MASAIE is a common project with University Paul Verlaine of Metz 1. The team has been created on January the 1
^{st}, 2008.

The overall objective of MASAIE is to develop and apply methods and tools of control theory and dynamical systems for the mathematical modeling in epidemiology and immunology. The problem at issue is twofold. The first objective is to obtain a better understanding of epidemiological and immunological systems. The second objective is to mathematically study problems arising naturally when addressing questions in the fields of epidemiology and immunology. In our opinion our two endeavors operate in a synergic way : new problems will appear in control theory and their study will give new tools to epidemiology and immunology.

In this regard the first step is modeling. Modeling has always been a strong activity in control theory, however modeling in epidemiology and immunology has some specificities not encountered in engineering. The systems are naturally complex and have highly nonlinear parts. A second characteristic is the paucity of data. These data, when existing, are often imprecise or corrupted by noise. Finally rigorous laws seldom exists, this is a major difference with engineering. In this situation modeling is a back and forth process between the “mathematician" and the “biologist." When modeling, it is necessary to decide what is important and what can be neglected. This is not an easy task. A term or a structure, that can be discarded by the biologist modeler, turns out to give an unexpected behavior to the system. On the other side the biologist wants the more complete model possible, which can be difficult for the mathematical tractability. In MASAIE a close collaboration with researchers in epidemiology and immunology (IMTSSA, INRA, IRD, Institut Pasteur, University of Tübingen) is essential and will be developed.

Beyond the stage of modeling we have the validation, simulation and mathematical analysis of the models. This is also a part of modeling. For example some models can be rejected for inappropriate behavior while others are accepted for their agreement with data. Once again the role of data and the collaboration with researchers in these fields are certainly crucial, but the mathematical analysis cannot be neglected.

Emerging and reemerging diseases have led to a revived interest in infectious diseases and immunology. Our final objective is to propose and study epidemiological and immunological models for

analysis of the spread and control of infectious disease,

a better understanding of the dynamics and behavior of epidemics,

clarification of hypotheses, variables and parameters,

proposition of conceptual results (thresholds, sensitivity analysis ...),

simulation as an experimental tool for building and testing theories,

effective evaluation of field and outbreak data,

planning and evaluation of intervention campaigns.

Building models in epidemiology and immunology. Studies of models and their global behavior. We will concentrate primarily on models for disease transmitted by blood-sucking insect vectors (malaria, dengue, chikungunya, yellow fever) but we will also consider some diseases for which we have collaborations and data such as Ebola haemorrhagic fever, Hepatitis B or Meningitis.

Modeling and model validation guided by field data.

Design of observers (software sensors for biological systems): observers are auxiliary dynamical systems that use the model together with the available measurement data in order to reconstruct the unobservable variables (that are not measured directly) and to estimate some parameters of the system. Observers are related to observability and, therefore, also determine data collection plans.

Establishing control strategies for the considered systems that can help to determine some policies in public health and fishery.

In our project, Africa has a special place:

Our research focuses on infectious diseases caused by bacteria, parasites in humans and animals. The populations of less developed countries are specially affected by these diseases. "End users" with whom we work are specialists in tropical diseases. This explains the interest in our project for African collaborations. A strong partnership exists with the network EPIMATH in central Africa. The objective of EPIMATH is to promote collaboration between different communities: Specialists in Health Sciences on the one hand and modellers, mathematicians, computer and automation on the other. Another objective is to encourage mathematicians from Africa to work in the field of mathematical epidemiology. This partner explains the strong set of data we have and also the number of Phd's students coming from subsaharian Africa.

Intra-host models for malaria.

Metapopulation models considering the dynamics of
*Plasmodium falciparum*causing tropical malaria in human populations, and the development of drug resistance.

Modeling the dynamics of immunity in human populations in endemic areas. Models describing the intra-host parasite dynamics, considering the development and loss of immunity.

Spread of epidemics of arbovirus diseases (dengue, chikungunya ...)

Disease leading to structured model to allow to take in account the effect of asymptomatic carriers, differential infectivity or differential susceptibility (HBV, Meningitis ...)

One of the challenge of the project is to ensure the relevance of these models. It is Important to closely involve the “end users" (specialists in the fields, experimenters, observers,
physicians, epidemiologists, entomologists, etc.) and “providers" (Mathematicians, numerical, statisticians, computer scientists,...). Users are able to bring a critical evaluation on the
quality of results, to validate them or exploit them further. For example we want to understand the genetic diversity and structure of African
*Plasmodium falciparum*population. The spread of drug resistance is due to gene flow and the scale of
*P. falciparum*population structure. A better understanding of
*P. falciparum*population genetics is necessary to adjust control measures. The findings of Rogier et al
provide evidence for support structured
*P. falciparum*populations in Africa, and suggest that malaria epidemiology in urban areas depends on local transmission, geographic isolation, and parasite flow between the city and the
surrounding rural areas. The molecular geneticists use many different statistical measure of distance. (For example
F_{st}, Nei's distance ...). It is important in our modeling process to understand how these measures can be obtained as output of our models. This explains why our team is composed of
"control theorist" "applied mathematician" and "statisticians" (A. Maul, B. Cazelles).

Our conceptual framework is that of Control Theory : the system is described by state variables with inputs (actions on the system) and outputs (the available measurements). Our system is either an epidemiological or immunological system or a harvested fish population. The control theory approach begins with the mathematical modeling of the system. When a “satisfying" model is obtained, this model is studied to understand the system. By “satisfying", an ambiguous word , we mean validation of the model. This depends on the objectives of the design of the model: explicative model, predictive model, comprehension model, checking hypotheses model. Moreover the process of modeling is not sequential. During elaboration of the model, a mathematical analysis is often done in parallel to describe the behavior of the proposed model. By behavior we intend not only asymptotic behavior but also such properties as observability, identifiability, robustness ...

Problems in epidemiology, immunology and virology can be expressed as standard problems in control theory. But interesting new questions do arise. The control theory paradigm, input-output systems built out of simpler components that are interconnected, appears naturally in this context. Decomposing the system into several sub-systems, each of which endowed with certain qualitative properties, allow the behavior of the complete system to be deduced from the behavior of its parts. This paradigm, the toolbox of feedback interconnection of systems, has been used in the so-called theory of large-scale dynamic systems in control theory . Reasons for decomposing are multiple. One reason is conceptual. For example connection of the immune system and the parasitic systems is a natural biological decomposition. Others reasons are for the sake of reducing algorithmic complexities or introducing intended behavior ...In this case subsystems may not have biological interpretation. For example a chain of compartments can be introduced to simulate a continuous delay , . Analysis of the structure of epidemiological and immunological systems is vital because of the paucity of data and the dependence of behavior on biological hypotheses. The issue is to identify those parts of models that have most effects on dynamics. The concepts and techniques of interconnection of systems (large-scale systems) will be useful in this regard.

In mathematical modeling in epidemiology and immunology, as in most other areas of mathematical modeling, there is always a trade-off between simple models, that omit details and are designed to highlight general qualitative behavior, and detailed models, usually designed for specific situations, including short-terms quantitative predictions. Detailed models are generally difficult to study analytically and hence their usefulness for theoretical purposes is limited, although their strategic value may be high. Simple models can be considered as building blocks of models that include detailed structure. The control theory tools of large-scale systems and interconnections of systems is a mean to conciliate the two approaches, simple models versus detailed systems.

Many dynamical questions addressed by Systems Theory are precisely what biologist are asking. One fundamental problem is the problem of equilibria and their stability. To quote J.A. Jacquez

A major project in deterministic modeling of heterogeneous populations is to find conditions for local and global stability and to work out the relations among these stability conditions, the threshold for epidemic take-off, and endemicity, and the basic reproduction number

The basic reproduction number is an important quantity in the study in epidemics. It is defined as the average number of secondary infections produced when one infected individual is introduced into a host population where everyone is susceptible. The basic reproduction number is often considered as the threshold quantity that determines when an infection can invade and persist in a new host population. To the problem of stability is related the problem of robustness, a concept from control theory. In other words how near is the system to an unstable one ? Robustness is also in relation with uncertainty of the systems. This is a key point in epidemiological and immunological systems, since there are many sources of uncertainties in these models. The model is uncertain (parameters, functions, structure in some cases), the inputs also are uncertain and the outputs highly variable. That robustness is a fundamental issue and can be seen by means of an example : if policies in public health are to be taken from modeling, they must be based on robust reasons!

The concept of observer originates in control theory. This is particularly pertinent for epidemiological systems. To an input-output system, is associated the problem of reconstruction of the state. Indeed for a given system, not all the states are known or measured, this is particularly true for biological systems. This fact is due to a lot of reasons : this is not feasible without destroying the system, this is too expensive, there are no available sensors, measures are too noisy ...The problem of knowledge of the state at present time is then posed. An observer is another system, whose inputs are the inputs and the outputs of the original system and whose output gives an estimation of the state of the original system at present time. Usually the estimation is required to be exponential. In other words an observer, using the signal information of the original system, reconstructs dynamically the state. More precisely, consider an input-output nonlinear system described by

where
is the state of the system at time
t,
is the input and
is the measurable output of the system.

An observer for the the system ( ) is a dynamical system

where the map
ghas to be constructed such that: the solutions
x(
t)and
of (
) and (
) satisfy for any initial conditions
x(0)and

or at least converges to zero as time goes to infinity.

The problem of observers is completely solved for linear time-invariant systems (LTI). This is a difficult problem for nonlinear systems and is currently an active subject of research. The problem of observation and observers (software sensors) is central in nonlinear control theory. Considerable progress has been made in the last decade, especially by the “French school", which has given important contributions (J.P. Gauthier, H. Hammouri, E. Busvelle, M. Fliess, L. Praly, J.L. Gouze, O. Bernard, G. Sallet ) and is still very active in this area. Now the problem is to identify relevant class of systems for which reasonable and computable observers can be designed. The concept of observer has been ignored by the modeler community in epidemiology, immunology and virology. To our knowledge there is only one case of use of an observer in virology ( Velasco-Hernandez J. , Garcia J. and Kirschner D. ) in modeling the chemotherapy of HIV, but this observer, based on classical linear theory, is a local observer and does not allow to deal with the nonlinearities.

Another crucial issue for biological systems is the question of delays. Delays, in control theory, are traditionally discrete (more exactly, the delays are lags) whereas in biology they usually are continuous and distributed. For example, the entry of a parasite into a cell initiates a cascade of events that ultimately leads to the production of new parasites. Even in a homogeneous population of cells, it is unreasonable to expect that the time to complete all these processes is the same for every cell. If we furthermore consider differences in cell activation state, metabolism, position in the cell cycle, pre-existing stores of nucleotides and other precursors needed for the reproduction of parasites, along with genetic variations in the parasite population, such variations in infection delay times becomes a near certainty. The rationale for studying continuous delays are supported by such considerations. In the literature on dynamical systems, we find a wealth of theorems dealing with delay differential equations. However they are difficult to apply. Control theory approaches (interconnections of systems), is a mean to study the influence of continuous delays on the stability of such systems. We have obtained some results in this direction .

We are considering general classes of models to address some epidemiological peculiarity. For example we consider and analyze a class of models , under the general form

where
represents the concentration of susceptible individuals or target cells,
represents the different class of latent, infectious and removed individuals. The matrix
Cis a nonzero
k×
nnonnegative matrix,
is a positive vector,
Pdenotes a linear projection,
Ais a stable Metzler matrix and
denotes a scalar product in
. The function
describes the recruitment (or the demography) of susceptible individuals or cells and the quantity
represents the infection transmission. For some diseases, a bilinear infection transmission function
is not adequate so we have to replace in equation (
) the expression
Cyby a more general non-linear incidence function
Cf(
y). The parameter
utakes only the value 0 or 1.

which gives the differential equation

where
is an input flow (or a recruitment rate) which is supposed to be constant,
is the natural death rate of the population. For each
jthe parameter
_{j}is the contact rate, i.e., the rate at which susceptibles meet infectious individuals belonging to the class
j, the parameter
_{j}is the disease-related death rate of the class
jand

Similarly the SP model can be represented by

The parameter
_{j}denotes the fractional rate of transfer of infected from the stage
jto the stage
j+ 1. The dynamical progression of the disease can be represented by the differential equation:

The DISP is the combination of these two structures. These models are easily put under the general form. ( ).

where the variable
xdenotes the concentration of uninfected RBC, the variable
y_{j}is the concentration of parasitized red blood cell (PRBC) of class
j, and
mis the concentration of the free merozoites in the blood. The example of malaria gives an example where stages in modeling are created for biological reasons. We have seen before that
continuous delays are important to be modeled. The process of converting time-delay integro-differential equations in a set of ODE is coined by MacDonald
as the linear chain trick. In other community this is also known as
the method of stages. Actually any distribution can be approximated by a combination of stages in series and in parallel (Jacquez). This process consists to insert stages in the model. This
is an example of stages created to take into account a behavior. This added stages have no biological meaning. Our general model is also well suited for this process.

The general model ( ) can take into account the case of different strains for the parasites and can be adapted to cope with vector transmitted diseases. Then we have a building block to model complex systems. System ( ) describes the basic model which can be extended, by introducing interconnections of blocks of the form ( ), to describe more complex systems : more classes of susceptible can be introduced, the recruitment of susceptible individuals can be replaced by an output of an explicit model of the population dynamics, each sub-system describes what happens in a patch, inflows and outflows can be introduced to model the population movement between patches, different strains for the pathogen can be introduced, others systems can bring input in these models (e.g. the immune system) ...

This general form will be used to model some well-identified diseases for which we have data and expert collaborators (e.g. malaria, dengue, Ebola ...). This form has
to be tailored to the particular case considered. For example the matrix
Arepresents connections and the structure of this matrix
A(triangular, Hessenberg, sparse ...) depends on the disease.

In modeling the reaction of the immune system to a
*Plasmodium falciparum*infection. Malaria infection gives rise to host responses which are regulated by both the innate and acquired immune system as well as by environmental factors.
Acquired immunity is species- and stage-specific. A malaria infection initiates a complicated cascade of events. The regulation of this complex system with numerous feedbacks is intricately
balanced. The objective is to build a computer model which allows to test the dynamics of malaria infection. This research is conducted in collaboration with immunologists. We collaborate also
with B. Cazelles and J.F. Trape of the research unity 77 “Afro-tropical epidemiology" of IRD in Sénégal. The steady increase of
*Plasmodium falciparum*resistance to cheap first line antimalarials over the last decades has resulted in a dramatic increase in malaria-associated morbidity and mortality in sub-Saharan
Africa. Research in recent years has established that resistance to chloroquine (CQ), pyrimethamine has been controlled and constantly monitored for more than a decade, coinciding to the time
period of expansion of CQ- and SP-resistance across Africa.

The longitudinal active case detection study launched in Dielmo in 1990 by the UR77 of IRD, a rural Senegalese village, is probably the only place where drug use has been controlled and constantly monitored for more than a decade, coinciding to the time period of expansion of CQ- and SP-resistance across Africa. This is an unprecedented opportunity to quantify the impact of a strictly controlled use of antimalarials on drug resistance. Furthermore, first line treatment was changed in 1995, allowing to explore its consequences on dynamics of spreading of drug resistance.

Heterogeneity plays an important role in many infectious disease processes. For instance, spatial heterogeneity is a strong determinant of host-parasite relationships. In modeling spatial or
geographic effects on the spread of a disease, a distinction is usually made between diffusion and dispersal models. In diffusion models, spread is to immediately adjacent zones, hence the
phenomenon of traveling waves can appear. These models traditionally use partial differential equations. However, there are some important situations that cannot be modeled by PDE. This is the
case when the space considered is discrete. For example, when we have to consider sparsely populated regions, the human population is located in patches. The organization of human-hosts into
well-defined social units such as families, villages or cities, are good examples of patches. Another examplearises in the study of the human African Trypanosomiasis. The vector is the tse-tse
fly, and it is known that flies take fewer blood meals in villages than in coffee plantations where the villagers work during the day. For such situations where human or vectors can travel a
long distance in a short period of time, dispersal models are more appropriate. These models consider migration of individuals between patches. The infection does not take place during the
migration process. The situation is that of a directed graph, where the vertices represent the patches and the arcs represent the links between patches. Recently, there has been increased
interest in these deterministic metapopulation disease models. We have generalized to
npatches the Ross-Macdonald model which describes the dynamics of malaria. We incorporate in our model the fact that some patches can be vector free. We assume that the hosts can migrate
between patches, but not the vectors. The susceptible and infectious individuals have the same dispersal rate. We compute the basic reproduction ratio
. We prove that if
, then the disease-free equilibrium is globally asymptotically stable. When
, we prove that there exists a unique endemic equilibrium, which is globally asymptotically stable on the biological domain minus the disease-free equilibrium.

We are studying intra-host models for malaria. We have obtained the complete stability analysis for these models .

We have introduced a general epidemic model with different susceptible compartments and different infective compartments. This model addresses the problem of differential susceptibility as well as of differential infectivity. It also takes into account the different flows between different compartments. This model encompasses and generalizes previous models of the literature. The rationale to introduce differential infectivity and susceptibility is motivated by the heterogeneity, concerning the mode of infection, for the individuals. For many reasons difference in susceptibility to infection can occur : genetic variations, different social behaviors, different states of immunization, different vaccines ...The infection by HBV, hepatitis B, is typically a disease where simultaneously differential susceptibility and infectivity appear. We have computed the basic reproduction ratio for this model and have studied the dynamical properties .

We have have shown how observers can be helpful to deal with stock estimation for some renewable resource models , .

MASAIE is involved with the SARIMA project (Soutien aux Activités de Recherche en Informatique et Mathématiques en Afrique). G. Sallet and A. Iggidr have given lectures in Saint-Louis at master level.

A “AIRES-SUD" projet has been accepted with MASAIE and the LANI (Laboratoire d'Analyse Numérique et Informatique) laboratory of the university Gaston Berger of Saint-Louis for 2008-2011.

G. Sallet and M. Tchuente have been selected to hold a chair “crossed" between the university of Yaoundé, MASAIE and the university of Metz during the period 2008-2009.

G. Sallet with J. Arino (university of Manitoba) and M. Tchuente have organized a
**MITACS-CDM-INRIA Summer School**on Mathematical Epidemiology "Mathematical Modeling of ÊInfectious Diseases" in Yaoundé, Cameroon, August 10-20, 2009. This school has been funded by
MITACS-Centre for Disease Modelling, IRD, University of Yaoundé.
http://

A. Iggidr has been invited to the workshop "Dynamique de populations marines et halieutique". Safi, Morocco, June 2-5 2009.

A. Iggidr has been invited to the workshop "Analyse Non Linéaire et Application aux Rythmes complexes du vivant" organized by the "MIA" and "Fédération de Recherche en Environnement pour le Développement Durable". University of La Rochelle, December 14-16 2009.

G. Sallet has given a 20 hours lecture in Saint-Louis at master 2 level.

G. Sallet has given a 10 hours lecture in University of Ouagadougou at master 2 level.

A. Iggidr has given a 20 hours lecture in Saint-Louis at master 2 level.

D. Ngom defended his Phd thesis "Observation et régulation de certains modèles discrets d'écosystèmes" on December 15th 2008.

A. Guiro defended his Phd thesis "Sur quelques problèmes d'observateurs, Application à certains modèles d'écosystèmes aquatiques" on Marsh 12th 2009.

J. Mbang defended his Phd thesis "Analyse de la stabilité des modèles intra-hôtes avec retards. Applications à des modèles intra-hétes de paludisme et de VIH-1" on Marsh 20th 2009.