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This paper is concerned with the efficient stochastic simulation of multiple scenarios of an infectious disease as it propagates through a population. In particular, we propose a simple "green" method to speed up the simulation of disease transmission as we vary the probability of infection of the disease from scenario to scenario. After running a baseline scenario, we incrementally increase the probability of infection, and use the common random numbers variance reduction technique to avoid re-simulating certain events in the new scenario that would not otherwise have changed from the previous scenario. A set of Monte Carlo experiments illustrates the effectiveness of the procedure. We also propose various extensions of the method, including its use to estimate the sensitivity of propagation characteristics in response to small changes in the infection probability. [ABSTRACT FROM AUTHOR]