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 exist, 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.

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.

Our main research works can be summarized as follows:

Building models in epidemiology and immunology. Studies of models and their global behavior. We 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 hemorrhagic 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 modelers, mathematicians, computer scientists and specialists of control theory on the other. Another objective is to encourage mathematicians from Africa to work in the field of mathematical epidemiology. This partnership explains the strong set of data we have and also the number of Phd's students coming from sub-Saharan Africa.

We are interested in applying the research themes described above to the following epidemiological problems:

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 into account the effect of asymptomatic carriers, differential infectivity or differential susceptibility (HBV, Meningitis ...)

One of the challenges 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, 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
*Plasmodium falciparum* population structure. A better understanding
of * Plasmodium falciparum* population genetics is necessary to adjust
control measures.
The findings of Rogier et al. 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 measures of distance. (For example

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, allows 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) is 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 are 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

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

An observer for the system () is a dynamical system

where the map

or at least

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. Gouzé, O. Bernard, G. Sallet ) and is still very active in this area. Now the problem is to identify relevant classes 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 one of the first use of an observer in virology was the work of Velasco-Hernandez J., Garcia J. and Kirschner D. in modeling the chemotherapy of HIV, but this observer, based on classical linearization method, 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 become a near certainty. The rationale for studying continuous delays is 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 approache (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 .

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 example arises 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.
During the last decade, there has been increased interest in these deterministic metapopulation disease models.
We have generalized to

MASAIE is developing, in the framework of the CAPES-COFECUB project (see international program), a metapopulation model for dengue. This model is for the state of Rio and is using the data of foundation FIOCRUZ.

The estimation of sequestered parasite population has been a challenge for the biologist and modeler, with many authors having studied this problem.
The difficulty is that the infected erythrocyte leaves the circulating peripheral blood
and binds to the endothelium in the microvasculature of various organs. A
measurement of Plasmodium falciparum parasitaemia taken from a blood smear therefore samples young parasites only and there is no clinical methods to measure the sequestered parasites.
We have developed a simple tool to estimate the sequestered parasites and hence the total parasite burden for *Plasmodium falciparum*
malaria patients. We have also given a method to estimate a crucial parameter in the model of infection. This parameter

We continued research on modeling the introduction of *Wolbachia* in a population of *Aedes Aegypti*. This research is done in collaboration with FGV (Fundação Getulio Vargas), Fiocruz (Fondation Oswaldo Cruz) and UFF (Universidade Federal Fluminense) in Rio de Janeiro (Brazil) .

*Wolbachia* is a bacteria which infects arthropod species, including a high proportion of insects (60% of species). Its interactions with its hosts are often complex, and in some cases it is considered as an endosymbiont.
The unique biology of *Wolbachia* has attracted a growing number of researchers interested in questions ranging from the evolutionary implications of infection through to the use of this agent for pest and disease control:
a public web site has been funded by the National Science Foundation of Australia, and a research in pubmeb (http://`wolbachia` gives 1889 results.

While *Wolbachia*
is commonly found in many mosquitoes it is absent from the species that are considered to be of major importance for the transmission of human pathogens.
The successful introduction of a life-shortening strain of *Wolbachia* into the dengue vector *Aedes
aegypti* that decreases adult mean life has recently been
reported.

Moreover it is estimated that the population of mosquitoes harboring *Wolbachia* is less efficient to transmit dengue
, , , .
Then it is considered that using *Wolbachia* can be a viable option for controlling the incidence of the dengue.

We consider an alternative infection (by Wolbachia) model which exhibits monotonous properties. This model is designed to take into account both the biology of this infection and any available data. The objective is to use this model for predicting the sustainable introduction of this bacteria. We provide a complete mathematical analysis of the model proposed and give the basic reproduction ratio *Wolbachia*. We observe a bistability phenomenon. Two equilibria are asymptotically stable: an equilibrium where all the population is uninfected and an equilibrium where all the population is infected. A third unstable equilibrium exists. We provide a lower bound for the basin of attraction of the desired infected equilibrium. We are in a backward bifurcation situation. The bistable situation occurs with natural biological values for the parameters. Our model is an example of an epidemiological model with only vertical transmission.

This infection model is then connected with a classical dengue model. We prove that for the complete model the equilibrium with *Wolbachia* for the mosquitoes and without dengue for the human is asymptotically stable. We prove that, if a sufficiently large population of infected (by Wolbachia) mosquitoes is introduced, dengue will disappear.

These results have been obtained in collaboration with Pierre-Alexandre Bliman (FGV, Inria); Moacyr Silva (FGV), Claudia Codeço (Fiocruz), Max Souza (UFF) and Jair Koiller (FGV).

Starting from the multi-scaled dengue system, we construct a pair of observers to estimate the dynamics of the disease. The nature of both the observers and the multi-scaled system allows to estimate both the number of susceptible and recovered hosts, as well as to provide information on the vector population, using only infected population data. Numerical simulations have been used to illustrate the performance of the observers.

We introduce a matrix that combines information about human circulation and the epidemiological situation at the nodes of a metapopulational model for an arboviral disease. Its singular value decomposition allows relationships between three basic reproduction numbers

The global mathematical analysis of a schistosomiasis infection model that involves human and intermediate snail hosts as well as an additional mammalian host and a competitor snail species has been done by constructing Lyapunov functions and using properties of K monotone systems.
We derived the basic reproduction number

The model SI (Susceptible-Infected) is one of the most important and used epidemiological models. We gave a complete analysis of the stability of the model with a non-linear incidence and two classes of infected individuals .

We have also studied SIS, SIR and MSIR models with bilinear incidence and varying population, with

A mathematical multi-patches model for highland malaria in Kenya has been developed and analysed in and .

MASAIE is the Inria EPI partner of GRIMCAPE (LIRIMA). It also has strong collaboration with M2IPE2S (LIRIMA). Two PhD students (Diaby and Diouf) are members of M2IPE2S.

G. Sallet participated to The "LIRIMA evaluation seminar", Paris, September 24-26, 2014.

MASAIE is the french correspondent in a cooperation program with Brazil.

We work with a Brazilian network that has been built in 2011, and which is composed of

FGV (Fundação Getulio Vargas ) Rio de Janeiro. Principal investigator: Jair Koiller.

UFF (Universidade Federal Fluminense) Rio de Janeiro. Principal investigator: Max Oliveira de Souza.

UNICAMP (Universidade Estadual de Campinas ) Campinas. Principal investigator: Hyun Mo Yang.

Fondation Oswaldo Cruz (Fiocruz, Rio). Principal investigator: Claudia Codeço.

IMPA Rio de Janeiro. Principal investigator: Jorge Zubelli.

This project, funded by CAPES-COFECUB, "new methods in epidemiology and early detection of events" began in January 2011 and finished in December 2013. However the collaboration with the Brazilian collaborators has continued and increased during 2014.

A research project on Bilharzia was submitted by the universities of Ouagadougou and Gaston Berger of Saint-Louis, in the framework of PAES(projet d'appui à l'enseignement supérieur) of UEMOA (Union Économique et Monétaire de l'Afrique de l'Ouest). MASAIE is an important component of this network. This project has been accepted in 2012 and funded with 30 000 000 CFA (XOF) (

The PhD thesis of M. Diaby (MASAIE and UGB St Louis) is part of this project.

Max Oliveira de Souza, professor at Universidade Federal Fluminense, Rio de Janeiro, Brazil, January 2014.

G. Sallet visited FGV and UFF (Rio de Janeiro) and UNESP (São José dos Campos, Sao Paulo) from November 10, 2014 to November 29, 2014. A. Iggidr visited FGV and UFF (Rio de Janeiro) from April 20, 2014 to May 11, 2014. A. Iggidr visited FGV and UFF (Rio de Janeiro) and UNESP (São José dos Campos, Sao Paulo) from November 7, 2014 to December 5, 2014.

PhD in progress: Mouhamadou Diaby, "Etude mathématique de l'évolution temporelle et spatiale de certaines épidémies. Applications à la Bilharziose (schistosomiase).", 2010, A. Iggidr.

PhD in progress: Mamadou Lamine Diouf, "Modélisation, observation et contrôle de la propagation de certaines épidémies en Afrique Subsaharienne.", 2010, A. Iggidr.