**prof. Jacky Cresson**(Université of Pau and Pays de l'Adour),

*How to construct stochastic models? Theory and numerics.*

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**General background : modelling of stochastic perturbations and validation of models**

The aim of this lecture is to present a set of mathematical tools and results permitting to construct a stochastic viable model from a known deterministic one and second to validate them via numerical simulations. Indeed, many dynamical systems in Biology or Physics are in a first approach modelled by a deterministic dynamical systems. We can cite for example the classical Hodgkin-Huxley model in Biology or the Landau-Lipshitz model in Physics. Then, a set of observations are made and lead to the introduction of a stochastic or random component. For the Hodgkin-Huxley model this comes from the fact that there exists an intrinsic stochastic bioelectrical activity of neurons which is observed experimentally. For the Landau-Lifshitz model this comes from the fact that the electromagnetic fields behaves very randomly. However, to take into account these stochastic effects is in general not an easy task.

The Lecture is made of two parts : the first one deals with the construction of the stochastic model. The second one is devoted to numerical methods designed in order to validate these models. All the mathematical tools and results will be illustrated by numerous examples coming from Biology, Physics, Astronomy and Celestial Mechanics.

**Part I – Admissible or viable stochastic models**

In a first part of this Lecture, we discuss such a modelling in the context of the theory of stochastic differential equations.

Returning to the initial deterministic model is always interesting. Indeed, it provides a set of constraints which are in general considered as fundamental by the scientists. This can be some fundamental law of Physics like conservation of energy, existence of some symmetries or invariance properties. At least these properties are in general respected by the deterministic model and a natural way to extend such a model in the stochastic framework is to construct a suitable stochastic perturbation respecting such a constraints in an appropriate sense.

In this Lecture, we will provide many results characterizing stochastic perturbations which preserve important properties : invariance of domain (in particular positivity), first integrals, variational structures (Lagrangian or Hamiltonian), symmetries, etc. These results will then be applied in various fields : Biology (behaviour of neurons, HIV population dynamics, Virus transmission models and models for the immune system, Cellular signaling networks, Population growth models, Tumor growth), Physics (Ferromagnetism), Astronomy and Celestial Mechanics (Two-body problem, Orbits of Satellites, Earth’s rotation).

**Part II – Numerical methods and validation of models**

In a second part of this Lecture, we discuss how to make numerical simulations in order to validate these stochastic models. The main difficulty which is not usually discussed in the literature, is to provide numerical methods respecting the constraints of the models. This problem is well known in Hamiltonian mechanics where the conservation of energy is an important feature of the models and has leaded to the theory of variational integrators. For invariance of domains, symmetries, etc, the state of the arts is not so clear even in the deterministic case.

In this Lecture, we will discuss the construction of variational integrators in the context of the theory of discrete embeddings ans secondly we will discuss the construction of topological numerical scheme which are reminiscent of a general program initiated by R. Mickens around non-standard numerical scheme. These methodes will be discussed in the deterministic and stochastic case with numerous examples.

**prof. József Z. Farkas**(University of Stirling),

*Structured models for mathematical epidemiology.*

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In recent years we have been working on the formulation an analysis of structured population models for infectious disease dynamics. In contrast to previous models, where for example the age of infection have been used as a structuring variable, we introduce structuring of the population with respect to infection (bacterium/virus) load and/or infectiousness.

In this talk we will focus on the models. First we will introduce the so called Wentzell (or Feller) boundary conditions in a structured population model with diffusion. The diffusion in our equation allows us to model random noise in a deterministic fashion. The power of Wentzell boundary condition is that it allows to incorporate a boundary state, which carries mass, namely the population of the uninfected individuals, in an elegant fashion. First we will consider a general linear model, then we will consider a nonlinear model which arises when modelling Wolbachia infection dynamics. We establish existence of solutions and consider the existence of positive steady states of the model. if time permits we will briefly mention a general framework we developed recently to treat models with 2-dimensional but non-monotone nonlinearities.

In the second part we will introduce and investigate an SIS-type model for the spread of an infectious disease, where the infected population is structured with respect to the different strain of the virus/bacteria they are carrying. Our aim is to capture the interesting scenario when individuals infected with different strains cause secondary (new) infections at different rates. Therefore, we consider a nonlinear infection process, which generalizes the bilinear process arising from the classic mass-action assumption. Our main motivation is to study competition between different strains of a virus/bacteria. From the mathematical point of view, we are interested whether the nonlinear infection process leads to a well-posed model. We use a semilinear formulation to show global existence and positivity of solutions up to a critical value of the exponent in the nonlinearity. Furthermore, we establish the existence of the endemic steady state for particular classes of nonlinearities.

**prof. Rafael Ortega**(Universidad de Granada, Spain),

*Periodic oscillations of a forced pendulum: from existence to stability.*

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Consider the differential equation

where is a positive parameter and is a -periodic function. This is a simple model frequently employed to illustrate the methods of Nonlinear Analysis. Results on the existence of periodic solutions are usually obtained by a combination of tools coming from Topology and Calculus of Variations. The goal of this talk is to show that these tools are also useful in the study of the stability of periodic solutions. Stability is understood in the Lyapunov sense. We will assume that the parameter satisfies and the function has zero average. The main result says that there exists a stable -periodic solution for almost every periodic function with zero average. The phrase „for almost every periodic function” is understood in the sense of prevalence. For this reason the notion of set of zero measure in a Banach space of infinite dimensions will play a role.The condition is sharp. The conclusion of the theorem is not valid „for all periodic functions”.