Chickenpox is a common acute and highly contagious disease in childhood; moreover, there is currently no targeted treatment. Carrying out an early warning on chickenpox plays an important role in taking targeted measures in advance as well as preventing the outbreak of the disease. In recent years, the infectious disease dynamic model has been widely used in the research of various infectious diseases. The logistic differential equation model can well demonstrate the epidemic characteristics of epidemic outbreaks, gives the point at which the early epidemic rate changes from slow to fast. Therefore, our study aims to use the logistic differential equation model to explore the epidemic characteristics and early-warning time of varicella. Meanwhile, the data of varicella cases were collected from first week of 2008 to 52nd week of 2017 in Changsha. Finally, our study found that the logistic model can be well fitted with varicella data, besides the model illustrated that there are two peaks of varicella at each year in Changsha City. One is the peak in summer–autumn corresponding to the 8th–38th week; the other is in winter–spring corresponding to the time from the 38th to the seventh week next year. The ‘epidemic acceleration week’ average value of summer–autumn and winter–spring are about the 16th week (ranging from the 15th to 17th week) and 45th week (ranging from the 44th to 47th week), respectively. What is more, taking warning measures during the acceleration week, the preventive effect will be delayed; thus, we recommend intervene during recommended warning weeks which are the 15th and 44th weeks instead.

We consider a network of interacting individuals, whose actions or transitions are determined by the states (behavior) of their neighbors as well as their own personal decisions. Specifically, we develop a model according to two simple decision-making rules that can describe the growth of the user population of a newly launched product or service. We analyze 22 sets of real-world historical growth data of a variety of products and services, and show that they all follow the growth equation. The numerical procedure for finding the model parameters allows the market size, and the relative effectiveness of customer service and promotional efforts to be estimated from the available historical growth data. We study the growth profiles of products and find that for a product or service to reach a mature stage within a reasonably short time in its user growth profile, the user growth rate corresponding to influenced transitions must exceed a certain threshold. Furthermore, results show that individuals in the group of celebrities having numerous friends become users of a new product or service at a much faster rate than those connected to ordinary individuals having fewer friends.

In this paper, semidefinite optimization method is proposed to estimate bounds on linear functionals defined on solutions of linear ordinary differential equations (ODEs) with smooth coefficients. The method can get upper and lower bounds by solving two semidefinite programs, not solving ODEs directly. Its convergence theorem is proved. The theorem shows that the upper and lower bounds series of linear functionals discussed can approach their exact values infinitely. Numerical results show that the method is effective for the estimation problems discussed. In addition, in order to reduce calculation amount, Cheybeshev polynomials are applied to replace Taylor polynomials of smooth coefficients in computing process.

We derive the explicit formula for the joint Laplace transform of the Wishart process and its time integral, which extends the original approach of Bru (1991). We compare our methodology with the alternative results given by the variation-of-constants method, the linearization of the matrix Riccati ordinary differential equation, and the Runge-Kutta algorithm. The new formula turns out to be fast and accurate.

In the first part of this paper a global Kushner–Clark theorem about the convergence of stochastic algorithms is proved: we show that, under some natural assumptions, one can ‘read' from the trajectories of its ODE whether or not an algorithm converges. The classical stochastic optimization results are included in this theorem. In the second part, the above smoothness assumption on the mean vector field of the algorithm is relaxed using a new approach based on a path-dependent Lyapounov functional. Several applications, for non-smooth mean vector fields and/or bounded Lyapounov function settings, are derived. Examples and simulations are provided that illustrate and enlighten the field of application of the theoretical results.