Mathematical Modeling of Random and Deterministic Phenomena

This book highlights mathematical research interests that appear in real life, such as the study and modeling of random and deterministic phenomena. As such, it provides current research in mathematics, with applications in biological and environmental sciences, ecology, epidemiology and social perspectives.

The chapters can be read independently of each other, with dedicated references specific to each chapter. The book is organized in two main parts. The first is devoted to some advanced mathematical problems regarding epidemic models; predictions of biomass; space-time modeling of extreme rainfall; modeling with the piecewise deterministic Markov process; optimal control problems; evolution equations in a periodic environment; and the analysis of the heat equation. The second is devoted to a modelization with interdisciplinarity in ecological, socio-economic, epistemological, demographic and social problems.

Mathematical Modeling of Random and Deterministic Phenomena is aimed at expert readers, young researchers, plus graduate and advanced undergraduate students who are interested in probability, statistics, modeling and mathematical analysis.

Solym Mawaki Manou-Abi is an Associate Professor at Centre Universitaire de Mayotte, France. He is a doctor of applied mathematics, and his research interests are in mathematics and applications-specifically probability, analysis and statistics.

Sophie Dabo-Niang is a Full Professor at the University of Lille, France. She is a doctor of statistics and her research program is focused on the study of non(semi)-parametric inference of functional and spatial data. She is interested in medical, environmental and hydrological studies from an applied perspective.

Jean-Jacques Salone is an Associate Professor at Centre Universitaire de Mayotte. He is a doctor of applied mathematics and education sciences, and his research interests are in didactics of mathematics and in modeling of social, natural or educational complex systems.

Preface xi

Acknowledgments xiii

Introduction xv
Solym Mawaki MANOU-ABI, Sophie DABO-NIANG and Jean-Jacques SALONE

Part 1. Advances in Mathematical Modeling 1

Chapter 1. Deviations From the Law of Large Numbers and Extinction of an Endemic Disease 3
Étienne PARDOUX

1.1. Introduction 3

1.2. The three models 5

1.2.1. The SIS model 5

1.2.2. The SIRS model 6

1.2.3. The SIR model with demography 7

1.3. The stochastic model, LLN, CLT and LD 8

1.3.1. The stochastic model 8

1.3.2. Law of large numbers 9

1.3.3. Central limit theorem 10

1.3.4. Large deviations and extinction of an epidemic 10

1.4. Moderate deviations 12

1.4.1. CLT and extinction of an endemic disease 12

1.4.2. Moderate deviations 13

1.5. References 29

Chapter 2. Nonparametric Prediction for Spatial Dependent Functional Data: Application to Demersal Coastal Fish off Senegal 31
Mamadou N’DIAYE, Sophie DABO-NIANG, Papa NGOM, Ndiaga THIAM, Massal FALL and Patrice BREHMER

2.1. Introduction 31

2.2. Regression model and predictor 34

2.3. Large sample properties 36

2.4. Application to demersal coastal fish off Senegal 39

2.4.1. Procedure of prediction 39

2.4.2. Demersal coastal fish off Senegal data set 40

2.4.3. Measuring prediction performance 41

2.5. Conclusion 48

2.6. References 49

Chapter 3. Space–Time Simulations of Extreme Rainfall: Why and How? 53
Gwladys TOULEMONDE, Julie CARREAU and Vincent GUINOT

3.1. Why? 53

3.1.1. Rainfall-induced urban floods 53

3.1.2. Sample hydraulic simulation of a rainfall-induced urban flood 54

3.2. How? 58

3.2.1. Spatial stochastic rainfall generator 58

3.2.2. Modeling extreme events 59

3.2.3. Stochastic rainfall generator geared towards extreme events 63

3.3. Outlook 64

3.4. References 66

Chapter 4. Change-point Detection for Piecewise Deterministic Markov Processes 73
Alice CLEYNEN and Benoîte DE SAPORTA

4.1. A quick introduction to stochastic control and change-point detection 73

4.2. Model and problem setting 76

4.2.1. Continuous-time PDMP model 77

4.2.2. Optimal stopping problem under partial observations 78

4.2.3. Fully observed optimal stopping problem 80

4.3. Numerical approximation of the value functions 82

4.3.1. Quantization 83

4.3.2. Discretizations 84

4.3.3. Construction of a stopping strategy 87

4.4. Simulation study 89

4.4.1. Linear model 89

4.4.2. Nonlinear model 91

4.5. Conclusion 92

4.6. References 93

Chapter 5. Optimal Control of Advection–Diffusion Problems for Cropping Systems with an Unknown Nutrient Service Plant Source 97
Loïc LOUISON and Abdennebi OMRANE

5.1. Introduction 97

5.2. Statement of the problem 99

5.2.1. Existence of a solution to the NTB uptake system 100

5.3. Optimal control for the NTB problem with an unknown source 102

5.3.1. Existence of a solution to the adjoint problem of NTB uptake system with an unknown source 103

5.4. Characterization of the low-regret control for the NTB system 107

5.5. Concluding remarks 110

5.6. References 111

Chapter 6. Existence of an Asymptotically Periodic Solution for a Stochastic Fractional Integro-differential Equation 113
Solym Mawaki MANOU-ABI, William DIMBOUR and Mamadou Moustapha MBAYE

6.1. Introduction 113

6.2. Preliminaries 115

6.2.1. Asymptotically periodic process and periodic limit processes 115

6.2.2. Sectorial operators 117

6.3. A stochastic integro-differential equation of fractional order 118

6.4. An illustrative example 137

6.5. References 138

Chapter 7. Bounded Solutions for Impulsive Semilinear Evolution Equations with Non-local Conditions 141
Toka DIAGANA and Hugo LEIVA

7.1. Introduction 141

7.2. Preliminaries 142

7.3. Main theorems 144

7.4. The smoothness of the bounded solution 151

7.5. Application to the Burgers equation 156

7.6. References 159

Chapter 8. The History of a Mathematical Model and Some of Its Criticisms up to Today: The Diffusion of Heat That Started with a Fourier “Thought Experiment” 161
Jean DHOMBRES

8.1. Introduction 161

8.2. A physical invention is translated into mathematics thanks to the heat flow 163

8.3. The proper story of proper modes 164

8.3.1. Mathematical position of the lamina problem 165

8.3.2. Simple modes are naturally involved 166

8.3.3. A remarkable switch to proper modes 167

8.4. The numerical example of the periodic step function gives way to a physical interpretation 169

8.4.1. A calculation that a priori imposes an extension to the function f at the base of the lamina 169

8.4.2. A crazy calculation 170

8.4.3. Fourier is happily confronted with the task of finding an explanation for the simplicity of the result about coefficients 174

8.4.4. Criticisms of the modeling 175

8.5. To invoke arbitrary functions leads to an interpretation of orthogonality relations 177

8.5.1. Function is a leitmotiv in Fourier’s intellectual career 180

8.6. The modeling of the temperature of the Earth and the greenhouse effect 181

8.7. Axiomatic shaping by Hilbert spaces provides a good account for another dictionary part in Fourier’s theory, and also to its limits, so that his representation finally had to be modified to achieve efficient numerical purposes 184

8.7.1. Another dictionary: the Fourier transform for tempered distributions 184

8.7.2. Heisenberg inequalities may just be deduced from the existence of a scalar product 185

8.7.3. Orthogonality and a quick look to wavelets 187

8.8. Conclusion 187

8.9. References 189

Part 2. Some Topics on Mayotte and Its Region 191

Chapter 9. Towards a Methodology for Interdisciplinary Modeling of Complex Systems Using Hypergraphs 193
Jean-Jacques SALONE

9.1. Introduction 193

9.1.1. The ARESMA project 193

9.1.2. Towards a methodology of interdisciplinary modeling 194

9.2. Systemic and lexicometric analyses of questionnaires 195

9.2.1. Complex systems 195

9.2.2. Methodology 198

9.2.3. Results 199

9.2.4. Conclusion of the section 205

9.3. Hypergraphic analyses of diagrams 205

9.3.1. Hypergraphs and modeling of a complex system 205

9.3.2. Methodology 208

9.3.3. Results 208

9.3.4. Conclusion of the section 212

9.4. Discussion and perspectives 212

9.5. Appendix 214

9.5.1. Other properties of a connected hypergraph 214

9.5.2. Metric over an FHT 214

9.6. References 217

Chapter 10. Modeling of Post-forestry Transitions in Madagascar and the Indian Ocean: Setting Up a Dialogue Between Mathematics, Computer Science and Environmental Sciences 221
Dominique HERVÉ

10.1. Introduction 221

10.2. Interdisciplinary exploration of agrarian transitions 223

10.2.1. Exploration of post-forestry transitions in rainforests of Madagascar 223

10.2.2. Applications to dry forests in southwestern Madagascar 228

10.2.3. Viability 229

10.3. Community management of resources, looking for consensus 232

10.3.1. Degradation, violation, sanction 232

10.3.2. Local farmers’ maps and conceptual graphs 234

10.4. Discussion and conclusion 237

10.5. References 240

Chapter 11. Structural and Predictive Analysis of the Birth Curve in Mayotte from 2011 to 2017 245
Julien BALICCHI and Anne BARBAIL

11.1. Introduction 245

11.1.1. Motivation 245

11.1.2. Context 246

11.1.3. About the literature on the birth curve in Mayotte 247

11.1.4. Objective of ARS OI 248

11.2. Origin of the data 248

11.3. Methodologies and results 248

11.3.1. Methodological approach 248

11.3.2. Annual trend 249

11.3.3. Monthly trend 249

11.3.4. Characterization of the explosion risk of the number of births 250

11.3.5. Autocorrelation 252

11.3.6. Modeling by an ARIMA process (p, d, q) 253

11.3.7. Predictions for the year 2018 256

11.4. Discussion 257

11.5. Conclusion 259

11.6. References 259

Chapter 12. Reflections Upon the Mathematization of Mayotte’s Economy 261
Victor BIANCHINI and Antoine HOCHET

12.1. Introduction 261

12.2. Justifying the mathematization of economics 263

12.2.1. The ontological and linguistic arguments 264

12.2.2. Towards a naturalization of modeling in economics 265

12.2.3. A number of caveats 267

12.3. For a reasonable mathematization of economics: the case of Mayotte 268

12.3.1. The trend towards the mathematization of the economics of Mayotte 269

12.3.2. From Mayotte’s formal economy to its informal one 269

12.3.3. When the formal economy interacts with the informal one: some issues for the modelization of complex systems 270

12.4. Concluding remark 273

12.5. References 273

List of Authors 279

Index 281

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