Imperfect vaccine aggravates the long-standing dilemma of voluntary vaccination

Abstract

Achieving widespread population immunity by voluntary vaccination poses a major challenge for public health administration and practice. The situation is complicated even more by imperfect vaccines. How the vaccine efficacy affects individuals' vaccination behavior has yet to be fully answered. To address this issue, we combine a simple yet effective game theoretic model of vaccination behavior with an epidemiological process. Our analysis shows that, in a population of self-interested individuals, there exists an overshooting of vaccine uptake levels as the effectiveness of vaccination increases. Moreover, when the basic reproductive number, R0, exceeds a certain threshold, all individuals opt for vaccination for an intermediate region of vaccine efficacy. We further show that increasing effectiveness of vaccination always increases the number of effectively vaccinated individuals and therefore attenuates the epidemic strain. The results suggest that 'number is traded for efficiency': although increases in vaccination effectiveness lead to uptake drops due to free-riding effects, the impact of the epidemic can be better mitigated.

Figure 1. The vaccination behavior on the basic reproductive ratio and the effectiveness .

Here , where . See main text for details.

Figure 2. Fractions of the vaccinated and the effective vaccinated for a disease with a moderate infectiveness.

The upper panel shows the stationary frequency of the vaccinated with respective to the effectiveness. No one takes vaccination until it is sufficiently efficient, . Then the vaccine uptake level increases with the effectiveness. When the effectiveness exceeds a threshold, , however, the vaccination level decreases with the effectiveness. The lower panel shows the stationary abundance of the effectively vaccinated individuals with respect to the effectiveness. It is shown the efficient vaccinated individual increases with the effectiveness all the time. Thus the behavior of vaccination and the impact of the vaccination against epidemic are not in agreement: for high effectiveness, even though vaccination rate is decreasing, the number of effectively vaccinated individual increases as the effectiveness increases. Here , satisfying .

Figure 3. Fractions of the vaccinated and the effective vaccinated for a serious disease.

The upper panel shows the stationary frequency of the vaccinated with respective to the effectiveness. Compared to Fig. (2), the whole population could take vaccination provided the effectiveness is moderate, . The lower panel indicates the stationary abundance of the effectively vaccinated individuals with respect to the effectiveness. Compared with Fig. (2), the frequency of the effective vaccinated individual also increases with the effectiveness, but it is higher than that of Fig. (2). Here , satisfying .

Figure 4. Final epidemic size for the flu and the measles.

The final epidemic size here is the average abundance of the infected individual in the long run. For both the flu and the measles, the final epidemic size decreases with the effectiveness. Here is the birth rate of the population. For the measles, , and ; For the flu , and , .