The mathematics of vaccination and hospitalization

 Vaccination and hospitalization :

a mathematical study

 

Suppose that with the Healthy (H), the probability of being vaccinated is 0.65, and the vaccine is 0.9 effective against hospitalization.

Suppose that with the Weak (W), i.e. those with serious health and immunological issues, the probability of being vaccinated is 0.95, and the vaccine is 0.77 effective against hospitalization.

Suppose that, without the vaccine, 1000 healthy people (H) and 4000 weak people (W) would have been hospitalized.

The table below shows the prob (columns 2,3) and numbers (columns 4,5) of H and W that are unvaccinated, effectively and ineffectively vaccinated, and hospitalized, given the vaccine.

The last row in the table is the prob and numbers of people being hospitalized, being the sum of those unvaccinated and those ineffectively vaccinated.

	Prob with the  Healthy (f)	Prob with the Weak (g)	1000 × f	4000 x g
Unvaccinated (~v)	0.35	0.05	350	200
Vaccine effective (v,e)	0.65 × 0.9 = 0.585	0.95 × 0.77 = 0.7315	585	2926
Vaccine ineffective (v,~e)	0.65 × 0.1 = 0.065	0.95 × 0.23 = 0.2185	65	874
Hospitalized  = (~v) + (v,~e)	0.415	0.2685	415	1074

The total number hospitalized (H)

= 415 + 1074

= 1489

The total number of vaccinated hospitalized (VH)

= 65 + 874

= 939

VH ÷ H = 939 ÷ 1489 = 63.06 %

63 % of the hospitalized were vaccinated.

To generalize, let

R  = (N, the number of weak people hospitalized without vaccine) 

÷ (M, the number of healthy people hospitalized without vaccine)

In the previous example,

R  = 4000 ÷ 1000 = 4

VH ÷ H =  (0.065 + 4 x 0.2185)

                   ÷ (0.415 + 4 x 0.2685)

             = 0.939 ÷ 1.489

             = 0.6306

In general, 

Z  = VH ÷ H =  (0.065 + R x 0.2185)

                   ÷ (0.415 + R x 0.2685)

 R can assume any non-negative real value, from 0 (when N = 0), to

∞ (when M = 0)

The value of Z monotonically increases from 0.157 (= 0.065/0.415) when R=0  to 0.814 (= 0.2185/0.2685) when R=∞.

Therefore, Z, the percentage of the vaccinated among the hospitalized can vary from 15.7 % to 81.4 %, depending on the ratio of N to M.

To further generalize,  0.157 (0.814 resp.) is the percentage of the vaccinated among the healthy (weak resp.) hospitalized.

Therefore, Z, the percentage of the vaccinated among the hospitalized can take on any value between

• the percentage of the vaccinated among the healthy hospitalized

and 

• the percentage of the vaccinated among the weak hospitalized, 

depending on the ratio of N to M.

Finally, we generalize Y, the percentage of the vaccinated among the healthy/weak hospitalized.

Let us consider the healthy hospitalized. (The case of the weak hospitalized is identical.)

Let v

= probability of a healthy person being vaccinated

= proportion of the vaccinated among the healthy

= vaccination rate among the healthy

Let k = prob of the vaccine being ineffective with a healthy person

Y = ( v x k ) ÷ ((1- v ) + v x k )

In the example above, 

v = 0.65, k = 0.1, 

Y = 0.065 ÷ (0.35 + 0.065)

[1] Y as a function of v

Given a fixed k , Y monotonically increases (as v increases) from 0 (when v =0) to 1 (when v =1).

Given any vaccine efficacy, the proportion of the vaccinated among the hospitalized can be any value between 0% and 100%, depending on the proportion of the vaccinated among the general population in question.

[2] Y as a function of k

Given a fixed v , Y monotonically increases (as k increases) from 0 (when k =0) to v (when k =1).

Given any vaccination rate v, the proportion of the vaccinated among the hospitalized can be any value between 0% and v, depending on the vaccine efficacy.

□♡♤