Section: Scientific Foundations
Application fields and collaborations with biologists
We present here collaborations on specific biological modeling problems.
Epidemiology
(B. Ainseba, A. Ducrot, M. Langlais)
Brucellosis
This is a collaboration with CHU of the university of Tlemcen (Algeria). Brucellosis is a highly contagious infectious disease in domestic livestock and many other species and is communicable to humans by contact with infected animals or by infected products (milk, meat, ...). This disease is not transmitted between humans but is a major disease in developing countries because of its severity in human cases and the economically caused damage to the livestock. Our goal is to study the disease within an ovine population. Infection usually occurs after contact with tissues, urine, vaginal discharges, aborted fetuses and placentas,... When infected for a first time the female aborts and the infected fetus remains in the environment still highly contaminating for several months. The pioneering works on the subject focused only on direct transmission mechanisms and did not take in account the indirect transmission by the contaminated environment.
HIV1 Infection in tissue culture
Since the 80's there has been a big effort made in the mathematical modeling of the human Immunodeficiency Virus type 1, the virus which causes AIDS. The major targets of HIV1 infection is a class of lymphocytes or white blood cells known as CD4^{ + } Tcells which are the most abundant white blood cells in the immune system. It is thought that HIV1, although attacking many different cells, wreaks the most havoc on the CD4^{ + } Tcells by causing their destruction and decreasing the body's ability to fight infection. Many mathematical models have been introduced to describe the dynamics in HIV1 infection in the bloodstream (see the works of Leenheer et al. , Nowak et al. , Kirshner, May, Perelson et al. , ....). For tissue culture (lymph nodes, brain, ...) the cell to cell mode contact is much more important for the infection than the cellfree viral spread (see Culshaw et al. , Philips, Dimitrov, ...). Following these pioneering works we propose a model of the SI type with delay, modeling the interaction between healthy cells, infected cells, and infected cells that are still not infectious .
Toxoplasma gondii
Toxoplasma gondii (T. gondii) is a most successful parasite infecting a wide range of intermediate hosts (mammals, rodents and birds). Up to 30Toxoplasmosis can cause lifethreatening encephalitis in immunocompromised persons (AIDS patients, recipients of organ transplants, cancer chemotherapy). Infection acquired during pregnancy may cause severe problems to the fetus (baby’s eyes, nervous system, skin, and ears). Toxoplasmosis may also lead to neuropsychiatric disorders, e.g. schizophrenia.
This is a pluridisciplinary and collaborative work with Ecole Vétérinaire de Lyon, CERFE and French National Reference Center for Toxoplasmosis at University of Limoges and Reims, funded by AFSSET. A multipatch model coupling SIR epidemic models and predatorprey systems is currently under study.
Blue Tongue Virus
Bluetongue (BT) is a vectorborne disease of ruminants transmitted by biting midges (Culicoïdes). BT virus (BTV) serotypes spreading in southern Europe before 2006 could infect ruminants, cattle being subclinically infected. Since 2006, BTV8 infected cattle may show clinical signs. Current epizootics have a large socioeconomic impact on the international trade of animals and their productivity. Strategies to control their consequences and the spread of the virus were set up such as vaccination. In Europe, vaccination against BTV8 was recommended in 2008. However, a partial coverage and waning immunity weaken the efficacy of such a strategy.
This is a pluridisciplinary and collaborative work with Ecole Vétérinaire de Nantes, funded by INRA. A large ODE model at a single cattle herd scale is currently studied before tackling the more realistic spatial problem.
Blood cells
(M. Adimy, B. Ainseba, A. Ducrot, A. Noussair) Due to the departure of M. Adimy to Lyon to create the team MAIA, this research is decreasing in our team.
Generating process for blood cells (Hematopoiesis)
Cellular population models have been investigated intensively since the 1960's (see, for example, Rubinow and Lebowitz [58] ) and still interest a lot of researchers. This interest is greatly motivated, on one hand, by the medical applications and, on the other hand, by the biological phenomena (such as oscillations, bifurcations, traveling waves or chaos) observed in these models and, generally speaking, in the living world (Mackey and Glass [53] ).
Hematopoiesis is the process by which primitive stem cells proliferate and differentiate to produce mature blood cells. It is driven by highly coordinated patterns of gene expression under the influence of growth factors and hormones. The regulation of hematopoiesis is about the formation of blood cell elements in the body. White and red blood cells and platelets are produced in the bone marrow whence they enter the blood stream. Abnormalities in the feedback are considered as major suspects in causing periodic hematological diseases, such as autoimmune hemolytic anemia, cyclical neutropenia, chronic myelogenous leukemia.
Cell biologists classified stem cells as proliferating cells and resting cells (also called G_{0} cells) (see Mackey [54] ). Proliferating cells are committed to undergo mitosis a certain time after their entrance into the proliferating phase. Mackey supposed that this time of cytokinesis is constant, that is, it is the same for all cells.
Based on [37] , [38] , [39] , , we propose a more general model of hematopoiesis. We take into account the fact that a cell cycle has two phases, that is, stem cells in process are either in a resting phase or actively proliferating. However, we do not suppose that all cells divide at the same age, because this hypothesis is not biologically reasonable. We obtain a system of two nonlinear partial differential equations. Due to cellular replication, the system exhibits a retardation of the maturation variable and temporal delay depending on this maturity.
Malignant proliferation of hematopoietic stem cells
Hematological diseases have attracted a significant amount of modeling attention because a number of them are periodic in nature . Some of these diseases involve only one blood cell type and are due to the destabilization of peripheral control mechanisms, e.g., periodic autoimmune hemolytic anemia. Such periodic hematological diseases involve periods between two and four times the bone marrow production/maturation delay. Other periodic hematological diseases, such as cyclical neutropenia , involve oscillations in all of the blood cells and very long period dynamics on the order of weeks to months and are thought to be due to a destabilization of the pluripotent stem cell compartment from which all types of mature blood cells are derived.
We focus, in particularly, on chronic myelogenous leukemia (CML), a cancer of the white cells, resulting from the malignant transformation of a single pluripotential stem cell in the bone marrow . Oscillations can be observed in patients with CML, with the same period for white cells, red blood cells and platelets. This is called periodic chronic myelogenous leukemia (PCML). The period of the oscillations in PCML ranges from 30 to 100 days , depending on patients.
We have studied in a delay model that describes the dynamics of a pluripotent stem cell population involved in the blood production process in the bone marrow. The delay describes the cell cycle duration. We established stability conditions for the model independent of the delay. We have also observed oscillations in the pluripotent stem cell population through Hopf bifurcations. With parameter values given by Mackey [54] , our calculations indicate that the oscillatory pluripotent stem cell population involves a period of 46 days.
It will be interesting to study the dynamics of the hematopoietic cells throughout different compartments modeling various stages of the maturation of cells. This resarch is joint with INSERM teams E 217 in Bordeaux 2 and U 590 in Lyon.
Sociobiological activities of the ImmuneSystem cells
In recent years, much effort has been put into the mathematical foundation of models of TumorImmune System Interaction. The aim is the description of the cell distribution as a function of time and a physiological state which includes both mechanical and sociobiological activities. . This new class of models of population dynamics with stochastic interation, is characterized by a mathematical structure similar to the one of the Boltzmann equations. In this theory it is assumed that the system under consideration consists of a very large number of cells that can interact. This is the case in immunology problems and in particular in the competition between tumors and immune system. The motivation is that the stage of the early growth of a tumor belongs to the socalled free cells regime, in which the tumor cells are not yet condensed in a macroscopically observable spatial structure and the interactions between tumor and immune system, occur at a cellular level. This makes the kinetic approach particularly appropriate.
However the development of numerical schemes which gives precise calculations of these class of models, is desirable and hopefully the work envisaged here is a step towards obtaining such algorithms (see B. Aylaj and A. Noussair State trajectory analysis of a nonlinear kinetic model of multispecies population dynamics MCM in press doi:10.1016/j.mcm.2008.07.022). The question of how to discretize a given model, in particular the treatment of the discrete encounter operator together with the special treatment of the nonlocal boundary condition, represents an interesting part. Convergence properties of numerical schemes for these models seem to be a rather unexplored area. Least squares technique must be developed for identifying unknown parameters of the models. Convergence results for the parameters must be investigated and established. Ample numerical simulations and statistical evidence will be provided to demonstrate the feasibility of this approach.
Modeling in viticulture; collaboration with INRA
(B. Ainseba, J.B. Burie, J. Henry, M. Langlais, A. Noussair)
This is a joint research with different groups of UMR “Santé végétale” of INRA, Villenave d'Ornon.
Integrated Pest Management in viticulture.
Integrated Pest Management (IPM) is an approach to solving pest problems by using knowledge on the pest to prevent them from damaging crops. Under an IPM approach, actions are taken to control insects, disease or weed problems only when their numbers exceed acceptable levels. The goal is to promote and coordinate research on integrated control strategies in viticulture which reduce inputs of pesticides and maximize the effects of natural enemies, thereby minimizing impacts on the environment.
A first objective of our work here is to progress in the risk assessment of the moth Lobesia botrana. Host plant and grape varieties eaten by the larvae modify the protandry between males and females, the female fecundity, the egg fertility and thus the demography of the offspring, with its consequences on the temporal dynamics of oviposition and thus grape damages see [55] , [61] , and [62] . We are developing models and numerical methods including parameter estimation procedures to follow the level of the population in a vineyard.
A second objective to develop a numerical code to model a " Mating disruption technic" for insect control: Pheromone are volatile chemical scents involved in communication between individuals of the same species. One type that is used in pest management is called sex pheromone. Individuals of one gender produce and liberate the chemical to attract individuals of the other sex. One novel insect control approach, ”pheromone mediated mating disruption”, interrupts the reproductive cycle so that no eggs are produced. The main consequence of mating disruption is a decrease of female active space. A last biological control is narrowly defined here as the use of predators, parasites, pathogens, competitors, or antagonists to control a pest.
Spreading of a fungal disease over a vineyard
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 INRAENITA 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 [45] .
In a more distant future this study will give rise to new developments within the projectteam:

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
(B. Ainseba, J. Henry)
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.
MEGEEG inverse problem
One of the goals of MEGEEG is to reconstruct human functional brain activity images with a much better time resolution than functional MRI. Starting from electric potential and magnetic field external measurements, it consists in recovering internal electric dipoles which generated them. The Odyssée projectteam animates a multidisciplinary group on this research theme. We intend to go on participating in these researches especially with respect to the methodology of inverse problem. On the one hand one has to set the problem in an as less instable as possible form. On the other hand the factorization method can be used twice : the optimality system for the inverse problem set as a control is linear and includes two coupled elliptic problems. By a space invariant embedding as previously, one can obtain a factorization in first order Cauchy problems and decouple state and adjoint state as in [49] at the same time. Due to the linearity of the problem, the resolution of the optimality system is transformed in a Cauchy problem on a family of surfaces starting from the electrodes and sweeping over the domain to the surface of dipoles. At each time an inverse problem is to be solved for the measured data and so there is a family of inverse problem to be solved indexed by time. The factorization computation being done once for all, the method should be quite efficient. This research is carried on within the Enée 06 associated team with A. Ben Abda at LAMSIN in Tunis.
Modeling populations of neurons by a population density approach
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 sensorimotor 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.