Impulsive control of a sequence of rumour processes

Abstract

In this paper we introduce an impulsive control model for a sequence of rumour processes evolving in a given population. Each rumour process begins with a broadcast, the recipients of which begin to spread that rumour. The recipients of the first broadcast are termed the subscribers. The second and subsequent broadcasts are either to the subscribers (Scenario 1) or to all individuals who have at any time to date been spreaders (Scenario 2). The objective is to time the second and subsequent broadcasts so as to minimise the final proportion of ignorants. It is shown that with either scenario the optimal time for each broadcast after the first is when the proportion of spreaders in the rumour process begun by the previous broadcast reaches zero. Results are presented concerning dependence on initial conditions, as well as graphical illustration of the controlled rumour processes under each scenario.