Estimating the proportion of infectees in the population

Now, after having seen what we can tell about a single individual chosen at random and of which we have no information about possible symptoms, contacts or behavior, let us see what we can tell about the proportion $p$ of infected in the population, based on the tests performed on the sampled individuals. The first idea is to solve Eqs. () and () with respect to $p$, from which it follows

$$p = \frac{n_P - \pi_2\cdot m}{(\pi_1-\pi_2)\cdot m}\,.$$ (5)

Applying this formula to the 2060 positives got in our numerical example we re-obtain the input proportion of 10%, somehow getting reassured about the correctness of the reasoning. If, instead, we get more positives, for example 2500, 3000 or 3500, then the proportion would rise to 15.1%, 20.1% and 26.7%, respectively, which goes somehow in the `right direction'. If, instead, we get less, for example 2000 or 1500, then the proportion lowers to 9.3% and 3.5%, respectively, which also seems to go into the right direction.

However, keeping lowering the number of positives something strange happens. For $n_P= 1200$ Eq. () vanishes and it becomes even negative for smaller numbers of positives, which is something concerning, indicating that the above formula is not valid in general. But why did it nicely give the exact result in the case of 2060 positives? And what is the reason why it yields negative proportions below 1200 positives? Moreover, Eq. () has a worrying behavior of diverging for $\pi_1=\pi_2$, even though irrelevant in practice, because such a test would be ridiculous - the same as tossing a coin to tag a person Positive or Negative (but in such a case we would expect to learn nothing from the `test', certainly not that the real proposition of infectees diverges!).

Let us see the limits of validity of the equation.

• The lower limit $p \ge 0$ implies, as we have already seen in the numerical example, $n_P \ge \pi_2 \cdot m$ and $\pi_1 > \pi_2$.⁹
• The upper limit $p \le 1$ is reflected in the condition $n_P\le \pi_1\cdot m$ (and $\pi_1 > \pi_2$). In our numeric example this would mean to have less than 9800 positives in our sample of 10000. But this ignores the fact that the proportion of infectees in the sample could be higher than that in the population.

Anyway, it is clear that when the model contemplates probabilistic effects we have to use sound methods based on probability theory.