Mathematics in Epidemiology with SIR model

When I flick through the blog posts, one thing that I find missing is the content related to epidemics. Yes, it is now April 2020, and COVID-19 is definitely a thing worth paying attention to – you must have been living under the rock if you are not tracking this destructive disease that is wreaking havoc throughout all nations around the world. Of course, a disease of such a scale requires us to bring out some big guns, and mathematics fans would surely enjoy the fact that modeling has played a crucial part in updating the world with the information it needs to effectively combat the dire situation. So today we are going to look at the basics of models related to this field – real-life models would be much more complicated – by analyzing, in my opinion, the simplest one out there that is also the father of more complex variations, the SIR compartmental model.

1. What is the SIR model?

The SIR (acronym for Susceptible – Infected – Recovered) model is, as mentioned above, arguably the simplest of all the models you can use to predict the statistics related to a specified epidemic. It is classified as a compartmental model, which means that it divides the population into compartments using their respective statuses. In this case, there will be three compartments, three “boxes”, and a person will belong to one of them:

• The “Susceptible” compartment includes any person who has not caught the disease yet and hence is prone to it. Basically, this equals the entire population minus any person who has caught the disease currently or in the past;
• The “Infected” compartment includes any person who is currently having the disease;
• The “Recovered” compartment includes any person which has contracted the disease and has recovered from it.

With such categorization, a person’s status would change in one of the following ways:

• Stay in the Susceptibles; or
• Change from Susceptibles to an Infected and stay there; or
• Change from Susceptibles to Infected and to Recovered.

It is important, however, to note the assumptions, which are also limitations of this model. The model does not account for birth and death rates, which is, for the most part, a tolerable error due to being very minor compared to the disease spreading rates. It also assumes that a Recovered person will not be prone to secondary infections of the disease, and therefore would not return to being a Susceptible. The last assumption would be that no person would die of the disease. Therefore, this model would work best for a rapid, non-fatal disease with no chance of reinfection (via acquired immunity).

2. Model analysis:

The model measures the number of people in each compartment as a function of time by using a set of 3 differential equations for the rate of change of the three compartments:

In which:

• , , and are the number of people in the Susceptible, Infected and Recovered compartments respectively;
• is the population. Naturally, ;
• is the number of contacts an infected person have per day that carries the disease;
• is the rate of recovery.

Surprisingly, for such an important model in science, the differential equations can be easily expressed through common sense. For the first equation:

It is relatively well-known that the disease spreads through contact between an infected and a susceptible person, and here the model is trying to measure those contacts. Since is the fraction of the population that is susceptible, would be the number of contacts aimed at Susceptibles for each infected individual. Multiply that by the number of infected people and we would have the number of susceptible-turn-infected people, which is what represents (the quantity is negative because the number of susceptibles is reducing).

For the third equation:

it is even more straightforward: the number of recovered individuals would equal the recovery rate multiplied by the number of infected people.

Moreover, since we assume an unchanging population,

then the second equation can be followed from the first two.

3. Model variations:

As mentioned above, the models have various setbacks of not mentioning natural birth and death rates as well as assuming that the disease is non-fatal and not reinfectious. The majority of the diseases will not possess all three of these traits, and therefore variations have been developed to take into account those factors:

• The SIRD model tackles fatal diseases by adding another Deceased compartment into the model. The infected person would go into either the Deceased or the Recovered compartment;
• The SIS model tackles reinfectious diseases by creating a loop: a person, after recovering from the disease, is immediately prone to secondary infection and thus will change the status from Infected back to Susceptible;
• Vital dynamics (birth/death rates) can be incorporated into the differential equations;
• All of the aforementioned models can be combined into a complex one.

With that, we would end our discussion of the SIR model and its variations. Until the next time, math on!