Time-Inhomogeneous Feller-Type Diffusion Process in Population Dynamics

The time-inhomogeneous Feller-type diffusion process, having infinitesimal drift α(t)x+β(t) and infinitesimal variance 2r(t)x, with a zero-flux condition in the zero-state, is considered. This process is obtained as a continuous approximation of a birth-death process with immigration. The transition probability density function and the related conditional moments, with their asymptotic behaviors, are determined. Special attention is paid to the cases in which the intensity functions α(t), β(t), r(t) exhibit some kind of periodicity due to seasonal immigration, regular environmental cycles or random fluctuations. Various numerical computations are performed to illustrate the role played by the periodic functions.

On-Demand Public Transit: A Markovian Continuous Approximation Model

With recent advances in mobile technology, public transit agencies around the world have started actively experimenting with new transportation modes, many of which can be characterized as on-demand public transit. Design and efficient operation of such systems can be particularly challenging, because they often need to carefully balance demand volume with resource availability. We propose a family of models for on-demand public transit that combine a continuous approximation methodology with a Markov process. Our goal is to develop a tractable method to evaluate and predict system performance, specifically focusing on obtaining the probability distribution of performance metrics. This information can then be used in capital planning, such as fleet sizing, contracting, and driver scheduling, among other things. We present the analytical solution for a stylized single-vehicle model of first-mile operation. Then, we describe several extensions to the base model, including two approaches for the multivehicle case. We use computational experiments to illustrate the effects of the inputs on the performance metrics and to compare different modes of transit. Finally, we include a case study, using data collected from a real-world pilot on-demand public transit project in a major U.S. metropolitan area, to showcase how the proposed model can be used to predict system performance and support decision making.

Continuous Approximation Model for Hybrid Flexible Transit Systems with Low Demand Density

Flexible transit systems are a way to address challenges associated with conventional fixed route and fully demand responsive systems. Existing studies indicate that such systems are often planned and designed without established guidelines, and optimization techniques are rarely implemented on actual flexible systems. This study presents a hybrid transit system where the degree of flexibility can vary from a fixed route service (with no flexibility) to a fully flexible transit system. Such a system is expected to be beneficial in areas where the best transit solution lies between the fixed route and fully flexible systems. Continuous approximation techniques are implemented to model and optimize the stop spacing on a fixed route corridor, as well as the boundaries of the flexible region in a corridor. Both user and agency costs are considered in the optimization process. A numerical analysis compares various service areas and demand densities using input variables with magnitudes similar to those of real-world case studies. Sensitivity analysis is performed for service headway, percent of demand served curb-to-curb, and user and agency cost weights in the optimization process. The analytical models are evaluated through simulations. The hybrid system proposed here achieves estimated user benefits of up to 35% when compared with fixed route systems, under different case scenarios. Flexible systems are particularly beneficial for serving corridors with low or uncertain demand. This provides value for corridors with low demand density as well as communities in which transit ridership has dropped significantly because of the COVID-19 pandemic.

On mathematical analysis of a discrete electrical lattice with nonlinear dispersion

In this paper, we consider the discrete electrical lattice with nonlinear dispersion described by Salerno equation, Fig. 1. Stability of equilibrium points, limit cycles and flip and Hopf bifurcations of the system are discussed. New exact solutions of a continuous approximation of the discrete system in the upper forbidden band gap are obtained by two methods, namely, [Formula: see text]-expansion function method and [Formula: see text] expansion method. Numerical simulation is used to follow the dynamics of the system and to investigate its physical properties.

COVID-19 dynamical evolution prediction in Mexico, decision making and social implementation: mid/low income countries study

A normal distribution approach is implemented to predict the evolution of the COVID-19 epidemic. The fit to the COVID-19 daily cases in Mexico, in the rising stage of the epidemic, is a very good continuous approximation to the data with R2 = 0.976. The derivative of this function provides a measure of the increase/decrease or acceleration of new cases per day that are otherwise buried in the noise of the raw data. The predictions are depicted in a novel 3D way, so as to convey the evolution of the forecasts as data becomes available. The estimations are in accordance within standard deviation, with the logistic and Gompertz functions fitted to the corresponding epidemic models. This scheme can be used to model the epidemic and use it as an ancillary for decision making at a municipal or regional level. Simplicity with robust prediction is favoured, so that the model can be understood and implemented by local government advisors or personnel not familiar with specialized statistical methods.