A network-based explanation of why most COVID-19 infection curves are linear

Abstract

Many countries have passed their first COVID-19 epidemic peak. Traditional epidemiological models describe this as a result of nonpharmaceutical interventions pushing the growth rate below the recovery rate. In this phase of the pandemic many countries showed an almost linear growth of confirmed cases for extended time periods. This new containment regime is hard to explain by traditional models where either infection numbers grow explosively until herd immunity is reached or the epidemic is completely suppressed. Here we offer an explanation of this puzzling observation based on the structure of contact networks. We show that for any given transmission rate there exists a critical number of social contacts, Dc, below which linear growth and low infection prevalence must occur. Above Dc traditional epidemiological dynamics take place, e.g., as in susceptible–infected–recovered (SIR) models. When calibrating our model to empirical estimates of the transmission rate and the number of days being contagious, we find Dc∼7.2. Assuming realistic contact networks with a degree of about 5, and assuming that lockdown measures would reduce that to household size (about 2.5), we reproduce actual infection curves with remarkable precision, without fitting or fine-tuning of parameters. In particular, we compare the United States and Austria, as examples for one country that initially did not impose measures and one that responded with a severe lockdown early on. Our findings question the applicability of standard compartmental models to describe the COVID-19 containment phase. The probability to observe linear growth in these is practically zero.

A primer on using mathematics to understand COVID-19 dynamics: Modeling, analysis and simulations

“…Network models are often used when the research question to be addressed involves heterogeneous populations that are connected through specific mechanisms. A number of network models, such as those in ( Firth, Hellewell, Klepac, & Kissler, 2020 ; Thurner, Klimek, & Hanel, 2020 ; Xue et al., 2020 ), have been used to study the transmission dynamics and control of COVID-19.…”

Exact solution of infection dynamics with gamma distribution of generation intervals

“…At short times the outbreak grows as a power law, t s −1 , where the exponent is determined by the shape parameter of the gamma distribution. Therefore a power law growth is not incompatible with the homogeneous mixing approximation as previously claimed [17]. Furthermore, this power law should not be confused with the long-time power law induced by the truncation of the disease transmission at a maximum generation, as expected from an imposed lockdown for example [7, 9].…”

Modeling Growth, Containment and Decay of the COVID-19 Epidemic in Italy

“…Among others, the number of confirmed infections is influenced by the number of tests performed each day, an aspect that was only weakly incorporated into the model via the (constant) fraction of reported cases ϕ r . Even though many countries experienced initial difficulties with the testing capacity, the robustness and universality of the observed epidemic trends [19] strongly suggest that the scaling laws at play are primarily driven by the described containment-related mechanisms, with the scale of testing playing a minor role in the process [22]. Furthermore, the COVID-19 pandemic has been shown to have a strong geographical character with localized outbreaks, due to socalled superspreaders or family clusters [42].…”

“…Network effects and their relationship with COVID-19 epidemic trends are further discussed in Ref. 22.…”