Mathematical Analysis of Infectious Diseases

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Mathematical Analysis of Infectious Diseases
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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
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