Advances in Epidemiological Modeling and Control of Viruses

Contents
Contributors
Preface
1 Global dynamics of a delayed reaction-diffusion viral infection in a cellular environment
	1.1 Introduction
	1.2 Global convergence to the homogeneous solution
	1.3 Numerical simulations
		1.3.1 Example 1
		1.3.2 Example 2
	1.4 Spatiotemporal pattern formation
		1.4.1 Maximum maps
		1.4.2 Bifurcations and branches of solutions
	1.5 Conclusion and open problems
	Acknowledgments
	References
2 Hepatitis B virus transmission via epidemic model
	2.1 Introduction
	2.2 Model formulation
	2.3 Stability analysis
	2.4 Simulation and concluding remarks
	References
3 Global dynamics of an HCV model with full logistic terms and the host immune system
	3.1 Introduction
	3.2 Previous works
		3.2.1 Modelling virus-immune system interaction
		3.2.2 Modelling virus infection with full logistic terms and antivirus treatment
	3.3 Mathematical preliminaries
		3.3.1 Linearization
		3.3.2 Lyapunov functions
		3.3.3 Bifurcation analysis
		3.3.4 Li and Muldowney\'s geometric approach
	3.4 Modelling virus-immune system interaction with full logistic terms in both uninfected and infected cells
		3.4.1 Model construction
	3.5 Analysis of the model
		3.5.1 Dissipativity, basic reproduction number, and equilibria
			3.5.1.1 Dissipativity
			3.5.1.2 Basic reproduction number
			3.5.1.3 Existence of endemic equilibrium
		3.5.2 Local and global stability analysis
			3.5.2.1 Local stability analysis of the disease-free equilibrium
			3.5.2.2 Local stability analysis of the endemic equilibrium
			3.5.2.3 Global stability analysis of the disease-free equilibrium
			3.5.2.4 Uniform persistence
			3.5.2.5 Global stability analysis of the endemic equilibrium
	3.6 Numerical simulations
	3.7 Conclusion and discussion
	Acknowledgments
	References
4 On a Novel SVEIRS Markov chain epidemic model with multiple discrete delays and infection rates: modeling and sensitivity analysis to determine vaccination effects
	4.1 Introduction
	4.2 Description of the SVEIRS epidemic and the delays in the disease dynamics
	4.3 Discretization of time and decomposition of the SVEIRS population over time
		4.3.1 Decomposition of the total human population over discrete time intervals
		4.3.2 One-state-at-a-time decomposition of the population states over the finite delay times
		4.3.3 Joint state decomposition of the population over the finite delay times
	4.4 The SVEIRS stochastic process
		4.4.1 The SVEIRS Markov chain
	4.5 Some special SVEIRS epidemic models
		4.5.1 The SVEIRS model with correlated vaccination and infection rates
		4.5.2 The SVEIRS model with no correlation between vaccination and infection rates
	4.6 Numerical study: some prototype SVEIRS epidemic models and sensitivity analysis to determine the effects of infection and vaccination
		4.6.1 The general algorithm for the simulations
		4.6.2 The prototype SVEIRS model with correlated vaccination and infection rates
			4.6.2.1 Sensitivity of the SVEIRS model when p = ϕ
			4.6.2.2 Sensitivity analysis of the SVEIRS model when either p or ϕ is fixed and the other parameter continuously changes
		4.6.3 The prototype SVEIRS model with uncorrelated vaccination and infection rates
	4.7 Conclusion
	References
5 Hopf bifurcation in an SIR epidemic model with psychological effect and distributed time delay
	5.1 Introduction
	5.2 Model
	5.3 Direction of bifurcation and stability of periodic solution
	5.4 Example: a truncated exponential distribution
	5.5 Numerical simulation
	5.6 Discussion
	Acknowledgments
	5.A Matlab code for Fig. 5.8(b)
	References
6 Modeling of the effects of media in the course of vaccination of rotavirus
	6.1 Introduction
	6.2 Epidemic modeling
	6.3 Existence of equilibria of system Υ1
		6.3.1 Equilibria of system Υ2
	6.4 Stability of the equilibria
	6.5 Optimal control problem
		6.5.1 Existence of optimal control
		6.5.2 Characterization of optimal control
	6.6 Efficacy analysis
	6.7 Numerical simulations
	6.8 Discussion
	References
7 Mathematical models of early stage Covid-19 transmission in Sri Lanka
	7.1 Introduction
	7.2 Mathematical model to estimate initial parameters
		7.2.1 Analysis of the model
		7.2.2 Estimation of initial parameters
		7.2.3 Optimization
		7.2.4 Numerical results
	7.3 Mathematical models with heterogeneity of cases
		7.3.1 Analysis of the model
			7.3.1.1 Basic reproduction number
			7.3.1.2 Stability analysis of the disease-free equilibrium
		7.3.2 Introducing optimal control measures
			7.3.2.1 Mathematical model with control parameters
			7.3.2.2 Mathematical analysis of the model
		7.3.3 Numerical results
			7.3.3.1 Algorithm for the optimal control problem
		7.3.4 Simulation of the COVID 19 dynamic system without control
		7.3.5 Simulation of the optimal control problem
			7.3.5.1 Scenario 1
			7.3.5.2 Scenario 2
			7.3.5.3 Scenario 3
	7.4 Mathematical model with imported cases
		7.4.1 Sensitivity of the control measures (NPIs)
		7.4.2 Sensitivity of the control with overseas exposed cases
		7.4.3 Sensitivity of the timing of implementing combined control measures
	7.5 Conclusion
	References
8 Global stability of a diffusive HTLV-I infection model with mitosis and CTL immune response
	8.1 Introduction
		8.1.1 Mathematical models
			8.1.1.1 Model with latent HTLV-infected cells
			8.1.1.2 Model with delay
			8.1.1.3 Model with mitosis
			8.1.1.4 Model with CTL immune response
			8.1.1.5 Model with mitosis and CTL immune response
			8.1.1.6 Model with diffusion
	8.2 Model formulation
	8.3 Well-posedness of solutions
	8.4 Steady state analysis
	8.5 Global stability analysis
	8.6 Numerical simulations
	8.7 Conclusion and discussion
	References
9 Mathematical tools and their applications in dengue epidemic data analytics
	9.1 Introduction
	9.2 Fourier transformation
		9.2.1 Discrete time Fourier transform
		9.2.2 Fast Fourier transform
		9.2.3 Dengue epidemic data analysis
		9.2.4 Study areas
		9.2.5 Numerical results and discussion
	9.3 Wavelet analysis
		9.3.1 Wavelet transform
		9.3.2 Basic wavelet functions
			9.3.2.1 Haar wavelet
			9.3.2.2 Meyer wavelet
			9.3.2.3 Morlet wavelet
		9.3.3 Wavelet power spectrum
		9.3.4 Wavelet coherency and phase difference
		9.3.5 Statistical significance
		9.3.6 Cone of influence
	9.4 Wavelet analysis in epidemiology and dengue
		9.4.1 Analysis of dengue incidents in urban Colombo
		9.4.2 Effect of climate
		9.4.3 Analysis of dengue incidents in urban Colombo
		9.4.4 Effect of human mobility: a case study
	9.5 Conclusion
	References
10 Covid-19 pandemic model: a graph theoretical perspective
	10.1 Introduction
	10.2 Preliminaries
		10.2.1 Graph theory terminology
		10.2.2 Epidemiological terminology
	10.3 A survey of mathematical models on diseases
		10.3.1 Epidemic/pandemic models on Covid-19
		10.3.2 Network models
	10.4 SEIRD model on Covid-19
	10.5 Some results
		10.5.1 Effects of R<1 on a network
		10.5.2 Effects of R>1 on a network
	10.6 Conclusion and recommendation
	References
11 Towards nonmanifest chaos and order in biological structures by means of the multifractal paradigm
	11.1 Introduction
	11.2 Mathematical model
		11.2.1 Short reminder on the multifractal theory of motion
		11.2.2 Stationary nonlinear behaviors through Schrödinger-type ``regimes\'\' as ``synchronization modes\'\'
		11.2.3 Nonstationary nonlinear behaviors through Schrödinger-type ``regimes\'\' as ``synchronization modes\'\'
		11.2.4 Space-time ``synchronization modes\'\' and nonmanifest scenarios towards chaos
	11.3 Conclusions
	References
12 Global stability of epidemic models under discontinuous treatment strategy
	12.1 Impact of discontinuous treatments on disease dynamics in an SIR epidemic model
		12.1.1 Introduction
		12.1.2 Model description and preliminaries
		12.1.3 Equilibria and their stability
		12.1.4 Global convergence in finite time
		12.1.5 Conclusion and discussion
	12.2 Global stability of an SIS epidemic model with discontinuous treatment strategy
		12.2.1 Introduction
		12.2.2 Model and preliminaries
		12.2.3 Positivity and boundedness
		12.2.4 Stability of equilibria
		12.2.5 Simulation
		12.2.6 Conclusion
	12.3 Global stability of an SEIR epidemic model with infectious force in latent period and infected period under discontinuous treatment strategy
		12.3.1 Introduction
		12.3.2 Model and preliminaries
		12.3.3 Positivity and boundedness
		12.3.4 Stability of equilibrium
		12.3.5 Global convergence in finite time
		12.3.6 Simulations
	References
Index