  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.
