Evolutionary dynamics of voluntary vaccination for imperfect multi-efficacy vaccines

Voluntary vaccination is crucial for public health. The likelihood that the vaccinated get fully immune is typically referred to as the vaccine efficacy. In addition to this, the efficacy of vaccines also depends on inhibiting the transmission rates and protecting patients from critical symptoms. It has yet to be answered how the multi-efficacy of vaccines affects the vaccination behaviour. Here, we propose a game theoretical model of vaccination behaviour with a heterogeneous-transmission epidemiological process. Intuitively, individuals are likely to take vaccination for either high-efficient vaccines or epidemics with a large basic reproductive number. We find, however, a low economic discount rate can promote the uptake level for low-effectiveness vaccines and epidemics with small basic reproductive numbers. This result suggests that the perceived cost is more important than the risk of infection and the effectiveness of vaccines. In addition, we develop an analytical approximation to address the challenge arising from the coupled dynamics of vaccination and disease spread. This study shows how the interplay among three types of vaccine efficacies (transmission blocking, symptom mitigating and full immunity protecting) affects individual vaccination decisions and epidemic outcomes, and the model can be insightful for epidemic control.