Introduction⌗

This post is an assignment to one of my maths classes.

Based on a huge simplified mathematical model, you will examine the propagation of an epidemic in a population by implementing the model in a Python program. The model calculates the number of susceptible, contagious and immune number of individuals from one day to another.

The States⌗

The state of an individual, at a given time, in the population is exactly one of

  • $S$: Susceptible
  • $C$: Contagious
  • $I$: Immune

An individual not been contagious is susceptible, a contagious individual is not susceptible and an immune individual is neither susceptible nor contagious. An individual reaches immunity after a given time of beeing contagious.

The Data Representation⌗

  • $N$: Number of individuals in the population
  • $a$: The time an individual is contagious (days, in this example)
  • $b$: A constant being related to the infectivity of the disease (the higher value, the more infectivity)

The Model⌗

The number of individuals whith each status after $k$ days is denoted $S_k$, $C_k$ and $I_k$. The number develops from day to day following the model

  • $S_{k+1}=S_k - b\cdot C_k\cdot S_k$
  • $C_{k+1}=C_k + b\cdot C_k\cdot S_k-C_k/a$
  • $I_{k+1}=I_k+C_k/a$

Suppose the starting values $N=1\,000\,000$, $S_0=999\,999$, $C_0=1$, $a=7$ and $b=2.0\cdot 10^{-7}$. Note that this is a huge simplified model which may not be applicable in real world.

The Tasks⌗

  1. Describe each equation in the model above.
  2. Implement the model in Python and present the outcome as three diagrams, respectively showing graphs of

  • the development of the number of individuals with each status as $b=2.0\cdot 10^{-7}$

  • the development of the number of individuals with each status as $b=1.7\cdot 10^{-7}$

  • the development of the number of individuals with each status as $b=2.3\cdot 10^{-7}$

    for a suitable number of days. Comment these graphs.

  1. Present the progress of the number of individuals of each state with respect to time as $b=2\cdot 10^{-7}$, but with variation of the number of days an individual is contagious. Let $a$ take the value 6, 7 and 8, one value at a time. Comment the result.

  2. Epidemiologs talk about “flattening the curve”, example in this article. Why is this important? Based on your results above, what is important to achieve?