Are We #StayingHome to Flatten the Curve?

The recent spread of COVID-19 across the U.S. led to concerted efforts by states to ``flatten the curve" through the adoption of stay-at-home mandates that encourage individuals to reduce travel and maintain social distance. Combining data on changes in travel activity with COVID-19 health outcomes and state policy adoption timing, we characterize nationwide changes in mobility patterns and isolate the portion attributable to statewide mandates. We find evidence of dramatic nationwide declines in mobility prior to adoption of any statewide mandates. Once states adopt a mandate, we estimate further mandate-induced declines between 2.1 and 7.0 percentage points across methods that account for states' differences in travel behavior prior to policy adoption. In addition, we investigate the effects of stay-at-home mandates on changes in COVID-19 health outcomes while controlling for pre-trends and observed pre-treatment mobility patterns. We estimate mandate-induced declines between 0.13 and 0.17 in deaths (5.6 to 6.0 in hospitalizations) per 100 thousand across methods. Across 43 adopting states, this represents 23,366-30,144 fewer deaths (and roughly one million averted hospitalizations) for the months of March and April - which indicates that death rates could have been 42-54% higher had states not adopted statewide policies. We further find evidence that changes in mobility patterns prior to adoption of statewide policies also played a role in reducing COVID-19 mortality and morbidity. Adding in averted deaths due to pre-mandate social distancing behavior, we estimate a total of 48-71,000 averted deaths from COVID-19 for the two-month period. Given that the actual COVID-19 death toll for March and April was 55,922, our estimates suggest that deaths would have been 1.86-2.27 times what they were absent any stay-at-home mandates during this period. These estimates represent a lower bound on the health impacts of stay-at-home policies, as they do not account for spillovers or undercounting of COVID-19 mortality. Our findings indicate that early behavior changes and later statewide policies reduced death rates and helped attenuate the negative consequences of COVID-19. Further, our findings of substantial reductions in mobility prior to state-level policies convey important policy implications for re-opening.Take Away Link https://are.berkeley.edu/sites/are.berkeley.edu/files/PolicyTakeAway_Web.pd

The football player and the infinite series

This is the text of an expository talk given at the May 1997 Detroit meeting of the American Mathematical Society. It is a tale of a famous football player and a subtle problem he posed about the uniform convergence of Dirichlet series. Hiding in the background is the theory of analytic functions of an infinite number of variables

Mocposite functions

Traditional mathematical notation can lead to confusion. Expressions that appear to define composite functions sometimes do not. A particular example with engineering applications is studied in detail.Comment: To appear in the American Mathematical Monthl

The Master Ward Identity

In the framework of perturbative quantum field theory (QFT) we propose a new, universal (re)normalization condition (called 'master Ward identity') which expresses the symmetries of the underlying classical theory. It implies for example the field equations, energy-momentum, charge- and ghost-number conservation, renormalized equal-time commutation relations and BRST-symmetry. It seems that the master Ward identity can nearly always be satisfied, the only exceptions we know are the usual anomalies. We prove the compatibility of the master Ward identity with the other (re)normalization conditions of causal perturbation theory, and for pure massive theories we show that the 'central solution' of Epstein and Glaser fulfills the master Ward identity, if the UV-scaling behavior of its individual terms is not relatively lowered. Application of the master Ward identity to the BRST-current of non-Abelian gauge theories generates an identity (called 'master BRST-identity') which contains the information which is needed for a local construction of the algebra of observables, i.e. the elimination of the unphysical fields and the construction of physical states in the presence of an adiabatically switched off interaction.Comment: 73 pages, version to appear in Rev. Math. Phy