NOCIME

Summary.

Automatic control is very little used in epidemiology, whereas central questions of estimation, prediction and supervision naturally arise in this domain in terms of input-output systems. These questions are made difficult by the non-linearity of the models. The NOCIME project aims at studying new problems related to identification, observation and optimal control, posed by mathematical epidemiology. The most critical issues that will be examined relate to estimating the short- and long-term dynamics of a novel pathogen after its detection; and the control of the epidemic by minimizing "crisis" situations. Several types of compartmental models from the literature (based on ODE systems) will be considered: models with direct or vector transmission, intra-hosts and/or groups, with acquired or temporary immunity. The consortium has identified two major challenges to be conducted, which will be considered for a general class of models: 

1. State and parameter estimation for non-globally identifiable/observable dynamics. Typically, at the onset of an epidemic, the state of the system is still close to disease-free equilibria, which are points of non-identifiability and non-observability. This renders the conventional estimators and observers inefficient. This type of situation has been studied very little in the literature. We will first analyze identification and observability according to various possible measurements, including incidence, number of reinfections and seroprevalence. For the design and tuning of estimators/observers, we will study several approaches: local (nonlinear) transformation and approximation; integral observers extending those obtained for “batch” processes; and interval observers. 

2. Optimal control for unconventional criteria. The classical theory considers integral and/or terminal costs. However, minimizing the epidemic peak or the prevalence duration above a certain threshold (related to hospital capacity) turn out to be more relevant criteria. However, they cannot be expressed under classic form, or present a lack of regularity. These difficulties have recently been tackled by different approaches: approximation techniques leading to numerical procedures; and equivalent reformulations in higher dimension. We will apply and generalize these results to a class of problems rich enough to include epidemiological models. We will focus on the synthesis of optimal or sub-optimal state feedbacks with guaranteed value, by combining analytical and numerical approaches.

Observers coupled with these control laws will also be tested. An originality of the project is to consider random models of population and measurement noise, and their deterministic approximations, for which observers will be established. The performances of the observers (and of the control laws) will then be tested in a more realistic framework of noisy data.

The consortium brings together researchers familiar with automatic control, optimal control and epidemiological models. All of them contributed on at least one of the three themes considered, and carried out together several collaborations around mathematical epidemiology. The funding requested mainly corresponds to the recruitment of two post-docs dealing with the two challenges above, missions and the organization of an international workshop, organized with the aim of popularizing the tools of Automation in Epidemiology.

Workplan.

T1. Model and data from Epidemiology

T1.a: Study of classical epidemiological compartmental models

T1.b: Exploration of existing techniques of semi-analytical approximations of SDE

T1.c: Use of simulations of stochastic version of ODE models developed

T2. Identification and observation with observer synthesis

T2.a: Observability and identifiability of various relevant epidemiological systems

T2.b: Design of estimators

T2.c: Consideration of sampled measurements

T2.d: Estimators/observers for semi-analytical approximations of SDE

T3: Optimal control problems with non-conventional criteria

T3.a: Time crisis criterion and regularizations

T3.b: L∞ criterion and reformulations in higher dimensions

T3.c: Feedback synthesis

Consortium.

• INRIA: Pierre-Alexandre Bliman (Paris) , Denis Efimov (Lille), Rosane Ushirobira (Lille), Abderrahman Iggidr (Metz)

• INRAE: Alain Rapaport (Montpellier), Patrice Loisel (Montpellier), Tewfik Sari (Montpellier), Claude Lobry (Nice)

• IRD: Bernard Cazelles (Paris), Gauthier Sallet (Metz)