Bernard P. Brooks

Linear stability conditions for a first-order three-dimensional discrete dynamic

Linear stability conditions for a first-order three-dimensional discrete dynamic are derived in terms of the trace, determinant, and sum of principle minors of the Jacobian evaluated at the equilibrium.

The coefficients of the characteristic polynomial in terms of the eigenvalues and the elements of an n×n matrix

Abstract The coefficients of the characteristic polynomial of an n × n matrix are derived in terms of the eigenvalues and in terms of the elements of the matrix. The connection between the two expressions allows the sum of the products of all sets of k eigenvalues to be calculated using cofactors of the matrix.

The Brouwer Fixed Point Theorem applied to rumour transmission

The results of the Brouwer Fixed Point Theorem are extended to continuous dynamical systems. It is shown that if there exists a compact convex positive invariant set for the dynamical system, then this convex positive invariant set contains an equilibrium point. The existence of an interior equilibrium is shown for a general model of rumour transmission.

Rumour Propagation on Social Networks as a Function of Diversity

In the case of a rumour propagating across a social network comprised of two interconnected groups, a majority group and a minority group, the effect on the rumour propagation of the minority’s distribution in the social network is investigated. The rumour is derogatory towards the minority group and its transmission is simulated using the GBN-Dialogue model of rumour propagation on realistic social networks. The integration of the minority group into the entire social network is measured by the Minority Integration Metric (MIM). Monte Carlo simulations show that rumour penetration into the minority subgroup is an increasing linear function of the MIM and thus minority members of the social network will have a higher level of belief in the minority-derogatory rumour if the minority is more integrated into the social network. A threshold MIM value can be estimated below which the rumour fails to penetrate the minority subgroup.

A Multidisciplinary Predictive Model for Managing Critical Infrastructure Disruptions

When communities are subjected to disruptive events, their response structure is composed of two interconnected systems: (i) a formal professional system that includes emergency services and auxiliary services professionals; and (ii) an ad hoc system formed by community members when the professional response is delayed or is inadequate. The community system typically persists until the professional system is able to take over completely. As the role of the community as responder is not well understood, community systems are often underutilized or even discouraged; this reduces the overall response efficacy. Improved understanding of the interplay between these systems could help ensure an effective overall response to disruptions.

Linear Stability Conditions for a First Order 4-Dimensional Discrete Dynamic

Linear stability conditions for a first order 4-dimensional discrete dynamic are derived in terms of the trace, sum of minors, sum of their minors, and the determinant of the Jacobian evaluated at the equilibrium.

Modelling communication network effect on emergency evacuation times: public vs. personal

Unexpected, catastrophic events initially result in anxiety and confusion. The congestion of pedestrians and vehicles further exacerbates the already turbulent environment. This uncertainty results in a chaotic site of victims, curiosity seekers, emergency responders, and mere transient individuals. Initial public announcements are often delayed, inaccurate, and met with skepticism and disbelief. However, interpersonal communications, although potentially less accurate, may have a greater influence on behaviour. With this in mind, an agent-based model was developed that included vehicle movement on roadways with traffic lights and stop signs, a disaster site, evacuation sites, and communication systems consisting of both public broadcast and personal communication networks. As an extension of earlier work, we explored the effect of the public announcement control variables and contrasted the results with those from a personal communication network. In addition, the model is thoroughly explored, explained, and shown to demonstrate the potential for improved site evacuation via personal communication networks, i.e., cell phones.