Clive Best

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.

Social networks derived from news reports concerning the Iraq Insurgency in 2007 (my work). This just illustrates how large variations in network size can be.

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

UK cumulative deaths compared to a slowing in R with average social network size

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.

Comparison of a reducing network model  with Sweden’s  accumulated deaths per million people.

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.

We constantly hear at the UK daily Covid press conferences that we must keep R <1 to avoid a second wave of infections. Prof. Chris Witty warns us that if R goes above 1 then we will see an “exponential” increase in cases. However any second wave will depend critically on the exact value of R, because R  defines both the speed of the outbreak and the total number of infections needed before it naturally ends. This is  because it runs out of (an R -dependent)  pool of susceptible (i.e. non-infected) people. This finally results in so-called  “herd immunity”. Figure 1 shows  some examples of how the rate of infections develop in time and severity for different values of R.

Fig 1. Infections for different values of R0. (cases for UK)

The initial rise is not really “exponential” but a power series where is the number of infection cycles. This strong dependence on R suggests that one possible strategy would be to restart the UK economy after lockdown but just try to keep R slightly above 1 – say 1.1. This would minimise total deaths in the long run and eventually achieve herd immunity,  unless of course a vaccine can be found to achieve this sooner (see fig. 2). The value of R becomes a pay off between the length of an outbreak and its severity. Measuring R would depend  on widespread public testing.

Fig 2. An epidemic with R=1.1 moves very slowly peaking only after 1.5 years but would minimise UK deaths  (this assumes IFR = 0.5%)

This strategy appears to be exactly that being adopted in Sweden, as explained by Professor Johan Giesecke in his lockdown TV interview. Sweden has adopted a limited social distancing policy but kept primary schools, bars  and restaurants open. They are planning for herd immunity while at the same time minimising deaths.

These are incredibly difficult decisions for any government to take. Apparently successful countries like Taiwan, New Zealand, Hong Kong and Iceland may succeed to eradicate Covid-19 entirely.  However afterwards they would still need to isolate themselves from the rest of the world indefinitely in order to stop any possible reinfection. This is unsustainable long term. The global economy depends on trade and international travel. Any resultant collapse in global living standards would likely then cause more deaths than the virus itself.

Let’s hope a vaccine appears as soon as possible!