The SEIR Model
- The standard model for the spread of a virus is the Susceptible, Exposed (infected, but not yet infectious),
Infectious (now can infect others), Removed (SEIR) model.
- This is a system of nonlinear Ordinary Differential Equations (ODEs), which must
be solved numerically.
- You can download the Matlab files main_corona.m
and seir.m, and try running this yourself. The most
important parameter is R_zero as a function of time.
- If you don't have access to Matlab, this code also runs under the open source
Octave package .
- Here a link to the Python
version (thanks to Kees de Graaf).
- Here is a link to the R version
(thanks to Pierre Chausse).
- Using parameters from the literature, the model shows that the peak number of cases
in the Wuhan area should be about 140,000 by the end of March, 2020. Current data
shows a much smaller number of cases. This could be due to (i) very successful
efforts to control spreading (i.e. reduction of "R_zero"), (ii) undereporting
of cases, or (iii) incorrect data for the SEIR model.
- The important input is Rzero vs time. Rzero is the average number of
people infected by a single infectious person. Rzero greater than 1, means
that infection will spread until herd immunity reached. Rzero less than 1,
means that infection will die out, without infecting a large proportion
of susceptible population.
- How do you change Rzero? It appears that the corona virus (without any
mitigation) has an Rzero somewhere between 2.0-3.0.
See paper in Cell Discovery
Social distancing and quarantines can reduce Rzero to less than one.
- Made small change in R_zero vs. time, this has a very large effect. Now
results are closer to observed data in Wuhan. This indicates that these models are very
sensitive to this data. Total active cases now peaks at around 90 days. Cumulative
total cases at 120 days is about 80,000.
- The bad news. The population of Wuhan is about 11 million. After 180 days, at most
100,000-200,000 will have been infected, recovered, and are now immune. This hardly makes
a dent in the susceptible population. This means that any new outbreak will start the
process all over again. Can economies stand several waves of social distancing
The Next Wave
Suppose that we assume the data from Wuhan. Now, suppose that social distancing is relaxed beginning
at 120 days. A gradual lifting of mitigation occurs between 120 days and 240 days, so that
at 240 days, Rzero is now 1.75. As we shall see, this causes a second wave of infection,
so that once again, social distancing is imposed, at day 240, with Rzero returning to 0.5 at day 360.
The second wave actually peaks higher than the first wave. This assumes that the beginning
of the next wave is recognized early, but the reintroduction of social distancing takes
longer the second time around (i.e. more people need to keep working).
All Models are Wrong, Some are Useful
This is a quote from the statistican George Box. There has been much criticism in recent days about model predictions, and
the widely varying forecasts for the number of coronavirus cases. As you can see from the above, small changes in the R_zero
input have a large effect on the output. With no social mitigation efforts, this number is already hard to determine. Now, you
also have to determine how social distancing and quarantines effects R_zero, and how fast this occurs. You should try different
scenarios for R_zero yourself, to get a feel for how difficult it will be to make reliable forecasts.
But, as Box points out, it is useful to play around with these models, since it gives you an understanding of the factors at play.
The SEIR model assumes a "well mixed" homogeneous population. This is obviously
not true over a large geographic region. One way to extend this model is to assume that (S,E,I,R)
are spatially distributed. Then, the quantities which appear on the rhs of the ODE
system are spatially smoothed. This will give rise to a system of Partial Integro-Differential Equations (PIDEs).
See Archiv paper arXiv:1909.01330.
Another approach is to use a network based SEIR model, i.e. nodes in the network represent
different populations. Contacts between the populations occur through a network
See Archiv paper arXiv:2006.06939.
In addition, there is a large amount of randomness associated with disease spread, so a better model
uses stochastic differential equations to simulate pandemics.
The GLEAM model includes both spatial and stochastic effects.
Webinar April 29, 2020 at 2:30pm (GMT+1)
Mathematics of the COVID19 crisis – In the eye of the storm