Week 4: Closure

Week 4: Closure#

Syllabus#

Looping over a sequence (known number of iterations), the keywords for and in and the range() function
Understanding that indexing variable in the for loop is reassigned with each iteration
Looping as long as a condition is true (unknown number of iterations), the keyword while
Understanding that the variables used in the condition need to be defined before the while loop
Understanding that the variables used in the condition should be changed in the while loop body for loop to terminate
The keyword break
Solve simple problems that require a lot of computation e.g. simulations and population models
The concept of pseudocode

Checkpoints#

Checkpoint 4.1: Disease Simulation#

We investigate an SIR (susceptible, infectious, recovered) model for an infectious disease. The model is based on the following assumptions:

People can be in one of three states: susceptible \(S_t\), infectious \(I_t\), or recovered \(R_t\). These values change over time \(t\), but the total population size \(S_t + I_t + R_t\) remains constant.
The infection rate \(\beta\) determines how many susceptible people an infectious person will infect each day. The number of people becoming infectious each day is \(\beta S_t I_t\).
The recovery rate \(\gamma\) determines how many infectious people will recover each day. The number of people recovering each day is \(\gamma I_t\).

The model is:

\[\begin{split} \begin{aligned} \nonumber S_{t+1} &= S_t - \beta S_t I_t \\ \nonumber I_{t+1} &= I_t + \beta S_t I_t - \gamma I_t \\ \nonumber R_{t+1} &= R_t + \gamma I_t \end{aligned} \end{split}\]

Write a script that simulates the spread of a disease over 100 days. You should always start with a population where only one person is infectious, and the rest are susceptible. You shouldn’t round the calculations, but you should print the rounded values.

For example, try the following parameters:

population = 1000
infectious = 1
infection_rate = 0.0004
recovery_rate = 0.2
days = 100

Print the number of susceptible, infectious, and recovered people every tenth day. The first few lines of the output should look like this:

Day 10, Susceptible: 989, Infectious: 6, Recovered: 5
Day 20, Susceptible: 930, Infectious: 34, Recovered: 36
Day 30, Susceptible: 699, Infectious: 125, Recovered: 176

Checkpoint 4.2: Which Tetrahedral#

The \(n^\text{th}\) tetrahedral number is given by

\[ T_n = \frac{n(n+1)(n+2)}{6} \]

and a sequence of tetrahedral numbers is

\[ 1,\, 4,\, 10,\, 20,\, 35,\, 56,\, 84,\, 120,\, 165,\, 220, \ldots \]

For an integer \(t\), you are interested in knowing whether \(t\) is a tetrahedral number, and if so, which number in the sequence it is. Write a script that finds the \(n\) such that \(T_n = t\) for a given \(t\). If \(t\) is not a tetrahedral number, the script should print that \(t\) is not a tetrahedral number.

For example, the script starting with t = 121 should print:

121 is not a tetrahedral number

Given other starting values, your script should print the following:

171700 is a tetrahedral number with n = 100

227920 is a tetrahedral number with n = 110

222439000 is not a tetrahedral number