This article examines large time behaviour and the second eigenvalue problem for Markovian mean-field interacting particle systems with jumps. Our first main result is on the time required for convergence of the empirical measure process of the particle system to its invariant measure; we show that, there is a constant Λ ≥ 0 such that, when there are N particles in the system and when time is of the order exp{N (Λ + O(1))}, the process has mixed well and is very close to its invariant measure. We then obtain large-N asymptotics of the second eigenvalue of the generator associated with the empirical measure process when it is reversible with respect to its invariant measure. We show that its absolute value scales like exp{−N Λ}. The main tools used in establishing our results are the large deviation properties of the empirical measure process from its large-N limit. As an application of the study of large time behaviour, we also show convergence of the empirical measure of the system of particles to a global minimum of a certain entropy function when particles are added over time in a controlled fashion. The controlled addition of particles is analogous to the cooling schedule associated with the search for a global minimum of a function using the simulated annealing algorithm.
This paper studies large deviations of a "fully coupled" finite state mean-field interacting particle system in a fast varying environment. The empirical measure of the particles evolves in the slow time scale and the random environment evolves in the fast time scale. Our main result is the path-space large deviation principle for the joint law of the empirical measure process of the particles and the occupation measure process of the fast environment. This extends previous results known for two time scale diffusions to two time scale mean-field models with jumps. Our proof is based on the method of stochastic exponentials. We characterise the rate function by studying a certain variational problem associated with an exponential martingale.
COVID-19 vaccination is being rolled out among the general population in India. Spatial heterogeneities exist in seroprevalence and active infections across India. Using a spatially explicit age-stratified model of Karnataka at the district level, we study three spatial vaccination allocation strategies under different vaccination capacities and a variety of non-pharmaceutical intervention (NPI) scenarios. The models are initialised using on-the-ground datasets that capture reported cases, seroprevalence estimates, seroreversion and vaccine rollout plans. The three vaccination strategies we consider are allocation in proportion to the district populations, allocation in inverse proportion to the seroprevalence estimates, and allocation in proportion to the case-incidence rates during a reference period.
The results suggest that the effectiveness of these strategies (in terms of cumulative cases at the end of a four-month horizon) are within 2% of each other, with allocation in proportion to population doing marginally better at the state level. The results suggest that the allocation schemes are robust and thus the focus should be on the easy to implement scheme based on population. Our immunity waning model predicts the possibility of a subsequent resurgence even under relatively strong NPIs. Finally, given a per-day vaccination capacity, our results suggest the level of NPIs needed for the healthcare infrastructure to handle a surge.
This paper considers the family of invariant measures of Markovian mean-field interacting particle systems on a countably infinite state space and studies its large deviation asymptotics. The Freidlin-Wentzell quasipotential is the usual candidate rate function for the sequence of invariant measures indexed by the number of particles. The paper provides two counterexamples where the quasipotential is not the rate function. The quasipotential arises from finite horizon considerations. However there are certain barriers that cannot be surmounted easily in any finite time horizon, but these barriers can be crossed in the stationary regime. Consequently, the quasipotential is infinite at some points where the rate function is finite. After highlighting this phenomenon, the paper studies some sufficient conditions on a class of interacting particle systems under which one can continue to assert that the Freidlin-Wentzell quasipotential is indeed the rate function.
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