BackgroundThe coronavirus disease 2019 (COVID-19) epidemic, caused by severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2), began in Wuhan city, Hubei province, in December, 2019, and has spread throughout China. Understanding the evolving epidemiology and transmission dynamics of the outbreak beyond Hubei would provide timely information to guide intervention policy. MethodsWe collected individual information from official public sources on laboratory-confirmed cases reported outside Hubei in mainland China for the period of Jan 19 to Feb 17, 2020. We used the date of the fourth revision of the case definition (Jan 27) to divide the epidemic into two time periods (Dec 24 to Jan 27, and Jan 28 to Feb 17) as the date of symptom onset. We estimated trends in the demographic characteristics of cases and key time-to-event intervals. We used a Bayesian approach to estimate the dynamics of the net reproduction number (R t ) at the provincial level. FindingsWe collected data on 8579 cases from 30 provinces. The median age of cases was 44 years (33-56), with an increasing proportion of cases in younger age groups and in elderly people (ie, aged >64 years) as the epidemic progressed. The mean time from symptom onset to hospital admission decreased from 4•4 days (95% CI 0•0-14•0) for the period of Dec 24 to Jan 27, to 2•6 days (0•0-9•0) for the period of Jan 28 to Feb 17. The mean incubation period for the entire period was estimated at 5•2 days (1•8-12•4) and the mean serial interval at 5•1 days (1•3-11•6). The epidemic dynamics in provinces outside Hubei were highly variable but consistently included a mixture of case importations and local transmission. We estimated that the epidemic was self-sustained for less than 3 weeks, with mean Rt reaching peaks between 1•08 (95% CI 0•74-1•54) in Shenzhen city of Guangdong province and 1•71 (1•32-2•17) in Shandong province. In all the locations for which we had sufficient data coverage of Rt, Rt was estimated to be below the epidemic threshold (ie, <1) after Jan 30. Interpretation Our estimates of the incubation period and serial interval were similar, suggesting an early peak of infectiousness, with possible transmission before the onset of symptoms. Our results also indicate that, as the epidemic progressed, infectious individuals were isolated more quickly, thus shortening the window of transmission in the community. Overall, our findings indicate that strict containment measures, movement restrictions, and increased awareness of the population might have contributed to interrupt local transmission of SARS-CoV-2 outside Hubei province.

The use of non-Hamiltonian dynamical systems to perform molecular dynamics simulation studies is becoming standard. However, the lack of a sound statistical mechanical foundation for non-Hamiltonian systems has caused numerous misconceptions about the phase space distribution functions generated by these systems to appear in the literature. Recently, a rigorous classical statistical mechanical theory of non-Hamiltonian systems has been derived, [M. E. Tuckerman, , Europhys. Lett. 45, 149 (1999)]. In this paper, the new theoretical formulation is employed to develop the non-Hamiltonian generalization of the usual Hamiltonian based statistical mechanical phase space principles. In particular, it is shown how the invariant phase space measure and the complete sets of conservation laws of the dynamical system can be combined with the generalized Liouville equation for non-Hamiltonian systems to produce a well defined expression for the phase space distribution function. The generalization provides a systematic, controlled procedure for designing non-Hamiltonian molecular dynamics algorithms which can be used to generate nonmicrocanonical ensembles, stationary nonequilibrium flows, and/or the dynamics of constrained systems. In light of this new general analysis, molecular dynamics algorithms for the canonical and isothermal-isobaric ensembles are examined, potential difficulties are illuminated, and the limitations of previous theoretical treatments are elucidated. (C) 2001 American Institute of Physics

A new method for generating the canonical ensemble via continuous dynamics is presented. The new method is based on controlling the fluctuations of an arbitrary number of moments of the multidimensional Gaussian momentum distribution function. The equations of motion are non-Hamiltonian, and hence have a nonvanishing phase space compressibility. By applying the statistical mechanical theory of non-Hamiltonian systems recently introduced by the authors ͓M. E. Tuckerman, C. J. Mundy, and G. J. Martyna, Europhys. Lett. 45, 149 ͑1999͔͒, the equations are shown to produce the correct canonical phase space distribution function. Reversible integrators for the new equations of motion are derived based on a Trotter-type factorization of the classical Liouville propagator. The new method is applied to a variety of simple one-dimensional example problems and is shown to generate ergodic trajectories and correct canonical distribution functions of both position and momentum. The new method is further shown to lead to rapid convergence in molecular dynamics based calculations of path integrals. The performance of the new method in these examples is compared to that of another canonical dynamics method, the Nosé -Hoover chain method ͓G. J. Martyna, M. L. Klein, and M. E. Tuckerman, J. Chem. Phys. 97, 2635 ͑1992͔͒.The comparison demonstrates the improvements afforded by the new method as a molecular dynamics tool. Finally, when employed in molecular dynamics simulations of biological macromolecules, the new method is shown to provide better energy equipartitioning and temperature control and to lead to improved spatial sampling over the Nosé -Hoover chain method in a realistic application.

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BackgroundThe COVID-19 epidemic originated in Wuhan City of Hubei Province in December 2019 and has spread throughout China. Understanding the fast evolving epidemiology and transmission dynamics of the outbreak beyond Hubei would provide timely information to guide intervention policy. MethodsWe collected individual information on 8,579 laboratory-confirmed cases from official publically sources reported outside Hubei in mainland China, as of February 17, 2020. We estimated the temporal variation of the demographic characteristics of cases and key time-to-event intervals. We used a Bayesian approach to estimate the dynamics of the net reproduction number (Rt) at the provincial level.