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Modelling the Spread of a Virus

I am currently in lockdown owing to the coronavirus pandemic that is sweeping the World.  It is apt therefore to present a model of epidemiology that shows how disease spreads through a population. Models of this type, but much more complex than the one presented here, are being used to inform national policies in order to reduce the devastating impact that the coronavirus is having on our societies.

The model here is based on a very simple idea.  You have a population that starts off with a single individual who is infected with a virus and sets off the chain of infection.  That person then infects a number of other people according to the transmission rate (more on this later).  Those people then infect other people and so on. Initially the rate of growth of the number of people infected is exponential, that is a doubling of people with the disease for every round of infections, which demonstrates just how difficult it is to contain a virus once is has broken out. But as people get infected and recover they then become immune to the virus.  The spread of the virus then plateaus and then reduces.  With an unvaccinated population a virus will rip through a population until many have been infected and gained immunity.  With a population that is vaccinated, then the spread of the virus is subdued much more rapidly even though not all people are inoculated against the virus.  The so called "herd immunity" is important in helping protect entire populations from outbreaks of disease.

The computer model work in this way.  We have a population of individuals that have 1 of 3 states.  Firstly, individuals can have no disease and no immunity and are susceptible to infection. Secondly, individuals can have the disease and spread it as well. Thirdly individuals can be permanently immune either because they have been inoculated against the disease, or have had the disease but recovered from it.  Each time step corresponds to a day and at each time step those individuals with the disease will infect a number of other people according to the transmission rate, R0.  R0 is an important concept in epidemiological modelling and determines the rate of spread of the disease.  The whole purpose of lockdown and associated measures such as social distancing, hand washing and so in is to reduce R0.  For coronavirus under business-as-usual circumstances R0 is said to be around 2.5.  That is each individual will on average infect between 2 and 3 people.  This is considered a highly infections disease.  Under lockdown, in the UK it though that R0 has been reduced to around 0.6, meaning that each infected person will infect on average 0 or 1 other person.  As time progresses more and more people will get the disease and recover with immunity, so the likelihood of meeting someone who is not immune will diminish over time. So too the likelihood of infecting someone else
will reduce.

Just what the impact of reducing R0 is illustrated nicely in the two plots below.  Initially R0 is set to 2.5 and the number of infections at the peak is more than twice that where R0 is 1.5.


By reducing R0 we are limiting the number of people who will become infected at a given time thereby "flattening the curve" and giving our NHS a fighting chance to treat all serious cases without being overwhelmed.

The code can be downloaded from https://github.com/wmfgrey/epidemiology.  I have presented this idea to my Year 12 students and they got on well with it developing their own models.  Feel free to tinker, improve and extend.

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