Research Spotlights

Abstract

The first of the two papers appearing in this issue's Research Spotlights section is “A Gillespie Algorithm for Non-Markovian Stochastic Processes,” by Naoki Masuda and Luis E. C. Rocha. This paper is concerned with stochastic dynamical modeling of interacting sequences of discrete events. Various applications include modeling of earthquake occurrences, modeling of networks of spiking neurons, and modeling of epidemic formation on networks. In this paper, the authors propose the Laplace Gillespie algorithm, which is distinguishable from previous variants of the Gillespie algorithm as a computationally efficient approach applicable to realistic, non-Poissonian renewal processes while being accurate for an arbitrary number of ongoing renewal processes. The easy-to-digest five-step algorithm is presented first. This is followed by presentation of the necessary and sufficient conditions for a renewal process to be simulated by the Laplace Gillespie algorithm and a long list of examples of distributions of interevent times for which the algorithm can be used. The authors demonstrate their new approach by performing exact simulations of an epidemic process. The codes used in the numerical simulations section are made available in the supplementary materials to readers who want to test this algorithm on their own application. The work in the second paper, “The Ghosts of Departed Quantities in Switches and Transitions,” by Mike R. Jeffrey, represents an addition to the literature on piecewise-dynamical systems theory. The author offers a derivation of and motivation for a new uniform switching model that allows for the study of dynamics at discontinuities. The key feature of the model is the inclusion of a term that is hidden outside the so-called switching layer. This term represents the ghosts, or artifacts, due to the transition, and it requires careful attention. The author shows how analyzing the hidden dynamics inside this layer from what is encoded in this term is extremely important to understanding local and global behavior of solutions. In one example, for instance, the author shows that even determining whether a system will cross through a switch cannot be done without considering the effects of nonlinearity at the switch. The paper concludes by offering the reader suggestions for future research and by making an appeal for the development of numerical solution methods aimed at “capturing the ghosts of switching” through the proposed model.