Research Matters is happy to bring you this article as part of the series on Mathematical Modeling and Data Analysis by the Mathematical Modeling team of Indian Scientists' Response to Covid-19 (ISRC). The second part of this series is on SIR and SEIR Models of Infectious Diseases.
SIR models are commonly used to study the number of people having an infectious disease in a population. The model categorizes each individual in the population into one of the following three groups :
Susceptible (S) – people who have not yet been infected and could potentially catch the infection.
Infectious (I) – people who are currently infected (active cases) and could potentially infect others they come in contact with.
Recovered (R) – people who have recovered (or have died) from the disease and are thereby immune to further infections.
These compartments contain a certain number of people on each day. However, that number changes from day to day, as individuals move from one compartment to another. For instance, individuals in compartment S will move to the compartment I, if they are infected. Similarly, infected people, I will move to the recovered R compartment once they recover or die from the disease.
The total population across the three compartments (S+I+R) is assumed to remain the same at all times. This is just the total population of the country (or state/region) we are considering. This means that everyone exists in one of these 3 compartments. This ignores the fact that in the natural course of things (epidemic or not), births and deaths continue to happen in the country. But for short epidemics that last a few months, this is a reasonable assumption to make! For modelling other diseases like childhood infectious diseases, such as measles, that recur regularly, natural birth and death rates of the population will also have to be taken into account.
As in the current epidemic, from authorized sources such as ministries , one can find the numbers of active cases (I) and the number of recovered or dead (R). Also reported is the total number of infected people to date, which, if one thinks about it, is nothing but the sum I+R.
Our goal is to find out how the number of people in each compartment changes with time. In order to do that, we make two simple hypotheses on what drives the movement of people between these compartments.
The first hypothesis: Let us suppose you have not been infected at this point in time. So, you would belong to the S compartment. You can be exposed to the virus only when you come in contact with an infected person. The greater the number of infected people in the general population, the higher the chance that you will come in contact with an infected individual. This same principle which applies to you, applies equally to every other susceptible individual in the population. Therefore, the rate at which susceptible people become infected, i.e., the rate at which people are transferred from the S to the I compartments on a given day is proportional to the size of the I compartment as well as to the size of the S compartment on that day.
The second hypothesis: Infected people will either recover or die of the disease. On each day, a certain fraction of infected individuals will recover or die. This fraction is taken to be a constant, independent of the number of susceptible, infected, or recovered individuals on that given day. This fraction is somehow “intrinsic” to the specific pathogen and captures the average human body’s recovery time for that particular disease.
What mathematical modellers do is to write the above hypothesis in terms of mathematical equations which tell you how the number of susceptible, infected and recovered individuals change with time. In the language of mathematics, such equations are referred to as differential equations. These equations are solved by a process called integration, and these solutions will allow us to calculate, for example, the number of infected people for any time in future.
For diseases such as CoVID-19, we need to consider another compartment called ‘Exposed’ (E). This consists of individuals who might have the virus (due to travel, direct/indirect with an already positively tested person), but do not show any symptoms. For example, if your cousin travelled to Wuhan and came back she is more susceptible than you – because she has been around the virus. In other words, they are between the susceptible and infected compartments. However, despite not showing any symptoms, these (asymptomatic) individuals can still transmit the disease to susceptible individuals. One can add more compartments, for example, ‘Quarantined’ or ‘Isolated’, to better capture ongoing disease control measures. The modelling proceeds in the same way as in the previous case, with assumptions on the rates at which people move between these compartments. The solution allows us to calculate the number of infectious people at any future time.
Coming soon — Disease Transmission and Containment
Editor's Note: This was originally published by Indian Scientists Response to Covid-19 (ISRC). Republished here with permission.