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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.

Mathematical Modeling of Random and Deterministic Phenomena

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

Etienne 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 Benoite 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

Loic 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 HERVE





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