One should always be cautious of entering a field you know nothing about! However I wanted to understand better how epidemic models work since these are now the primary driver affecting government policy during the Coronavirus emergency.

The classic model of infectious diseases dates back to the 1920s and is called the SIR (**S**usceptible, **I**nfectious & **R**ecovered) model. This is well described in this article plus.maths.org on which I base this post. Here is my (probably naive) interpretation of the SIR model:

During any epidemic there are 3 groups of persons within a population.

S – Susceptible persons to infection

I – Those sick and infectious to others

R – Those who have recovered from infection or died

The rate of change of each group is as given by the following differential equations.

where

B is the birth rate (births/day)

g is the recovery rate so that

is the infectious period in number of days

is the contact rate (people/day)

D is the death rate (deaths/day)

One important number describes overall how serious any outbreak becomes – the basic reproduction rate – , which is equal to the average number of new infections passed on by each infected person at the beginning of an outbreak. This value naturally changes as the epidemic evolves. It can also be changed by taking such measures as quarantining, and Social Distancing to reduce the contact rate, as currently imposed in the UK.

If is > 1 then the disease will spread rapidly at first but will then peter out when <1 . Note that as more people get infected the initially large susceptible group reduces and as a consequence so does until it eventually falls below zero and the peak quickly decays. That is why eventually epidemics will always end.

So let’s try to run this model for the UK under different scenarios. To make life easy we assume that the birth rate B and death rate D can be ignored as they are small compared to the overall population (60 million). We assume.

First infected person arrived in the UK say on February 1st 2020

- The infectious period is 5 day
- The contact rate is 0.5/day
- Therefore is 2.5

I coded up this model and started the simulation from day 1. I used Python even though I hate it. This is so that anyone else can run a simulation. Here are the initial results.

There is a rapid exponential rise in cases following an initial long quiet stage. The peak of the epidemic occurs about 62 days after the first cases but then rapidly collapses as the number of susceptible people quickly decline. The final number of deaths reaches a total of 271,500. At the end of the epidemic most of the population would have been infected but some lucky people remain unscathed by the virus. Out of a population of 60 million people about 5 million might escape being infected at all.

Now suppose the government imposes a near lockdown in order to suppress the epidemic at day 30. This reduces the contact rate to 0.15/day shifting < 1. This has a dramatic effect on new cases but the long tail of infections is extended for several months. This means that any relaxation of lockdown measures will simply increase again within the next month thereby triggering a second epidemic and inducing a second lockdown etc.

Can our economies survive such a series of stop go epidemic lockdowns until a vaccine becomes available in optimistically 10 months from now ?

Let’s hope that our governments actions are based on far better scientific advice than mine, because otherwise their only alternative exit strategy is to simply let the epidemic run its course, but yet more painfully slowly by turning on and off lockdowns.

P.S. I hope I am totally wrong on this !

One possible motivation for lockdowns is to buy time in order to deploy more effective anti-viral drugs to greatly reduce the death rate.