Lehrstuhl fur¨ Angewandte Mathematik / Numerische Analysis Fachgruppe Mathematik und Informatik Fakultat¨ fur¨ Mathematik und Naturwissenschaften Bergische Universitat¨ Wuppertal Masterarbeit Mathematical Modelling and Nonstandard Schemes for the Corona Virus Pandemic zur Erlangung des Grades Master of Science betreut von Prof Dr. rer.nat. Matthias Ehrhardt Prof. Dr. med. Helmut Brunner Vorgelegt von: Sarah Marie Treibert Studiengang: Master Wirtschaftsmathematik Abgabe: 06.04.2021 Contents 1 Introduction 1 1.1 Thematic Background . 1 1.2 Methods, Structure and Objectives . 3 1.2.1 Methods . 3 1.2.2 Structure . 3 1.2.3 Objectives . 5 2 The Severe Acute Respiratory Syndrome Corona Virus Type 2 6 2.1 Worldwide Incidence of the Novel Corona Virus . 6 2.2 Aetiology and Transmission Paths . 13 2.3 Disease Process and Symptomatology . 15 2.3.1 Progression of the Corona Virus Disease 2019 . 15 2.3.2 Symptoms of the Corona Virus Disease 2019 . 17 2.4 Policies of Containing the Corona Virus Disease 2019 . 18 2.4.1 Clinical Diagnosis . 18 2.4.2 Governmental Measures in Sweden and Germany . 20 2.4.3 The Stringency Index . 23 2.4.4 Vaccines and Vaccination Strategies . 24 2.4.5 Mutations . 27 2.5 Lethality in Corona Virus Disease 2019 Cases . 29 2.5.1 Case-fatality in Corona Virus Disease 2019 Cases . 30 2.5.2 Uncertainty in Case-fatality Computations . 36 2.5.3 Excess Mortality . 41 3 The SIR Model in Epidemic Modelling 45 3.1 The Basic SIR Model . 45 3.1.1 Definition of the Basic SIR Model . 46 3.1.2 Maximum number of infections in the Basic SIR Model . 48 3.1.3 Incidence Rates . 49 3.2 Simple Enhancements to the Basic SIR Model . 51 3.2.1 SIRD Model . 51 3.2.2 SEIR Model . 51 3.2.3 SEIS Model . 52 3.2.4 Stages Related to Disease Progression and Control Strategies . 52 4 The SARS-CoV-2-fitted SEIR Model 54 4.1 Differences to the Basic SIR Model and Model Assumptions . 54 4.2 Compartments and Transitions in the SARS-CoV-2-fitted SIR Model . 56 4.2.1 Sojourn Times . 57 i 4.2.2 Distributions of the Incubation Period and the Serial Interval . 57 4.2.3 The Susceptible Compartments . 61 4.2.4 The Exposed Compartments . 62 4.2.5 The Asymptomatic Infectious Compartments . 63 4.2.6 The Symptomatic Compartments . 63 4.2.7 The Hospitalized Compartment . 65 4.2.8 The Intensive Care Unit Compartment . 65 4.2.9 The Recovered Compartment . 66 4.2.10 The Compartment of Deceased Individuals . 67 4.2.11 The Compartment of Vaccinated Individuals . 68 4.2.12 Overview of Compartments . 70 4.3 Transmission in the SARS-CoV-2-fitted Model . 72 4.3.1 Transmission Risk . 72 4.3.2 Contact, Quarantine and Isolation Rates . 76 4.3.3 Transmission Rates . 79 4.3.4 Time Delay . 82 5 Model Specifications 84 5.1 Model Variants . 84 5.1.1 Exclusion of Quarantine or the Quarantine Compartment . 84 5.1.2 Pooling of Isolated and not Isolated Compartments . 85 5.1.3 Exclusion of Unconfirmed Infected Cases . 85 5.1.4 Pooling of Infected Compartments . 85 5.2 Systems of Ordinary Differential Equations . 86 5.2.1 Formulation of the Initial Value Problem . 87 5.2.2 The SV IHCDR Model and the SV A~I II HCDR Model . 87 5.2.3 The SARS-CoV-2-fitted Model . 91 5.2.4 Age Group Model . 92 5.3 Reproduction Numbers . 95 6 Parameter Estimation in MAT LAB 101 6.1 Problem Formulation and Implementation Process . 102 6.1.1 The Nonlinear Least Squares Approach to Compartment Models . 102 6.1.2 The Implementation Process . 104 6.2 Parameter Definitions and Bounds . 105 6.2.1 Parameters in the SIHCDR Model . 105 6.2.2 Parameters in the SV ID Age Group Model . 109 6.3 Compartment Size Predictions with the SIHCDR Model . 110 6.3.1 Prediction of the Second Wave . 112 6.3.2 Prediction of the Third Wave . 119 6.4 Compartment Size Predictions with the SV ID Age Group Model . 135 6.5 Application of Non-Standard Solvers . 140 6.5.1 Definition of Nonstandard Finite Difference Schemes . 140 6.5.2 Implementation of a Nonstandard Finite Difference Scheme for the SIHCDR Model . 143 7 Markov Chain Epidemic Models 148 7.1 Multi-state Models . 148 7.2 The Bayesian Inference Approach to Compartment Models . 151 7.3 The Metropolis-Hastings Algorithm . 153 ii 8 R´esum´e 155 8.1 Conclusion . 155 8.2 Future Work . 158 Appendix 161 A The Trust-Region Method 161 B MAT LAB Scripts 163 B.1 Implementation of the system of ODEs (for Germany and the prediction of the third wave) . 163 B.2 Implementation of the NSFD Scheme (for Germany and the prediction of the third wave) . 164 B.3 Implementation of the system of ODEs for the SV ID Age Group Model . 165 B.4 Algorithm for the Computation of the Discrete `2-Error . 166 B.5 Error Minimization (SIHCDR Model) . 167 B.6 Error Minimization (SV ID Age Group Model) . 170 B.7 Compartment Size Prediction (SIHCDR Model) . 171 B.8 Compartment Size Prediction (SV ID Age Group Model) . 177 B.9 Computation of the Basic Reproduction Number and Sensitivity Indices . 180 iii List of Figures Chapter 2 2.1 Daily total newly confirmed cases in Europe, North America, South America and Asia between the calendar week 5 in 2020 and 12 in 2021 on a linear scale (upper graph) and logarithmic scale (lower graph). 7 2.2 Daily newly confirmed cases per million people between the calendar week 10 in 2020 and 12 in 2021 on a linear scale (upper graph) and logarithmic scale (lower graph). 9 2.3 Weekly number of German newly confirmed SARS-CoV-2-infected cases by age group per 100; 000 people, for 7 age groups. ..