Social networks and epidemics

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 \beta 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 \beta 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.

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Herd Immunity

Note: My definition of Herd Immunity is the total % of the population who get infected once the epidemic has completely finished. Most people define it as just the % of the population infected once  the peak is reached. This gives much lower figures – 60% rather than 90% !

When a new disease enters into a community without prior immunity it will spread  if R0 > 1 . R0 is simply the number of people the first person with the disease infects before he/she recovers (or dies). If R0 < 1 then the disease fades quickly away and has no lasting effect,  while if R0 >> 1 the faster the disease spreads. Avian Flu was an example of a lethal disease but fortunately for us had R0 < 1. People got infected by birds but did not easily pass it on to others before they died. As a result there was no pandemic, despite WHO warnings of one at the time.

In the UK there are 65 million people all initially susceptible to any new infectious disease arriving in the country. It is estimated for Covid-19 that  R0 probably was  around 2.4 in Wuhan. However for simplicity let’s consider a general case where some disease has R0 = 4 and that the infectious period is 5 days.

At the start of the epidemic one infected person arrives in UK  and infects 4 others before recovering. These then each infect 4 others so that the infected population will increase as 4^n every 5 days.


It is probably only after ~1 month that anyone really notices that there is a problem, but by then the epidemic is already increasing “exponentially” out of control. However there is a safety catch.  Eventually each new infected person begins to meet some of those also infected or already recovered, so now cannot pass it on to 4 new cases. The reservoir of susceptible people is quickly running out and R is now diminishing fast. It reduces to 1 at the peak of the outbreak and then falls dramatically as the epidemic collapses. The population is then said to have reached “herd immunity”, and the UK is afterwards immune to any new infections arriving from abroad. One interesting fact is that herd immunity is always reached before everyone in the country is infected.  A percentage of the population will always escape any infection, but this percentage depends critically on the initial value of R0. Here are two examples.

Herd immunity for R0 = 1.2.  Only 30% get infected before Herd Immunity is reached.

However if R0 is much larger than 2 it is a different story.

Herd immunity is only reached after 90% of the population have been infected, but at least the epidemic ends much faster!

Governments can reduce “R” through social distancing measures, but this is a tricky process because to return to “normal” life infections would essentially have to drop to zero, perhaps through track & trace. Alternatively we could eventually reach herd immunity with maintaining R=1.2 but that would take 9 months, and even then may not be sustainable while infection rates outside the country remain at R=2.4.  So In both cases international travel might still need to be controlled indefinitely.

The only certain way out of this dilemma remains either a rapid development of an effective vaccine, or a drug treatment which  renders the disease no worse than a cold.


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The R value

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 R^\tau where \tau 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!

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