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Abstract The purpose of this work is to make a case for epidemiological models with fractional exponent in the contribution of sub-populations to the incidence rate. More specifically, we question the standard assumption in the literature on epidemiological models, where the incidence rate dictating propagation of infections is taken to be proportional to the product between the infected and susceptible sub-populations; a model that relies on strong mixing between the two groups and widespread contact between members of the groups. We contend, that contact between infected and susceptible individuals, especially during the early phases of an epidemic, takes place over a (possibly diffused) boundary between the respective sub-populations. As a result, the rate of transmission depends on the product of fractional powers instead. The intuition relies on the fact that infection grows in geographically concentrated cells, in contrast to the standard product model that relies on complete mixing of the susceptible to infected sub-populations. We validate the hypothesis of fractional exponents (1) by numerical simulation for disease propagation in graphs imposing a local structure to allowed disease transmissions and (2) by fitting the model to the JHU CSSE COVID-19 Data for the period Jan-22-20 to April-30-20, for the countries of Italy, Germany, France, and Spain. 

We consider damped stochastic systems in a controlled (time-varying) potential and study their transition between specified Gibbs-equilibria states in finite time. By the second law of thermody- namics, the minimum amount of work needed to transition from one equilibrium state to another is the difference between the Helmholtz free energy of the two states and can only be achieved by a reversible (infinitely slow) process. The minimal gap between the work needed in a finite-time transition and the work during a reversible one, turns out to equal the square of the optimal mass transport (Wasserstein- 2) distance between the two end-point distributions times the inverse of the duration needed for the transition. This result, in fact, relates non-equilibrium optimal control strategies (protocols) to gradient flows of entropy functionals via the Jordan-Kinderlehrer-Otto scheme. The purpose of this paper is to introduce ideas and results from the emerging field of stochastic thermodynamics in the setting of classical regulator theory, and to draw connections and derive such fundamental relations from a control perspective in a multivariable setting. 

The classical Monge–Kantorovich (MK) problem as originally posed is concerned with how best to move a pile of soil or rubble to an excavation or fill with the least amount of work relative to some cost function. When the cost is given by the square of the Euclidean distance, one can define a metric on densities called the Wasserstein distance . In this note, we formulate a natural matrix counterpart of the MK problem for positive-definite density matrices. We prove a number of results about this metric including showing that it can be formulated as a convex optimisation problem, strong duality, an analogue of the Poincaré–Wirtinger inequality and a Lax–Hopf–Oleinik–type result. 

We seek network routing towards a desired final distribution that can mediate possible random link failures. In other words, we seek a routing plan that utilizes alternative routes so as to be relatively robust to link failures. To this end, we provide a mathematical formulation of a relaxed transport problem where the final distribution only needs to be close to the desired one. The problem is cast as a maximum entropy problem for probability distributions on paths with an added terminal cost. The entropic regularizing penalty aims at distributing the choice of paths amongst possible alternatives. We prove that the unique solution may be obtained by solving a generalized Schro ̈dinger system of equations. An iterative algorithm to com- pute the solution is provided. Each iteration of the algorithm contracts the distance (in the Hilbert metric) to the optimal solution by more than 1/2, leading to extremely fast convergence. 

We propose unbalanced versions of the quantum mechanical version of optimal mass transport that is based on the Lindblad equation describing open quantum systems. One of them is a natural interpolation framework between matrices and matrix-valued measures via a quantum mechanical formulation of Fisher-Rao information and the matricial Wasserstein distance, and the second is an interpolation between Wasserstein distance and Frobenius norm. We also give analogous results for the matrix-valued density measures, i.e., we add a spatial dependency on the density matrices. This might extend the applications of the framework to interpolating matrix-valued densities/images with unequal masses.