Rare Event Simulation in a Dynamical Model Describing the Spread of Traffic Congestions in Urban Network Systems

In this chapter, we present a mathematical framework that provides a new insight for understanding the spread of traffic congestions in an urban network system. In particular, we consider a dynamical model, based on the well-known susceptible-infected-recovered (SIR) model from mathematical epidemiology, with small random perturbations, that describes the process of traffic congestion propagation and dissipation in an urban network system. Here, we provide the asymptotic probability estimate based on the Freidlin-Wentzell theory of large deviations for certain rare events that are difficult to observe in the simulation of urban traffic network dynamics. Moreover, the framework provides a computational algorithm for constructing efficient importance sampling estimators for rare event simulations of certain events associated with the spread of traffic congestions in the dynamics of the traffic network.

SIR Model with Homotopy to Predict Corona Cases

In this chapter, we will discuss SIR model to study the spread of COVID-2019 pandemic of India. We will give the prediction of corona cases using homotopy method. The HM is a method for solving the ordinary differential equations. The SIR model consists of three ordinary differential equations. In this study, we have used the data of COVID-2019 Outbreak of India on 20 Jan 2021. In this data, Recovered is 102656163, Active cases are 189245 Susceptible persons are 189347782 for the experimental purpose. Data about a wide variety of infectious diseases has been analyzed with the help of SIR model. Therefore, this model has been already well tested for infectious diseases by various scientists and researchers.

Perturbation Methods to Analysis of Thermal, Fluid Flow and Dynamics Behaviors of Engineering Systems

This chapter presents the applications of perturbation methods such as regular and homotopy perturbation methods to thermal, fluid flow and dynamic behaviors of engineering systems. The first example shows the utilization of regular perturbation method to thermal analysis of convective-radiative fin with end cooling and thermal contact resistance. The second example is concerned with the application of homotopy perturbation method to squeezing flow and heat transfer of Casson nanofluid between two parallel plates embedded in a porous medium under the influences of slip, Lorentz force, viscous dissipation and thermal radiation. Additionally, the dynamic behavior of piezoelectric nanobeam embedded in linear and nonlinear elastic foundations operating in a thermal-magnetic environment is analyzed using homotopy perturbation method which is presented in the third example. It is believed that the presentation in this chapter will enhance the understanding of these methods for the real world applications.