Section: Application Domains

Application fields and collaborations with biologists

We present here collaborations on specific biological modeling problems.

Epidemiology

Participants : Bedr'Eddine Aïnseba, Arnaud Ducrot, Michel Langlais, Pierre Magal.

Nosocomial diseases

Nosocomial infections caused by antibiotic-resistant pathogens are a global public health problem, in both developed and developing countries, including China, France, and USA. For example in USA, every year approximately 2 million people acquire a clinically significant hospital infection, which cause about 20,000 deaths and cost hospitals $20 billions per year. A recent estimate showed that there were 18,650 deaths in patients with invasive methicillin-resistant Staphylococcus aureus (MRSA) in the US in 2005, exceeding the total number of deaths due to HIV/AIDS in the same year.

In the last 15 years, various mathematical models have been proposed to describe the transmission dynamics of antibiotic-resistant bacteria in hospitals or communities, we refer to the survey on this topic [26] . In [30] , we formulated a two-level population model to quantify key elements in nosocomial infections. The objectives were to study the effect of antibiotic treatments of the dynamic elements of nonresistant and resistant bacteria strains in hospital environments and to provide understanding of measures to avoid the endemicity of resistant antibiotic strains in hospitals in USA (and also in the rest of the world). The theoretical analysis was reported in [24] . Most recently, in [23] we used an individual based model (so called Monte-Carlo simulations in mathematics) which is formulated by individuals and includes the interaction with the healthcare workers. The main point in this paper is that we derive a model in which all the parameters are expressed in terms of observable quantities. An important conclusion for this paper is that the average time of visit of patient by healthcare workers plays a key role in such a problem, while this parameter has not been considered before.

In China, one special groups of health care workers (HCWs), volunteers, are widely used in many tertiary care hospitals. The relationship between this kind of healthcare mode and the transmission of antibiotic-resistant bacteria has not been investigated in detail. Proper infection control measures are needed to attenuate the nosocomial infection involving volunteers. Our goal in this part is to construct some mathematical models which are suitable tools to study the transmission dynamics of antibiotic-resistant bacteria in hospitals and to design effective infection control programs in China.

Modeling in viticulture : Spreading of a fungal disease over a vineyard (collaboration with INRA)

Participants : Jean-Baptiste Burie, Michel Langlais.

This is a joint research with different groups of UMR “Santé végétale” of INRA, Villenave d'Ornon.

This part is mostly an application of section 3.2.1 . We aim at investigating the spreading of powdery mildew upon vine within a growing season to help having a better management of the disease. Indeed fungicide treatments have a financial and environmental cost. This is a collaborative work with A. Calonnec and P. Cartolaro from INRA in Villenave d'Ornon (UMR INRA-ENITA en santé végétale). The ultimate goal is to provide a diagnosis tool to help the vine producer treating the disease.

Until now a mechanistic model has been built that takes into account the interaction between host growth, pathogen development and climatic conditions. This mechanistic model is being extended at the vineyard scale using the knowledge in high performance computations of some INRIA ScAlApplix members: G. Tessier and J. Roman.

But still disease features have to be investigated at a higher level. This will be done thanks to epidemiological models based on ODE or PDE systems that will focus on a particular characteristic of the disease propagation mechanism. These models will also be used to quantify key parameters of the infection using outputs of the mechanistic model or directly with the real field data available. In particular we are currently investigating the interaction between the date of primary infection and growth of the host, the role of a dual short and long range dispersal of the disease and the effects of the spatially periodic structure of vineyards [7] . Moreover in the 1D spatial case we have developped new tools to exhibit traveling fronts for complex models [22] .

In a more distant future this study will give rise to new developments within the project-team:

• compare delay equation models with epidemiological models based on classical ODEs in the phytopathologic domain;

• in the spatial case improve the code by the use of transparent boundary conditions to simulate an unbounded domain;

• include the effects of fungicide treatments in the models;

• use homogeneization techniques for the mathematical study of the disease spreading in periodic environnments;

• extend these models to the study of diseases in other examples of periodic environnments such as orchards.

Modeling in neurobiology

Participants : Jacques Henry, Gregory Dumont, Oana Tarniceriu.

As an other medical field of application of mathematical modeling we have chosen neurophysiology. Our interest is at two levels : the global electric and magnetic activities generated by the cortex as measured by EEG and MEG. At this level we are mainly interested by the inverse problem which is also studied by the Odyssée and Apics teams. Our approach is based on the factorization methods described in section 3.3.2 . We are also interested in modeling the neural activity at the level of interacting populatios of neurons. Our main collaborations is with the “Basal Gang” team of UMR 5227 at the Bordeaux 2 university.

Our approach for modeling neuron populations is based on structured population dynamics and gives a description of the activity of the tissue at a higher level, through the density function of neurons in the state space. It is based on realistic models at the level of the neuron: each neuron is described by a 2D Izhikevich model. The synchronization or desynchronization of neurons can be represented in this description. This modeling has the advantage of being insensitive to the number of neurons (as opposed to a direct simulation). Whether this kind of modeling can give insight into the functioning of the sensori-motor pathways in the brain has still to be investigated. This methodology has not been fully utilized in computational neurosciences and we believe that classical tools in population dynamics, as for instance the renewal process formulation, could be applied with benefit. Will they help to build a bridge using aggregation techniques with models used at a larger scale in time and space as firing rate models? This would give a basis at the neuron level for these models.

Modeling in electrocardiology

Participants : Jacques Henry, Bedr'Eddine Aïnseba, Simon Labarthe, Alejandro Lopez Rincon.

This is a new field of application we are starting to develop this year. This new orientation is mainly due to the initiative of Pr Michel Haissaguerre an internationally renowned cardiologist, head of the cardiology department at the hospital “Haut Leveque” in Pessac near Bordeaux. He is applying for the creation in Bordeaux of an IHU (a new prestigious structure for medical research). His proposal “Liryc” will mix intimately research in cardiac rhythmology and mathematical modeling and computer simulation. Based on a previous experience of J. Henry in the field, we decided to reshape a part of the research activity of the team to this domain. The main objectives of this institute in which we will be involved are the improvement of the management of cardiac arrhytmias. More specifically Pr Haissaguerre's team made important progresses in the discovery of the role of pulmonary veins in the triggering of atrial fibrillation and its curative treatment by thermoablative therapy. Nevertheless there remain many open questions to fully understand the mechanisms at the origin of the atrial fibrillation and to improve its treatment. For example it is important to make the distinction between paroxystic and permanent fibrillation. The institute will also tackle the prevention of sudden death by ventricular fibrillation. This needs efficient diagnosis tools. Electrocardiographic imaging consists in reconstructing an epicardial map of potential from measurements on torso. From the mathematical viewpoint this is an inverse problem. Our targets in that domain are i) lead theoretical investigations on the nature of fibrillation ii) improve the existing monodomain and bidomain models to have a more realistic modeling of the cardiac tissue and its inhomogeneities which will be applied in a first step at the atrial level iii) improve the numerical methods to solve these models iv) enhance the precision of the resolution of the ECG inverse problem. These researches will be lead in collaboration with the teams of the former Cardiosense 3D project of INRIA.