Example: A Flu Epidemic
Example: A Flu Epidemic
Influenza (commonly called “flu”) is caused by the influenzavirus, which infects the respiratory tract (nose, throat, lungs). It can cause mild to severeillness that most of us get during winter time. The influenza virus is spread from person to person in respiratory droplets of coughs and sneezes. This is called “dropletspread”. This can happen when droplets from a cough or sneeze of an infected person are propelled through the air and deposited on the mouth or nose of people nearby. Itis good practice to cover your mouth when you cough or sneeze so as not to infect others around youwhen you have the flu. Regular hand washing is an effective way to prevent the spread of infection and illness.
Assume that you have the flu virus, and you forgot to cover your mouth when two friends came tovisit while you were sick in bed. They leave, and the next day they also have the flu. Let's assumethat each friend in turn spreads the virus to two of their friends by the same droplet spread the followingday. Assuming this pattern continues and each sick person infects 2 other friends, we can representthese events in the following manner:
Each person infects two more people with the flu virus.
We can tabulate the events and formulate an equation for the general case:
|
Day (n) |
No. of newly-infected people |
|
\(\text{1}\) |
\(2 =2\) |
|
\(\text{2}\) |
\(4=2\times 2=2\times {2}^{1}\) |
|
\(\text{3}\) |
\(8=2\times 4=2\times 2\times 2=2\times {2}^{2}\) |
|
\(\text{4}\) |
\(16 =2\times 8=2\times 2\times 2\times 2=2\times {2}^{3}\) |
|
\(\text{5}\) |
\(32 =2\times 16=2\times 2\times 2\times 2\times 2=2\times {2}^{4}\) |
|
\(\vdots\) |
\(\vdots\) |
|
\(n\) |
\(2\times 2\times 2\times 2\times \cdots \times 2=2\times {2}^{n-1}\) |
The above table represents the number of newly-infected people after \(n\) days since youfirst infected your \(\text{2}\) friends.
You sneeze and the virus is carried over to \(\text{2}\) people who start the chain (\({a}=2\)). The next day, each one then infects \(\text{2}\) of their friends. Now \(\text{4}\) people are newly-infected. Each of them infects \(\text{2}\) people the third day, and \(\text{8}\) new people are infected, and so on. These events can be written as a geometric sequence:
\[2; 4; 8; 16; 32; \ldots\]
Note the constant ratio (\(r=2\)) between the events. Recall from the linear arithmetic sequence how the common difference between terms was established. In the geometric sequence we can determine the constant ratio (\(r\)) from:
\[\cfrac{{T}_{2}}{{T}_{1}}=\cfrac{{T}_{3}}{{T}_{2}}=r\]
More generally,
\[\cfrac{{T}_{n}}{{T}_{n-1}}=r\]
This lesson is part of:
Sequences and Series