Patterns of seasonal epidemics for 12 seasons with 100 independent simulation runs when vaccines are randomly distributed. Colorbars represent the total number of infections for each season; 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. Vaccine efficacy in prevention of infection is 80%, and 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.