One-dimensional Voter Model Interface Revisited

We consider the voter model on Z, starting with all 1's to the left of the origin and all 0's to the right of the origin. It is known that if the associated random walk kernel p has zero mean and a finite r-th moment for any r>3, then the evolution of the boundaries of the interface region between 1's and 0's converge in distribution to a standard Brownian motion (B_t)_{t>0} under diffusive scaling of space and time. This convergence fails when p has an infinite r-th moment for any r<3, due to the loss of tightness caused by a few isolated 1's appearing deep within the regions of all 0's (and vice versa) at exceptional times. In this note, we show that as long as p has a finite second moment, the measure-valued process induced by the rescaled voter model configuration is tight, and converges weakly to the measure-valued process 1_{x0.Comment: revised versio

Critical age-dependent branching Markov processes and their scaling limits

This paper studies: (i) the long-time behaviour of the empirical distribution of age and normalized position of an age-dependent critical branching Markov process conditioned on non-extinction; and (ii) the super-process limit of a sequence of age dependent critical branching Brownian motions

Infinite dimensional stochastic differential equations of Ornstein-Uhlenbeck type

We consider the operator \sL f(x)=\tfrac12 \sum_{i,j=1}^\infty a_{ij}(x)\frac{\del^2 f}{\del x_i \del x_j}(x)-\sum_{i=1}^\infty \lam_i x_i b_i(x) \frac{\del f}{\del x_i}(x). We prove existence and uniqueness of solutions to the martingale problem for this operator under appropriate conditions on the $a_{ij}, b_i$, and \lam_i. The process corresponding to \sL solves an infinite dimensional stochastic differential equation similar to that for the infinite dimensional Ornstein-Uhlenbeck process

Survival of contact processes on the hierarchical group

We consider contact processes on the hierarchical group, where sites infect other sites at a rate depending on their hierarchical distance, and sites become healthy with a constant recovery rate. If the infection rates decay too fast as a function of the hierarchical distance, then we show that the critical recovery rate is zero. On the other hand, we derive sufficient conditions on the speed of decay of the infection rates for the process to exhibit a nontrivial phase transition between extinction and survival. For our sufficient conditions, we use a coupling argument that compares contact processes on the hierarchical group with freedom two with contact processes on a renormalized lattice. An interesting novelty in this renormalization argument is the use of a result due to Rogers and Pitman on Markov functionals.Comment: Minor changes compared to previous version. Final version. 30 pages. 1 figur

Second round statewide sentinel-based population survey for estimation of the burden of active infection and anti-SARS-CoV-2 IgG antibodies in the general population of Karnataka, India, during January-February 2021

Objective Demonstrate the feasibility of using the existing sentinel surveillance infrastructure to conduct the second round of the serial cross-sectional sentinel-based population survey. Assess active infection, seroprevalence, and their evolution in the general population across Karnataka. Identify local variations for locally appropriate actions. Additionally, assess the clinical sensitivity of the testing kit used on account of variability of antibody levels in the population. Methods The cross-sectional study of 41,228 participants across 290 healthcare facilities in all 30 districts of Karnataka was done among three groups of participants (low, moderate, and high-risk). The geographical spread was sufficient to capture local variations. Consenting participants were subjected to real-time reverse transcription-polymerase chain reaction (RT-PCR) testing, and antibody (IgG) testing. Clinical sensitivity was assessed by conducting a longitudinal study among participants identified as COVID-19 positive in the first survey round. Results Overall weighted adjusted seroprevalence of IgG was 15.6% (95% CI: 14.9–16.3), crude IgG prevalence was 15.0% and crude active infection was 0.5%. Statewide infection fatality rate (IFR) was estimated as 0.11%, and COVID-19 burden estimated between 26.1 to 37.7% (at 90% confidence). Further, Cases-to-infections ratio (CIR) varied 3-35 across units and IFR varied 0.04–0.50% across units. Clinical sensitivity of the IgG ELISA test kit was estimated as ≥38.9%. Conclusion We demonstrated the feasibility and simplicity of sentinel-based population survey in measuring variations in subnational and local data, useful for locally appropriate actions in different locations. The sentinel-based population survey thus helped identify districts that needed better testing, reporting, and clinical management. The state was far from attaining natural immunity during the survey and hence must step up vaccination coverage and enforce public health measures to prevent the spread of COVD-19.The Institutional Ethics Committee (IEC) of the Indian Institute of Public Health - Bengaluru campus reviewed and approved the study (vide. IIPHHB/TRCIEC/174/2020) and the subsequent change of protocol to perform the longitudinal study (vide PHFI/IIPH-BLR/076/2020-21). We informed the participants of the purpose of the surveys, how the samples would be taken and requested them to respond to the screening questions. After obtaining informed consent, we noted basic demographic details, exposure history, symptoms observed in the previous month, and clinical history. Participants' test results were shared with them by the concerned healthcare facility.Scopu