Additionally, represents a constant birth and death rate, ␤ is the contact rate, ␣ is the rate of infection, so that 1 / ␣ is the mean latency period, and ␥ is the rate of recovery, so that 1 / ␥ is the mean infectious period. Although the contact rate ␤ could be given by a time-dependent function ͑e.g., due to seasonal fluctuations͒, for simplicity, we assume ␤ to be constant. Throughout this article, we use the following parameter values: = 0.02͑yr͒ −1 , ␤ = 1575.0͑yr͒ −1 , ␣ =1/ 0.0279͑yr͒ −1 , and ␥ =1/ 0.01͑yr͒ −1 . Disease parameters correspond to typical measles values. 26, 27 Note that any other biologically meaningful parameters may be used as long as the basic reproductive rate R 0 = ␣␤ / ͓͑␣ + ͒͑␥ + ͔͒ Ͼ 1. The interpretation of R 0 is the number of secondary cases produced by a single infectious individual in a population of susceptibles in one infectious period.