The Imperial College Modeling Of The Coronavirus

The Imperial College study, “Impact of non-pharmaceutical interventions (NPIs) to reduce COVID19 mortality and healthcare demand,” from the Imperial College COVID-19 Response team, is said to have influenced both the United Kingdom and the United States governments.

It models how the SARS-CoV-2 virus and its resulting disease, COVID-19, might spread through those two countries. The numbers of cases of COVID-19 and resulting deaths are high, as are the amounts of time predicted necessary for social distancing. It looks at interventions that can bring the numbers down or spread them out in time.

All models are wrong, but some are useful, it is said. It can be misleading to simply take those bottom-line numbers, though. What models are good for is looking at what changes can be put into the system and which are most important.

This is going to be somewhat wonky, so I’ll interpolate, in italics, tl;drs that will summarize the following material.

tl;dr: This is a description of the model. If you don’t read anything else in this section, read this and look at the demonstrations. They are how the model works, but it applies different numbers to the probabilities that a red dot will turn a blue dot red.

The model used is an individual-based simulation model. That means that it follows simulated individuals through time as they participate in communities within the household, at school, in the workplace and in the wider community. It then combines the histories of many individuals to give an overall outcome.

Calculating these interactions requires:

  • Age and household distribution size – from census data
  • Population of schools distributed proportional to local population density – from data on average class sizes and staff-student ratios
  • Size and geographical location of workplace.

These data are generally available and reasonably reliable.

The virus is transmitted through contacts between susceptible and infectious individuals in those four places. Transmission in the community depends on spatial distance between contacts, and contacts in schools are taken to be double those elsewhere.

From these calculations, the paper concludes that “approximately one third of transmission occurs in the household, one third in schools and workplaces and the remaining third in the community. These contact patterns reproduce those reported in social mixing surveys.”

That’s the groundwork for the rest of the analysis. Now come the numbers we don’t know well at all. A model like this is made up of equations that represent how likely it is that an interaction between a susceptible and an infectious individual will transmit the disease. Initially, with a new virus like this, everyone is susceptible. When a person is infected, they are no longer susceptible. Likewise, they may remain immune to the virus after they recover. We don’t know that this is the case for SARS-CoV-2, but there are some indications it may be, and that is the assumption in the model.

Because the virus is new, none of these numbers is well known. Sources are given in parentheses. If no source is listed, it is assumed by the authors.

  • Incubation period 5.1 days (from references)
  • Infectiousness from 12 hours prior to the onset of symptoms for those that are symptomatic and from 4.6 days after infection in those that are asymptomatic with an infectiousness profile over time that results in a 6.5-day mean generation time (no reference given)
  • R0 = 2.4 but values between 2.0 and 2.6 were examined. R0 is the number of other people infected by one infectious person. (Based on fits to the early growth-rate of the epidemic in Wuhan)
  • Symptomatic individuals are 50% more infectious than asymptomatic individuals (no reference given)
  • Infectiousness is assumed to be variable, described by a gamma distribution with mean 1 and shape parameter α = 0.25
  • On recovery from infection, individuals are assumed to be immune to re-infection in the short term. (analogy to seasonal influenza)
  • Start date: early January 2020
  • Doubling time: 5 days
  • Rate of seeding calibrated to give local epidemics which reproduced the observed cumulative number of deaths in GB or the US seen by 14th March 2020.
  • two-thirds of cases are sufficiently symptomatic to self-isolate (if required by policy) within 1 day of symptom onset
  • mean delay from onset of symptoms to hospitalization of 5 days
  • The age-stratified proportion of infections that require hospitalisation and the infection fatality ratio (IFR) were obtained from an analysis of a subset of cases from China. These estimates were corrected for non-uniform attack rates by age and when applied to the GB population result in an IFR of 0.9% with 4.4% of infections hospitalized
  • 30% of those that are hospitalized will require critical care (invasive mechanical ventilation or ECMO) (based on early reports from COVID-19 cases in the UK, China, and Italy)
  • 50% of those in critical care will die and an age-dependent proportion of those that do not require critical care die (expert clinical opinion)
  • Bed demand numbers based on a total duration of stay in hospital of 8 days if critical care is not required and 16 days (with 10 days in ICU) if critical care is required. With 30% of hospitalised cases requiring critical care, the overall mean duration of hospitalization is 10.4 days. (consistent with general pneumonia admissions)

All these numbers look similar to numbers I’ve seen tumbling out in scientific papers and other places. None look wildly off. But there are reservations about some that I’ll discuss later.

Put those numbers into the model, and you can calculate numbers of hospital beds needed, deaths over time, when the worst of the epidemic will come, when it will die down, and other numbers we’ve all had questions about. This is the case without intervention. These are the worst numbers you will see in media reports of this study.

To get a feel for how the model works, look at this wonderful visualization by Harry Stevens at Washington Post. The Imperial College model sets probabilities that a red dot will collide with a blue dot and turn it red. It also sets up four different areas – household, school, workplace and the wider community – as the second Washington Post simulation has two. The Washington Post simulation is much simpler than the Imperial College model, but it shows how it works.


Intervention Scenarios

tl;dr: Five interventions involving various degrees of social distancing show improvements over the base case. The best case is to combine them all.

The conditions can be varied and the calculations repeated for different scenarios. The paper calls them non-pharmaceutical interventions because they don’t involve treatment drugs or vaccines, but rather behavior modifications.

Five interventions were studied:

  • Case isolation – People with symptoms isolate themselves at home and reduce interaction with family members
  • Voluntary home quarantine – People with symptoms remain at home but do not isolate themselves from family members, who remain at home
  • Social distancing of those over 70 years – Reduce contact outside the home, but contact within the home increases
  • Social distancing of the entire population – Same as the previous, but school contact rates remain the same
  • Closure of schools and universities – Closure of all schools and 75% of universities. Contact increases at home and in the community.

Policies are assumed to be in force for 3 months in four scenarios, 4 months for social distancing of those over 70.

The authors say that their assumptions for the scenarios are pessimistic with regard to the changes in contacts, which seems to mean that actual mitigation by these interventions may be better than the results given in the paper.



tl;dr: The epidemic will peak this summer, with or without intervention. There may be a secondary peak later. Things may not stabilize until a vaccine is available.

This is the base case, with no interventions: a half million dead in the UK and over 2 million dead in the US, peaking in late June and July, tailing off to August.

Imperial College 200319

Critical care bed capacity could be exceeded as early as the second week in April. Simulations were also run for the individual American states, with broadly similar results. The graph is hard to read, but you can find it in the paper.

The aim of the interventions is to flatten the curve. That also means that they will increase the duration and make the peak later.

Imperial College 200319a


Obviously, all of the interventions are needed. This is probably the most significant finding of the study, and one that should hold up despite any reservations I’ll express later.

The paper goes into more complex strategies as well. If infections go down and controls are released, there is likely to be a second, smaller peak in the fall. Some controls may continue to be necessary until a vaccine is available.



  1. The shapes of the curves are probably more reliable than absolute numbers or dates. Interventions are important in lessening the effect of the virus.
  2. It’s going to be a difficult twelve to eighteen months. After the biggest danger period from the virus is past, we are going to have to pick up the pieces of the economy.
  3. ALL of the interventions are necessary to flatten the curve and save lives.

Some thoughts on the model and its parameters:

The model should be reliable. It’s been used before, and models of this type are much less complex than, say, climate models. Some of the parameters, like those derived from the Chinese experience, may be pessimistic, but others may not. Overall, the choices of parameters seem to be reasonable, given the newness of the virus and thus the lack of information on its behavior.

The experiences in Wuhan and Italy have been particularly virulent, so basing parameters on them may give a pessimistic result. It would be useful to get comparable numbers from Singapore and South Korea. Unfortunately, the United States seems to be on a trajectory more like Italy’s. For doubling time, the model uses 5.1 days, but data I see looks like the doubling time in the United States is 3 days; others say 1.9 days. That’s going to make everything happen faster. The lower number could be because of increased testing, but I’m not convinced of that.

Although indications are that people who recover have some immediate immunity to reinfection, this is not well known. There is still no test for antibodies, although several laboratories are working on one. Immunity makes a big difference in whether there will be a second wave and how big it will be.

I would like to see a sensitivity analysis of the model. That would show which parameters are most important and how much uncertainty in them affects the results. To some degree, the calculations of the interventions are an implicit sensitivity analysis, and the paper has a few comments about parameter sensitivities. Seems to be nothing surprising, with no particular intervention more effective than others. No silver bullet.

The biggest uncertainty is not having testing data. We need those tests for many reasons. We need tests to monitor whether seemingly well people are carrying the virus so that they can isolate themselves. We need tests so that people in early stages of the illness can be treated if a treatment becomes available. We need tests to see if people become immune to the virus, and then we need to test people so that they can go back to work if they are immune. More in a Twitter thread.

We will need patience. We will not see results from actions we take now for weeks. Things will get worse no matter what we do, but they will be less worse if we do these interventions conscientiously. Authorities in the states have the right idea and are taking action. We will see things start to improve in the summer.

Addendum: Columbia University has done a modeling study that shows results that are similar qualitatively and has the same takeaways I’ve given here. The big difference is in the numbers. The New York Times gives the results of that study, but it’s not possible from what is in that article to say what the differences are between the two. Hopefully Columbia will publish the study.


Because people who don’t know me are likely to read this post: I’m a chemist who has worked with modelers of chemical reactions. The mathematics of an epidemic are very similar to the mathematics of chemical reaction kinetics.  I’ve developed the basis for chemical models and understand how models are put together and used. I’ve managed a project in which chemical models were used to get practical results. Check me out on Google Scholar to see some of my papers.


Cross-posted at Balloon Juice


  1. Dan Connelly · March 21, 2020

    Thanks for the great review. My take on the paper was it suggested R0 reduction could come only with oppressive general isolation, and was inherently unsustainable. But testing (COVID19, but also other screens) is used in Asia to target interventions, making them much more sustainable, especially once general interventions knock the cases down 99.9% or so. Thus approaches assuming the inevitability of natural herd immunity, or some pulsed introduction of increased exposure for social relief, may be pessimistic to the point of distorting policy decisions. I’m not an epidemiologist either but wonder what you thought about that.


    • Cheryl Rofer · March 21, 2020

      That’s a good comment. The study doesn’t really consider the effect of testing, and I’m not sure, off the top of my head, how that could be factored in. I can think of at least three or four other things that could be mitigating as well, which could lead to a better outcome.

      I’m comfortable with having a worst case out there. And, if the interventions are successful, the complaint afterward will be that we did too much. But we’ll never know for sure.


  2. CraigM · March 21, 2020

    There are a lot of things that could come into play to make a second wave less severe. If there is extreme heterogeneity in the effectiveness of people in transmitting the virus once infected, and the most efficient spreaders tend to have interconnected social networks (which is probably true across situations and countries – think of elementary schools, old folks residential facilities and other places with lots of people whose self-care skills are not at their peak) – then these folks would tend to get disproportionately infected relatively early in the first wave. This would lead to artificially high initial estimates of population R0, and more importantly, if there is even medium-term post-recovery immunity, the second wave would be missing many of the super-spreaders and the effective R0 would be lower than average, possibly much lower. This is a case where the assumptions about the shape of the distribution of individual infectivity may be as important as the assumptions about its mean value.

    Or perhaps the second wave could be more severe, if immunity is limited-term and the damage from a prior round leaves one more vulnerable to death from a second infection. Or the virus could mutate into a more lethal form, which would require stronger isolation measures.

    Obviously, these dynamics apply to influenza too, but the impact increases with the magnitude of R0 and with the variability in individual infectivity, which is driven by both biology and behavior. Other disease models may not have needed to capture this effect, so our intuitions may be wrong in some crucial ways.


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