The SIR model was created by W. O. Kermack and A. G. McKendrick in 1927. The model was presented in order to explain the fast increase and decline in epidemics such as Ebola in large populations. Examples include the plague (London 1665-1666, Bombay 1906) and cholera (London 1865). The name of the model is derived from the fact that it involves equations relating to the number of people susceptible to the disease S, the number of people infected I, and the number of people recovered R.
The SIR model is used to comprehend an epidemic over a period of time, and so the independent variable is time, which will be represented by the letter t, measured by the number of days.
The dependent variables, however, are two related sets. These sets are proportional to each other, and so either will aid us with the same information about the development of the epidemic. The reason for using two related sets is that it may be easier to conduct calculations using the fractions.
The first set of dependent variable count the number of people that are split across three groups. These are all functions of time and change according to a system of differential equations.
St; the number of individuals who are susceptible to the disease at time t, but can become infected.
It; the number of individuals who are infected with the disease at time t. The individuals can transmit the disease to the susceptible individuals.
Rt; the number of individuals who have been infected and recovered from the disease, either by death or immunization. The individuals in this category are not able to become infected again or transmit the disease to others.
The second set of dependent variables represent the fraction of total N, in the three groups aforementioned.
, the susceptible fraction of the population.
, the infected fraction of the population.
, the recovered fraction of the population.
The flow of the model can be considered to be S à I à R.
At any time during the epidemic, an individual is either susceptible S, infected and infectious I or removed R.
In the SIR model, underlying assumptions are made, they are:
– Closed population.
This means that throughout the time period that the SIR model is presenting, the total population N, remains constant. No one is added to the susceptible group S, as we are ignoring births, immigration, and emigration. The only way to leave the susceptible group S is by becoming infected and joined the infected group I.
– Homogenous mixing.
This assumption is made in order to make the mathematics tractable. It is that each pair of individuals has an equal probability of coming in contact with each other. Any susceptible individuals have the same probability of becoming infected as those who have already been infected or removed. However, this theory is rarely justified as it is in a population, there are lots of small subgroups that are very unlikely to mix with a lot of people outside their group.
– The final assumption is that an individual becomes infected, they are instantly infectious and can pass the disease on to others. There is no in-between stage between the susceptible and infected populations.
Law of Mass Action/ Mass Action Principle.
The law of Mass Action originates from the chemistry, and is the proposition that the rate of reaction is proportional to the product of the masses of the reacting substances. And so applying it to this situation, it becomes the rate at which two type of individuals meet (S and I), is proportional to the product of populations of the respective sub-populations, at the give time.
Rate of two types of individuals meeting ? (St) * (It)