Mathematics of Public Health: Mathematical Modelling from the Next Generation (Fields Institute Communications, 88)

This volume addresses SDG 3 from a mathematical standpoint, sharing novel perspectives of existing communicable disease modelling technologies of the next generation and disseminating new developments in modelling methodologies and simulation techniques. These methodologies are important for training and research in communicable diseases and can be applied to other threats to human health. The contributions contained in this collection/book cover a range of modelling techniques that have been and may be used to support decision-making on critical health related issues such as: Resource allocation Impact of climate change on communicable diseases Interaction of human behaviour change, and disease spread Disease outbreak trajectories projection Public health interventions evaluation Preparedness and mitigation of emerging and re-emerging infectious diseases outbreaks Development of vaccines and decisions around vaccine allocation and optimization The diseases and public health issues in this volume include, but are not limited to COVID-19, HIV, Influenza, antimicrobial resistance (AMR), the opioid epidemic, Lyme Disease, Zika, and Malaria. In addition, this volume compares compartmental models, agent-based models, machine learning and network. Readers have an opportunity to learn from the next generation perspective of evolving methodologies and algorithms in modelling infectious diseases, the mathematics behind them, the motivation for them, and some applications to supporting critical decisions on prevention and control of communicable diseases. This volume was compiled from the weekly seminar series organized by the Mathematics for Public Health (MfPH) Next Generation Network. This network brings together the next generation of modellers from across Canada and the world, developing the latest mathematical models, modeling methodologies, and analytical and simulation tools for communicable diseases of global public health concerns. The weekly seminar series provides a unique forum for this network and their invited guest speakers to share their perspectives on the status and future directions of mathematics of public health. Preface Contents 1 Mathematical Models: Perspectives of Mathematical Modelers and Public Health Professionals 1.1 Natural History of Disease in Humans 1.2 Introduction to Mathematical Epidemiology 1.3 Model Formulation and Examples of Some Communicable Disease Models 1.3.1 Simple SIR Compartmental Models 1.3.2 Simple Endemic Models 1.3.3 Agent-Based Models 1.3.4 Network Models 1.3.5 Machine Learning Models 1.3.5.1 Estimating Parameters 1.3.5.2 Estimating Hidden States 1.4 Qualitative Analysis of Selected Models 1.4.1 Epidemic Model 1.4.2 Endemic Model 1.4.3 Network Model 1.5 Quantitative Analysis 1.6 Review of Mathematical Models of Selected Communicable Diseases and Their Impacts on Policy- and Decision-Making 1.6.1 SARS 2003 Pandemic Models 1.6.2 Pandemic Influenza Models 1.6.3 SARS-CoV-2 Pandemic Models 1.6.4 HIV Models 1.6.5 HCV Models 1.7 Model Algorithms for a Simple SIR Model 1.7.1 Python Code 1.7.2 Julia Code 1.7.3 R Code 1.7.4 MATLAB Code 1.8 Human Epidemiology Data, Model Fitting, and Parameter Estimation 1.9 Conclusion References 2 Discovering First Principle of Behavioural Change in Disease Transmission Dynamics by Deep Learning 2.1 Introduction 2.2 Expert-Based Behavioural Change Transmission Dynamics Models 2.2.1 Calculation of the Final Epidemic Size 2.2.2 Applications to the Ontario's First COVID-19 Pandemic Wave 2.3 Two-Step Recovering-Explaining Framework 2.3.1 Universal Differential Equations 2.3.2 Data-Driven Methods or Equation-Searching Methods 2.3.2.1 Symbolic Regression 2.3.2.2 Sparse Identification of Nonlinear Dynamics (SINDy) 2.3.3 Two-Step Recovering-Explaining Methods 2.4 Deep Learning-Based Behavioural Change Transmission Dynamics Models 2.4.1 The Behavioural Change Laws 2.5 Discussions and Conclusions References 3 Understanding Epidemic Multi-wave Patterns via Machine Learning Clustering and the Epidemic Renormalization Group 3.1 Introduction 3.2 Renormalization Group Epidemiology: From eRG to CeRG 3.2.1 The Single-Wave eRG Approach 3.2.2 The Multi-wave CeRG Approach 3.3 A Machine Learning Approach to the Wave Pattern 3.3.1 The Status of Variants 3.3.2 Method 3.3.2.1 Cluster Algorithm 3.3.2.2 Emerging Variants as Persistent Time-Ordered Cluster Chains 3.3.3 Application to COVID-19 Data 3.4 An Epidemiological Theory of Variants: The MeRG Framework 3.4.1 The Model 3.4.2 Flow Among Variants: Fixed Points and (Ir)relevant Operators 3.4.3 Connecting Variant Dynamics to the CeRG 3.4.4 Fitting the Real Data 3.5 Conclusion References 4 Contact Matrices in Compartmental Disease Transmission Models 4.1 Introduction 4.2 Motivating Example 4.3 Defining Contact Matrices 4.3.1 What Is a Contact? 4.3.2 Sources of Contact Data 4.3.3 Assumptions and Parametric Forms 4.3.4 Example 4.4 Properties of Contact Matrices 4.4.1 Balancing Contact Matrices 4.4.2 Intrinsic Connectivity 4.4.3 Example 4.5 Restratifying Contact Matrices 4.5.1 Intuition and Equations for Restratification 4.5.2 Example 4.6 Mobility in Contact Matrices 4.6.1 Mobility Data and Mobility Matrices 4.6.2 Contact Matrices from Mobility Matrices 4.6.3 Integrating Age Mixing and Mobility Data in Contact Matrices 4.6.4 Example References 5 An Optimal Control Approach for Public Health Interventions on an Epidemic-Viral Model in Deterministic and Stochastic Environments 5.1 Introduction 5.1.1 A Fast Time Scale Viral Model 5.1.2 SIQR Epidemic Model with a Coupled Viral Model 5.1.3 Qualitative Analysis of the Coupled Model 5.2 Optimal Control Analysis 5.2.1 Investigation of the Deterministic Optimal Control 5.2.2 Investigation of the Stochastic Optimal Control 5.3 Numerical Simulations 5.4 Conclusion References 6 Modeling Airborne Disease Dynamics: Progress and Questions 6.1 Introduction 6.2 Viral Matter in an Infectious Individual 6.3 Aerosol Size Distribution in Human Exhalations 6.4 Airborne Transmission of Aerosols 6.5 Transmission Through Fomites 6.6 Infection Probability of a Susceptible 6.7 Probability Distribution for Number of Secondary Infections Z 6.8 Conclusion References 7 Modeling Mutation-Driven Emergence of Drug-Resistance: A Case Study of SARS-CoV-2 7.1 Introduction 7.2 Methods 7.2.1 Model Structure 7.2.2 Model Equations 7.2.3 Reproduction Number 7.3 Results 7.3.1 Baseline Scenario 7.3.2 Waning Immunity and Reinfection 7.4 Discussion References 8 A Categorical Framework for Modeling with Stock and Flow Diagrams 8.1 Introduction 8.2 The Syntax of Stock-Flow Diagrams 8.3 The Semantics of Stock-Flow Diagrams 8.3.1 ODEs (Ordinary Differential Equations) 8.3.2 Causal Loop Diagrams 8.3.3 System Structure Diagrams 8.4 Composing Open Stock-Flow Diagrams 8.5 Stratifying Typed System Structure Diagrams 8.6 ModelCollab: A Graphical Real-Time Collaborative Compositional Modeling Tool 8.7 Conclusion References 9 Agent-Based Modeling and Its Trade-Offs: An Introduction and Examples 9.1 Introduction 9.2 Characteristics of Agent-Based Models 9.2.1 Parameters 9.2.2 State, Actions, and Rules 9.2.3 Environment 9.2.4 Outputs and Emergent Behavior 9.2.5 Stochastics 9.2.6 Interventions 9.3 Example: Chickenpox 9.3.1 Chickenpox and Shingles 9.3.2 Model Scope 9.3.3 Statecharts 9.3.4 Model Fit 9.3.5 Costs and QALYs 9.3.6 Suitability of ABM 9.3.7 Choice of AnyLogic as a Tool 9.4 Example: Pertussis 9.4.1 Pertussis 9.4.2 Model Scope 9.4.3 Model Structure 9.4.4 Model Fit 9.4.5 Scenarios 9.4.6 Suitability of ABM 9.5 Trade-Offs Between ABMs and Aggregate Models 9.6 Summary References 10 Mathematical Assessment of the Role of Interventions Against SARS-CoV-2 10.1 Introduction 10.2 Formulation of Vaccination Model for COVID-19 10.2.1 Data Fitting and Parameter Estimation 10.2.2 Basic Qualitative Properties 10.3 Existence and Asymptotic Stability of Equilibria 10.3.1 Disease-Free Equilibrium 10.3.1.1 Local Asymptotic Stability of DFE 10.3.1.2 Existence of Backward Bifurcation 10.3.1.3 Global Asymptotic Stability of DFE: Special Cases 10.3.2 Existence and Stability of Endemic Equilibria: Special Case 10.3.2.1 Existence 10.3.2.2 Local Asymptotic Stability 10.3.3 Vaccine-Induced Herd Immunity Threshold 10.3.4 Global Parameter Sensitivity Analysis 10.4 Numerical Simulations 10.4.1 Effect of Masking as a Singular Control and Mitigation Intervention 10.4.2 Assessing the Combined Impact of Vaccination and Masks on Herd Immunity Threshold 10.4.3 Assessing the Combined Impact of Vaccination and Masks on Daily New Cases 10.5 Discussion and Conclusions Appendix 1: Proof of Theorem 3 Computation of Left and Right Eigenvectors of Jβp* Computation of Backward Bifurcation Coefficients, a and b Appendix 2: Proof of Theorem 4 Appendix 3: Proof of Theorem 5 Proof of Positive Invariance and Attractivity of Ω** Next-Generation Matrices for the Second Special Case of the Model Proof of Theorem 5 Appendix 4: Proof of Theorem 7 Case 1: θ= 0 Case 2: θ≠0 References 11 Long-Term Dynamics of COVID-19 in a Multi-strain Model 11.1 Introduction 11.2 Methodology 11.2.1 Model Description 11.2.2 Parameter Estimation 11.2.3 Data Sources 11.3 COVID-19 Long-Term Scenarios Modelling 11.4 Results 11.5 Discussion 11.6 Conclusion 11.7 Supplementary Information References Correction to: Contact Matrices in Compartmental Disease Transmission Models