**Program**

The CIMPA school « Probabilistic and statistical modeling in epidemiology and environment » will take place at the University of Fianarantsoa from May 10 to 19, 2021.

# Introductory courses

##### Course 1: On the study of markov processes and extensions (Solym MANOU-ABI)

The purpose of this course is to introduce basics notions of stochastic modeling. The main mathematical objects are continuous time stochastic processes. Firstly, we will study the discrete-time Markov chains, the Poisson process and some examples in population dynamics. We will also present the continuous-time Markov chains, namely the poisson process, and more generally Markov jump processes. Finally we will consider diffusion processes. The course will be accompanied by a practical session work in order to illustrate the concepts covered. This course will allow the participants to better understand the advanced courses: Course 4 and Course 5.

##### Course 2 : Introduction to statistical analysis of spatial data (Amir ABOUBACAR)

In several areas, when we want to study a random phenomenon, we notice that it occurs somewhere. With the development of technological tools, it is easy to record observations in a database, the coordinates of the places where these random phenomena occur. Thus, most of the databases studied have a spatial component. The purpose of the course is to ask ourselves what can we do with this spatial data. The geographic location of a random event is an important explanatory variable in many fields such as econometrics, epidemiology, environment, agronomy, road and air safety, meteorology, image processing. It is therefore essential to include this information as much as possible in the statistical models. This course is an introduction to spatial data analysis. We will learn what types of statistical problems arise with spatial data and the basic techniques for dealing with them. We will see how to use lots of points on a map to bring out useful information and make predictions on the evolution of phenomena. After a general review of spatial statistics as a whole, as well as the usual methods of description and representation of spatial data, we will focus on a point analysis. More specifically, we will learn to recognize and test different types of spatial models.

⁃ Stationary process and notion of covariance

⁃ Spatial dependence and notion of variogram

⁃ Spatial prediction by kriging

⁃ Analysis and estimation of a variogram (if possible)

⁃ Validation of a variogram model (if possible)

##### Course 3 : Introduction to R Software and simulation methods (Angelo RAHERINIRINA)

In this course, we will introduce the R software:

creation of vectors, matrix, lists, data bases, writing a function, programming with R; and a presentation of some classical simulation methods. This course will allow participants to acquire a solid foundation of preliminary notions of simulation in order to perform practical sessions of courses 6 and 7.

# Advanced Courses

##### Course 4 : Stochastic and deterministic models of epidemics (Etienne PARDOUX)

##### Course 5 : Piecewise deterministic Markov processes: modeling and simulation (Benoîte De Saporta)

When a phenomenon is too complex to modeled by a deterministic dynamic system of reasonable dimension, one can use systems which include several regimes. Each complex regime is controlled and the system is allowed to switch from one regime to another at random and at random times. These hybrid stochastic systems form the class of piecewise deterministic Markov processes (PDMP). They have many advantages, such as a flexible and easily interpretable parametric form, and are the subject of many recent developments related to their simulation and optimization.

##### Course 6 : Statistics and image processing (Mame Diarra FALL)

Nowadays, image processes are omnipresent in our daily life, whether static (pictures) or dynamic (videos). More and more fields (environment, agriculture, medicine, risk management and prevention etc.) are using image process as an information tool. However, the relevant information is not always directly accessible in the raw images produced by the sensors and specific treatments must be applied. Statistical methods can be used for image processing, both for improving the visual quality (denoising, deflashing etc.) and for extracting information. In this course, we are interested on the methodology and application of statistical models to image processing to young researchers and PhD students. The course will be accompanied by practical sessions works with R in order to illustrate the concepts covered.

##### Course 7 : Statistics of extremes and applications to the environment (Gwladys TOULEMONDE and Jean-Noël BACRO)

The course offers an introduction to the theory of univariate and multivariate extreme values.

When we try to assess a dike height with only a very low probability of being exceeded or

when we are interested in the maximum rainfall height expected on average once every hundred years,

only the Distribution tails of the variables or vectors considered are informative.

To make the desired extrapolation and estimate extreme quantiles, conventional statistical approaches

based on the characterization of behavior on average are not suitable. The theory of extreme values offers

the mathematical concepts and the statistical tools allowing to approach these problems in a relevant way.

The laws of the maxima of random variables or vectors or the crossing of high thresholds and their

properties are presented. These concepts introduced will be implemented using the R software:

program implementations and case studies (risk study, etc.). Finally we will place ourselves in a spatial framework

and present tools and models adapted to extremes.

**Reference :**

Beirlant, J., Goegebeur, Y., Segers, J. et Teugels, J. (2004) : Statistics of extremes: Theory and Applications Coles, S. G. (2001) : Introduction to statistical modelling of extreme value. Springer. de Haan, L., Ferreira, A. (2006) : Extreme value theory : an introduction. Springer

##### Course 8 : Modeling of insect pest dynamics (Bedreddine AINSEBA)

We aim to introduce mathematical models for the study and understanding of the population dynamics of insect pests in their ecosystem. The mathematical tools of interest are systems of partial differential equations which describe the numerical variations over time of the population as a function of the stages of development, the sex of individuals and environmental conditions. Food resources, temperature, humidity and predation are the main environmental factors explaining the fluctuations in the number of individuals over time. The differences in development that exist in the cohorts are also modeled to refine the model’s predictions. In this course we will cover the following sessions:

- Session 1 (1h): Using degree days to predict the dates of emergence of insects + 30 minutes of practical work.
- Session 2 (1h): Use of finite differences to solve the problem of monitoring insect cohorts + 30 minutes of practical work
- Session 3 (1h): estimation of demographic parameters from experimental data + 30 minutes of practical work