Measure valued solutions of sub-linear diffusion equations with a drift term

In this paper we study nonnegative, measure valued solutions of the initial value problem for one-dimensional drift-diffusion equations when the nonlinear diffusion is governed by an increasing $C^1$ function $\beta$ with $\lim_{r\to +\infty} \beta(r)<+\infty$. By using tools of optimal transport, we will show that this kind of problems is well posed in the class of nonnegative Borel measures with finite mass $m$ and finite quadratic momentum and it is the gradient flow of a suitable entropy functional with respect to the so called $L^2$-Wasserstein distance. Due to the degeneracy of diffusion for large densities, concentration of masses can occur, whose support is transported by the drift. We shall show that the large-time behavior of solutions depends on a critical mass ${m}_{\rm c}$, which can be explicitely characterized in terms of $\beta$ and of the drift term. If the initial mass is less then ${m}_{\rm c}$, the entropy has a unique minimizer which is absolutely continuous with respect to the Lebesgue measure. Conversely, when the total mass $m$ of the solutions is greater than the critical one, the steady state has a singular part in which the exceeding mass ${m} - {m}_{\rm c}$ is accumulated.Comment: 30 page

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

Parabolic problems with mixed variable lateral conditions: an abstract approach

We study the initial value problem for parabolic second order equa- tions with mixed and time-dependent boundary conditions obtaining optimal regularity results under weak assumptions on the data and on the geometrical behavior of the boundary. An approximation approach to abstract evolution equations on variable domains is the basic tool we develop; an application to parabolic problems in non-cylindrical domains is also given

On the regularity of the positive part of functions

The paper studies the regularity properties of the truncation operator max(u,0) in Sobolev and Besov spaces and the space of functions with gradient of bounded variation on a Lipschitz open set

Regularity and perturbation results for mixed second order elliptic problems

We study a mixed boundary value problem for elliptic second order equations obtaining optimal regularity results under weak assumptions on the data. We also consider the dependence of the solution with respect to perturbations of the boundary sets carrying the Dirichlet and the Neumann conditions

Gradient flows and diffusion semigroups in metric spaces under lower curvature bounds

We present some new results concerning well-posedness of gradient flows generated by λ-convex functionals in a wide class of metric spaces, including Alexandrov spaces satisfying a lower curvature bound and the corresponding L2 -Wasserstein spaces. Applications to the gradient flow of Entropy functionals in metric-measure spaces with Ricci curvature bounded from below and to the corresponding diffusion semigroup are also considered. These results have been announced during the workshop on “Optimal Transport: theory and applications” held in Pisa, November 2006

Weak solutions and maximal regularity for abstract evolution inequalities

We study the regularity and the approximation of the solution of a parabolic evolution inequality in the framework of a Hilbert triple. We give a weak formulation which allows for data with weak regularity and we obtain new existence and regularity results. We also prove an optimal error estimate for the backward Euler discretization and we apply these results to the porous medium and the Stefan problem

Approximation and regularity of evolution variational inequalities

In the framework of a Hilbert triple {V, H, V′} we study the approximation and the regular- ity of parabolic variational inequalities, by a time discretization by means of the backward Euler scheme. Under suitable regularity hypotheses on the data, we prove that the order of convergence in H1(0, T ; H) is 1/2 and the solution belongs to Hs(0, T ; H), ∀ s < 3/2. Moreover, in the case of a symmetric linear operator with L2(0, T ; H) data, we prove the H1/2(0, T ; V )-regularity of the solution with the same error estimate in the “energy norm” of L2(0,T;V)∩L∞(0,T;H)