Bayes Theorem, Vaccination and Media mis-interpretations

Recently I read an article in a reputed magazine discussing the resurgence of COVID in Israel, a country that had vaccinated a major fraction of its population. The article mentioned that 50% of the new COVID cases comprised of individuals who were already vaccinated. Subsequently, it raised the question of the effectiveness of the vaccine. But on second thought, it is not just about this article or Israel but such argumentations take place on day to day basis and across countries. Frankly speaking, I won't blame individuals for misunderstanding is due to the nuances involved. However, I have better expectations from journalists. 

Anyways, the situation, as is discussed, actually does not provide any evidence to dilute vaccine effectiveness. Instead, it's a result of misinterpretation arising due to lacking understanding of the Bayes Theorem. To know more, look at the math:

So I hope the math gave you some idea about what I am talking about. Presently, there exists cross-sectional heterogeneity in vaccination drives and a fraction of the population vaccinated across nations. Therefore, it would be a good exercise to look at the effect of different initial conditions on the potential misinterpretations. 

For this purpose, I took different combinations of - vaccinated population fraction (x) and positive rates (q) covering possible scenarios of x > q, x < q, x=q. Now the rate of vaccination might vary as well and so this scenario was considered. Next, the resurgence of cases might happen at different rates, that is, the rate of increase of q might change as well, so I formed this into a scenario too. Hence, in total, I checked for 5 scenarios. To do this, I formed a simple model using python code (find below the link). The model takes initial conditions as input, that is, x, q and rate of increase of x (say, dx) and q (say, dq). It then yields us P(I|V) and P(V|I), therefore, helps us understand the extent of differences that might exist between these two. Subsequently, it would aid us in understanding the difference between these two terms, which are usually thought and reported interchangeably.

I have shown some scenarios and the results which the model yielded. I assumed vaccine effectiveness to be normally distributed in each run with mean 80% (0.8) and sd 2% (0.02). The conditions and scenarios were as follows,

a. x = 4.5, q = 4.5, dx = dq = 0.5

b. x = 50, q = 4.5,  dx = dq = 0.5

c. x = 4.5, q = 50,  dx = dq = 0.5

d. x = 4.5, q = 4,5,  dx = 5, dq = 0.5

e. x = 4.5, q = 4,5,  dx = 0.5, dq = 5

Note that y-axis in the below plots tell us what fraction of new cases fall under the specific regime (P(V|I) and P(I|V)) out of all new cases 

Two cases here are quite interesting, case (b) and (e). The former is the case in Israel while the latter potentially may occur in India or is already happening. So, now you know if and when Indian media starts reporting a rise in cases amongst vaccinated individuals, you should ask yourself if their math is really correct!

Click here for the python code