Optimal design of lock-down and reopening policies for early-stage epidemics through SIR-D models.

The diffusion of COVID-19 represents a real threat for the health and economic system of a country. Therefore the governments have to adopt fast containment measures in order to stop its spread and to prevent the related devastating consequences. In this paper, a technique is proposed to optimally design the lock-down and reopening policies so as to minimize an aggregate cost function accounting for the number of individuals that decease due to the spread of COVID-19. A constraint on the maximal number of concomitant infected patients is also taken into account in order to prevent the collapse of the health system. The optimal procedure is built on the basis of a simple SIR model that describes the outbreak of a generic disease, without attempting to accurately reproduce all the COVID-19 epidemic features. This modeling choice is motivated by the fact that the containing measurements are actuated during the very first period of the outbreak, when the characteristics of the new emergent disease are not known but timely containment actions are required. In fact, as a consequence of dealing with poor preliminary data, the simplest modeling choice is able to reduce unidentifiability problems. Further, the relative simplicity of this model allows to compute explicitly its solutions and to derive closed-form expressions for the maximum number of infected and for the steady-state value of deceased individuals. These expressions can be then used to design static optimization problems so to determine the (open-loop) optimal lock-down and reopening policies for early-stage epidemics accounting for both the health and economic costs.

A time-varying SIRD model for the COVID-19 contagion in Italy.

The purpose of this work is to give a contribution to the understanding of the COVID-19 contagion in Italy. To this end, we developed a modified Susceptible-Infected-Recovered-Deceased (SIRD) model for the contagion, and we used official data of the pandemic for identifying the parameters of this model. Our approach features two main non-standard aspects. The first one is that model parameters can be time-varying, allowing us to capture possible changes of the epidemic behavior, due for example to containment measures enforced by authorities or modifications of the epidemic characteristics and to the effect of advanced antiviral treatments. The time-varying parameters are written as linear combinations of basis functions and are then inferred from data using sparse identification techniques. The second non-standard aspect resides in the fact that we consider as model parameters also the initial number of susceptible individuals, as well as the proportionality factor relating the detected number of positives with the actual (and unknown) number of infected individuals. Identifying the model parameters amounts to a non-convex identification problem that we solve by means of a nested approach, consisting in a one-dimensional grid search in the outer loop, with a Lasso optimization problem in the inner step.

Boolean network analysis through the joint use of linear algebra and algebraic geometry.

Among the various phenomena that can be modeled by Boolean networks, i.e., discrete-time dynamical systems with binary state variables, gene regulatory interactions are especially well known. Therefore, the analysis of Boolean networks is critical, e.g., to identify genetic pathways and to predict the effects of mutations on the cell functionality. Two methodologies (i.e., the semi-tensor product and the Gröbner bases over finite fields) have recently been proposed to tackle the problem of determining cycles and attractors (with the corresponding basin of attraction) for such systems. Here, it is shown that, by suitably coupling methodologies taken from these two fields (i.e., linear algebra and algebraic geometry), it is not only possible to determine cycles and attractors, but also to find closed-form solutions of the Boolean network. Such a goal is pursued by finding an immersion that recasts the Boolean dynamics in a linear form and by computing the closed-form solution of the latter system. The effectiveness of this technique is demonstrated by fully computing the solutions of the Boolean network modeling the differentiation of the Th-lymphocyte, a type of white blood cells involved in the human adaptive immune system.

Data-Driven Policy Iteration for Nonlinear Optimal Control Problems.

The design of optimal control laws for nonlinear systems is tackled without knowledge of the underlying plant and of a functional description of the cost function. The proposed data-driven method is based only on real-time measurements of the state of the plant and of the (instantaneous) value of the reward signal and relies on a combination of ideas borrowed from the theories of optimal and adaptive control problems. As a result, the architecture implements a policy iteration strategy in which, hinging on the use of neural networks, the policy evaluation step and the computation of the relevant information instrumental for the policy improvement step are performed in a purely continuous-time fashion. Furthermore, the desirable features of the design method, including convergence rate and robustness properties, are discussed. Finally, the theory is validated via two benchmark numerical simulations.

Output Feedback Q-Learning for Linear-Quadratic Discrete-Time Finite-Horizon Control Problems.

An algorithm is proposed to determine output feedback policies that solve finite-horizon linear-quadratic (LQ) optimal control problems without requiring knowledge of the system dynamical matrices. To reach this goal, the Q -factors arising from finite-horizon LQ problems are first characterized in the state feedback case. It is then shown how they can be parameterized as functions of the input-output vectors. A procedure is then proposed for estimating these functions from input/output data and using these estimates for computing the optimal control via the measured inputs and outputs.

A Universal Approximation Result for Difference of Log-Sum-Exp Neural Networks.

We show that a neural network whose output is obtained as the difference of the outputs of two feedforward networks with exponential activation function in the hidden layer and logarithmic activation function in the output node, referred to as log-sum-exp (LSE) network, is a smooth universal approximator of continuous functions over convex, compact sets. By using a logarithmic transform, this class of network maps to a family of subtraction-free ratios of generalized posynomials (GPOS), which we also show to be universal approximators of positive functions over log-convex, compact subsets of the positive orthant. The main advantage of difference-LSE networks with respect to classical feedforward neural networks is that, after a standard training phase, they provide surrogate models for a design that possesses a specific difference-of-convex-functions form, which makes them optimizable via relatively efficient numerical methods. In particular, by adapting an existing difference-of-convex algorithm to these models, we obtain an algorithm for performing an effective optimization-based design. We illustrate the proposed approach by applying it to the data-driven design of a diet for a patient with type-2 diabetes and to a nonconvex optimization problem.

Log-Sum-Exp Neural Networks and Posynomial Models for Convex and Log-Log-Convex Data.

In this paper, we show that a one-layer feedforward neural network with exponential activation functions in the inner layer and logarithmic activation in the output neuron is a universal approximator of convex functions. Such a network represents a family of scaled log-sum exponential functions, here named log-sum-exp ( LSET ). Under a suitable exponential transformation, the class of LSET functions maps to a family of generalized posynomials GPOST , which we similarly show to be universal approximators for log-log-convex functions. A key feature of an LSET network is that, once it is trained on data, the resulting model is convex in the variables, which makes it readily amenable to efficient design based on convex optimization. Similarly, once a GPOST model is trained on data, it yields a posynomial model that can be efficiently optimized with respect to its variables by using geometric programming (GP). The proposed methodology is illustrated by two numerical examples, in which, first, models are constructed from simulation data of the two physical processes (namely, the level of vibration in a vehicle suspension system, and the peak power generated by the combustion of propane), and then optimization-based design is performed on these models.

A new metric for understanding hidden political influences from voting records.

Inspired by the increasing attention of the scientific community towards the understanding of human relationships and actions in social sciences, in this paper we address the problem of inferring from voting data the hidden influence on individuals from competing ideology groups. As a case study, we present an analysis of the closeness of members of the Italian Senate to political parties during the XVII Legislature. The proposed approach is aimed at automatic extraction of the relevant information by disentangling the actual influences from noise, via a two step procedure. First, a sparse principal component projection is performed on the standardized voting data. Then, the projected data is combined with a generative mixture model, and an information theoretic measure, which we refer to as Political Data-aNalytic Affinity (Political DNA), is finally derived. We show that the definition of this new affinity measure, together with suitable visualization tools for displaying the results of analysis, allows a better understanding and interpretability of the relationships among political groups.