Mathematical Modeling of Random and Deterministic Phenomena

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 xiAcknowledgments xiiiIntroduction xvSolym Mawaki MANOU-ABI, Sophie DABO-NIANG and Jean-Jacques SALONEPart 1. Advances in Mathematical Modeling 1Chapter 1. Deviations From the Law of Large Numbers and Extinction of an Endemic Disease 3Étienne PARDOUX1.1. Introduction 31.2. The three models 51.2.1. The SIS model 51.2.2. The SIRS model 61.2.3. The SIR model with demography 71.3. The stochastic model, LLN, CLT and LD 81.3.1. The stochastic model 81.3.2. Law of large numbers 91.3.3. Central limit theorem 101.3.4. Large deviations and extinction of an epidemic 101.4. Moderate deviations 121.4.1. CLT and extinction of an endemic disease 121.4.2. Moderate deviations 131.5. References 29Chapter 2. Nonparametric Prediction for Spatial Dependent Functional Data: Application to Demersal Coastal Fish off Senegal 31Mamadou N’DIAYE, Sophie DABO-NIANG, Papa NGOM, Ndiaga THIAM, Massal FALL and Patrice BREHMER2.1. Introduction 312.2. Regression model and predictor 342.3. Large sample properties 362.4. Application to demersal coastal fish off Senegal 392.4.1. Procedure of prediction 392.4.2. Demersal coastal fish off Senegal data set 402.4.3. Measuring prediction performance 412.5. Conclusion 482.6. References 49Chapter 3. Space–Time Simulations of Extreme Rainfall: Why and How? 53Gwladys TOULEMONDE, Julie CARREAU and Vincent GUINOT3.1. Why? 533.1.1. Rainfall-induced urban floods 533.1.2. Sample hydraulic simulation of a rainfall-induced urban flood 543.2. How? 583.2.1. Spatial stochastic rainfall generator 583.2.2. Modeling extreme events 593.2.3. Stochastic rainfall generator geared towards extreme events 633.3. Outlook 643.4. References 66Chapter 4. Change-point Detection for Piecewise Deterministic Markov Processes 73Alice CLEYNEN and Benoîte DE SAPORTA4.1. A quick introduction to stochastic control and change-point detection 734.2. Model and problem setting 764.2.1. Continuous-time PDMP model 774.2.2. Optimal stopping problem under partial observations 784.2.3. Fully observed optimal stopping problem 804.3. Numerical approximation of the value functions 824.3.1. Quantization 834.3.2. Discretizations 844.3.3. Construction of a stopping strategy 874.4. Simulation study 894.4.1. Linear model 894.4.2. Nonlinear model 914.5. Conclusion 924.6. References 93Chapter 5. Optimal Control of Advection–Diffusion Problems for Cropping Systems with an Unknown Nutrient Service Plant Source 97Loïc LOUISON and Abdennebi OMRANE5.1. Introduction 975.2. Statement of the problem 995.2.1. Existence of a solution to the NTB uptake system 1005.3. Optimal control for the NTB problem with an unknown source 1025.3.1. Existence of a solution to the adjoint problem of NTB uptake system with an unknown source 1035.4. Characterization of the low-regret control for the NTB system 1075.5. Concluding remarks 1105.6. References 111Chapter 6. Existence of an Asymptotically Periodic Solution for a Stochastic Fractional Integro-differential Equation 113Solym Mawaki MANOU-ABI, William DIMBOUR and Mamadou Moustapha MBAYE6.1. Introduction 1136.2. Preliminaries 1156.2.1. Asymptotically periodic process and periodic limit processes 1156.2.2. Sectorial operators 1176.3. A stochastic integro-differential equation of fractional order 1186.4. An illustrative example 1376.5. References 138Chapter 7. Bounded Solutions for Impulsive Semilinear Evolution Equations with Non-local Conditions 141Toka DIAGANA and Hugo LEIVA7.1. Introduction 1417.2. Preliminaries 1427.3. Main theorems 1447.4. The smoothness of the bounded solution 1517.5. Application to the Burgers equation 1567.6. References 159Chapter 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” 161Jean DHOMBRES8.1. Introduction 1618.2. A physical invention is translated into mathematics thanks to the heat flow 1638.3. The proper story of proper modes 1648.3.1. Mathematical position of the lamina problem 1658.3.2. Simple modes are naturally involved 1668.3.3. A remarkable switch to proper modes 1678.4. The numerical example of the periodic step function gives way to a physical interpretation 1698.4.1. A calculation that a priori imposes an extension to the function f at the base of the lamina 1698.4.2. A crazy calculation 1708.4.3. Fourier is happily confronted with the task of finding an explanation for the simplicity of the result about coefficients 1748.4.4. Criticisms of the modeling 1758.5. To invoke arbitrary functions leads to an interpretation of orthogonality relations 1778.5.1. Function is a leitmotiv in Fourier’s intellectual career 1808.6. The modeling of the temperature of the Earth and the greenhouse effect 1818.7. 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 1848.7.1. Another dictionary: the Fourier transform for tempered distributions 1848.7.2. Heisenberg inequalities may just be deduced from the existence of a scalar product 1858.7.3. Orthogonality and a quick look to wavelets 1878.8. Conclusion 1878.9. References 189Part 2. Some Topics on Mayotte and Its Region 191Chapter 9. Towards a Methodology for Interdisciplinary Modeling of Complex Systems Using Hypergraphs 193Jean-Jacques SALONE9.1. Introduction 1939.1.1. The ARESMA project 1939.1.2. Towards a methodology of interdisciplinary modeling 1949.2. Systemic and lexicometric analyses of questionnaires 1959.2.1. Complex systems 1959.2.2. Methodology 1989.2.3. Results 1999.2.4. Conclusion of the section 2059.3. Hypergraphic analyses of diagrams 2059.3.1. Hypergraphs and modeling of a complex system 2059.3.2. Methodology 2089.3.3. Results 2089.3.4. Conclusion of the section 2129.4. Discussion and perspectives 2129.5. Appendix 2149.5.1. Other properties of a connected hypergraph 2149.5.2. Metric over an FHT 2149.6. References 217Chapter 10. Modeling of Post-forestry Transitions in Madagascar and the Indian Ocean: Setting Up a Dialogue Between Mathematics, Computer Science and Environmental Sciences 221Dominique HERVÉ10.1. Introduction 22110.2. Interdisciplinary exploration of agrarian transitions 22310.2.1. Exploration of post-forestry transitions in rainforests of Madagascar 22310.2.2. Applications to dry forests in southwestern Madagascar 22810.2.3. Viability 22910.3. Community management of resources, looking for consensus 23210.3.1. Degradation, violation, sanction 23210.3.2. Local farmers’ maps and conceptual graphs 23410.4. Discussion and conclusion 23710.5. References 240Chapter 11. Structural and Predictive Analysis of the Birth Curve in Mayotte from 2011 to 2017 245Julien BALICCHI and Anne BARBAIL11.1. Introduction 24511.1.1. Motivation 24511.1.2. Context 24611.1.3. About the literature on the birth curve in Mayotte 24711.1.4. Objective of ARS OI 24811.2. Origin of the data 24811.3. Methodologies and results 24811.3.1. Methodological approach 24811.3.2. Annual trend 24911.3.3. Monthly trend 24911.3.4. Characterization of the explosion risk of the number of births 25011.3.5. Autocorrelation 25211.3.6. Modeling by an ARIMA process (p, d, q) 25311.3.7. Predictions for the year 2018 25611.4. Discussion 25711.5. Conclusion 25911.6. References 259Chapter 12. Reflections Upon the Mathematization of Mayotte’s Economy 261Victor BIANCHINI and Antoine HOCHET12.1. Introduction 26112.2. Justifying the mathematization of economics 26312.2.1. The ontological and linguistic arguments 26412.2.2. Towards a naturalization of modeling in economics 26512.2.3. A number of caveats 26712.3. For a reasonable mathematization of economics: the case of Mayotte 26812.3.1. The trend towards the mathematization of the economics of Mayotte 26912.3.2. From Mayotte’s formal economy to its informal one 26912.3.3. When the formal economy interacts with the informal one: some issues for the modelization of complex systems 27012.4. Concluding remark 27312.5. References 273List of Authors 279Index 281