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, [Formula: see text], below which linear growth and low infection prevalence must occur. Above [Formula: see text] 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 [Formula: see text] 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.

Keywords: COVID-19; compartmental epidemiological model; mean-field (well mixed) approximation; network theory; social contact networks.

Fig. 1.

Cumulative numbers of positively tested cases normalized to the last day (8 May 2020). Countries, even though many followed radically different strategies in response to the pandemic, seem to belong to one of three groups: (A) countries with a remarkably extended linear increase of the cumulated number of positively tested cases, including the United States, the United Kingdom, and Sweden, and (B) countries with an extended linear increase that tends to level off and enter a regime with a smaller slope. B, Inset shows an extended regime after the peak (cases per population size).

Fig. 2.

Schematic demonstration of the model. Nodes are connected in a Poissonian small-world network. Locally close neighbors resemble the family contacts, and long links to different regions represent contacts to others, such as people at work. (A) Initially, a subset of nodes is infected (blue), and most are susceptible (green). (B) At every timestep, infected nodes spread the disease to any of their neighbors with probability $r$. After $d$ days infected nodes turn into “recovered” and no longer spread the disease. (C) The dynamics end when no more nodes can be infected and all are recovered. (D) Infection curve $P\left(t\right)$ (blue dots) for the model on a dense Poissonian small-world network, $D=8$. The daily cases (red) first increase and then decrease. For comparison, we show the recovered cases, $R\left(t\right)$, of the corresponding SIR model with $\gamma =1/d$, and $\beta =rD/N$ (green). The mean-field conditions are obviously justified to a large extent. (E) Situation for the same parameters except for a lower average degree, $D=3$. The infection curve now increases almost linearly; daily increases are nearly constant for a long time. The dynamics reach a halt at about 17% infected. The discrepancy to the SIR model (green) is now obvious.

Fig. 3.

(A) Order parameter for the transition from linear to S-shaped infection curves as a function of degree, $D$, for transmission rates $r=0.05$ (blue), 0.1 (red), and 0.2 (orange). The transition happens at the critical points, $D_c$, where the order parameter starts to diverge (arrows) (Table 1). The asterisks in Fig. 3A denote the degree, $D_{\mathrm{s}\mathrm{i}\mathrm{r}}$, at which the SIR model would show a linear curve, $D_{\mathrm{s}\mathrm{i}\mathrm{r}}=1/dr$. Colors correspond to the respective $r$. (B) Infection curves (20 realizations) for three scenarios for a network with $D=10$. Red scenario: At a transmission rate of $r=0.1$ we see S-shaped curves reaching herd immunity at about 75%. Black scenario: For the same network with a lower transmission rate, $r=0.05$, we fall below the critical degree $D_c$ and consequently observe linear growth; note the convergence of infected at levels of 1 to 4%, which are very much below herd immunity (75%). Turquoise scenario (lockdown): We start with the same network with $r=0.1$, as in the red scenario. After 5% of the population (black bar) is infected there is a lockdown that changes the network to one of degree $D_2=3$, from one day to the next. The S-shaped growth immediately stops and levels off at about 10% infected. Other parameters: $d=2$, $ϵ=0.3$, and $N=\mathrm{10,000}$; 10 initially infected.

Fig. 4.

Model infection curves (red) when calibrated to the COVID-19 curves of positively tested in (A) the United States and (B) Austria. Five realizations with different sets of initially infected are shown. The simulation starts when more than 0.1% of the population tested positive. The situation in the United States assumes a Poissonian small-world network with average (daily) degree $D=5$. The lockdown scenario in Austria that has been in place from 16 March to 15 May 2020 is modeled with social contacts limited to households, $D=2.5$. For the choice of the other model parameters, see main text. The model clearly produces the correct type of infection curves.