In this paper, we present a spatialized extension of a SIR model that accounts for undetected infections and recoveries as well as the load on hospital services. The spatialized compartmental model we introduce is governed by a set of partial differential equations (PDEs) defined on a spatial domain with complex boundary. We propose to solve the set of PDEs defining our model by using a meshless numerical method based on a finite difference scheme in which the spatial operators are approximated by using radial basis functions. Such an approach is reputed as flexible for solving problems on complex domains. Then we calibrate our model on the French department of Isère during the first period of lockdown, using daily reports of hospital occupancy in France. Our methodology allows to simulate the spread of Covid-19 pandemic at a departmental level, and for each compartment. However, the simulation cost prevents from online short-term forecast. Therefore, we propose to rely on reduced order modeling to compute short-term forecasts of infection number. The strategy consists in learning a time-dependent reduced order model with few compartments from a collection of evaluations of our spatialized detailed model, varying initial conditions and parameter values. A set of reduced bases is learnt in an offline phase while the projection on each reduced basis and the selection of the best projection is performed online, allowing short-term forecast of the global number of infected individuals in the department. The original approach proposed in this paper is generic and could be adapted to model and simulate other dynamics described by a model with spatially distributed parameters of the type diffusion-reaction on complex domains. Also, the time-dependent model reduction techniques we introduced could be leveraged to compute control strategies related to such dynamics.

In this paper we investigate feedback control techniques for the COVID-19 pandemic which are able to guarantee that the capacity of available intensive care unit beds is not exceeded. The control signal models the social distancing policies enacted by local policy makers. We propose a control design based on the bang-bang funnel controller which is robust with respect to uncertainties in the parameters of the epidemiological model and only requires measurements of the number of individuals who require medical attention. Simulations illustrate the efficiency of the proposed controller.

In our previous work, we investigated detectability of discrete event systems, which is defined as the ability to determine the current and subsequent states of a system based on observation. For different applications, we defined four types of detectabilities: (weak) detectability, strong detectability, (weak) periodic detectability, and strong periodic detectability. In this paper, we extend our results in three aspects. (1) We extend detectability from deterministic systems to nondeterministic systems. Such a generalization is necessary because there are many systems that need to be modeled as nondeterministic discrete event systems. (2) We develop polynomial algorithms to check strong detectability. The previous algorithms are based on observer whose construction is of exponential complexity, while the new algorithms are based on a new automaton called detector. (3) We extend detectability to D-detectability. While detectability requires determining the exact state of a system, D-detectability relaxes this requirement by asking only to distinguish certain pairs of states. With these extensions, the theory on detectability of discrete event systems becomes more applicable in solving many practical problems.

This paper studies monotone tridiagonal systems with negative feedback. These systems possess the Poincaré-Bendixson property, which implies that, if orbits are bounded, if there is a unique steady state and this unique steady state is asymptotically stable, and if one can rule out periodic orbits, then the steady state is globally asymptotically stable. Two different approaches are discussed to rule out period orbits, one based on direct linearization and another one based on the theory of second additive compound matrices. Among the examples that illustrate the theoretical results is the classical Goldbeter model of the circadian rhythm.

Our previous work considers detectability of discrete event systems which is to determine the current state and subsequent states of a system based on event observation. We assume that event observation is static, that is, if an event is observable, then all its occurrences are observable. However, in practical systems such as sensor networks, event observation often needs to be dynamic, that is, the occurrences of same events may or may not be observable, depending on the state of the system. In this paper, we generalize static event observation into dynamic event observation and consider the detectability problem under dynamic event observation. We define four types of detectabilities. To check detectabilities, we construct the observer with exponential complexity. To reduce computational complexity, we can also construct a detector with polynomial complexity to check strong detectabilities. Dynamic event observation can be implemented in two possible ways: a passive observation and an active observation. For the active observation, we discuss how to find minimal event observation policies that preserve four types of detectabilities respectively.

After stimulation by chemoattractant, Dictyostelium cells exhibit a rapid response. The concentrations of several intracellular proteins rise rapidly reaching their maximum levels approximately 5-10 seconds, after which they return to prestimulus levels. This response, which is found in many other chemotaxing cells, is an example of a step disturbance rejection, a process known to biologists as perfect adaptation. Unlike other cells, however, the initial first peak observed in the chemoattractant-induced response of Dictyostelium cells is then followed by a slower, smaller phase peaking approximately one to two minutes after the stimulus. Until recently, the nature of this biphasic response has been poorly understood. Moreover, the origin for the second phase is unknown. In this paper we conjecture the existence of a feedback path between the response and stimulus. Using a mathematical model of the chemoattractant-induced response in cells, and standard tools from control engineering, we show that positive feedback may elicit this second peak.