Herd immunity and epidemic size in networks with vaccination homophily

We study how the herd immunity threshold and the expected epidemic size depend on homophily with respect to vaccine adoption. We find that the presence of homophily considerably increases the critical vaccine coverage needed for herd immunity and that strong homophily can push the threshold entirely out of reach. The epidemic size monotonically increases as a function of homophily strength for a perfect vaccine, while it is maximized at a nontrivial level of homophily when the vaccine efficacy is limited. Our results highlight the importance of vaccination homophily in epidemic modeling.

Figure 1

Critical coverage ${\pi }_{\mathrm{v}}^{\mathrm{c}}$ of a perfect vaccine required for herd immunity as a function of homophily strength $h$ for different values of basic reproduction number $R_0$. Positive and negative values of $h$ imply homophily and heterophily, respectively. The area shaded in gray represents the parameter region where the network is unrealizable.

Figure 2

Epidemic size in Poisson networks as a function of homophily strength $h$ and vaccine coverage ${\pi }_{\mathrm{v}}$. Top row: Two-dimensional heat maps representing the epidemic size. The solid red line in each panel denotes the vaccination threshold. We represent contours of the epidemic size at 0.1 intervals by different colors and solid black lines. Bottom row: Epidemic size divided by the size of the unvaccinated population. Theoretical predictions (in lines) are compared with the giant component sizes obtained by simulating networks of size $N={10}^5$ (in symbols). The details of the network simulation can be found in the Supplemental Material [24]. (a) and (b) show the results for $R_0=1.5$ and a perfect vaccine, (c) and (d) are for $R_0=3$ and a perfect vaccine, and (e) and (f) are for $R_0=3$ and an imperfect vaccine with $f_{\mathrm{S}}=0.75$. If the vaccine is perfect, only the unvaccinated individuals contract the disease; thus, the vertical axis in (b) and (d) corresponds to the fraction of the unvaccinated population that will be infected. The cross symbols in (f) indicate the maximum of each curve. Note that the homophily strength at which the epidemic size takes the maximum is independent of ${\pi }_{\mathrm{v}}$.

Figure 3

Effects of vaccination homophily for imperfect vaccines. (a) The sizes of vaccinated population $s_{\mathrm{v}}$ and unvaccinated population $s_{\mathrm{u}}$ in the epidemic of size $s$. The parameters are $R_0=3,{\pi }_{\mathrm{v}}=0.8,f_{\mathrm{S}}=0.75$. (b) The homophily strength $h^{*}$ maximising $s$ as a function of $f_{\mathrm{S}}$ and $R_0$.