Figure 4. Optimal prevalence rate of vaccination program: Increasing of the loss of the protection effectiveness engenders a catastrophic change in the optimal prevalence rate.

The optimal rate increases as the loss increases if the loss of the protection effectiveness is small (0≤σ≤σ_o). This implies that a small loss of the protection effectiveness can be compensated by a high optimal prevalence rate of the vaccination program. On the other hand, if the loss is large (σ_o≤σ≤1), the optimal rate decreases as the loss of the protection effectiveness increases. This eventuality implies that a large loss of the protection effectiveness is no longer compensated by the high optimal prevalence rate of the vaccination program. Therefore, a low prevalence rate, which does not engender the emergence of a vaccine-resistant strain becomes optimal because the poor vaccine engenders the increase of final size of the epidemic because of the spread of the resistant strain.