#### 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.

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?