DISEASE CONTROL
The use of mathematical models to study spread of infectious diseases dates back to the 18th century. At this time small pox was a major threat to public health and a controversial inoculation program had been put in place. We consider the effectiveness of inoculation in what follows -the model works for any disease which once contracted gives lifetime immunity to that disease. We will refer to the disease as the dreaded 1 mgr.
Consider the set of individuals born in a given year (t = 0) and let n(t) denote the number of these individuals that are still alive t years later.
1. What is a reasonable graph of the function n (t) in the absence of the dreaded lurgy?
Also let x(t) be the number of individuals of this set who have not had the dreaded lurgy by time t, and who are therefore still susceptible.
2. What is a reasonable graph of the function x(t) in the absence of the dreaded lurgy?
Let b be the rate at which susceptible contract the disease, and let be the rate at which people who contract the disease die from the disease. Note that we assume that band, are independent of the age of the individual. Finally let m(t) be the death rate from all causes other than the dreaded funny.
3. What is a reasonable graph of the function m(t)?
4. Use a rate of change argument to give a differential equation that he scribes how the number of susceptibles decline.
5. Use a rate of change argument to give a differential equation that describes how the death rate of the entire population.
6. Under the unreasonable assumption that m(t) is independent of t take the quotient of your answers in Id and. to give a differential equation of the form
dn/dx = f(n, x)
7. Solve the differential equation in #6.
8. What conclusions can you make regarding the solution to.7?
9. What does the variable z = z/n represent, and why might we be interested in the iitial condition z(0) = 1?
10. Combine the differential equations from #4 and #5 to arrive at the following differential equation:
dz/dt =-bz(1 - vz)
11. The differential equation in #10 is autonomous. Draw the phase diagram for this differential equation, and indicate the position of the initial condition z(0) = 1 on the diagram. What conclusions can you make?
12. What does this model have to say about the eventual spread of the dreaded lurgy?