OConsider the SIR model equations given below. dS -aSI dt dI aSI – bI dt dR bỊ dt (a) Show that S(t) +I(t) + R(t) = N, a constant, by using the above equations to show that i (S(t) + I(t) + R(t)) = 0. (b) Note that by the chain rule and implicit differentiation, dI b 1 -1+ a S dI aSI – bI aSI bI dt dS + aSI -aSI dS -aSI dt so that b 1 -1+ a S dI dS Show that I(t) + S(t) – ° In(S(t)) = k (1) a where k is a constant of integration by integrating with respect to t. (c) Verify your work in the previous problem by computing (1) + s() - m(5() d In(S(t)) dt a using the chain rule and the SIR equations to show that this is zero for all S. ||
OConsider the SIR model equations given below. dS -aSI dt dI aSI – bI dt dR bỊ dt (a) Show that S(t) +I(t) + R(t) = N, a constant, by using the above equations to show that i (S(t) + I(t) + R(t)) = 0. (b) Note that by the chain rule and implicit differentiation, dI b 1 -1+ a S dI aSI – bI aSI bI dt dS + aSI -aSI dS -aSI dt so that b 1 -1+ a S dI dS Show that I(t) + S(t) – ° In(S(t)) = k (1) a where k is a constant of integration by integrating with respect to t. (c) Verify your work in the previous problem by computing (1) + s() - m(5() d In(S(t)) dt a using the chain rule and the SIR equations to show that this is zero for all S. ||
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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letter C
Transcribed Image Text:OConsider the SIR model equations given below.
dS
-aSI
dt
dI
aSI – bI
dt
dR
bỊ
dt
(a) Show that S(t) +I(t) + R(t) = N, a constant, by using the above equations to show that
i (S(t) + I(t) + R(t)) = 0.
(b) Note that by the chain rule and implicit differentiation,
dI
b 1
-1+
a S
dI
aSI – bI
aSI
bI
dt
dS
+
aSI
-aSI
dS
-aSI
dt
so that
b 1
-1+
a S
dI
dS
Show that
I(t) + S(t) – ° In(S(t)) = k
(1)
a
where k is a constant of integration by integrating with respect to t.
(c) Verify your work in the previous problem by computing
(1) + s() - m(5()
d
In(S(t))
dt
a
using the chain rule and the SIR equations to show that this is zero for all S.
||
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