1. Consider the initial value problem dy DE : = ayx = 0.5y(1 – y), IC : y(0) = Yo > 0. (1) dt Notice that a = 0.5 and x =1- y. (a) Find the critical points of the DE. Discuss whether the critical point an asymp- totically stable solution, or an unstable equilibrium solution? (b) Without solving the DE, plot y versus t for the DE.
1. Consider the initial value problem dy DE : = ayx = 0.5y(1 – y), IC : y(0) = Yo > 0. (1) dt Notice that a = 0.5 and x =1- y. (a) Find the critical points of the DE. Discuss whether the critical point an asymp- totically stable solution, or an unstable equilibrium solution? (b) Without solving the DE, plot y versus t for the DE.
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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Question
Please answer question A and B
Transcribed Image Text:4.
Consider the initial value problem
dy
ayx = 0.5y(1 – y), IC : y(0) = Yo > 0.
DE :
dt
(1)
Notice that a =
0.5 and x =
1- y.
(a) Find the critical points of the DE. Discuss whether the critical point an asymp-
totically stable solution, or an unstable equilibrium solution?
(b) Without solving the DE, plot y versust for the DE.
(c) Suppose the initial value problem (1) is a simple epidemic model with no mitiga-
tion, where y is the proportion of infectious individuals, x = 1-y is the proportion
= 0.5 is the proportionality factor. Let yo = 0.01 be
of susceptible individuals, a =
the initial proportion of infectious individuals and t be time in year.
i. Find y(t) and determine t so that y(t) > 0.8, where 0.8 is the threshhold for
the so-called herd immunity.
ii. Based on this model and your calculation, determine whether the strategy of
herd immunity is reasonable, if no mitigation is applied.
iii. Argue that without a mitigation, y(t) → 1 as t → xo is inevitable, which
means that every one in community will be infected.
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