Investigating Stochastic Differential Equations Modelling for Levodopa Infusion in Patients with Parkinson's Disease.

BACKGROUND AND OBJECTIVES: Levodopa concentration in patients with Parkinson's disease is frequently modelled with ordinary differential equations (ODEs). Here, we investigate a pharmacokinetic model of plasma levodopa concentration in patients with Parkinson's disease by introducing stochasticity to separate the intra-individual variability into measurement and system noise, and to account for auto-correlated errors. We also investigate whether the induced stochasticity provides a better fit than the ODE approach. METHODS: In this study, a system noise variable is added to the pharmacokinetic model for duodenal levodopa/carbidopa gel (LCIG) infusion described by three ODEs through a standard Wiener process, leading to a stochastic differential equations (SDE) model. The R package population stochastic modelling (PSM) was used for model fitting with data from previous studies for modelling plasma levodopa concentration and parameter estimation. First, the diffusion scale parameter (σw), measurement noise variance, and bioavailability are estimated with the SDE model. Second, σw is fixed to certain values from 0 to 1 and bioavailability is estimated. Cross-validation was performed to compare the average root mean square errors (RMSE) of predicted plasma levodopa concentration. RESULTS: Both the ODE and the SDE models estimated bioavailability to be approximately 75%. The SDE model converged at different values of σw that were significantly different from zero. The average RMSE for the ODE model was 0.313, and the lowest average RMSE for the SDE model was 0.297 when σw was fixed to 0.9, and these two values are significantly different. CONCLUSIONS: The SDE model provided a better fit for LCIG plasma levodopa concentration by approximately 5.5% in terms of mean percentage change of RMSE.

Geographical Clusters of Rape in the United States: 2000-2012.

BACKGROUND: While rape is a very serious crime and public health problem, no spatial mapping has been attempted for rape on the national scale. This paper addresses the three research questions: (1) Are reported rape cases randomly distributed across the USA, after being adjusted for population density and age, or are there geographical clusters of reported rape cases? (2) Are the geographical clusters of reported rapes still present after adjusting for differences in poverty levels? (3) Are there geographical clusters where the proportion of reported rape cases that lead to an arrest is exceptionally low or exceptionally high? METHODS: We studied the geographical variation of reported rape events (2003-2012) and rape arrests (2000-2012) in the 48 contiguous states of the USA. The disease Surveillance software SaTScan™ with its spatial scan statistic is used to evaluate the spatial variation in rapes. The spatial scan statistic has been widely used as a geographical surveillance tool for diseases, and we used it to identify geographical areas with clusters of reported rape and clusters of arrest rates for rape. RESULTS: The spatial scan statistic was used to identify geographical areas with exceptionally high rates of reported rape. The analyses were adjusted for age, and in secondary analyses, for both age and poverty level. We also identified geographical areas with either a low or a high proportion of reported rapes leading to an arrest. CONCLUSIONS: We have identified geographical areas with exceptionally high (low) rates of reported rape. The geographical problem areas identified are prime candidates for more intensive preventive counseling and criminal prosecution efforts by public health, social service, and law enforcement agencies Geographical clusters of high rates of reported rape are prime areas in need of expanded implementation of preventive measures, such as changing attitudes in our society toward rape crimes, in addition to having the criminal justice system play an even larger role in preventing rape.