This example comes from an outbreak of influenza in a boarding school for boys in England...

 

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This example comes from an outbreak of influenza in a boarding school for boys in England in 1978. The case was initially reported in the British Medical Journal, 4 March, 1978.¹ The total number of boys at the boarding school was N = 763, and the daily numbers of infectious boys are available and presented in Table 1. Only one boy was infectious when students returned to school after a break and therefore I(0) = 1. Using the data, the removal rate is estimated to be y 0.476, and the infectivity rate B 0.002342. In this instance there were no deaths and all the infectious boys recovered. = 0 4 70 13 25 27 2 25 75 6 7 8 9 10 296 258 236 192 126 |H|2|3|4 71 28 11 Q.2.1. h=2, Q.2.2. h=1. Q.2.3. h=0.1. Table 1: Number of infectious boys on day t. Q.1. Give the initial conditions S(0), I(0), and R(0) for this specific problem. Q.2. Use Euler's method to answer the following questions. For each value of h, calculate approximations for t = 3,4,5,6,7,8,9, 10, 11, 12, 13, 14 and plot the approximations for S and I as a function of t, and the given data on the same axis. Optional: you may also use the more accurate 4th order Runge Kutta method described on pages 293-295 of your textbook, [5], and optional video-lecture 04.3. How do the approximations change as h decreases? Comment on the smoothness and accuracy of the approximate solutions in comparison to the data in Table 1. Q.3. Would you say the approximate solution to (1) with h = 0.1 provides a reasonable description of the flu data? Is the approximation less accurate on some time intervals than other intervals? If so, give an example. Q-4. How many boys did not get infectious during the first 20 days of the outbreak? Use h=0.1 for your approximation and round to the nearest integer. This example comes from an outbreak of influenza in a boarding school for boys in England in 1978. The case was initially reported in the British Medical Journal, 4 March, 1978.¹ The total number of boys at the boarding school was N = 763, and the daily numbers of infectious boys are available and presented in Table 1. Only one boy was infectious when students returned to school after a break and therefore I(0) = 1. Using the data, the removal rate is estimated to be y 0.476, and the infectivity rate B 0.002342. In this instance there were no deaths and all the infectious boys recovered. = 0 4 70 13 25 27 2 25 75 6 7 8 9 10 296 258 236 192 126 |H|2|3|4 71 28 11 Q.2.1. h=2, Q.2.2. h=1. Q.2.3. h=0.1. Table 1: Number of infectious boys on day t. Q.1. Give the initial conditions S(0), I(0), and R(0) for this specific problem. Q.2. Use Euler's method to answer the following questions. For each value of h, calculate approximations for t = 3,4,5,6,7,8,9, 10, 11, 12, 13, 14 and plot the approximations for S and I as a function of t, and the given data on the same axis. Optional: you may also use the more accurate 4th order Runge Kutta method described on pages 293-295 of your textbook, [5], and optional video-lecture 04.3. How do the approximations change as h decreases? Comment on the smoothness and accuracy of the approximate solutions in comparison to the data in Table 1. Q.3. Would you say the approximate solution to (1) with h = 0.1 provides a reasonable description of the flu data? Is the approximation less accurate on some time intervals than other intervals? If so, give an example. Q-4. How many boys did not get infectious during the first 20 days of the outbreak? Use h=0.1 for your approximation and round to the nearest integer.

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