How to run a campaign: Optimal control of SIS and SIR information epidemics

Abstract

Information spreading in a population can be modeled as an epidemic. Campaigners (e.g. election campaign managers, companies marketing products or movies) are interested in spreading a message by a given deadline, using limited resources. In this paper, we formulate the above situation as an optimal control problem and the solution (using Pontryagin's Maximum Principle) prescribes an optimal resource allocation over the time of the campaign. We consider two different scenarios-in the first, the campaigner can adjust a direct control (over time) which allows her to recruit individuals from the population (at some cost) to act as spreaders for the Susceptible-Infected-Susceptible (SIS) epidemic model. In the second case, we allow the campaigner to adjust the effective spreading rate by incentivizing the infected in the Susceptible-Infected-Recovered (SIR) model, in addition to the direct recruitment. We consider time varying information spreading rate in our formulation to model the changing interest level of individuals in the campaign, as the deadline is reached. In both the cases, we show the existence of a solution and its uniqueness for sufficiently small campaign deadlines. For the fixed spreading rate, we show the effectiveness of the optimal control strategy against the constant control strategy, a heuristic control strategy and no control. We show the sensitivity of the optimal control to the spreading rate profile when it is time varying.Information can be communicated to the population directly by the campaigner (direct recruitment of individuals to spread the message). However, recruiting individuals and direct information communication comes with a cost (such as placing advertisement in the mass media). A campaigner may also provide incentives to individuals for spreading the message. Such an incentive is termed as a word-of-mouth incentive and it rewards an individual who refers a product or a piece of information to others.The campaigner possesses limited resources and is unable to communicate information to the entire population. Not only resource allocation among different strategies, but also the timing of direct recruitment of individuals and giving out word-of-mouth incentives are crucial for maximizing the information epidemic. We model the information spreading process as a Susceptible-Infected-Susceptible (SIS) and Susceptible-Infected-Recovered (SIR) epidemic process with time varying information spreading rate, and formulate an optimal control problem which aims to minimize the campaign cost over a given period of time.SIS and SIR epidemic processes are suitable for modeling information epidemics due to the similarities in the ways disease spreads in a biological network and this information spreads in social networks. When susceptible and infected individuals interact, the topic of interest may come up with some probability, which will lead to transfer of information from infected to susceptible individual. This process is very similar to the way in which a communicable disease spreads in ...

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