3*. Consider the following SIS epidemic model with variable population size (birthrate is different from death rate):dSbN – BS+Yl – µSdtNdIIBS(y+µ)I.Ndta. Check thatdN= rN,dtwhere r= (b– u), and that consequently, N(t) = N(0)e".b. Rescale the above system by introducing the new variablesS(t)y(t) =N(t)I(t)x(1)N(t)1.4 Qualitative Analysis17Note that1 dSN dtdxS dNdtN² dtand that x(t) + y(t) = 1 for all t.c. Verify thatdy(B – (y+b))y ( 1ydt1(y+b)d. Let Ro = B/(y+b). Can you interpret Ro? What is the qualitative behaviorof the above equation?e. What is the meaning in terms of the original variables S and I as y → 0 andy → y* € (0, 1).

On Adding the two equations of the SIS epidemic model, we get…

. Consider the following SIS epidemic model with variable population size (birth rate is different from death rate): dS = bN – BS÷+Yl – µS dt dI - ( γ+ μ). dt a. Check that dN = rN, dt (b – µ), and that consequently, N(t) = N(0)e". b. Rescale the above system by introducing the new variables where r = S(t) N(t) I(t) y(t) = N(t)' x(1) Qualitative Analysis 17 Note that S dN 1 dS N² dt dx dt N dt and that x(t) +y(t)=1 for all t. c. Verify that dy (ß – (y+b))y | 1 dt 1 – (r+b)

. Consider the following SIS epidemic model with variable population size (birth rate is different from death rate): dS = bN – BS÷+Yl – µS dt dI - ( γ+ μ). dt a. Check that dN = rN, dt (b – µ), and that consequently, N(t) = N(0)e". b. Rescale the above system by introducing the new variables where r = S(t) N(t) I(t) y(t) = N(t)' x(1) Qualitative Analysis 17 Note that S dN 1 dS N² dt dx dt N dt and that x(t) +y(t)=1 for all t. c. Verify that dy (ß – (y+b))y | 1 dt 1 – (r+b)

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

3*. Consider the following SIS epidemic model with variable population size (birth
rate is different from death rate):
dS
bN – BS+Yl – µS
dt
N
dI
I
BS
(y+µ)I.
N
dt
a. Check that
dN
= rN,
dt
where r= (b– u), and that consequently, N(t) = N(0)e".
b. Rescale the above system by introducing the new variables
S(t)
y(t) =
N(t)
I(t)
x(1)
N(t)
1.4 Qualitative Analysis
17
Note that
1 dS
N dt
dx
S dN
dt
N² dt
and that x(t) + y(t) = 1 for all t.
c. Verify that
dy
(B – (y+b))y ( 1
y
dt
1
(y+b)
d. Let Ro = B/(y+b). Can you interpret Ro? What is the qualitative behavior
of the above equation?
e. What is the meaning in terms of the original variables S and I as y → 0 and
y → y* € (0, 1).

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