Heterogeneity in SIR epidemics modeling: superspreaders and herd immunity
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RESEARCH
Applied Network Science
Open Access
Heterogeneity in SIR epidemics modeling: superspreaders and herd immunity Istvan Szapudi*
*Correspondence: [email protected] Institute for Astronomy, University of Hawaii, 2680 Woodlawn Drive, Honolulu, HI 96822, USA
Abstract Deterministic epidemic models, such as the Susceptible-Infected-Recovered (SIR) model, are immensely useful even if they lack the nuance and complexity of social contacts at the heart of network science modeling. Here we present a simple modification of the SIR equations to include the heterogeneity of social connection networks. A typical power-law model of social interactions from network science reproduces the observation that individuals with a high number of contacts, “hubs” or “superspreaders”, can become the primary conduits for transmission. Conversely, once the tail of the distribution is saturated, herd immunity sets in at a smaller overall recovered fraction than in the analogous SIR model. The new dynamical equations suggest that cutting off the tail of the social connection distribution, i.e., stopping superspreaders, is an efficient non-pharmaceutical intervention to slow the spread of a pandemic, such as the Coronavirus Disease 2019 (COVID-19). Keywords: COVID-19, Epidemiology, SIR equations, Disease dynamics, Network science, Superspreaders, Herd immunity
Introduction According to recent results on the spread of COVID-19, a small fraction of the population is responsible for most infections (Adam et al 2020; Schuchat 2020). Clusters in care facilities, restaurants and bars, workplaces, and music events (Furuse et al 2020), or even choir practice (Hamner et al. 2020), dance events (Jang et al 2020) are a hallmark of superspreading (Lloyd-Smith et al. 2005). The most wide-spread deterministic Kermack–McKendrick equations (Kermack and McKendrick 1927), or SIR equations, are not designed to model superspreading. Superspreading is introduced as an additional dispersion of the secondary infections (Lloyd-Smith et al. 2005) using a negative binomial distribution, possibly in the context of random network theory (Hébert-Dufresne et al 2020), or modeled with stochastic Markov Chain methods (Allen 2017). The simplest way to modify the SIR equations to include superspreaders is by adding a new class of susceptible individuals, P, superspreaders (Mkhatshwa and Mummert 2010) to the equations. However, network science (Barabási and Albert 1999) suggests a more complex picture. If we map individuals into vertices of a graph and their connections
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