Non-Pharmaceutical Interventions Reshape Network Immunization Outcomes

TL;DR

A reversal in the relative effectiveness of disease- versus vaccine-induced immunization schemes is uncovered, highlighting the average number of contacts as a critical determinant of emerging herd immunity in sparse geometric networks with limited degree heterogeneity.

Abstract

Herd immunity is shaped not only by the infection capacity of a spreading epidemic or the contact structure of the hosting population, but also by how and under what circumstances individuals acquire immunity. Immunization strategies may interact with ongoing non-pharmaceutical interventions, which commonly aim to reduce social contact numbers. We demonstrate that these interactions can induce unexpectedly strong and counterintuitive effects on herd immunity. We explore these phenomena on spatially embedded contact networks and uncover a reversal in the relative effectiveness of disease- versus vaccine-induced immunization schemes, highlighting the average number of contacts as a critical determinant of emerging herd immunity. In sparse geometric networks with limited degree heterogeneity, uniform vaccination proves most effective; however, as average contact numbers increase, naturally acquired immunity ultimately becomes the better strategy. We show that this phenomenon may emerge not only in synthetic networks but also in real-world mixing networks, observed during non-pharmaceutical intervention periods across multiple states of the United States.

Figures (18)

• Figure 1: Dependence of natural and random immunization effects on network connectivity. Panel a) shows the daily number of new confirmed COVID-19 cases per million people in Hungary (solid line) together with the average number of contacts of people, as recorded in the MASZK questionnaire (dashed green line), between March 2020 and March 2022. Panels b-c and e-f show the outcomes of natural (left-red) and random (right-blue) immunization on modeled Random Geometric Graphs (RGGs) (modeled as special cases of GIRGs), respectively with $\langle k \rangle = 6$, and $\langle k \rangle = 11$. In all these networks, immune individuals are consistently shown in gray, with their fraction fixed at $f \approx 0.43$. Darker shades highlight the largest residual component, while lighter colors represent the indirectly protected smaller clusters. Panel d) depicts the schematic comparison of natural and random immunization outcomes and the $C$ size of their largest residual component as the function of $f$ fraction of immunized nodes.
• Figure 2: Effectiveness of disease- and vaccine-induced immunity across scale-free contact networks with varying average degree $\left<k\right>$ and fixed degree-decay exponent $\gamma = 4$. Panels a) and b) display the size of the largest residual component under disease-induced (red curves) and vaccine-induced (blue curves) immunization, denoted by $C_N$ and $C_R$, respectively, both plotted as a function of immunization coverage $f$. Results are shown for contact networks with varying average degrees $\left<k\right>$, indicated by different color tones on the gray-scaled colorbar. Panel a) shows the results for non-geometric networks (GIRGs with $\alpha\to 1$), while panel b) displays the corresponding results for their geometric counterparts, generated at $\alpha = 10^6$. Each curve represents averages over five independent network realizations, with 200 SIR processes per network evaluated at 150 transmission probabilities. Panels c) and d) depict the difference in the fraction of indirectly protected individuals between the two immunization strategies, $\delta\pi_{NR}$, for the same types of contact networks shown as a function of both immunization coverage $f$ and the network’s average degree $\langle k\rangle$. Surface points are colored according to the magnitude and sign of this difference using the blue–red colormap below: red indicates greater indirect protection under natural immunity, whereas blue indicates the opposite. Panel c) showcases results for the configuration-model limit, and panel d) for strongly geometric GIRG networks. Each surface point is obtained from 200 SIR simulations evaluated at 100 evenly spaced transmission probabilities.
• Figure 3: Effects of immunization strategies in geometric and non-geometric scale-free networks. Cumulative structural immunity $\Pi$ (panels a, b), average degree of immunized nodes $K$ (upper insets), and interface area between susceptible and immunized nodes $P$ (lower insets) are shown as functions of the average contact number $\langle k \rangle$ for natural (red) and random (blue) immunity. Panel a) corresponds to strongly geometric GIRGs ($\alpha = 10^6$, $\gamma=4$), while panel b) depicts non-geometric limits ($\alpha \to 1$, $\gamma=4$). The displayed metrics are evaluated across all immunity coverage levels $f \in [0,1]$, and each point shows an average over five independent network realizations.
• Figure 4: The relative effectiveness of disease- and vaccine-induced immunization on state-level U.S. mobility networks before and during lockdowns. Panels a) and b) display the $\Delta\Pi_{NR}$ relative difference in effectiveness values obtained for each state, highlighting the spatial distribution of the effective immunization strategy before and during pandemic interventions. Specifically, panel b) corresponds to the pre-lockdown period in April 2019, while panel b) depicts results during the lockdown period in April 2020. The insets show the layouts of the mobility networks corresponding to California in the two time periods. In all panels, colors indicates the magnitude of $\Delta\Pi_{NR}$, according to the color scale on the right of panel c). Red and blue shows the better effectiveness of natural and vaccine-induced immunity, respectively. Data are displayed for a state-wise optimal threshold value $w^{*}_{\text{thres}}$ (see Methods and Supplementary Information S4). Panel c) shows a bar chart illustrating the differences in effectiveness between natural and vaccine-induced immunity for each U.S. state mobility network with circles representing the pre-lockdown period and triangles indicating the during-lockdown period.
• Figure S1: Visualization of averaging methods and the relationship between transmission probability and the final fraction of immunized individuals under natural immunization. Panels a), b) depict the relationship between the transmission probability $T$ of the immunity-inducing epidemics and the stationary fraction of immunized individuals $f$. Panels a), c) shows results for a strongly geometric GIRG ($\alpha = 10^6$), while panels b), d) corresponds to the configuration model limit of GIRGs ($\alpha = 1$). Networks have size $N = 10^4$ and degree exponent $\gamma = 4$. Scatter points show the averages computed using Method A, whereas solid lines represent the outcomes of Method B, highlighting how the choice of averaging procedure affects the binning of the order parameter $f$. Green color correspond to sparser networks ($\langle k \rangle = 6$) while purple color to denser networks ($\langle k \rangle = 12$), illustrating the influence of contact density on the final fraction of immunized individuals.