The Mathematics Behind Modeling Infectious Diseases

SIR Model

The SIR Model is a mathematical model that calculates the population of those susceptible (S), infected (I), or recovered (R) from a disease or epidemic over a period of time. The difference between the SIR model and an exponential model is that an exponential model has no limits, because it is exponential, and does not take into account that people can move among susceptible, infected, and recovered population groups. Typically, variables are rearranged to find the population of either the susceptible, infected, or recovered population depending on what the goal of the model. Then, different values, known as rates, are assigned to the populations or variables to find an outcome. However, people can return to the susceptible population after being recovered because viruses are particularly sensitive to the altering of DNA, thus creating a new strand of the same infection. This model is more realistic in predicting COVID-19 because people can become infected and recover. Additionally, people can be born into the population and die out of the population. This model demonstrates if no precautions were taken to prevent the spread of COVID-19. The data is used to help professionals such as epidemiologists predict the spreading of a disease and take action. For our purposes, the population is the number of people in the state of Ohio.

Here is how SIR models are represented:

Here is what each variable means:


Graph of a SIR Model:

With a relatively small infection rate, this model shows that the peak of the infection occurs around 10 days. The infection curve appears bell shaped and does not drastically increase nor decrease.
If the infection is doubled from the first example, we can see that the time it takes for the infection to peak is cut in half, and the peak of the cases has doubled. The number of deaths also noticeably increases.
If the infection rate grows again, the peak of the infection occurs more rapidly than in the first two cases. The peak of the infection occurs near four days and the slope of the line grows exponentially, unlike in the first case where the slope of the line appears to grow linearly.


INTERACTIVE MATLAB SIR MODEL:

  1. Open the following link
  2. Login with with MATLAB account.
  3. Open folder “COVID-19 Interactive Models”
  4. Find the file “COVIDBasic.m”
  5. Right click the file and hit “Run”

Next, check out the Age Population Model or Social Distancing Model.
These models are variations of the SIR Model, but with additional variables. 


Check out the Current COVID-19 Cases tab for current data in the state of Ohio and internationally.

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