Mathematical Analysis of Infectious Diseases

Description

Mathematical Analysis of Infectious Diseases updates on the mathematical and epidemiological analysis of infectious diseases. Epidemic mathematical modeling and analysis is important, not only to understand disease progression, but also to provide predictions about the evolution of disease. One of the main focuses of the book is the transmission dynamics of the infectious diseases like COVID-19 and the intervention strategies. It also discusses optimal control strategies like vaccination and plasma transfusion and their potential effectiveness on infections using compartmental and mathematical models in epidemiology like SI, SIR, SICA, and SEIR. The book also covers topics like: biodynamic hypothesis and its application for the mathematical modeling of biological growth and the analysis of infectious diseases, mathematical modeling and analysis of diagnosis rate effects and prediction of viruses, data-driven graphical analysis of epidemic trends, dynamic simulation and scenario analysis of the spread of diseases, and the systematic review of the mathematical modeling of infectious disease like coronaviruses. Front Cover Mathematical Analysis of Infectious Diseases Copyright Contents Contributors Preface 1 Spatiotemporal dynamics of the first wave of the COVID-19 epidemic in Brazil 1.1 Introduction 1.2 Materials and methods 1.2.1 The SEIR model 1.2.2 Model integration 1.2.3 Model calibration 1.2.4 Parameters 1.2.5 Case study: Brazil 1.2.6 Obtaining a network structure from Rt data 1.3 Results 1.3.1 Spatiotemporal dynamics of coronavirus spread 1.3.2 Coronavirus spread and area coverage of the epidemic 1.4 Discussion 1.4.1 Early spread of coronavirus in Brazil 1.4.2 Spread velocity 1.4.3 Brazil's failure in containing the pandemic at early stages 1.5 Final remarks References 2 Transport and optimal control of vaccination dynamics for COVID-19 2.1 Introduction 2.2 Vaccine transport model 2.3 Initial mathematical model for COVID-19 2.4 Mathematical model for COVID-19 with vaccination 2.5 Optimal control 2.6 Numerical results 2.7 Conclusion Acknowledgments References 3 COVID-19's pandemic: a new way of thinking through linear combinations of proportions 3.1 Introduction 3.2 Estimation of linear combinations of proportions 3.3 Material and methods 3.3.1 Datasets 3.3.2 Linear functions of proportions 3.3.3 Inference 3.3.4 Graphical procedure 3.4 Results and discussion 3.4.1 Linear combination L1(t) = ω1 p1(t)-p2(t)+p3(t)/2 3.4.2 Linear combination L2(t)= ω1 p1(t)-p2(t) 3.4.3 Linear combination L3(t)= ω2 p2(t)-p3(t) 3.4.4 Linear combination L4(t)= ω4 p4(t)-p5(t) 3.4.5 Linear combination L5(t)= ω4 p4(t) - p3(t)+p5(t)/2 3.5 Conclusion Acknowledgments References 4 Stochastic SICA epidemic model with jump Lévy processes 4.1 Introduction 4.2 Existence and uniqueness of a global positive solution 4.3 Extinction 4.4 Persistence in the mean 4.5 Numerical results 4.6 Conclusion Acknowledgments References 5 Examining the correlation between the weather conditions and COVID-19 pandemic in Galicia 5.1 Introduction 5.2 Fuzzy sets 5.3 Results 5.4 Conclusions References 6 A fractional-order malaria model with temporary immunity 6.1 Introduction 6.2 Preliminaries on fractional calculus 6.3 Model description 6.3.1 Classical integer model 6.3.2 Fractional order mathematical model 6.4 Basic properties of the ABC malaria model 6.4.1 Existence and uniqueness 6.4.2 Invariant region and attractivity 6.4.3 Positivity and boundedness 6.5 The analysis 6.5.1 Stability analysis 6.5.1.1 Local stability of DFE point E0 via R0 6.5.1.2 Global stability of DFE point E0 via R0 6.6 Numerical solution of fractional malaria model 6.7 Discussion 6.8 Conclusion References 7 Parameter identification in epidemiological models 7.1 Introduction 7.2 SEIJR models for closed systems 7.3 Uncertainty quantification by Bayesian techniques 7.3.1 Bayesian formulation for SEIJR coefficients 7.3.2 Prior selection 7.3.3 Markov Chain Monte Carlo sampling 7.4 Effect of nonpharmaceutical actions 7.4.1 Piecewise SEIJR system: lockdown 7.4.2 Piecewise SEIJR system: release 7.4.3 Equilibrium 7.5 SEIJR model including migration 7.6 Optimization approach to control 7.7 Conclusions Acknowledgments References 8 Lyapunov functions and stability analysis of fractional-order systems 8.1 Introduction 8.2 Preliminaries 8.3 Useful fractional derivative estimates 8.4 An application 8.5 Conclusion Acknowledgments References 9 Some key concepts of mathematical epidemiology 9.1 Introduction 9.2 A short historical introduction 9.3 Equilibria, the basic reproduction number and final size relation 9.3.1 Equilibria and stability 9.3.2 The basic reproduction number 9.3.3 The final size relations 9.4 Sojourn time, delay, and incidence forms 9.4.1 Sojourn time 9.4.2 Delays in epidemiological models 9.4.3 Different forms of incidence 9.5 Numerical simulations 9.5.1 Numerical methods 9.5.2 Agent-based simulation modeling 9.5.3 Neural networks A long short time memory ANN with a modified SEIR compartmental model 9.6 Herpes modeling 9.7 Conclusion Acknowledgment References 10 Analytical solutions and parameter estimation of the SIR epidemic model 10.1 Introduction 10.2 The SIR model 10.3 Second-order systems equivalent to SIR 10.3.1 A second order differential equation for the i-variable 10.3.2 A second order differential equation for the s-variable 10.4 Indeterminate analytical solution 10.4.1 The s-variable 10.4.2 The i-variable 10.4.3 The r-variable 10.5 Inverse parametric solution 10.5.1 Peak value parametrization 10.5.2 Initial value parametrization 10.6 Analysis of the incidence variable 10.7 Asymptotic analysis of the SIR model 10.8 Numerical approximation 10.9 Cast study I: application to influenza A 10.10 Cast study II: application to COVID-19 10.11 Discussion and conclusions 10.A The Lambert W function and related integrals 10.B Differential fields References 11 Global stability of a diffusive SEIR epidemic model with distributed delay 11.1 Introduction 11.2 Mathematical model 11.3 Analysis of the model 11.3.1 Well-posedness 11.3.2 Equilibria and the basic reproduction number 11.3.3 Global stability of the disease free equilibrium 11.3.4 Global stability of the endemic equilibrium 11.4 Numerical simulations 11.5 Concluding remarks Acknowledgment References 12 Application of fractional order differential equations in modeling viral disease transmission 12.1 Introduction 12.2 Preliminaries 12.3 Mathematical model of the AH1N1/09 influenza transmission 12.4 Equilibrium points 12.4.1 Stability of equilibrium point 12.5 Existence of solution 12.5.1 Existence of solution by the Picard-Lindelof approach 12.6 Optimal control approach 12.7 Numerical results 12.7.1 Numerical method 12.7.2 Numerical simulation 12.7.3 Reproduction number sensitivity 12.8 Conclusion References 13 Role of immune effector responses during HCV infection: a mathematical study 13.1 Introduction 13.2 The mathematical model 13.2.1 Basic properties of the model 13.2.2 Local stability analysis 13.2.3 Global stability of the disease free equilibrium 13.3 Optimal control problem 13.4 Numerical simulations 13.5 Discussion and conclusion References 14 Modeling the impact of isolation during an outbreak of Ebola virus 14.1 Introduction 14.2 Mathematical model 14.3 Mathematical analysis of the model 14.3.1 Positivity and boundedness 14.3.2 Reproduction number and equilibrium points 14.4 Numerical simulation 14.5 Optimal control of the spread of the virus 14.6 Conclusions References 15 Application of the stochastic arithmetic to validate the results of nonlinear fractional model of HIV infection for CD8+T-cells 15.1 Introduction 15.2 Preliminaries 15.3 Nonlinear model of HIV infection for CD8+T cells 15.4 Existence of solution 15.4.1 Existence of solution by the Picard-Lindelof approach 15.5 Special solution via iteration approach 15.5.1 Fixed point theorem for stability analysis of the iteration method 15.6 Application of the HATM to solve the model 15.7 Control of accuracy by the CESTAC method and the CADNA library 15.8 Numerical results 15.9 Conclusion References 16 Existence of solutions of modified fractional integral equation models for endemic infectious diseases 16.1 Introduction and preliminaries 16.2 Fixed point theorems 16.3 Coupled fixed point theorems 16.4 Application Acknowledgments References 17 Numerical solution of a fractional epidemic model via general Lagrange scaling functions with bibliometric analysis 17.1 Introduction 17.1.1 A brief bibliometric analysis on ``epidemic models'' 17.1.2 Chapter outline 17.2 Preliminaries 17.2.1 Fractional calculus 17.2.2 Lagrange polynomials and general Lagrange scaling functions 17.2.2.1 General Lagrange scaling functions 17.2.2.2 Approximation based on GLSFs 17.3 GLSF Riemann-Liouville pseudo-operational matrix 17.3.1 Transformation matrix of GLSFs to PTFs 17.3.2 PTF Riemann-Liouville pseudo-operational matrix 17.3.3 GLSF Riemann-Liouville pseudo-operational matrix 17.4 Computational method 17.5 Error analysis 17.6 Numerical results and discussion 17.7 Conclusion References Index Back Cover