Epidemiology Model (Differential Equations) Part I

An Epidemiology Model (Differential Equations) Part I

Scientists use mathematical functions to assess the spread of diseases. Here is a simple model for the number of people infected with a particular disease, say chicken pox, in a closed environment, like a school.

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

where:

t is time measured in days.

I(t) is the number of infected people with the disease at time t. Note that I is a function of t.

r > 0 is a parameter measuring the rate at which people are infected by an infectious person, (how many people an infected person infects per day) and

a > 0 is a parameter measuring the rate of removal, that is, the rate at which people become not infectious any more; they either recover or die (1/the number of days a person is infectious).

r-a can be considered as a parameter of this equation.

Exploration 1

1. Solve the equation for I(t). That is, find the general solution. Your solution will have a constant resulting from integration, as well as the parameter (r-a). Verify that your solution is correct by substituting it back into the equation (1).

2. The character of this solution depends on r-a. Describe this dependence mathematically and physically. (In other words, for what values of r-a does the disease spread? Why?) Draw the slope field for this equation, for two characteristically different values of r-a. Draw the corresponding two phase lines.

3. Now consider initial conditions. At time = 0, we start with a certain number of sick people, I_0. For each of those characteristically different values of r-a from part 2:

- Choose several initial conditions, and graph the particular solution (using the general solution in part 1).

- For those same initial conditions, use Euler's method with a large and a small step size to solve the equation. Graph the numerical solutions on the same axes (slope field, even) as the analytical particular solutions.

Exploration 2

The model of the spread of a disease involves several parameters. We can see what happens in a particular set of circumstances by plugging in some numbers for these parameters. Think of t=0 as the start of time that we get involved. t is negative in the past, and t is positive in the future. But I, the population, can never be negative.

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. I₀ is the number of sick people that we start with. What happens to I(t) as t increases if I₀ is zero? What do we call that? Explain its meaning in words. From now on, let's assume I₀=1.

4. Using r=1/2 day^-1, a=1/7 day^-1, and I₀=1, write out the solution equation. Draw the graph of I(t) vs. t. What does the model predict will happen to I(t)? (Hint: the future is represented as increasing t, so look at I(t) for large values of t.) 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?

Exploration 3

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 (and start over with I₀=1) draw the graph of I(t) vs. t. What does the model predict will happen to I, the number of infectious people?

2.* Administrators see that the better hygiene measures at the Country School are making a difference in the spread of chicken pox. Try graphing the equation with parameter values as in the City School, and then after three days, change them to the Country School values. Can the spread of chicken pox be curbed?

3. It is called an epidemic if dI/dt is positive. What relationship between r and a causes an epidemic? (Look back at Exploration 1, #2.) 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.

Exploration 4

1. Use the first set of parameters given (City School) 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 2, questions 1 & 2). Write out the solution equation using the original chicken pox values, and then using your new HIV values.

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.) Graph both of your equations (chicken pox and HIV) on the same axes. Is the progress of the spread of the disease different? (That is, do the curves look different for middle values of t?) Is the end result different? (That is, do we get an epidemic in both cases?) Explain how your different numbers gave the same or different results. (Hint: Use Exploration 3, part 3.)

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