases. The studies provided here will be very helpful for conservation strategies in forestry habitats, and the epidemic model studies that will be invaluable for public health policymakers for controlling the rapid outbreak of infectious diseases.

Mathematical biology is a fast-growing, well-recognized field and is one of the most exciting modern applications of mathematics. The study of population dynamics and epidemiology is a vast domain that provides correlation of mathematics with biology. Population biology involves the study of the dynamics of biological populations and their interactions with the environment. This volume demonstrates the usefulness of mathematics in biology and provides the background needed to interpret, construct, and analyze a wide variety of mathematical models. Most of the techniques presented here can be readily applied to model other phenomena, in biology as well as in other disciplines.

Since mathematical modelling plays a major role in population dynamics and epidemiology, it is a very valuable and useful experimental tool in making practical predictions, building and testing theories, answering specific questions, determining sensitivities of the parameters, making control strategies, and much more.

Chapter 1: General Introduction
1.1 Introduction
1.1.1 Prey-predator interactions
1.1.2 Discrete generations
1.1.3 Epidemics
1.1.4 Stage-structure
1.1.5 Time delay
1.1.6 Immunity
1.1.7 Vaccination
1.1.8 Non-pharmaceutical interventions through media awareness
1.2 Mathematical Preliminaries
1.2.1 Euler’s scheme for discretization
1.2.2 Center manifold theorem
1.2.3 Bifurcation Theory
1.2.4 Equilibria of linear and non-linear systems
1.2.5 Stability of steady-state solutions and bifurcations
1.2.6 Routh Hurwitz criteria
1.2.7 Geometric stability switch criteria in delay differential equations with delay dependent
1.2.8 Hopf bifurcation
1.2.9 Next generation operator method
1.2.10 Sensitivity analysis

Chapter 2: Discrete-Time Bifurcation Behavior of a Prey-Predator System with Generalized Predator
2.1 Introduction
2.2 Formulation of Mathematical Model
2.2.1 Stability of fixed points of Model
2.3 Bifurcation Behavior of Model
2.3.1 Flip Bifurcation
2.3.2 Hopf Bifurcation
2.4 Numerical Simulations
2.5 Summary

Chapter 3: Prey-Predator Dynamics with Disease in Prey and Predator Depends on Alternative Resources
3.1 Introduction
3.2 Formulation of Model
3.3 Positivity and Boundedness of the system
3.4 Dynamical Behavior of the system at the equilibrium points
3.5 Sensitivity Analysis
3.6 Numerical Simulation
3.7 Summary

Chaprter 4: Dynamics of Childhood Disease with Maturation Delay and Latent Period of Infection
4.1 Introduction
4.2 Formulation of Mathematical Model
4.3 Non-negative equilibria and permanence of the system
4.4 Stability Analysis
4.5 Sensitivity Analysis
4.6 Numerical Simulation
4.7 Summary

Chapter 5: Bifurcation in Disease Dynamics with Latent Period of Infection and Media Awareness
5.1 Introduction
5.2 Formulation of Mathematical Model
5.3 Dynamical Behavior of the System
5.4 Sensitivity Analysis
5.5 Numerical Simulation
5.6 Summary

Chapter 6: Continuous and Discrete Dynamics of SIRS Epidemic Model with Media Awareness
6.1 Introduction
6.2 Formulation of Mathematical Model
6.3 Dynamical Behaviour of the System
6.4 Discrete time system
6.5 Stability of fixed points
6.6 Bifurcation behavior
6.6.1 Flip bifurcation
6.6.2 Hopf Bifurcation
6.7 Sensitivity Analysis
6.8 Numerical Simulations
6.9 Summary

Chapter 7: Dynamics of SEIRVS Epidemic Model with Media Awareness
7.1 Introduction
7.2 Formulation of Mathematical Model
7.3 Dynamical Behaviour of the System
7.4 Sensitivity Analysis
7.5 Numerical Simulation
7.6 Summary

Chapter 8: Future Scope of Research

References/ Bibliography

About the Authors / Editors:
Harkaran Singh, PhD
Khalsa College of Engineering and Technology, Punjab, India

Joydip Dhar, PhD
ABV-Indian Institute of Information Technology and Management, Madhya Pradesh, India

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