Spread of Disease Simulation for MATH 2411 (Calculus II)
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Date:
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INTRODUCTION
Epidemiology is the study of the spread of disease. Often, when a small percentage of a population
becomes infected, the rate at which the disease spreads is often given by the first-order autonomous
ordinary differential equation:
[Write the ODE here. Do not include the initial conditions yet.]
We measure t in convenient time units. Depending on the disease spreads, this might be measured in
days, weeks, or months.
y(t) = the number of infected so far at time t. Once someone is infected, we assume that they stay
infected for the duration.
dy/dt = [Write something here that makes sense!]
L = the total population which will eventually be infected (sometimes known as the carrying capacity).
A = y(0) = the initial number of infected (at t = 0).
k = a constant which describes how fast the disease will spread.
SIMULATION
We attempted to simulate the spread of a disease, “Calculosis”, among a class of students. The two
instructors were initially infected, so we had y(0) = 2.
Including the instructors, the total population was L = [enter the number here].
Thus, initially, there were (L – 2) uninfected students.
The instructor arbitrarily chose an initial rate for the spread of the disease at t = 0.
dy/dt = +2 infections/this turn
Our model equation determines the value of k.
[Write some algebra here which shows how we determined k.]
We used this value throughout the remainder of the simulation.
Thus, the expected number of infections on any particular turn is k*y*(L – y), which for our simulation
(in terms of L and y only) is
[Write that expression here.]
On any given turn, an uninfected individual’s probability of becoming infected is
(Expected # of new infections)/(Current # of non-infected)
and this is equal to
[Write the simplified expression here.]
We used the following procedure to determine new infections on each turn:
1. Calculate the new probability that an uninfected participant will be infected this turn.
2. Find the decimal equivalent of this number.
3. Allow each uninfected participant to use the “rand” function on the TI 83/84 to determine if the
participant has become infected on this turn. If the random number is less than the critical
value from Step #2, then the individual has become infected.
4. Total the number of new infections on this turn and add it to the number of currently infected.
We halted the simulation when there were 2 or fewer participants uninfected.
[You may either create a results table here, or just attach the worksheet provided during the simulation.
If you are attaching the worksheet, then you should state that here. We hope it is filled out neatly.]
THEORETICAL SOLUTION
[If you can stand to type out the mathematics here, then do so, else you may just say that you are
attaching your solution (on Engineering Pad) to:
dy/dt = k*y*(L – y), y(0) = 2, y’(0) = 2.]
[Use technology to sketch a graph for t between 0 and 10. State that you are attaching the graph to the
report.]
Based on this logistic curve, an epidemiologist can predict the following:
[Work out the answers to these and include the results here:]
1. During which turn should we see the GREATEST number of new infections. In other words,
when is dy/dt the largest?
2. Can we explain this another way? Yes.
[Graph dy/dt vs. y. That is, for the values of y between 0 and L, let y be represented on the
horizontal axis and the quadratic k*y*(L – y) be represented on the vertical axis.]
According to this graph, for what value of y do we get the largest value of dy/dt?
What is the associated value of t?
3. What do we call the point on the logistic curve which is associated with the largest derivative?
4. If every non-infected participant had a risky encounter with every infected participant (exactly
once) on this turn, then how many risky encounters would there be this turn?
According to our model, what PERCENT of those encounters result in new infections?
5. Approximately how many turns should it take us until there are only 2 uninfected participants?
COMPARING OUR DATA TO THEORY
[Answer the following in the form of a paragraph:]
1. When we superimpose our data onto the theoretical logistic curve, are there many large
deviations from the theoretical values? If so, where?
2. Compare when the simulation saw the greatest number of new infections to your answer from
above. Was there a great deviation there? If so, do you have any ideas why this occurred?
3. Do you think the simulation yielded a curve is similar to the logistic curve? If not, why not?
UNDERSTANDING WHY THINGS AREN’T ALWAYS PERFECT
If we toss 10 fair coins, we say that we expect 5 heads and 5 tails. We know that we do not always
achieve that result, but in the long run, if we toss a million fair coins, we know that we get
approximately [HOW MANY?] heads and that the probability of heads is basically [???]%.
[During the simulation, find any turns for which the actual number of new infections deviated from the
theoretical number. How did this affect the simulation?]
CONCLUSIONS
[Answer the following in paragraph form:]
1. Was it a reasonable simulation?
2. If the public were totally ignorant during the first part of the epidemic, do you think the logistic
curve is a reasonable model for the spread of disease? If not, then what real-life factors
influence how a real epidemic curve would look. If you have a specific reference, then cite it
here. You do not need to write a bibliography for this report.
3. Research: Can you find a population (might be an animal population) where at least half of the
population was infected with some disease? If so, was the logistic curve or some similar curve
cited as a reference?