A major limitation of the discussed model is that it assumes that a second or subsequent epidemic cannot occur. The decline in the epidemic curve shows that the pathogen is running short of susceptible people to infect, or there is an increase in the rate of treating the infected people, introduction of a vaccine or people have turned out to be immune. To have a new infection in the future, it means that we must have a source of susceptible people.
An expansion of the SIR model to include births per unit time and mortality rate denoted by B and µ respectively brings the new equations.
Through out the formulation of this model, it has been an assumption that the transmission rate β is a constant. A careful scrutiny of this reveals that the transmission rate is a product of the probability that a susceptible person when contacted by an infected is infected and the rate of contact. However, contact rate among individuals is not constant all through; it varies with season and place. Thus, it is right to infer that the transmission rate is not constant as held.
Fig 2 is a representation of the data on reported cases on measles infection in New York City over a period of 3 months. Measles has been in New York City for decades, and shows no sign of extinction. In the year 1961, measles was already in the city, and thus the epidemic did not start simply due to an infected individual thus the assumptions that I (0) = 1 and S (0) = N do not hold.
The introduction of the birth rate in the SIR model gives a clearer understanding of the same. The changes in the rate of birth have similar effects to changes in the rate of transmission. Thus, equivalence is essential as it makes the explanation of historical case report data of different areas possible. It is deducible from this report that successful mathematical modeling for biological systems is systematic. Starting from simple steps we can make a model of the present information. Where this does not capture fully the present phenomena, additional details are added to the model.
In modeling recurrent epidemics, Kermack and McKendrick models, which are an SIR model, are indispensable. These models make not only help in the data analysis, interpretation, but also in the understanding. Starting from the data collection, the SIR model helps in making a model that will enable graphing, making it presentable. The model is made initially from the simple data available. The model also makes addition of more biological data possible, thus fully capturing the phenomena. Despite their help in making epidemic analysis possible, they are based on axioms that rarely hold. Many of the assumptions that are the fabric holding the model have been proven useless in other epidemics, such as the Measles epidemic in New York.