Spatially Inhomogeneous Evolutionary Games

We introduce and study a mean-field model for a system of spatially distributed players interacting through an evolutionary game driven by a replicator dynamics. Strategies evolve by a replicator dynamics influenced by the position and the interaction between different players and return a feedback on the velocity field guiding their motion. One of the main novelties of our approach concerns the description of the whole system, which can be represent-dimensional state space (pairs (x, σ) of position and distribution of strategies). We provide a Lagrangian and a Eulerian description of the evolution, and we prove their equivalence, together with existence, uniqueness, and stability of the solution. As a byproduct of the stability result, we also obtain convergence of the finite agents model to our mean-field formulation, when the number N of the players goes to infinity, and the initial discrete distribution of positions and strategies converge. To this aim we develop some basic functional analytic tools to deal with interaction dynamics and continuity equations in Banach spaces that could be of independent interest. © 2021 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC

Development and validation of a clinical risk score to predict the risk of SARS-CoV-2 infection from administrative data: A population-based cohort study from Italy

Background The novel coronavirus (SARS-CoV-2) pandemic spread rapidly worldwide increasing exponentially in Italy. To date, there is lack of studies describing clinical characteristics of the people at high risk of infection. Hence, we aimed (i) to identify clinical predictors of SARSCoV-2 infection risk, (ii) to develop and validate a score predicting SARS-CoV-2 infection risk, and (iii) to compare it with unspecific scores. Methods Retrospective case-control study using administrative health-related database was carried out in Southern Italy (Campania region) among beneficiaries of Regional Health Service aged over than 30 years. For each person with SARS-CoV-2 confirmed infection (case), up to five controls were randomly matched for gender, age and municipality of residence. Odds ratios and 90% confidence intervals for associations between candidate predictors and risk of infection were estimated by means of conditional logistic regression. SARS-CoV-2 Infection Score (SIS) was developed by generating a total aggregate score obtained from assignment of a weight at each selected covariate using coefficients estimated from the model. Finally, the score was categorized by assigning increasing values from 1 to 4. Discriminant power was used to compare SIS performance with that of other comorbidity scores. Results Subjects suffering from diabetes, anaemias, Parkinson’s disease, mental disorders, cardiovascular and inflammatory bowel and kidney diseases showed increased risk of SARSCoV-2 infection. Similar estimates were recorded for men and women and younger and older than 65 years. Fifteen conditions significantly contributed to the SIS. As SIS value increases, risk progressively increases, being odds of SARS-CoV-2 infection among people with the highest SIS value (SIS = 4) 1.74 times higher than those unaffected by any SIS contributing conditions (SIS = 1). Conclusion Conditions and diseases making people more vulnerable to SARS-CoV-2 infection were identified by the current study. Our results support decision-makers in identifying high-risk people and adopting of preventive measures to minimize the spread of further epidemic waves

Detecting early signals of COVID-19 outbreaks in 2020 in small areas by monitoring healthcare utilisation databases: first lessons learned from the Italian Alert_CoV project

During the COVID-19 pandemic, large-scale diagnostic testing and contact tracing have proven insufficient to promptly monitor the spread of infections.AimTo develop and retrospectively evaluate a system identifying aberrations in the use of selected healthcare services to timely detect COVID-19 outbreaks in small areas. Methods: Data were retrieved from the healthcare utilisation (HCU) databases of the Lombardy Region, Italy. We identified eight services suggesting a respiratory infection (syndromic proxies). Count time series reporting the weekly occurrence of each proxy from 2015 to 2020 were generated considering small administrative areas (i.e. census units of Cremona and Mantua provinces). The ability to uncover aberrations during 2020 was tested for two algorithms: the improved Farrington algorithm and the generalised likelihood ratio-based procedure for negative binomial counts. To evaluate these algorithms' performance in detecting outbreaks earlier than the standard surveillance, confirmed outbreaks, defined according to the weekly number of confirmed COVID-19 cases, were used as reference. Performances were assessed separately for the first and second semester of the year. Proxies positively impacting performance were identified. Results: We estimated that 70% of outbreaks could be detected early using the proposed approach, with a corresponding false positive rate of ca 20%. Performance did not substantially differ either between algorithms or semesters. The best proxies included emergency calls for respiratory or infectious disease causes and emergency room visits. Conclusion: Implementing HCU-based monitoring systems in small areas deserves further investigations as it could facilitate the containment of COVID-19 and other unknown infectious diseases in the future

Existence and stability results for Fokker-Planck equations with log-concave reference measure

We study Markov processes associated with stochastic differential equations, whose non-linearities are gradients of convex functionals. We prove a general result of existence of such Markov processes and a priori estimates on the transition probabilities. The main result is the following stability property: if the associated invariant measures converge weakly, then the Markov processes converge in law. The proofs are based on the interpretation of a Fokker–Planck equation as the steepest descent flow of the relative entropy functional in the space of probability measures, endowed with the Wasserstein distance

Gradient flows in metric spaces and in the space of probability measures

Originating from lectures by L. Ambrosio at the ETH Zürich in Fall 2001, substantially extended and revised in cooperation with the co-authors, it serves as textbook and reference book on the topic and it is presented as much as possible in a self-contained way. The book contains new results that have never appeared elsewhere and it is devoted to a theory of gradient flows in spaces which are not necessarily endowed with a natural linear or differentiable structure. It consists of two parts, the first one concerning gradient flows in metric spaces and the second one devoted to gradient flows in the space of probability measures on a separable Hilbert space, endowed with the Kantorovich-Rubinstein-Wasserstein distance. The two parts have some connections, due to the fact that the space of probability measures provides an important model to which the "metric" theory applies. However, the book is conceived in such a way that the two parts can be read independently, the first one by the reader more interested in non-smooth analysis and analysis in metric spaces, and the second one by the reader more orientated towards the applications in partial differential equations, measure theory and probability. Written for: Graduate and postgraduate students, researcher

Stability of flows associated to gradient vector fields and convergence of iterated transport maps

In this paper we address the problem of stability of flows associated to a sequence of vector fields under minimal regularity requirements on the limit vector field, that is supposed to be a gradient. We apply this stability result to show the convergence of iterated compositions of optimal transport maps arising in the implicit time discretization (with respect to the Wasserstein distance) of nonlinear evolution equations of a diffusion type. Finally, we use these convergence results to study the gradient flow of a particular class of polyconvex functionals recently considered by Gangbo, Evans and Savin. We solve some open problems raised in their paper and obtain existence and uniqueness of solutions under weaker regularity requirements and with no upper bound on the jacobian determinant of the initial datum

A Functional Data Analysis Approach to Left Ventricular Remodeling Assessment

Left ventricular remodeling is a mechanism common to various cardiovascular diseases affecting myocardial morphology. It can be often overlooked in clinical practice since the parameters routinely employed in the diagnostic process (e.g., the ejection fraction) mainly focus on evaluating volumetric aspects. Nevertheless, the integration of a quantitative assessment of structural modifications can be pivotal in the early individuation of this pathology. In this work, we propose an approach based on functional data analysis to evaluate myocardial contractility. A functional representation of ventricular shape is introduced, and functional principal component analysis and depth measures are used to discriminate healthy subjects from those affected by left ventricular hypertrophy. Our approach enables the integration of higher informative content compared to the traditional clinical parameters, allowing for a synthetic representation of morphological changes in the myocardium, which could be further explored and considered for future clinical practice implementation

Gradient flows with metric and differentiable structures, and applications to the Wasserstein space

In this paper we summarize some of the main results of a orthcoming book on this topic, where we examine in detail the theory of curves of maximal slope in a general metric setting, and study in detail the case of the Wasserstein space of probability measures. In the first part we derive new general conditions ensuring convergence of the implicit time discretization scheme to a curve of maximal slope, the uniqueness, and the error estimates. In the second part we study in detail the differentiable structure of the Wasserstein space, to which the metric theory applies, and use this structure to give also an equivalent concept of gradient flow. Our analysis includes measures in infinite-dimensional Hilbert spaces and it does not require any absolute continuity assumption on the measures involved