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.

Auteur

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.

Contenu

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. SpaceTime 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 AdvectionDiffusion 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 st...