Maximizing Spread of a Message in the Susceptible-Infected-Recovered Process

In this work we borrow models from biology (epi-demics) to model spread of a message as a Susceptible-Infected-Recovered (SIR) process. We assume that the target population is large. Further, homogeneous mixing of population is considered. The campaigner enrolls people to spread the message to maximize its reach, this is in addition to the standard epidemic spread. We term this intervention by the campaigner as enrollment. Enrollment may be done by reaching out to people through advertisements, for example, in social media, in print or electronic media, etc. An appropriate cost function is chosen and the given situation is posed as a mathematical optimization problem, more specifically, an optimal control problem. The formulated problem is mathematically analyzed. To this end, the existence of a solution to the optimal control problem is explored. Further, we study the nature of state trajectories at the optimum. We provide insights that are useful in optimizing viral marketing strategies, political or social awareness campaigns, etc.