Spread a rumor math project

Spread of a Disease

A contagious disease such as a flu virus is spread throughout a community by people

encountering other people. Let y(t) denote the number of people who have infected

disease and x(t) denote the number of people who are healthy. It is reasonable to assume

that the rate 𝑑𝑦

𝑑𝑡 at which the disease spreads are proportional to the number of interaction

between those two groups of people. That means dy/dt is jointly proportional to x(t) and

y(t). Therefore, we have the equation 𝑑𝑦

𝑑𝑡 = 𝑘𝑥𝑦, where k is the proportional constant.

The diagram shows the rate of change of infected people is equal to the product of infection

rate k and the number of interaction of healthy people x(t) and infected people y(t).

Observation: Initially x(0) = 100, y(0) = 5 and infected rate is 0.05.

We approximate 𝑑𝑦

𝑑𝑡 by

𝑦(𝑡+ℎ)−𝑦(𝑡)

ℎ ≈ 0.05*x(t)*y(t).

y(t + h) ≈ y(t) + h*0.05*x(t)*y(t). ….. (a)

𝑑𝑥

𝑑𝑡 ≈

𝑥(𝑡+ℎ)−𝑥(𝑡)

ℎ ≈ -0.05*x(t)*y(t)

x(t + h) ≈ x(t) – 0.05*x(t)*y(t) ………… (b)

Equations (a) and (b) are the formula that we need to enter Excel Worksheet.

i. Initially set up Excel Worksheet row 1, row 2 and row 3 as the worksheet below.

ii. At row three enter initial conditions

iii. At cell A5, enter formula =A4+J$2; At B4 enter formula =ROUNDDOWN(B4-

H$2*J$2*B4*C4,0); At cell C4 enter formula = =ROUNDUP(C4+H$2*J$2*B4*C4,0).

iv. B4 and C4 are the formula (b) and (a) respectively and Round function to round decimals

do integers.

v. Copy formulas of B4 and C4 down until you see 0 at the column B.

vi. Select cells A3 to C13 as data for the graph.

Healthy

People

Infected

PeopleInfection

Infection Rate

Using EXCEI we get

Spread of a Disease

Initial Condition: x(0) = 100 y(0) = 5

infection

rate = 0.05 h = 0.2

time: t x(t) y(t)

0 100 5

0.2 95 10

0.4 85 20

0.6 68 37

0.8 42 63

1 15 90

1.2 1 104

1.4 0 105

1.6 0 105

1.8 0 105 0

20

40

60

80

100

120

1 2 3 4 5 6 7 8

Chart Title

time: t x(t) y(t)

Feed Back

1. Suppose a student carrying a flu virus returns to an isolated college campus of 1000

students. Determine a differential equation for the number of students y(t) who have

contracted the flu if the rate of change at which the flu spreads is proportional to the

number of interactions between the number of students who have the flu and the number

of student who have not yet been exposed to it. Without medication when will all students

have flu, assuming the rate of get infected due to interaction with infected people is 0.05?

Hint: This is a similar question as previous example with initial condition x(0) = 1000, and

y(0) = 1.

2. Suppose a small community has a fixed population of n people. If one person who wants to

spread a rumor is introduced into this community. If y(t) denote the number of people who

know the rumor, and x(t) denote the people who does not know the rumor. It is reasonable

to assume that the rate 𝑑𝑦

𝑑𝑡 at which the rumor spreads is proportional to the number of

encounters between the two groups of people. How to define the differential equation to

interpret this situation? Assume the proportional constant is 0.05, how long the rumor will

spread to everyone in the community?

Hint: This is a similar question as previous example with conditions x + y = n + 1, x(0) = n and

y(0) = 1.