PHYSICAL REVIEW E 101, 052309 (2020) Ring vaccination strategy in networks: A mixed percolation approach Lautaro Vassallo , 1, * Matías A. Di Muro, 1 Debmalya Sarkar , 2 Lucas D. Valdez , 3 and Lidia A. Braunstein 1, 3 1 Instituto de Investigaciones Físicas de Mar del Plata (IFIMAR-CONICET) and Departamento de Física, Facultad de Ciencias Exactas y Naturales, Universidad Nacional de Mar del Plata, 7600 Mar del Plata, Argentina 2 Department of Information and Communication Technology, Manipal Institute of Technology, Manipal Academy of Higher Education, Manipal 576104, India 3 Physics Department, Boston University, Boston, Massachusetts 02215, USA (Received 13 January 2020; accepted 20 April 2020; published 12 May 2020) Ring vaccination is a mitigation strategy that consists in seeking and vaccinating the contacts of a sick patient, in order to provide immunization and halt the spread of disease. We study an extension of the susceptible- infected-recovered (SIR) epidemic model with ring vaccination in complex and spatial networks. Previously, a correspondence between this model and a link percolation process has been established, however, this is only valid in complex networks. Here, we propose that the SIR model with ring vaccination is equivalent to a mixed percolation process of links and nodes, which offers a more complete description of the process. We verify that this approach is valid in both complex and spatial networks, the latter being built according to the Waxman model. This model establishes a distance-dependent cost of connection between individuals arranged in a square lattice. We determine the epidemic-free regions in a phase diagram based on the wiring cost and the parameters of the epidemic model (vaccination and infection probabilities and recovery time). In addition, we find that for long recovery times this model maps into a pure node percolation process, in contrast to the SIR model without ring vaccination, which maps into link percolation. DOI: 10.1103/PhysRevE.101.052309 I. INTRODUCTION The health of the population is one of the major concerns of governments and international health agencies globally. As reported by the World Health Organization, 7.5 trillion dollars was spent on health in 2016, equivalent to approximately 10% of the global gross domestic product [1]. In addition to the thousands of new pathogens discovered in the last decades, the reemergence of infectious diseases such as cholera, plague, and yellow fever is concerning [2]. Changes in lifestyles, coupled with environmental and biological changes, cause epidemics of infectious diseases to be more likely to occur and to spread farther and faster than ever before [2]. For this reason, significant efforts must be devoted to the study and control of infectious diseases, and the design of effective prevention and mitigation strategies is crucial. Complex networks [3–6] have proven to be an important tool in the study of the spread of epidemics since they capture some features of human societies [7–10]. The nodes of a complex network represent individuals, while links repre- sent interactions between them. In an epidemic model, the nodes adopt different states, such as susceptible or infected, while the links allow contagion between the nodes. This approach is useful when physical contact is the main route of transmission [11,12]. Epidemic models can be tested on networks with different topologies, which would represent different social structures, and in these models, the impact of different mitigation strategies, such as quarantine [13–15], * lvassallo@mdp.edu.ar isolation [13,14,16,17], and vaccination [16,18–22], can be estimated. In remote times, diseases spread as a diffusive process [23], while nowadays, due to modern means of transport, they can cover long distances in short times and be superdiffusive [24]. A model that allows weighting both behaviors is the Waxman model [25,26], which takes the distance-dependent cost into account when establishing connections between nodes (see Sec. IV). This model captures the fact that human interactions have a cost, such as time travel, which makes short-range interactions more frequent than long-range interactions. A wiring cost is also present in several other complex structures such as neural [27,28], telecommunication [29], Internet [30], and electricity networks [31]. Recently, the Waxman model was used to study the effects of the wiring cost on the univer- sality of critical phenomena [32]. A network built according to this model coincides with the Erdös-Rényi (ER) network [33] if the wiring cost tends to 0, which has a Poisson degree distribution of contacts. An epidemic model that is commonly used is the susceptible-infected-recovered (SIR) model [34–37], which reproduces the spread of nonrecurrent diseases such as in- fluenza and SARS [38]. In this compartmental model, nodes can be in three possible states: susceptible (S), infected (I), or recovered (R). When susceptible nodes are in contact with in- fected ones, they may become ill with probability β . Infected individuals recover after a period of time t r , remaining in this state as they acquire immunity against the disease. In the final state, there are only individuals in the S or R compartment. The fraction of recovered individuals R indicates the extent of the epidemic. The effective probability of transmission of the 2470-0045/2020/101(5)/052309(10) 052309-1 ©2020 American Physical Society