Note: Intro -- Preface -- Contents -- Modelling, Simulations, and Social Impact of Evolutionary Virus Pandemics -- 1 Aims and Plan of the Chapter -- 2 On the Contents of the Edited Book -- 3 Reasonings on Research Perspectives -- References -- Understanding COVID-19 Epidemics: A Multi-Scale ModelingApproach -- 1 Introduction -- 2 Mathematical Modeling Applied to Infectious Diseases: COVID-19 as a Case Study -- 2.1 The SIR and SHAR Models -- 2.2 The SHARUCD Modeling Framework -- 2.3 Modeling the Implementation of Control Measures -- 2.4 The Refined SHARUCD Model -- 2.4.1 Further Refinements: Detection Rate and Import -- 3 KTAP Modeling Framework -- 3.1 Modeling Contagion, Progression, and Recovery -- 3.2 Application of the KTAP Model to Selected Case Studies -- 3.2.1 Effect of Lockdown Measures and Restrictions Lifting -- 3.2.2 Effect of Heterogeneity -- 4 Discussion -- References -- Kinetic Modelling of Epidemic Dynamics: Social Contacts, Control with Uncertain Data, and Multiscale Spatial Dynamics -- 1 Introduction -- 2 Kinetic Modelling of Social Heterogeneity in Epidemic Dynamics -- 2.1 Modelling Contact Heterogeneity -- 2.1.1 Kinetic Model for Contact Formation -- 2.1.2 Quasi-Invariant Scaling and Steady States -- 2.1.3 The Macroscopic Social-SIR Dynamics -- 2.1.4 A Social-SIR Model with Saturated Incidence Rate -- 2.1.5 Extrapolation of the Shape of the Incidence Rate from Data -- 2.2 The Interplay Between Economy and the Pandemic -- 2.2.1 Wealth Exchanges in Epidemic Modelling -- 2.2.2 Fokker-Planck Scaling and Steady States -- 2.2.3 The Formation of Bimodal Wealth Distributions -- 2.2.4 The Increase of Wealth Inequalities -- 3 Social Control and Data Uncertainty -- 3.1 Control of Socially Structured Models -- 3.1.1 Optimal Control Formulation -- 3.1.2 Feedback Controlled Compartmental Models. , 3.1.3 Containment in Homogeneous Social Mixing Dynamics -- 3.2 Dealing with Data Uncertainty -- 3.2.1 Feedback Controlled and Socially Structured Models with Uncertain Inputs -- 3.2.2 Application to the COVID-19 Outbreak -- 4 Multiscale Transport Models -- 4.1 Spatial Dynamics on Networks -- 4.1.1 1D Hyperbolic Compartmental Model -- 4.1.2 Macroscopic Formulation and Diffusion Limit -- 4.1.3 Extension to Multi-Compartmental Modelling -- 4.1.4 Network Modelling -- 4.1.5 Effect of Spatially Heterogeneous Environments in Hyperbolic and Parabolic Configuration -- 4.1.6 Application to the Emergence of COVID-19 in Italy -- 4.2 Realistic Geographical Settings -- 4.2.1 2D Kinetic Transport Model -- 4.2.2 Macroscopic Formulation and Diffusion Limit -- 4.2.3 Extension to Multi-Compartmental Modelling -- 4.2.4 Application to the Spatial Spread of COVID-19 in Italy in Emilia-Romagna and Lombardy Region -- 5 Concluding Remarks and Research Perspectives -- 5.1 Data sources -- References -- The COVID-19 Pandemic Evolution in Hawai`i and New Jersey: A Lesson on Infection Transmissibility and the Role of HumanBehavior -- 1 Introduction -- 2 Mathematical Models -- 2.1 Agent-Based Models -- 2.1.1 COVID-19 Agent-Based Simulator (Covasim) -- 2.2 Compartmental SEIR Models and Variants -- 2.3 Comparison of Agent-Based and Compartmental Models -- 3 Archipelagos and Islands -- 3.1 March 2020-June 2021 -- 3.1.1 CM Model Fit from March 06, 2020 to January 15, 2021 -- 3.1.2 Comparing CM and ABM Models -- 3.2 July 2021-September 2021 -- 3.3 Discussion -- 4 The Pandemic Waves in New Jersey -- 4.1 Comparing New Jersey to the US -- 4.2 Spatial and Temporal Patterns in COVID-19 Cases in New Jersey -- 4.3 Sociodemographic Variables -- 4.4 Discussion -- 5 The Use of Compartmental Models in New Jersey -- 5.1 Time-Evolution of the Basic Reproduction Number. , 5.2 Infected Confirmed Cases, Hospitalizations, and Deaths -- 5.3 Discussion -- 6 Conclusion -- References -- A Novel Point Process Model for COVID-19: Multivariate Recursive Hawkes Process -- 1 Introduction -- 1.1 Hawkes Point Process Modeling of Infectious Diseases -- 1.2 Multivariate Hawkes Processes -- 1.3 Recursive Hawkes Processes -- 1.4 Outline -- 2 Theoretical Properties of Temporal Multivariate Recursive Hawkes Models -- 2.1 Existence -- 2.2 Mean -- 2.3 Variance -- 3 Parameter Fitting and Simulation Algorithms -- 3.1 Parameter Fitting Algorithms -- 3.1.1 Parametric (or Semi-parametric) Estimation -- 3.1.2 Temporal Version of Parameter Fitting Algorithms -- 3.2 Simulation Algorithm -- 4 Reconstruct Multivariate Point Process from Data with Imprecise Time -- 4.1 Time Reconstruction -- 4.2 Category Index Reconstruction -- 5 Numerical Experiments and Results -- 5.1 Synthetic Data Sets -- 5.1.1 Comparison Between Parametric Fitting and Non-parametric Fitting -- 5.1.2 Verification of the Parameter Fitting Algorithm -- 5.1.3 Experiments About Data Sets with Imprecise Time -- 5.2 Experiments on Real COVID-19 Data -- 5.2.1 Model Validation -- 5.2.2 Prediction Based on MRHP and Historical Information -- 6 Conclusion -- References -- Multiscale Aspects of Virus Dynamics -- 1 Introduction -- 1.1 On the Biology of the Virus -- 1.2 Modeling the Complexity of COVID-19 -- 2 Epistemic and Empirical Uncertainties in Compartmental and Individual-Based Models -- 2.1 SIR Model -- 2.2 Individual-Based Interpretation of λ -- 2.3 An Example of Modified SIR Model -- 2.4 Individuals Behind the Modified SIR Model -- 2.5 Time-Discretization -- 3 The Individual-Based Model of FlaLaFauciRiva -- 3.1 A Formula for the Parameter λ of Compartmental Models -- 3.2 Analysis of the Fluctuations -- 3.3 Simulations -- 3.4 Presence of Immunized Population and Virus Variants. , Appendix -- References -- Productivity in Times of Covid-19: An Agent-Based Model Approach -- 1 Introduction -- 2 Model -- 3 Mean Field Approximation -- 4 Setting the Model Functions -- 5 Simulations -- 6 Conclusion -- References -- Transmission Dynamics and Quarantine Control of COVID-19 in Cluster Community -- 1 Introduction -- 2 Mathematical Modeling -- 2.1 Stage 1: SEIR-Type Model Without Quarantine -- 2.2 Stage 2: Transmission-Quarantine (TQ) Model -- 3 Analytic Results and Case Study for Emerging Stage -- 3.1 Analytic Results -- 3.2 A Real World Case Study for Stage 1 -- 4 Case Study and Sensitivity Analysis for Quarantine Stage -- 4.1 A Real World Study for Stage 2 -- 4.2 Sensitivity Analysis -- 5 Discussion -- Appendix: Proofs of Theorems -- References -- A 2D Kinetic Model for Crowd Dynamics with Disease Contagion -- 1 Introduction -- 2 A Simplified Two-Dimensional Kinetic Model -- 3 Discretization in Space and Time -- 4 Numerical Results -- 4.1 Tests with v = 0 -- 4.2 Tests with Prescribed Walking Velocity -- 5 A More Complex 2D Kinetic Model -- 6 Conclusions -- References -- Multiscale Derivation of a Time-Dependent SEIRD Reaction-Diffusion System for COVID-19 -- 1 Introduction -- 2 Phenomenological Modeling of Diffusion Population Dynamics -- 3 From Kinetic Theory Model to SEIRD Reaction-Diffusion System -- 3.1 Kinetic Theory Model -- 3.2 Micro-Macro Formulation -- 4 Numerical Method -- 4.1 Semi-Implicit Time Discretization -- 4.2 Fully Discrete Asymptotic Preserving Numerical Scheme in 1D -- 4.3 Boundary Conditions -- 5 Numerical Results -- 5.1 Test 1: Asymptotic Preserving Numerical Scheme Property -- 5.2 Test 2: Diffusion Effect -- 5.3 Test 3: Role of the Transmission Function -- 6 Conclusion and Perspectives -- References.