Practice Problem: Modeling Disease Spread, Containment, and Prevention
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Since the beginning of the COVID-19 pandemic the world watched as various precautions and treatments were proposed, developed, and rolled out. We were able to see how various countries fared based on the measures they implemented.
Professionals who track diseases and offer advice often use variations of a type of mathematical model called an SIR model. In this type of model, we assume that at any given time, people in a population fall into one of three categories: Susceptible, Infected, or Recovered. The number of people in each category changes over time depending on the interactions among the people and the properties of the disease. This type of model focuses on the rates at which people transition from one category to the next. This website gives some basic information about the SIR model and provides an example of how you might get some numerical and graphical results using Excel.
Suppose we have a new novel disease, let’s call it ND22. There are things we can do to prevent the spread of the disease (preventative measures, such as wearing masks/PPE, cleaning practices and products, social distancing, vaccine) and there are things we can do in response when someone has contracted the disease (response measures, such as quarantine, prescription antiviral drugs that minimize the duration and/or severity of illness, medical treatments like hospitalization/use of ventilator). While response measures are typically provided to any infected patient in need (if they are available), leaders may be tasked with creating policies to determine how preventative measures are distributed. Ultimately, that is what you will be asked to do for ND22 by modifying the standard SIR model.
As you model, you should make assumptions about how easily the disease spreads and how long a person remains infectious. You may want to find examples of model parameters for other diseases (such as the flu, COVID-19, chicken pox) so your model has some realistic parameter values.
Q1—One way we can improve on the basic SIR model is by considering that the spread of the disease and the severity of outcomes are different for people of different ages. We can account for this in the SIR model by dividing the total population into subgroups based on age. That is, create a model that has multiple S compartments (one for each age group), multiple I compartments (one for each age group), and multiple R compartments (one for each age group). Since it is possible for a young person to spread the disease to an older person and vice versa, you will need to consider interactions among various age groups. Do not yet consider any preventative treatment or response measures. Apply your model to a specific country (one for which you can find population demographics) to demonstrate the spread of ND22.
Q2—Suppose the country decides to provide preventative measures under a distribution system that makes them available in the following order:
people with significant preexisting conditions that make the illness especially risky
people who work in high contact service industries
Adjust your model to allow for this preventative measure protocol and compare the results with the nonintervention strategy in Q1.
Q3—Develop an optimal strategy for dealing with an ND22 outbreak, taking into consideration multiple prevention and response measures. Compare with the results from Q1.
Here is an example of how you can implement a simple SEIR model in MATLAB. The Live Script provides a SEIR model overview and you can change the parameters using sliders and see the effect on the plot.
Problem Author: Dr. Karen Bliss, Applied Mathematics Faculty at Virginia Military Institute
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