An Epidemiology Model (Differential Calculus)

Model Differential Equation:

I'(t)=r I(t) - a I(t) (1)

where:

Questions:

1. What are the units of I(t), and what are the units of I'(t)?

2. What must the units of r and a be? Note that the units of the left and on the right should match.

Let us consider the case of chicken pox spreading through some particular (fictitious) schools, the Sunnyside Day Schools.

1. r, the infection rate, will vary based on several factors, including the nature of the disease and the nature of contact between people. Let's assume that in the Sunnyside City Day School, r=1/2 day^-1. So how many people are infected by an infectious person in a day? Does this make sense? State it in a way that does make sense.

2. a, the removal rate, also depends on several factors, including how fast people are diagnosed (so they can be told to stay home), and how quickly they get better (so they are no longer infectious). Interestingly, another way people can be "removed" from the infectious population is to die. It may sound callous, but it doesn't matter to the equations whether people are dying or getting better; if they're not spreading the disease for long, the removal rate is high. Assume at the Sunnyside City Day School that the removal rate on average is 1/7 day^-1. So how many days is a person infectious? How do you think that time period might be spent (how many days sick, at home, feeling well, at school, dead)?

3. Using r=1/2 day^-1, and a=1/7 day^-1, write out the model equation (1). Can you tell whether the number of infected people is increasing or decreasing? How? (Hint: What is the sign of the derivative?) What happens as t continues to increase? Can this be interpreted as the disease spreading throughout the school? Will absolutely everyone get chicken pox, according to the model? Is that realistic? Why or why not?

1. Now, suppose people are more careful about washing their hands at the Sunnyside Country Day School, and so r=1/10 day^-1. If a=1/7 day^-1 still, is the number of infected people increasing or decreasing? What does the model predict will happen to I, the number of infectious people, as t continues to increase? How do you interpret this outcome? Is it different from the case of the City School?

4. Use other values of r and a, and conjecture a general result based on their relationship. (Hint: write out an equality or inequality that predicts whether the disease will spread.) It is called an epidemic if I increases. What relationship between r and a causes an epidemic? Explain the significance of this relationship in words in terms of the specific meaning of r and a given above. Write a paragraph about specific changes in r and a (besides handwashing) which could prevent an epidemic. Make sure that you explain this so that parents (who took calculus a long time ago, so need terms defined, but don't accept just being told what to do) are persuaded.

5. (Special cases) What happens if r = a? What happens if I = 0? Interpret these cases.

1. You may have realized by now that equation (1) can be interpreted as "The derivative of I is a constant times I." What function do you know that behaves that way? Write out the function I(t) that satisfies the equation (1). Take its derivative to show that its derivative, I'(t), equals (r-a)I. (This may take a little trial-and-error.) Once you have an I function that works, we call that the solution function.

2. Now the population is not limited to one school, but might be considered, say, as the whole county of Sunnyside. (We will assume that the parameters are the same throughout the county's population. This is a big assumption, but maybe it's reasonable if we say that it's an average.) Use the first set of parameters (the City School) given for chicken pox, and make up relative numbers for HIV. (Is a bigger or smaller than for chicken pox? Is r bigger or smaller?) Justify the numbers you choose based on the definitions of the parameters (as you did in Exploration 1, questions 1 & 2). Write out the equation (1) using the original chicken pox values, and then using your new HIV values. Write out the solution function from the previous question using these two sets of values.