Simple SIR models essentially assume that the infection rate of any population is homogeneous, i.e. everyone stands an equal chance of getting infected. However this isn’t really true, because some people have very large social networks, while the majority vary from the the low tens, down to a handful of regular contacts. Politicians, business leaders and Scientists tend to have very large networks of regular contacts across the world, whereas most people have far smaller family and work related networks. The so-called super-spreaders are the first infected members of large social networks.
At the start of an infectious disease outbreak the probability of someone getting infected if they are a member of a large network and then spreading it is very high. These are the super spreaders which force the initial reproduction rate R0 to very high values. It is no real surprise that Boris Johnson, Mat Hancock, Chris Whitty, Dominic Cummins and Neil Fergusson all got infected together as they formed part of the government’s large COVID/SAGE network. However as an outbreak progresses, the larger networks become increasingly already infected, so that remaining networks for new infections will therefore tend to get smaller and smaller. As a direct result of this gradual process, R will naturally begin to reduce, even without imposing any extra social distancing policies. R is simply proportional to the size of network that the average infected person belongs to, so as mean network size diminishes so does R.
The Imperial model has a complex model of the distribution of UK population in cities, towns and villages and simulates travel and commuting. However as far as I can tell it does not include social networks nor could it easily do so. There are just too many types of network for them to model – entertainment, businesses, councils, sports clubs, etc. All we can really do is to estimate how R might reduce naturally with time as the susceptible network size reduces.
My model uses a contact rate which is the contact rate per day resulting in infection and 1/g which represents the average infectious period in days. Initially beta is 0.5 and 1/g = 5 but in order to simulate the decrease in social network size I reduce by 0.003 per day. As a result R slowly reduces from 2.5 on March 1st to 1.0 on June 2nd. Note that this does not include any government led extra social distancing measures.
Here is a comparison with this modified model with UK accumulated deaths
It is clear that the model doesn’t properly represents the shape of the UK data. It also seems clear that the first cases in UK also appeared probably 2 weeks before March 1st. One reason the curve can’t fit is because the UK also imposed a lockdown on March 23rd which changed the dynamics. Next we look at Sweden which did not impose a lockdown.
The model is almost a perfect fit, so I suspect that Sweden may well be on the right track. If this intuitive decrease in R due to ever decreasing social networks of the susceptible really is the driver, then by mid June the Swedish epidemic will be over through herd immunity.