Disease epidemics can be modelled by differential equations. Suppose we have two sub-populations of individuals in a city: individuals who are infected with some virus (1) and individuals who are susceptible to infection (S). We assume that if a susceptible person interacts with an infected person, there is a probability p that the susceptible person will become infected. Each infected person recovers from the infection at a rate r and becomes susceptible again. The differential equations that model these population sizes are d.S dt dl dt =TI - PSI, =pSI-rl. (a) Assuming no one dies from the virus, the total population N is a constant number defined as N = S+I. Show in this case that you can reduce the above system to the single differential equation for I. (b) Find the general solution I(t) to the ODE found in part (a). (c) If I(0) = lo, give the equation for the unknown constant in terms of Io, N, p and r.
Disease epidemics can be modelled by differential equations. Suppose we have two sub-populations of individuals in a city: individuals who are infected with some virus (1) and individuals who are susceptible to infection (S). We assume that if a susceptible person interacts with an infected person, there is a probability p that the susceptible person will become infected. Each infected person recovers from the infection at a rate r and becomes susceptible again. The differential equations that model these population sizes are d.S dt dl dt =TI - PSI, =pSI-rl. (a) Assuming no one dies from the virus, the total population N is a constant number defined as N = S+I. Show in this case that you can reduce the above system to the single differential equation for I. (b) Find the general solution I(t) to the ODE found in part (a). (c) If I(0) = lo, give the equation for the unknown constant in terms of Io, N, p and r.
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:Disease epidemics can be modelled by differential equations. Suppose we have
two sub-populations of individuals in a city: individuals who are infected
with some virus (I) and individuals who are susceptible to infection (S). We
assume that if a susceptible person interacts with an infected person, there is a
probability p that the susceptible person will become infected. Each infected
person recovers from the infection at a rate r and becomes susceptible again.
The differential equations that model these population sizes are
d.S
dt
dI
dt
=rI - PSI,
=pSI - TI.
(a) Assuming no one dies from the virus, the total population N is a constant
number defined as N = S+I. Show in this case that you can reduce the
above system to the single differential equation for I.
(b) Find the general solution I(t) to the ODE found in part (a).
=
(c) If I (0) Io, give the equation for the unknown constant in terms of
Io, N,p and r.
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are you sure about part 2? is the answer supposed to be i(t)=c+itp(N-i)-irt?
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