Georgia Institute of Technology College of Sciences

We will introduce a PDE model to investigate how epidemic metrics, such as the basic reproductive ratio R_0 and infection prevalence, depend on a pathogen's virulence. We define virulence as all harm inflicted on the host by the pathogen, so it includes direct virulence (increased host mortality and decreased fecundity) and indirect virulence (increased predation on infected hosts). To study these effects we use a Daphnia-parasite disease system. Daphnia are freshwater crustaceans that get infected while feeding, by consuming free-living parasite spores. These spores after they are ingested, they start reproducing within the host and the host eventually dies. Dead hosts decay releasing the spores they contain back in the water column. Visual predators, such as fish, can detect infected hosts easier because they become opaque, hence they prey preferentially on them. Our model includes two host classes (susceptible and infected), the free-living propagules, and the food resource (algae). Using experimental data, we obtain the qualitative curves for the dependence of disease-induced mortality and fecundity reduction on the age of infection. Among other things, we will show that in order the predator to keep the host population healthy, it needs to (i) detect the infected hosts very soon after they become infected and (ii) show very high preference on consuming them in comparison to the uninfected hosts. In order to address questions about the evolution of virulence, we will also discuss how we defined the invasion fitness for this compartmental model. We will finish with some pairwise invasibility plots, that show when a mutant strain can invade the resident strain in this disease system.

We consider a diffusion on a potential landscape which is given by a smooth Hamiltonian in the regime of small noise. We give a new proof of the Eyring-Kramers formula for the spectral gap of the associated generator of the diffusion. The proof is based on a refinement of the two-scale approach introduced by Grunewald, Otto, Villani, and Westdickenberg and of the mean-difference estimate introduced by Chafai and Malrieu. The Eyring-Kramers formula follows as a simple corollary from two main ingredients : The first one shows that the Gibbs measure restricted to a domain of attraction has a "good" Poincaré constant mimicking the fast convergence of the diffusion to metastable states. The second ingredient is the estimation of the mean-difference by a new weighted transportation distance. It contains the main contribution of the spectral gap, resulting from exponential long waiting times of jumps between metastable states of the diffusion. This new approach also allows to derive sharp estimates on the log-Sobolev constant. This is joint work with Andre Schlichting.

Database privacy has garnered a recent surge in interest from the theoretical science community following the seminal work of Dwork 2006, which proposed the strong notion of differential privacy. In this setting, each row of a database corresponds to the data owned by some (distinct) individual. An analyst submits a database query to a differentially private mechanism, which replies with a noisy answer guaranteeing privacy for the data owners and accuracy for the analyst. The mechanism's privacy parameter \epsilon is correlated negatively with privacy and positively with accuracy.This work builds a framework for creating and analyzing a market that 1) solves for some socially efficient value of \epsilon using the privacy and accuracy preferences of a heterogeneous body of data owners and a single analyst, 2) computes a noisy statistic on the database, and 3) collects and distributes payments for privacy that elicit truthful reporting of data owners' preferences. We present a market for database privacy in this new framework expanding on the public goods market of Groves and Ledyard, 1977.

A sandpile on a graph is an integer-valued function on the vertices. It evolves according to local moves called topplings. Some sandpiles stabilize after a finite number of topplings, while others topple forever. For any sandpile s_0 if we repeatedly add a grain of sand at an independent random vertex, we eventually reach a sandpile s_\tau that topples forever. Statistical physicists Poghosyan, Poghosyan, Priezzhev and Ruelle conjectured a precise value for the expected amount of sand in this "threshold state" s_\tau in the limit as s_0 goes to negative infinity. I will outline the proof of this conjecture in http://arxiv.org/abs/1402.3283 and explain the big-picture motivation, which is to give more predictive power to the theory of "self-organized criticality".