For proof of principle, we consider vaccination dynamics in an infinitely large well mixed population. In addition, we assume that individuals have a perfect knowledge on the effectiveness of the vaccination. In this case, there is only one parameter describing both the actual and the perceived effectiveness.

The vaccination game consists of two stages, the yearly vaccination campaign and an epidemic season. During the vaccination campaign, each individual decides whether or not to take vaccination. A vaccinated individual pays a cost

while an unvaccinated individual pays nothing. This cost

includes the time spent in taking the vaccination as well as its side effects. During the epidemic season, the population can be divided into two parts: one comprises effectively vaccinated individuals, and the rest is composed of unvaccinated individuals and the vaccinated ones whose vaccinations are not effective. Successfully vaccinated individuals are immune to the seasonal disease, and thus have no risk of getting infected. For the remaining individuals, however, they become infected with a probability

, where

is the frequency of effectively vaccinated individuals. In this case the infected bear a cost by

. This cost

includes expenses and time for health care as well as mortality. The larger the number of effectively vaccinated individuals is, the less likely an unvaccinated individual gets infected. Thus

is decreasing with

.

Let the effectiveness of the vaccination be

and the vaccine uptake level be

. The frequency of the effectively vaccinated individuals is

. The fraction of the vaccinated and healthy individuals is

, which is composed of two parts: these effectively vaccinated individuals (with frequency

) and those ineffectively vaccinated individuals (with frequency

) who are free from the infection (with frequency

). In this case, each effectively vaccinated individual gets payoff

. In analogy to this, the frequencies and payoffs for different individuals are given by .

| **Table 1**The fraction and the payoff for the four types of individuals in the population. |

When the epidemic season ends, i.e., the average abundance of infected individuals does not change, individuals adjust their strategies by imitation where successful individual's strategy is more likely to be followed

[33],

[34]. Here we employ the Fermi update rule to characterize such an imitation process

[35],

[36],

[8],

[37],

[38]: two individuals

and

are selected randomly;

learns to behave like

with probability

where

and

are the perceived payoffs for

and

, and

is the selection intensity indicating how strongly individuals are responsive to payoff difference.

The dynamics of the vaccination is governed by

[20],

[39]It has been suggested that the selection intensity for human imitation is rather weak

[21],

[34], i.e.

is sufficiently small. We perform the Taylor expansion of the r.h.s of Eq. (2) in the vicinity of

, then after a time rescaling which does not change the dynamics, Eq. (2) can be captured by a much more simple form

In what follows, we investigate how the vaccine uptake evolves by Eq. (3) for general function of infection risk

. To this end, we focus on how the effectiveness of vaccination has an impact on the the collective outcome of vaccination behavior and the effective vaccination level. Then we incorporate an epidemic dynamics to obtain a specific infection function. Based on this, we provide precise predictions for the two problems. Besides we also study how the effectiveness affects the final epidemic size in this case.

General infection function

For a general function of infection risk

, when

is valid for all

lying between zero and one, no one would take vaccination in the long run, i.e.

is the unique stable equilibrium for Eq. (3). Since

is a decreasing function,

is sufficient to ensure

. In analogy to this, when

is valid, the entire population ends up with full vaccination, i.e.

is the unique stable equilibrium. For

fulfilling

and

, by the monotonicity of

in

, there is a unique internal equilibrium,

Further,

is decreasing, the derivative at

, namely

, is negative. Thus

is stable, indicating the coexistence of the vaccinated and the unvaccinated. To show how

is affected by

requires the exact form of the function of infection risk. We will address it later.

The effective level of vaccination reads

By Eq. (5),

is an increasing function of the effectiveness,

. In other words, the effectively vaccinated level always increases with vaccine efficacy. This result only requires that

decreases with

. This is true for most, if not all, known infection functions

[22],

[23]. Therefore our predictions are robust with respect to variations in specific infection functions.

A specific infection function

In order to give precise predictions, we adopt a simple Susceptible-Infected-Recovered (SIR) model with demographical effects as presented in

[7]. In this model, the population is divided into three different compartments: susceptible, who are healthy but can catch the disease if exposed to infected individuals; Infective, who are infected and can pass the disease on to others; Recovered, who are recovered from the infection and gain immunity against the disease. The time evolution of the population states is governed by the following equations

where

is the birth rate and equal to the mortality rate (for simplicity, we only consider constant population size),

is the transmission rate,

is the recovery rate, and

is the fraction of effectively vaccinated individuals among newborns.

From Eq. (7), we derive the basic reproduction ratio

: if

, the time derivative of

is negative, suggesting that the disease cannot persist in the population. The equilibrium state of the population consists of

, with

,

and

. By setting

, we obtain the herd immunity needed to eradicate the disease,

.

Based on this stationary equilibrium, we calculate the probability that an unvaccinated individual gets infected in her life time. The waiting time to acquire infection follows an exponential distributions with rate

, and so does the waiting time to death but with rate

. Since infection and death are two independent processes, the probability that infection occurs before death event is the relative ratio of intensities,

. This probability gives the infection risk of an unvaccinated individual, namely,

which is a function of the population level of effective vaccine uptake

and holds for

. When

,

, i.e. the disease will be eradicated provided the effective level of vaccination exceeds the critical point

. Thus we have

Taking this specific infection function Eq. (9) into Eq. (3), we present the full dynamics analysis of the evolution of vaccination behavior in the long run (see ). Let the ratio of the vaccination cost versus the infection cost

be

. We have (For details, see

Text S1)

Case

: when

, all are unvaccinated for

.

Case

: when

; if

, all are unvaccinated, otherwise there is a unique internal stable equilibrium

.

Case

: when

; if

, all are unvaccinated, if

, there is a unique internal stable equilibrium

, if

, all are vaccinated, if

, there is a unique internal stable equilibrium

.

Where

,

.

Case

indicates that for a mild epidemic,

, vaccination behavior is impossible for any vaccination effectiveness. For a more serious epidemic, Case

shows, however, there is an overshooting of vaccine uptake: the coexistence of the vaccinated and the unvaccinated emerges as the effectiveness exceeds a threshold. Furthermore, interestingly, the increase in effectiveness does not always promote the vaccination behavior (see the upper panel of ). Intuitively, for the vaccinated, increasing the vaccination effectiveness does reduce the infection probability. For the unvaccinated, however, this leads to that they are protected by a even more effective herd immunity. Thus increasing the effectiveness of vaccination is beneficial both to the vaccinated and to the unvaccinated. The two strategies compete with each other and the more beneficial one is more likely to spread through imitation. The result shows, when the effectiveness is below the critical value, the more beneficial one is the vaccinated. When it exceeds the critical value, the more beneficial one is the unvaccinated. Mathematically, the non-monotonicity of

on

is induced from the non-monotonicity of

as discussed above. For an even more serious epidemic, Case

, the dynamics of the vaccination behavior is qualitatively identical to that of Case

. However, in contrast with Case

, full vaccination can be reached (see the upper panel of ).

Besides the vaccination behavior, by taking Eq. (9) into Eq. (5), the effective vaccination frequency,

is given by

Hence, the effective vaccination frequency increases as the effectiveness increases as predicted (See the lower panels of and ).

Further, it is of interest to investigate how the final epidemic size is influenced by the effectiveness of the vaccination. The final epidemic size

here refers to the average fraction of the infected individuals at the end of the epidemics. For the SIR model with vital dynamics discussed above, when the vaccine uptake reaches a stationary level

, the final epidemics size of the population is given by

Therefore,

is a decreasing function with

. That is to say, the more effective the vaccination is, the smaller proportion is infected eventually. In particular, we find this is true for the flu and the measles (see ).