## Mathematics & Science
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## Applications of Systems of Differential Equations## Epidemiology: The Spread of Disease## Theoretical Introduction
Let's consider modeling a new situation, namely, the way in which a
## Choosing VariablesBefore we can go very far in creating our model, we need to choose some variables to represent the quantities of interest. We'll need to talk about how many people have caught influenza already, how many people haven't caught it yet, and how many people have already recovered from the illness. Of course, we can think of each of these quantities as being a function of time. The following list of variable names should cover what we need: represents the number of people who have not yet caught the disease. The letter*S*(*t*)*S*is used because these people areto the illness.*susceptible*represents the number of people who currently have the disease. The letter*I*(*t*)*I*is used because these people areby the illness.*infected*represents the number of people who have*R*(*t*)from the illness. In the model we are creating, we consider patients that have already had the disease as having developed an*recovered***immunity**to re-infection. This means that recovery from the illness does**not**return you to the susceptible group.*(Note: the model would work in just the same way if the patients actually died from the illness. After all, once they die they are no longer susceptible or infected. Because of this fact, when some researchers work with this model they say that the**R*represents people that have beenfrom the disease-interacting population, either by death or recovery. But let's be positive—they recovered!)**removed**represents*t*.*time*
## Finding the Rates of ChangeNow let's think about how these sub-groups interact with one another. We'll consider the rates of change, (derivatives,) of each of our functions in turn.
S is changing is the same as the rate at which infection is taking place. The
underlying question, then, is "When does infection take place?"Infection occurs when the disease is passed from an infected person to a
susceptible person. This occurs during
where
a S I.
After all, the rate at which people are leaving the susceptible population is
exactly the same as the rate at which they are joining the infected population.
That's where they are going!But we have another factor to consider! There are also people Putting our infection and recovery rates together, we get a quantity representing the rate of change of the infected population:
Whether the infected group is growing or shrinking in size would be determined by the relative sizes of these two terms.
b I. The formula for the rate of change of the recovered population is
therefore:
Note that this quantity is positive, since the recovered group is growing all of the time as people overcome the illness. ## Putting it All TogetherSo far, our three rates of change have led to three differential equations. If we throw in three initial conditions to keep them company we should have a nicely formed initial value problem. The initial conditions are obvious. We'll use the sizes of the three groups
at whatever time we're choosing to call time zero—the beginning of our
analysis. Let's call these sizes
Make sure you understand why each part of the model makes sense. It's now
time to begin our computer analysis, so let's move on and see how we might implement these ideas in |
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ODE Laboratories: A Sabbatical Project by Christopher A. Barker©2017 San Joaquin Delta College, 5151 Pacific Ave., Stockton, CA 95207, USA e-mail: cbarker@deltacollege.edu |