# Under construction

## Modelling Disease Spread

Spread of infectious diseases can be investigated by means of mathematical models.

### The I-Model

The I-model is the simplest exponential model for disease spread.
It focuses only on the number *I* of infected people.
All infected people will also recover.

Let us assume that at day 0 one person - the so called patient zero - is ill. The patient 0 infects 2 person during 1 day and becomes healthy at the same time. So on day 1 there are 2 persons ill. Each of them infects ahain 2 persons during one day and recovers himself. So, on day 2 there are 4 persons ill. If this procedure continues the disease spreads as depicted in the figures below:

We see that that the number of infected people $I_n$ on day $n$ is: $$\begin{align} I_0 & = 1 \\ I_1 & = 2 \cdot 1 = 2 \\ I_2 & = 2 \cdot 2 = 4 \\ I_3 & = 2 \cdot 4 = 8 \end{align}$$ In general we have: $$\begin{align} I_0 & = 1 \tag {1a} \\ I_n & = 2 \cdot I_{n-1} \tag {1b} \\ I_n & = 2^n \tag {1c} \end{align}$$

The number of recovered people $R_n$ on day $n$ is: $$\begin{align} R_0 & = 0 \\ R_n & = I_0 + I_1 + ... + I_{n-1} = 1 + 2 + 2^2 + ... + 2^{n-1} \\ R_n & = 2^{n} - 1 , n \gt 0 \tag {2a} \\ R_n & = I_n - 1 , n \gt 0 \tag {2b} \end{align}$$

The number of infected people grows exponentially:

The grows at the beginning is slow and the spread is underestimated.
The spread becomes monstreous fast:

after 5 days 32 people are infected - a small group,

after 10 days 1 thousand (1,032) people are infected - a small town ,

after 20 days 1 million people (1,048,576) are infected - a city,

after 33 days 8 billion people (8,589,934,592) are infected - the whole world.

### References

- Julia Collins, Nadia Abdelal: Spread of Disease