Covid-19: How Modeling Infection Rates Highlights the Need for Social Distancing

A recent paper shows the impact of undocumented cases on the spread of covid-19.

This week, we are looking over a paper called Substantial undocumented infection facilitates the rapid dissemination of novel coronavirus (SARS-CoV2). It covers the rapid spread of covid-19 and was published March 16, 2020.

The paper models infection rates in China, retrospectively understanding the dynamics at play before and after control measures were put in place. In doing so, it can help inform policy decisions as nations attempt to proactively respond to the virus.

The scientists here only modeled the data; they did not collect it themselves. In the Supplementary Materials, they note National and Provincial Health Commissions in China as their data source for daily coronavirus cases.

The variables input into this model are as follows:

  1. θ – travel multiplicative factor to account for unreported travel
  2. Si, Ei, Iri, and Iui – the susceptible (S), exposed (E), documented infected (Ir), and undocumented infected (Iu) populations in some city i
  3. Z – average latent period
  4. D – average length of infection
  5. α – fraction of documented infections
  6. β – transmission rate from sick people documented with the illness
  7. μβ – transmission rate from sick people undocumented with the illness

Kalman filters are an algorithm for modeling systems where it is impossible to measure certain variables. So they don’t know the susceptible, the exposed, and the undocumented infected for any one city – but they could use this algorithm, and data from 375 Chinese cities, including their hospitalization and travel rates.

From there, they could start fiddling with the multipliers. What if I increase μ up or down? (This controls how contagious the undocumented individuals are.) As they fiddled, they compared the models’ predicted infection rates with the actual progression of covid-19 throughout China. They found that when μ = 0.55, the model fit the actual data most closely.

That suggests that undocumented individuals – people who are sick but show no symptoms or have not been tested – are about half as contagious as documented individuals. Even more startling, when they set their models to μ = 0 (what if undocumented individuals are not contagious at all?), the predicted infection rates fell to an almost flat line, with minimal growth completely contrary to observed rates (Fig. 2).

This alone demonstrates the need for social distancing. Most of the rapid spread we have seen cannot be accounted for by the documented cases alone. Individuals who carry it – even without their knowledge – account for much of the exponential spread of infection. By their estimates, 80-90% of the cases in these 375 cities were infected from undocumented individuals.

The scientists then repeated their model for infection rates after control measures began. In these 375 cities, control measures included government-mandated travel restrictions and self-quarantine, and rapid testing became available. What affect did this have on infection rates?

This required a re-start of the model with new estimations. Inter-city travel stopped by up to 98%, and self-quarantine stopping inner-city travel altogether. In the model, this helped to lower transmission rates (though in my opinion confidence intervals were a little wide for this one). The documented vs. undocumented cases in individuals also changed, with documented cases rising from 14% to 65% (with a much narrower and thus, more trustworthy, confidence interval).

So yes, it takes a while to see the effects of social distancing – everyone stays home, but you still see exponential growth. That’s to be expected, simply because more of the cases are becoming known even as less cases are being transmitted.

Bottom line: undocumented cases and high travel contributed to the rapid spread of disease observed in China earlier this year. Furthermore, the model predicted a nine day delay between infection and documentation; according to their best-fitting model, you can be an undocumented and infectious covid-19 carrier for nine days prior to diagnosis.

The scientists note that as this model was based off of Chinese cities, it does not necessarily hold for other countries with different social and travel habits. An epidemiological study includes human behavior, which can alter the variables in unexpected ways. Whether this paper accurately describes how quickly covid spreads in your town or not, you can take confidence in this: your staying home is making a difference.

Photo credit


Li, R., Pei, S., Chen, B., Song, Y., Zhang, T., Yang, W., & Shaman, J. (2020). Substantial undocumented infection facilitates the rapid dissemination of novel coronavirus (SARS-CoV2). Science.


  1. Thank you for your review, this was so informative. I am surprised to read that 80-90% cases is due to undocumented patients. This explains the rise of number of cases in Pakistan where people go untested due to lack of kits.

    Liked by 1 person

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