In this work, we consider a class of uniformly elliptic operators with a nonlocal term and mixed boundary conditions in bounded domains. We establish the existence of a principal eigenvalue and provide a result that offers both sufficient and necessary conditions for the validity of the maximum principle. As a consequence of these findings, we conduct a detailed study of an eigenvalue problem with an indefinite weight, as well as establish existence and uniqueness results for a logistic-type equation and prove some blow-up results.

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.

This paper is devoted to the study of the local existence, uniqueness, regularity, and continuous dependence of solutions to a logistic equation with memory in the Bessel potential spaces.

The underuse of cardiovascular recovery as an adjunct to reactivity may stem from a lack of research on how to assess the process reliably. We explore the test–retest reliability of three simple, intuitive approaches to measuring recovery, and of a more sophisticated curve-fitting technique. Eighteen young normotensive subjects experienced three stressors twice each, with 10-min baseline, 3-min task, and 20-min recovery periods and continuous monitoring of heart rate and blood pressure. Reactivity showed moderate reliability, but the three simple approaches to measuring recovery revealed essentially none. However, the curve-fitting approach, using a three-parameter (amount, speed, and level of recovery) logistic function was reliable. This approach, capturing the inherently dynamic process of cardiovascular recovery, may allow researchers to usefully add the assessment of recovery to paradigms exploring reactivity as a risk factor for cardiovascular disease.

Asymptotic approximations and numerical computations are used to estimate the accuracy of the diffusion approximation for the expected time to extinction for some stochastic processes. The results differ for processes with a continuant transition matrix (e.g. a birth and death process), and those with a noncontinuant transition matrix (e.g. a non-linear branching process). In the latter case, the diffusion equation does not hold near the point of exit. Consequently, high-order corrections do not result in substantial improvement over the diffusion approximation.