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We show that an $n$-uniform maximal intersecting family has size at most $e^{-n^{0.5+o(1)}}n^n$. This improves a recent bound by Frankl ((2019) Comb. Probab. Comput.28(5) 733–739.). The Spread Lemma of Alweiss et al. ((2020) Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing.) plays an important role in the proof.
Written by Sheldon Ross and Erol Peköz, this text familiarises you with advanced topics in probability while keeping the mathematical prerequisites to a minimum. Topics covered include measure theory, limit theorems, bounding probabilities and expectations, coupling and Stein's method, martingales, Markov chains, renewal theory, and Brownian motion. No other text covers all these topics rigorously but at such an accessible level - all you need is an undergraduate-level understanding of calculus and probability. New to this edition are sections on the gambler's ruin problem, Stein's method as applied to exponential approximations, and applications of the martingale stopping theorem. Extra end-of-chapter exercises have also been added, with selected solutions available.This is an ideal textbook for students taking an advanced undergraduate or graduate course in probability. It also represents a useful resource for professionals in relevant application domains, from finance to machine learning.
Clostridium botulinum causes infant botulism by colonising the intestines and producing botulinum neurotoxin in situ. Previous reports have linked infant botulism cases to C. botulinum spores in household dust, yet the baseline incidence of C. botulinum spores in residential households is currently unknown. Vacuum cleaner dust from 963 households in 13 major Canadian cities was tested for C. botulinum using a novel real-time PCR assay directed against all known subtypes of the botulinum neurotoxin gene. None of the samples tested positive for C. botulinum. Analysis of a random subset of samples by MALDI Biotyper revealed that the most common anaerobic bacterial isolates were of the genus Clostridium and the most common species recovered overall was Clostridium perfringens. Dust that was spiked with C. botulinum spores of each toxin type successfully produced positive real-time PCR reactions. These control experiments indicate that this is a viable method for the detection of C. botulinum spores in household dust. We make several recommendations for future work that may help discover a common environmental source of C. botulinum spores that could lead to effective preventative measures for this rare but deadly childhood disease.
Longitudinal studies on the variations of phenotypic and genotypic characteristics of K. pneumoniae across two decades are rare. We aimed to determine the antimicrobial susceptibility and virulence factors for K. pneumoniae isolated from patients with bacteraemia or urinary tract infection (UTI) from 1999 to 2022. A total of 699 and 1,267 K. pneumoniae isolates were isolated from bacteraemia and UTI patients, respectively, and their susceptibility to twenty antibiotics was determined; PCR was used to identify capsular serotypes and virulence-associated genes. K64 and K1 serotypes were most frequently observed in UTI and bacteraemia, respectively, with an increasing frequency of K20, K47, and K64 observed in recent years. entB and wabG predominated across all isolates and serotypes; the least frequent virulence gene was htrA. Most isolates were susceptible to carbapenems, amikacin, tigecycline, and colistin, with the exception of K20, K47, and K64 where resistance was widespread. The highest average number of virulence genes was observed in K1, followed by K2, K20, and K5 isolates, which suggest their contribution to the high virulence of K1. In conclusion, we found that the distribution of antimicrobial susceptibility, virulence gene profiles, and capsular types of K. pneumoniae over two decades were associated with their clinical source.
Rodents and shrews are major reservoirs of various pathogens that are related to zoonotic infectious diseases. The purpose of this study was to investigate co-infections of zoonotic pathogens in rodents and shrews trapped in four provinces of China. We sampled different rodent and shrew communities within and around human settlements in four provinces of China and characterised several important zoonotic viral, bacterial, and parasitic pathogens by PCR methods and phylogenetic analysis. A total of 864 rodents and shrews belonging to 24 and 13 species from RODENTIA and EULIPOTYPHLA orders were captured, respectively. For viral pathogens, two species of hantavirus (Hantaan orthohantavirus and Caobang orthohantavirus) were identified in 3.47% of rodents and shrews. The overall prevalence of Bartonella spp., Anaplasmataceae, Babesia spp., Leptospira spp., Spotted fever group Rickettsiae, Borrelia spp., and Coxiella burnetii were 31.25%, 8.91%, 4.17%, 3.94%, 3.59%, 3.47%, and 0.58%, respectively. Furthermore, the highest co-infection status of three pathogens was observed among Bartonella spp., Leptospira spp., and Anaplasmataceae with a co-infection rate of 0.46%. Our results suggested that species distribution and co-infections of zoonotic pathogens were prevalent in rodents and shrews, highlighting the necessity of active surveillance for zoonotic pathogens in wild mammals in wider regions.