  A Stackelberg duopoly model in which two firms compete to maximize their
market share is considered. The firms offer a service/product to customers that
are spread over several geographical regions (e.g., countries, provinces, or
states). Each region has its own characteristics (spreading and recovery rates)
of each service propagation. We consider that the spreading rate can be
controlled by each firm and is subject to some investment that the firm does in
each region. One of the main objectives of this work is to characterize the
advertising budget allocation strategy for each firm across regions to maximize
its market share when competing. To achieve this goal we propose a Stackelberg
game model that is relatively simple while capturing the main effects of the
competition for market share. {By characterizing the strong/weak Stackelberg
equilibria of the game, we provide the associated budget allocation strategy.}
In this setting, it is established under which conditions the solution of the
game is the so-called ``winner takes all". Numerical results expand upon our
theoretical findings and we provide the equilibrium characterization for an
example.
