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.
During an unfolding pandemic, countries may adopt different intervention strategies to balance between epidemic control, societal consequences, and economic burden [1][2][3] . Throughout the COVID-19 pandemic, several non-pharmaceutical interventions have been applied 4 , ranging from strict measures, such as national lock-downs, school closings and travel bans, to weaker mitigation strategies like mask-wearing and social distancing. All these measures essentially aim to decrease the number of social contacts between people to localize outbreaks and to reduce the possible transmission routes of the pathogen 5,6 . Such interventions alone, however, can hardly bring an epidemic to an end, as this requires building high immunity level within the population. This so called herd immunity can be achieved either by the distribution of vaccines, when available, or through the natural spread of the pathogen among the people 7,8 . While there is an ongoing debate whether vaccination or natural immunity is more efficient during a pandemic, like the COVID-19 pandemic 9 or HIV 10 , far less is known about how these immunization strategies perform when they interact with interventions 11 and the structure of the underlying social network 12 . We show that in certain cases, natural immunity could lead to better epidemic control of an upcoming outbreak, while vaccine induced immunity can be more effective if it is accompanied with non-pharmaceutical interventions, which effectively decrease social contacts.
We employ a network approach to provide a realistic and quantifiable insight into the outcome of different immunization strategies [13][14][15][16][17][18][19][20] . This approach assumes that the social network of people encodes all possible epidemic routes, highlighting the role of structural 21 , spatial 22,23 , and temporal 24 network properties in epidemic spreading. We simulate the epidemic as a spreading process on the network, and we model disease-induced immunity by assuming that infected nodes after recovery become immune to the disease, a process hereafter referred to as natural immunization. Vaccination is incorporated as an alternative immunization process, independent of the underlying network structure. Although vaccination behavior is known to depend on various social factors, we restrict our analysis to random immunization as a minimal model of vaccine-induced immunity 25 .
However, as found earlier, immunized individuals not only receive direct protection, but they simultaneously act as transmission barriers by blocking and shortening infection pathways 25,26 .Thereby, on one hand, they slow down or even prevent the spread of disease to others 25,27 ; and on the other hand, they create indirect collective protection for the non-immune community, known as the free rider effect [28][29][30] . This effect appears due to the unrolling immunization process, which gradually fragments the underlying network of susceptible individuals, resulting in many disconnected, smaller susceptible components. The magnitude of the free-rider effect, and thereby the effectiveness of the immunization strategy, can be measured by the size of the largest residual susceptible connected component after immunization, also known as the secondary outbreak size 20,31 . Intuitively, effective immunization strategies are able to fragment into small components, with the secondary outbreak size being a worst-case upper bound on the size of a subsequent second wave of the infection, seeded from a single node 20 .
Previous work, by Hiraoka et al., compared the random and natural immunization processes and found that they may fragment a social network in markedly different ways, giving rise to two competing network effects 20 . In one way, the epidemic tends to infect and thereby immunize highly connected individuals first, effectively turning natural immunization into a form of targeted immunization 13,14,20 . This mechanism enhances network fragmentation and strengthens disease-induced immunity, especially in networks with greater degree heterogeneity. On the other hand, since natural immunity develops along the pathways of the disease spread, the resulting immune nodes become highly localized, especially in clustered spatial networks. This localization effect weakens the effects of natural immunization, since it reduces the interface between immune and susceptible individuals, as compared to random immunization. As a consequence, the relative performance of the two strategies depends heavily on the heterogeneity of the degree (contact number) distribution and the geographic localization properties of the network. However, this description neglects the effects of non-pharmaceutical interventions, which commonly aim to decrease contact numbers, and this way could crucially change the properties of the underlying network 4 . Their effects on the social structure could lead to an overall drop of network density, to increased clustering,
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