Disease epidemics can be modelled by differential equations. Suppose we havetwo sub-populations of individuals in a city: individuals who are infectedwith some virus (I) and individuals who are susceptible to infection (S). Weassume that if a susceptible person interacts with an infected person, there is aprobability p that the susceptible person will become infected. Each infectedperson recovers from the infection at a rate r and becomes susceptible again.The differential equations that model these population sizes areTI - PSI,dSdtdIdtpSI - TI.(a) Assuming no one dies from the virus, the total population N is a constantnumber defined as N = S+I. Show in this case that you can reduce theabove 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 ofIo, N, p and r.

Given Information:Here, are constant that represents rate of infection and probability of the…

Answered: Disease epidemics can be modelled by…

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

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
TI - PSI,
dS
dt
dI
dt
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.

Expert Solution

Step 1: Analysis and Introduction

Given Information:

$fraction numerator d S over denominator d t end fraction equals r I minus p S I semicolon space fraction numerator d I over denominator d t end fraction equals p S I minus r l$

Here, $r comma space p$ are constant that represents rate of infection and probability of the susceptible person get infected.

To show:

a) Consider $N equals S plus I$ , reduce the given system into single differential equation on $I$ .

b) General solution of $I open parentheses t close parentheses$ .

c) At $I open parentheses 0 close parentheses equals I subscript 0$ , find the value of unknown constant.

Concept used:

Bernoulli's Differential Equation is of the form $y apostrophe plus p open parentheses x close parentheses space y equals q open parentheses x close parentheses space y to the power of n$ .

Substitute $u equals y to the power of 1 minus n end exponent$ to reduce the bernouli's equation to a linear equation.

The reduced linear equation is $fraction numerator d u over denominator d x end fraction minus open parentheses n minus 1 close parentheses p open parentheses x close parentheses u equals negative open parentheses n minus 1 close parentheses q open parentheses x close parentheses$ .

Integration Formula:

$table row cell integral open parentheses e to the power of a x end exponent close parentheses d x end cell equals cell e to the power of a x end exponent over a plus c end cell row cell integral a d x end cell equals cell a x plus c end cell end table$

Here, $a$ is any constants and $c$ is the constant of integration.