Infection curves on small-world networks are linear
only in the vicinity of the critical pointŁukasz Kuśmierza,1 and Taro Toyoizumia,b

In ref. 1, an epidemiological model based on the smallworld (2) contact network and the SIR (susceptibleinfected-recovered) infection dynamics is analyzed andcompared with the classical (well-mixed) SIR model. Theauthors claim that they observe a “a hitherto unobserved transition from linear growth to S-shaped infection curves.” This is rather surprising in the context ofprevious literature studying similar models (3–9). While
scale-free networks (10) were reported to give rise
to nonstandard features, including the absence of a
nontrivial epidemic threshold and an algebraic spreading growth (6–8), no such abnormalities were observedon small-world networks (3–5, 7).
Indeed, in the following, we show that the analysis
presented in the paper (1) is flawed. The problem liesin the order parameter O = SDðCðtÞÞ, the standard deviation of new daily cases (excluding days with no newcases), which is used to signal the postulated transition
between an S-shaped growth (O > 0) and a lineargrowth (O ≈ 0). In reality, O can attain values close tozero for two distinct reasons: 1) CðtÞ is constant (corresponding to a linear growth of the cumulative number of cases) as the authors claim, or 2) CðtÞ exhibits
low values, signaling that the epidemic is not able toeffectively spread in the population and the number ofcases drops exponentially, corresponding to the basic
reproduction number R0 taking a value below one.
Case 2, which is in line with the prediction of the classical SIR model, was never considered in ref. 1; boththeoretical and numerical calculations established theexistence of a critical point, but no evidence wasreported for the existence of the postulated linear
growth phase below the critical point.
We hypothesized that case 2 is the mechanism
yielding O ≈ 0 below the critical point. To test this, weindependently simulated the model introduced in ref.
1. First, we verified that O signals a continuous transition between two distinct phases (Fig. 1A). Second,
we calculated the basic reproduction number R0(Fig. 1B), which confirmed that, below the criticalpoint, an infection does not spread through the network, consistent with case 2. Third, we calculated theoutbreak duration Tend, which exhibits a sharp maximum at the critical point and drops especially quickly
below the critical point (Fig. 1C). Because Tend upperbounds the possible duration of a linear growth, thisresult indicates that prolonged linear growth is impossible even slightly below the critical point. Last, wecompared the evolution of the cumulative number of
infections obtained from the small-world model andthe classical SIR model (Fig. 1D). Already slightly below the critical point, the growth curve of the networkmodel is significantly sublinear, as expected from theclassical picture.
In conclusion, the presented results corroborate
our supposition that the linear growth in the cumulative number of infections is restricted to the vicinity of
the critical point. Furthermore, we checked that theresults are very similar on Erd}os–Rényi networks (Fig.
2). Thus, the model introduced in ref. 1 does not predict that nor explain why “most COVID-19 infection
curves are linear.”

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aLaboratory for Neural Computation and Adaptation, RIKEN Center for Brain Science, Saitama 351-0198, Japan; and bDepartment of MathematicalInformatics, Graduate School of Information Science and Technology, The University of Tokyo, Tokyo 113-8656, JapanAuthor contributions: Ł.K. designed research; Ł.K. performed research; Ł.K. and T.T. analyzed data; and Ł.K. and T.T. wrote the paper.
The authors declare no competing interest.
This open access article is distributed under Creative Commons Attribution License 4.0 (CC BY).
1To whom correspondence may be addressed. Email: nalewkoz@gmail.com.
Published February 26, 2021.

PNAS 2021 Vol. 118 No. 10 e2024297118

https://doi.org/10.1073/pnas.2024297118 | 1 of 2LETTERLETTER

ABCDFig. 1. Statistics obtained via computer simulations of the epidemic spread on Poisson small-world random contact networks, defined in ref. 1.
Parameters are the same as in figure 3A in ref. 1 (d = 4, e = 0.3, and N = 104). The simulation starts from 10 randomly infected nodes. All statistics
were obtained by averaging over M = 103 realizations of the process, each with an independent realization of the random graph. (A) The SD ofdaily cases; this panel confirms that we reproduce results from ref. 1 (see their figure 3A). In ref. 1, the low SDðCðtÞÞ regions are interpreted aslinear growth regimes. (B) The basic reproduction number as a function of the average degree of the network. The dashed line represents thecritical value R0 = 1. Below the critical point, the outbreak dies out exponentially; our numerical estimates confirm R0 < 1 in this regime. (C) Theoutbreak duration Tend, that is, the average time elapsed between the onset of an epidemic and its end (defined as the first time at which there
are no infected nodes in the population). We observe a sharp peak around the critical point, confirming that the slow dynamics, consistent with alinear growth, is only possible around the critical point. (D) The cumulative numbers of infections as a function of time (here r = 0.1). The dashedlines in D were obtained from the corresponding effective classical SIR models, which assume continuous time and homogeneous mixing.
Although the match is not perfect, the classical model can reproduce qualitative features of the network model studied in ref. 1.

ABCD

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Fig. 2. Statistics obtained via computer simulations of the epidemic spread on Erd}os–Rényi contact networks. The meaning of A–D andparameters used in the simulations are the same as in Fig. 1 (excluding the parameter e, which does not apply here).

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Kuśmierz and Toyoizumi
Infection curves on small-world networks are linear only in the vicinity of the critical point