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
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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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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