Optimal vaccine distribution for 12 seasons to achieve the minimum number of infections in each season. Vaccine efficacy in prevention of infection is 80%, and m
1 and m
2 represent the durations of immunity induced by natural infection and vaccination, respectively, in the absence of re-exposure or vaccination following priming. For each season, red circle, blue dot, and plus sign represent a possible scenario for vaccine distribution to susceptible, previously vaccinated, and previously recovered classes of individuals, respectively. The total fraction of the population vaccinated is 10% (a,b,c,d) and 20% (e,f,g,h). Other parameter values are given in Table 1.