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Consider the SIR model with the square root dynamics dS = A-µS - B√SI, dt B√SI - (+)1, dR = 71-µR, dt where the total population N(t) = S(t) + I(t) +R(t). (a) Show that the total population N(t) is not constant and determine the population steady state. (b) Use the answer above to determine N(t) explicitly if the initial condition is N(0) = No. (c) Take A = 0 and set u(t)=√S(t) and v(t) = √(t). Write the first two equations of the system above in terms of the new functions u(t) and v(t).

Consider the SIR model with the square root dynamics dS = A-µS - B√SI, dt B√SI - (+)1, dR = 71-µR, dt where the total population N(t) = S(t) + I(t) +R(t). (a) Show that the total population N(t) is not constant and determine the population steady state. (b) Use the answer above to determine N(t) explicitly if the initial condition is N(0) = No. (c) Take A = 0 and set u(t)=√S(t) and v(t) = √(t). Write the first two equations of the system above in terms of the new functions u(t) and v(t).

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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4. Consider the SIR model with the square root dynamics d.S = X-μS - B√SI, dt dI = B√SI - (μ+Y)I, dt dR - 7I - µR, dt where the total population N(t) = S(t) + I(t) +R(t). (a) Show that the total population N(t) is not constant and determine the population steady state. (b) Use the answer above to determine N(t) explicitly if the initial condition is N (0) = No. (c) Take A = 0 and set u(t)=√S(t) and v(t) = √I(t). Write the first two equations of the system above in terms of the new functions u(t) and v(t). (d) Eliminate the variable v(t) from (c) above to obtain a second-order ordinary differential equation. | |
Transcribed Image Text:4. Consider the SIR model with the square root dynamics d.S = X-μS - B√SI, dt dI = B√SI - (μ+Y)I, dt dR - 7I - µR, dt where the total population N(t) = S(t) + I(t) +R(t). (a) Show that the total population N(t) is not constant and determine the population steady state. (b) Use the answer above to determine N(t) explicitly if the initial condition is N (0) = No. (c) Take A = 0 and set u(t)=√S(t) and v(t) = √I(t). Write the first two equations of the system above in terms of the new functions u(t) and v(t). (d) Eliminate the variable v(t) from (c) above to obtain a second-order ordinary differential equation. | |
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