3. Drug Absorption

When a drug is administered to a pacient it follows a process of progressive absorption in the patient’s body. Its concentration increases each day according to a model that depends on the concentration of the drug in the body that day n (\(y_n\)), the absorption rate (\(k\)) and the initial drug concentration (\(y_0\)).

\[\begin{aligned} y_{i+1} &= y_i + k \cdot y_i \cdot (1 - y_i), \quad i>=1 \end{aligned}\]

For example, \(y_0 = 0.4\) and \(k = 0.5\) the first terms of the succession are:

\[\begin{split}\begin{align*} y_0 &= 0.4 \\ y_1 &= 0.52 \\ y_2 &= 0.6448 \\ y_3 &= 0.7593164800000001 \\ \end{align*}\end{split}\]

The concentration is considered stable when the drug concentration is equal to the previous day under the given tolerance.

You are asked to deliver the folowing function in the drug_abs module (drug_abs.py file):

drug_abs_days(y0, k, eps)

Parameters:

• y0 (float) – Initial drug concentration, \(0.0 < y0 < 1.1\).

• k (float) – Absorption rate, \(0.0 < k < 1.1\).

• eps (float) – Tolerance. \(eps <= 0.1\).

Returns:

Number of days needed by the drug to stabilize for the given tolerance.

Return type:

int

For example:

>>> drug_abs_days(0.4, 0.5, 0.1)
4

>>> drug_abs_days(0.4, 0.5, 0.01)
9

>>> drug_abs_days(0.4, 0.5, 0.001)
12

Doctests are available in the file drug_abs_days.test.