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

Edited by Nils C. Stenseth, University of Oslo, Oslo, Norway, and approved July 23, 2020 (received for review May 22, 2020)

## Significance

For many countries a plain-eye inspection of the COVID-19 infection curves reveals a remarkable linear growth over extended time periods. This observation is practically impossible to understand with traditional epidemiological models. These, to make them expressible in compact mathematical form, typically ignore the structure of real contact networks that are essential in the characteristic spreading dynamics of COVID-19. Here we show that by properly taking some relevant network features into account, linear growth can be naturally explained. Further, the effect of nonpharmaceutical interventions (NPIs), like national lockdowns, can be modeled with a remarkable degree of precision without fitting or fine-tuning of parameters.

## 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,

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

Textbook knowledge of epidemiology has it that an epidemic event comes to a halt when herd immunity in a population is reached (1, 2). Herd immunity levels depend on the disease. For influenza it is within the range of 33 to 44% of the population (3), for Ebola it is 33 to 60% (4), for measles it is 92 to 95% (5), and for the Severe Acute Respiratory Syndrome (SARS) levels between 50 and 80% are reported (6). For the current COVID-19 outbreak it is expected to be in the range of 29 to 74% (7, 8). On the way toward herd immunity, textbook knowledge teaches, the number of infected increases faster than linear (in early phases even exponentially) as long as the effective reproduction number is larger than 1. Once this threshold is passed, the daily increments in the number of infected start to decrease until they drop to zero (1, 9). Combining these two growth phases yields the characteristic S-shaped infection curves.

The COVID-19 outbreak shows a very different picture, however. Several countries have clearly passed a first maximum of the epidemic and are converging toward zero new cases per day. None of these countries are even close to herd immunity. In Austria at the first peak of the pandemic so far, a population-wide representative PCR study showed that only about 0.3% of the population tested positive (10). Similarly, in Iceland in a random-population screening the prevalence of positively tested was found to be 0.8% (11). Clearly, at this time the COVID-19 outbreak has been far from the uncontrolled case as many countries have implemented nonpharmaceutical interventions (NPIs) to reduce infection rates (12).

Maybe the most striking observation in the COVID-19 infection curves is that they exhibit linear growth for an extended time interval quite in contrast to the S-shaped curves expected from epidemiological models. For a wide range of countries regardless of size, demographic and ethnic composition, or geolocation, this linear growth pattern is apparent even by a plain-eye inspection of the number of positive cases, e.g., ref. 13. In Fig. 1*A* we show infection curves (number of confirmed positive cases) for the United States, the United Kingdom, Sweden, Finland, Poland, Indonesia, and a province of Canada. Clearly, after a short initial exponential phase, infection curves are practically linear for several weeks. For many other examples, see ref. 13. Many countries that implemented NPIs in response to the COVID-19 crisis (12) show a different pattern. They also show an extended linear growth; however, infection curves tend to bend and level off in response to the implemented measures (Fig. 1*B*). The extent of the linear regime depends on the onset of the measures (12). Many countries that are still in the early phase of the pandemic (8 May 2020) show the initial almost exponential growth (*SI Appendix*, Fig. 1). According to basic epidemiological concepts, growth patterns with extended linear regions are not to be expected. They can be observed only if the infection growth rate equals the recovery rate, giving an effective reproduction number,

The basic question of this paper is to clarify the mechanism that keeps *SI Appendix*). A simple explanation could be a limiting capacity of availability of test kits. If the daily number of tests is limited and assuming a fixed ratio of confirmed cases per test, linear growth in the number of positively tested would be the consequence. However, most European countries, even though experiencing initial difficulty with testing capacity, have, by now, enough tests available.

The rationale underlying social distancing efforts is that they lead to a reduction of contacts which essentially makes the social network sparser (12). Infections occur if 1) there is a social interaction between an infected and a susceptible person and 2) this contact is intense enough to lead to a disease transmission. For instance, given a basic reproduction number of

In classic epidemiology network effects have long been ignored in favor of analytical tractability (14). In that case epidemiological models can be formulated as differential equations, assuming that every person in principle can infect any other. This is called the well-mixed or mean-field approximation (*SI Appendix*). However, that fact that networks matter in epidemiology has been recognized for almost two decades and has led to extremely relevant contributions, such as the dependence of vaccination thresholds on network topology (15). Classic contributions such as refs. 16 and 17 were able to incorporate network topology into analytically solvable SIR models. There it is possible to solve the SIR model in terms of outbreak size and epidemic size; however, no focus was put on the details of infection curves below the epidemic limit. When dealing with structured networks, it might well be that the mean-field approximation does no longer hold, and details of the networks start to become crucial.

Since social networks are key to understand details of epidemic outbreaks, what do they look like? The answer is highly nontrivial since social networks are hard to define. In terms of network topology, it became clear that they are neither pure random graphs, nor small-world networks, nor purely scale-free. They are of a more involved structure, including multilevel organization (18); weak ties between communities (19); and temporal aspects that suggest a degree of fluidity, however, with stable social cores (20).

Here we try to understand the origin of the extended linear regime in infection curves, as currently observed in the number of positively tested cases in the COVID-19 pandemic across many countries. To this end we solve the SIR model on a simple social network and report a hitherto unobserved transition from linear growth to S-shaped infection curves. We show that for a given transmission rate there exists a critical degree below which linear growth is expected and above which the model reproduces the classical SIR results. Below the critical degree the mean-field approximation starts to fail. For the underlying social networks we use a Poissonian small-world network that tries to capture several empirical facts, including a heterogenous number of social links (degree), the small-world aspect, the fact that people tend to live in small groups (families), that these groups overlap, and that work and leisure relations can link distant groups (*Methods*). The framework allows us to model a lockdown as a change in social networks with a high degree to one with a degree that characterizes the members of a household. Based on data on household size in the European Union (21), on empirical estimates on how long individuals are contagious, and on transmission (or attack) rates we are able to calibrate the model to real countries. In particular, we compare the situation in the United States and Austria. These countries differ remarkably in size and the measures taken in response to the COVID-19 pandemic (12). While Austria imposed a lockdown relatively early on in combination with a number of other measures, the United States has implemented measures hesitantly with the consequence that the situation was “not under control,” as Dr. A. Fauci, an advisor to the Trump administration, stated on 12 May 2020 (22). The model reproduces the real infection curves to a remarkable degree. All parameters are empirically motivated; there are no fitted parameters involved.

### Model Dynamics.

We assume that there are N individuals connected by social links. If i and j are connected, *Methods*). The small-world aspect allows us to model transmission between local groups and “superspreaders” (23). As in a SIR model, every individual is in one of three possible states, susceptible (S), infected (I), and recovered (R). If an individual is infected, the individual will infect susceptible neighbors with a per-day transmission probability, r. This means that on every single day the probability of passing the infection to a susceptible neighbor is r, which is sometimes called the microscopic spreading rate (17). Once a person is infected that person stays infectious for d consecutive days. After this the person can no longer infect others and is called recovered. Once recovered the state will no longer change. The update rules of the corresponding model are as follows: Initialize all nodes as susceptible; select

At every timestep (day), t, we count the number of new cases, *SI Appendix*) by

## Results

### Infection Dynamics.

We demonstrate the model schematically in Fig. 2 *A*–*C*. In the limit of large degree D and large ϵ the model should approximately fulfill the mean-field conditions and should be close to a classical SIR model. This is seen in Fig. 2*D* where the trajectory of an infection curve,

The infection curve, *E*). Clearly, it increases almost linearly for a remarkable timespan, which is in marked contrast to the SIR expectation (green line). The situation already resembles the situation of many countries. Once the system converged to its final state, only about 17% of nodes were infected, which is far from the expected (SIR) herd immunity level of about 77%.

The change of the infection curve from the S-shaped to a linear behavior is clearly a network effect and indicates that the mean-field assumptions might be violated. To understand this better we next study the parameter dependence more systematically.

### Parameter Dependence and Phase Transition.

We are interested to see whether there is a critical degree, *Methods*). In Fig. 3*A* we show this order parameter as a function of the degree, D, of the network for three transmission rates *A*, arrows). It decreases with the transmission rate r; while for *SI Appendix*, Fig. 2. The asterisks in Fig. 3*A* denote the degree,

We checked that the position of the critical degrees is relatively robust under the size of the network and variations in topology. We find that for *SI Appendix*, Fig. 3).

For the Poissonian small-world network we are able to estimate the critical degree analytically by a “fuse model”*SI Appendix*, Text S6. This result slightly overestimates the simulation results (Table 1). However, for large values of r and ϵ, theoretical predictions and simulation results are in remarkable agreement. Indeed, the used second-order approximation systematically underestimates the spreading velocity, i.e., overestimates the number of infected in active regions of the network, which in turn leads to an overestimation of *SI Appendix*, Text S6). Also finite size effects in the simulation may add to the observed deviations.

The existence of critical degrees signals the presence of a hitherto overlooked transition between linear and S-shaped growth that is most likely due to the fact that the well-mixed or mean-field assumption breaks down below *B* shows 20 realizations of model infection curves for a network with

We confirm that the mechanism to obtain linear infection curves is present also for more realistic social contact networks (20, 24, 25), by running the algorithm on networks derived from actual contact networks that are publicly available (26). For the situation where the degree of these networks falls below *SI Appendix*, Fig. 7.

### Calibration.

We calibrate the model to the COVID-19 infection curves of two countries, the United States and Austria, to demonstrate its potential applicability for estimating the effects of NPIs. For this we have to make the following assumptions on the model parameters:

The viral dynamics of COVID-19 are highly heterogenous (27). Motivated by evidence that people carry viral loads and thus can be contagious for more than 20 d after disease onset (most people are contagious for shorter periods) (28, 29) and given that infectiousness can start 2 to 3 d before showing symptoms (28), we use

In 2019 the average household size in the European Union was 2.3 people (21). If we assume that at work and during leisure activity on average one meets 3 to 4 people more per day, we decide to use an average degree of *SI Appendix*, Text S7.

We use 100,000 nodes and 40 and 100 initially infected for the United States and Austria, respectively. Since it is not possible to compute every individual in the simulation, we decided to initiate the simulation at the point where 0.1% of the population tested positive, that is, 7 April for the United States and 3 April for Austria. For the respective population sizes we use United Nations data from 2019 (31).

In Fig. 4 we show the model infection curves in comparison to the number of positively tested persons (13) for the United States (Fig. 4*A*) and Austria (Fig. 4*B*). Solid blue lines mark the situation where more than 0.1% of the population tested positive; simulations are performed from that date on. Note that one case in the model represents many in reality. In the simulations relatively few cases are produced and the integer steps are still visible. Obviously, the model produces infection curves of the observed type.

We discuss the role of superspreaders in two additional simulations, one where we introduce superspreaders defined as individuals with a much higher degree than the population average and the other where superspreading comes from individuals which are much more contagious but have similar degrees to the rest of the population. Results are presented and discussed in *SI Appendix*, Fig. 4. Both types of superspreaders do shift the critical degree toward lower levels, as expected.

Finally, we performed a cross-correlation analysis of the actually measured *SI Appendix*, Figs. 9 and 10. We can exclude the a priori possibility (32) that mobility fluctuations influence *SI Appendix*, Text S9.

## Discussion and Conclusions

Here we offer an understanding of the origin of the extended linear region of the infection curves that is observed in most countries in the current COVID-19 crisis. This growth pattern is unexpected from mainstream epidemiological understanding. It can be understood as a consequence of the structure of low-degree contact networks and appears naturally as a hitherto unobserved (phase) transition from a linear growth regime to the expected S-shaped curves.

We showed that for any given transmission rate there exists a critical degree of contact networks below which linear infection curves must occur and above which the classical S-shaped curves appear that are known from epidemiological models. The model proposed here is based on a simple toy contact network that mimics features of a heterogenous degree, the small-world property, the fact that people tend to live in small groups that overlap, and the fact that distant groups are linked through work and leisure activities. We showed how the model can be used to simulate the effects of NPIs in response to the crisis by simply switching to low-degree networks that do not allow for linking of distant groups.

The model not only allows us to understand the emergence of the linear growth regime, but also explains why the epidemic halts much below the levels of herd immunity (given no in-flow of infected). Further, it allows us to explain the fact that in countries which are beyond the (first) maximum of the epidemic, a relatively small number of daily cases persist for a long time. This is because small alterations and rearrangements in the contact networks will allow for a very limited spread of infections.

We find that for the empirically motivated parameters used here, the critical degree is

The linear growth phase appears to be dominated by cluster transmission of the disease, meaning that new infections primarily appear in the “small-worlds” or local network neighborhood (households, workplaces, etc.) of infected individuals. In the superlinear (exponential) phase, sustained community transmission sets in where new cases cannot be traced to already known cases in their neighborhood. In this regime, transmission across the shortcuts in the network becomes more prevalent. This effective mixing of the population gives a dynamic that approaches the mean-field case of SIR-like models.

Finally, we calibrated the model to realistic network parameters, transmission rates, and the time of being contagious and showed that realistic infection curves (examples of the United States and Austria are shown) emerge without any fine-tuning of parameters. The onset of the NPI (lockdown)—and the associated reduction of the degree in the contact networks—determines the final size of the outbreak which is well below the levels of herd immunity. We demonstrate the importance of timing of the interventions in *SI Appendix*, Fig. 6. In *SI Appendix*, Text S7 we discuss the impact on the Austrian infection curve if the same NPI would have been implemented 10 d later (*SI Appendix*, Fig. 6*B*). An increase of about 30% of cases is observed. In the same spirit, in *SI Appendix*, Fig. 6*A* we demonstrate what could have happened if NPIs, with similar effects to those that were implemented in Austria, had been installed in the United States. The results indicate that about half of the cases could have been avoided (at the beginning of May 2020).

For a more detailed discussion of the applicability of the presented model in terms of modeling the effects of NPIs, refer to *SI Appendix*, Text S7. In *SI Appendix*, Text S8 we discuss three additional case studies of the infection curves of China (Hubei), Singapore, and South Korea. We implement temporal sequences of changes in the model parameters that roughly resemble the effects of the NPIs implemented in reality. We recover the basic features of the actual infection curves to a remarkable extent, at least qualitatively (*SI Appendix*, Fig. 8).

The two types of superspreading, the network based and transmission based, both lead to a clear finding that the presence of superspreaders shifts the critical degree toward lower levels. However, the presented mechanism to obtain linear infection curves remains fully intact. The message for Austria and the United States is that in both countries the density of superspreaders is not high enough in the considered observation period to shift the critical degree toward exponential growth, given the effective degrees in the populations.

Given the number of countries that entered linear growth phases, our results raise serious concerns regarding the applicability of standard compartmental models to describe the containment phase achieved by means of NPIs. SIR-like models show linear growth only after fine-tuning parameters and linear growth would be a mere statistical fluke. We argue that network effects must be taken into account to understand postintervention epidemic dynamics.

## Methods

### Poissonian Small-World Network.

For the network A we use a Poissonian small-world network, which generalizes the usual regular small-world network in the sense that the degree is not fixed, but is chosen from a Poissonian distribution, characterized by λ. The network is created by first imposing a Poissonian degree sequence on all nodes. Assume that nodes are arranged on a circle. Nodes are then linked to their closest neighboring nodes on the circle. This creates a situation where every person is a member of a small local community. As for real families, these communities strongly overlap. Finally, as for the conventional small-world network, with probability ϵ we relink the links of every node i to a new, randomly chosen target node j, which can be far away in terms of distance on the circle. ϵ is the fraction of an individual’s social contacts that are outside the local community (family). These links can be seen as links to colleagues at work or leisure activities and allow us to model the existence of superspreaders (23). Note that the actual average degree of the so-generated network is very close to the λ of the Poisson distribution,

### Order Parameter.

To distinguish the linear from the sigmoidal growth, we propose a simple “order parameter” as the standard deviation (SD) of all new daily cases (after excluding all days where there are no new cases),

## Data Availability.

All study data are included in this article and *SI Appendix*.

## Acknowledgments

We thank Christian Diem for helpful discussions, Georg Heiler and Tobias Reisch for the mobility data, and two anonymous referees for a large number of suggestions to improve the work. This work was supported in part by the Austrian Science Promotion Agency project under Grant FFG 857136, the Austrian Science Fund under Grant FWF P29252, the Wiener Wissenschafts und Technologiefonds under Grant COV 20-017, and the Medizinisch-Wissenschaftlichen Fonds des Bürgermeisters der Bundeshauptstadt Wien under Grant CoVid004.

## Footnotes

- ↵
^{1}To whom correspondence may be addressed. Email: stefan.thurner{at}muv.ac.at.

Author contributions: S.T. designed research; S.T. and P.K. conceived the work; S.T. and P.K. performed research; S.T., P.K., and R.H. contributed new reagents/analytic tools; S.T., P.K., and R.H. analyzed data; and S.T. wrote the paper.

The authors declare no competing interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at https://www.pnas.org/lookup/suppl/doi:10.1073/pnas.2010398117/-/DCSupplemental.

- Copyright © 2020 the Author(s). Published by PNAS.

This open access article is distributed under Creative Commons Attribution License 4.0 (CC BY).

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