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# Y k o h 4 2 h y k h 2 3 y k o h 4 y k o h 2 thus the

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y ′′′ k + O ( h 4 ) 2 h = y k + h 2 3! y ′′′ k + O ( h 4 ) = y k + O ( h 2 ) Thus, the accuracy for this method is O ( h 2 ). 7. Purpose : Computing project . Using packaged software to compute the solution to a system of ODEs representing an SIR epidemic model. Understanding the meaning of the model and experimenting with parameter values to determine different outcomes. From [Hea02, pp.418–419,#9.2] . Exercise : The Kermack-McKendrick model for the course of an epidemic in a population is given by the system of ODEs y 1 = - cy 1 y 2 y 2 = cy 1 y 2 - dy 2 y 3 = dy 2 where y 1 represents susceptibles, y 2 represents infectives in circulation, and y 3 respresents infectives removed by isolation, death, or recovery and immunity. The parameters c and d represent the infection rate and removal rate, respectively. Use a library routine to solve this system numerically, with the parameter values c = 1 and d = 5 , and initial values y 1 (0) = 95, y 2 (0) = 5, and y 3 (0) = 0 . Integrate from t = 0 to t = 1 . Plot each solution component on the same graph as a function of t. As expected with an epidemic, you should see the number of infectives grow at first, then diminish to zero. Experiment with other values for the parameters and initial conditions. Can you find values for which the epidemic does not grow, or for which the entire population is wiped out? Solution: A plot of each component on the same graph, as a function of t can be seen in Figure 21. In the plot, y 1 , the susceptibles is represented by the thin solid line, y 2 , the infectives in circulation represented by thin dashed line, and y 3 , the infectives removed is represented by the thick solid line. Examples of values for which the epidemic does not grow include c = 0 . 5 , d = 5 and c = 1 , d = 100. An example value for which the entire population is wiped out is c = 10 , d = 10. 8. Purpose : Computing project . Using packaged software to compare efficiency of various library routines. Note to Instructor: You may ask the students first to catalog the ODE solving routines available in whichever numerical package you choose to have them use. Or you may wish to specify a particular set of ODE solvers the students should implement. From [Hea02, p.419,#9.3] . 23

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0 20 40 60 80 100 0.2 0.4 0.6 0.8 1 t Figure 21: Phase portrait of the system in Exercise 4. Exercise : Experiment with several different library routines having automatic step-size selection to solve the ODE y = - 200 ty 2 numerically. Consider two different initial conditions, y (0) = 1 and y ( - 3) = 1 / 901 , and in each case compute the solution until t = 1 . Monitor the step size used by the routines and discuss how and why it changes as the solution progresses. Explain the difference in behavior for the two intial conditions. Compare the different routines with respect to efficiency for a given accuracy requirement. 9. Purpose : Computing project . Using packaged or self-written software to compare stiff and non-stiff numerical solution methods. Note to Instructor: You may choose to ask the students to compare forward Euler with backward Euler, in addition to comparing the packaged routines. From [Hea02, p.419,#9.5] . Exercise :
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