Introduction

This book, entitled "Mathematical Modeling of Random and Deterministic Phenomena", was written to provide details on current research in applied mathematics that can help to answer many of the modeling questions encountered in Mayotte. It is aimed at expert readers, young researchers, beginning graduate and advanced undergraduate students, who are interested in statistics, probability, mathematical analysis and modeling. The basic background for the understanding of the material presented is timely provided throughout the chapters.

This book was written after the international conference on mathematical modeling in Mayotte, where a call for chapters of the book was made. They were written in the form of journal articles, with new results extending the talks given during the conference and were reviewed by independent reviewers and book publishers.

This book discusses key aspects of recent developments in applied mathematical analysis and modeling. It also highlights a wide range of applications in the fields of biological and environmental sciences, epidemiology and social perspectives. Each chapter examines selected research problems and presents a balanced mix of theory and applications on some selected topics. Particular emphasis is placed on presenting the fundamental developments in mathematical analysis and modeling and highlighting the latest developments in different fields of probability and statistics. The chapters are presented independently and contain enough references to allow the reader to explore the various topics presented. The book is primarily intended for graduate students, researchers and educators; and is useful to readers interested in some recent developments on mathematical analysis, modeling and applications.

The book is organized into two main parts. The first part is devoted to the analysis of some advanced mathematical modeling problems with a particular focus on epidemiology, environmental ecology, biology and epistemology. The second part is devoted to a mathematical modelization with interdisciplinarity in ecological, socio-economic, epistemological, natural and social problems.

In Chapter 1, we present large population approximations for several deterministic and stochastic epidemic models. The hypothesis of constant population of susceptibles is explained through some realistic situations. After recalling the definition of SIS, SIRS and SIR models, a law of large numbers (LLN) is presented as well as a central limit theorem (CLT) to estimate the time of extinction of an epidemic and a principle of great deviation to estimate the error. This chapter then describes the principle of moderate deviations. These results are then used to deduce the critical population sizes for launching an epidemic. It explains how it can be used to predict the time taken for an epidemic to cease.

Chapter 2 is devoted to the study of non-parametric prediction of biomass of demersal fish in a coastal area, with a case study in Senegal. The inputs of the regression model are spatio-functional, i.e. the temperature and salinity of the water are depth curves recorded at different fishing locations. The prediction is done through a dual kernel estimator accounting the proximity between the temperature or salinity observations and locations. The originality of the approach lies in the functional nature of the exogeneous variables. Some theoretical asymptotic results on the predictor are provided.

Chapter 3 is concerned with the study of urban flood risk in urban areas caused by heavy rainfall, that may trigger considerable damage. The simulated water depths are very sensitive to the temporal and spatial distribution of rainfall. Besides, rainfall, owing in particular to its intermittency, is one of the most complex meteorological processes. Its simulation requires an accurate characterization of the spatio-temporal variability and intensity from available data. Classical stochastic approaches are not designed explicitly to deal with extreme events. To this end, spatial and spatio-temporal processes are proposed in the sound asymptotic framework provided by extreme value theory. Realistic simulation of extreme events raises a number of issues such as the ability to reproduce flexible dependence structure and the simulation of such processes.

In Chapter 4, we consider a problem of change-point detection for a continuous-time stochastic process in the family of piecewise deterministic Markov processes. The process is observed in discrete-time and through noise, and the aim is to propose a numerical method to accurately detect both the date of the change of dynamics and the new regime after the change. To do so, we state the problem as an optimal stopping problem for a partially observed discrete-time Markov decision process, taking values in a continuous state space, and provide a discretization of the state space based on quantization to approximate the value function and build a tractable stopping policy. We provide error bounds for the approximation of the value function and numerical simulations to assess the performance of our candidate policy. An application concerns treatment optimization for cancer patients. The change point then corresponds to a sudden deterioration of the health of the patient. It must be detected early, so that the treatment can be adapted.

The context of Chapter 5 is the nutrient transfer mechanism in croplands. The authors study the case of an additional nutrient which comes from a "service plant" (meaning a natural input), as a control function. The Nye-Tinker-Barber model is introduced with a perturbation as an unknown source of nutrient. An optimal control formulation of this problem is studied and adapted for the incomplete data case. A characterization of the low-regret optimal control is provided

In Chapter 6, basic stochastic evolution equations in long-time periodic environment are developed. Periodicity often appears in implicit ways in various phenomena. For instance, this is the case when we study the effects of fluctuating environments on population dynamics. Some classical books gave a nice presentation of various extensions of the concepts of periodicity, such as almost periodicity, asymptotically periodicity, almost automorphy, as well as pertinent results in this area. Recently, there has been an increasing interest in extending certain results to stochastic differential equations in separable Hilbert space. This is due to the fact that almost all problems in a real life situation, to which mathematical models are applicable, are basically stochastic rather than deterministic. In this chapter, we deal with a stochastic fractional integro-differential equation, for which a result of existence and uniqueness of an asymptotically periodic solution is given.

In Chapter 7, we study the existence of solutions in semilinear evolution equations with impulse, where the differential operator generates a strongly compact semi-group. The chapter generalizes a recent published work by one of the co-authors to the non-local initial condition case. In the previous work, the existence, stability and smoothness of bounded solutions for impulsive semilinear parabolic equations with Dirichlet boundary conditions, are obtained using the Banach fixed point theorem, under the classical Lipschitz assumptions.

In Chapter 8, we discuss the history and criticisms of a mathematical model, namely the diffusion of heat. The starting point is a "thought experiment" on the diffusion of heat through an infinite rectangular flat lamina. This is the path along which Fourier invented the representation of functions that bears his name; and we mainly treat the typical example of the periodic step function. Fourier thus invented the notion of proper modes, also known today as eigen modes, and found the orthogonality relations. Following Fourier, we then consider an example, the diffusion of heat in a sphere like the Earth, and come up with the required adaptation that, for the first time, allowed us to investigate the greenhouse effect. We then examine some of the criticisms related to Fourier's representation until functional analysis was created in the 20th Century, answering various questions. Still, an interesting creation came with a critique from quantum mechanics in the 1930s, perhaps not understood as such, but which led to wavelets as developed in the 21st Century, and a remarkable new tool that can be adapted to various situations. The text, in a story form, aims to combine mathematics, physics and also epistemology in a history that is rigorous with respect for original texts; it also tries to understand the meaning of a scientific posterity for the construction of science, as well as how a thought experiment has been transformed into a realistic modeling.

The second part is dedicated to the development of interdisciplinary modeling with mathematical approaches.

In Chapter 9, we present a methodology for interdisciplinary modeling of complex systems using hypergraphs. This project begins by setting out the research stakes related to the sustainable management of mangrove forests in Mayotte: Mangroves are coastal ecosystems that have undergone global upheavals while facing a number of issues regarding biodiversity, pressures for natural hazards and attractiveness for the socio-economic development of territories. The mangroves of Mayotte thus present high stakes of preservation and management. This sustainable management is conceived in a participatory framework where, "it seems necessary for the users of the mangrove and those involved in the management of these wetlands, to...