Interplay between cost and benefits triggers nontrivial vaccination uptake

The containment of epidemic spreading is a major challenge in science. Vaccination, whenever available, is the best way to prevent the spreading, because it eventually immunizes individuals. However, vaccines are not perfect, and total immunization is not guaranteed. Imperfect immunization has driven the emergence of antivaccine movements that totally alter the predictions about the epidemic incidence. Here, we propose a mathematically solvable mean-field vaccination model to mimic the spontaneous adoption of vaccines against influenzalike diseases and the expected epidemic incidence. The results are in agreement with extensive Monte Carlo simulations of the epidemics and vaccination coevolutionary processes. Interestingly, the results reveal a nonmonotonic behavior on the vaccination coverage that increases with the imperfection of the vaccine and after decreases. This apparent counterintuitive behavior is analyzed and understood from stability principles of the proposed mathematical model.

Figure 1

Selection gradient, $\dot{y}$ = $F\left(y\right)$, as a function of the vaccination coverage, $y$. The parameters are set to $c=0.1$, $\alpha =0.3$, $\beta =0.55$, $\gamma =0.1$, and $T=1.0$.

Figure 2

Vaccination coverage at equilibrium, $y^{*}$, as a function of effectiveness, $1-\gamma$ for two different values of the infection probability $\beta$. The dashed line indicates the fraction of effectively vaccinated agents. The other parameters are $\alpha =0.4$, $c=0.1$, and $T=1.0$. The maximum coverage $y_{max}^{*}$ is denoted by a point, and the dotted line delimits the tolerance range. The inset displays the value of the maximum coverage $y_{max}^{*}$ given in Eq. (10) as a function of infectivity $\beta$. The color highlights the region where vaccination takes place or not.

Figure 3

Vaccination coverage, $y_{\text{eq}}$ (a), and fraction of recovered individuals, $R_{\infty }$ (b), as a function of disease incidence in the previous season, $\alpha$, and infection probability, $\beta$. The red solid line corresponds to the vaccination threshold, $\tilde{\beta }$, separating the region where no vaccine uptake occurs from the one where nonzero vaccination could be observed. In panel (b), the white solid lines correspond to the epidemic thresholds in the absence of ($\beta =0.2$), and in presence ($\alpha =0.5$) of vaccination according to Eq. (13). The other variables are fixed to $c=0.1$, $T=1.0$, $\gamma =0.1$ and $\mu =0.2$. (c) $R_{\infty }$ as a function of $\beta$ for different costs of the vaccine, $c$. The inset presents the vaccination coverage, $y_{\text{eq}}$. Additionally, the critical values of $\beta$ are presented following Eq. (15) such as the purely epidemic threshold, ${\beta }_c$, the vaccination threshold, $\tilde{\beta }$, and the threshold for full vaccination, $\tilde{\beta }/\gamma$. The remaining variables are fixed to $T=1.0$, $\gamma =0.4$, $\alpha =0.4$, and $\mu =0.1$.

Figure 4

Differences between the quantities computed using Monte Carlo simulations and their analytical counterparts as a function of parameters $\alpha$ and $\beta$. Panel (a) presents the case of $y_{\text{eq}}$, while panel (b) the case of $R_{\infty }$, instead. The other parameters are set to $N=1000$, $\gamma =0.1$, $\mu =0.2$, and $T=1.0$, respectively.