ON SELF DIAGNOSING YOUR EXPECTED TIME TO A FIRST COVID INFECTION by Tom Leonard

                            

Omicron Alert: For a randomly chosen person in Scotland the chance of becoming infected with some sort of Covid on any particular day is currently (10th December 2021) about 1 in 1000 and rising. Please try to assess your own risk. The following procedure may help:

                                             A SIMPLE SELF DIAGNOSTIC TOOL

CONSIDER THE IDEALIZED ASSUMPTIONS THAT ON ANY PARTICULAR DAY  , YOU 

WILL, WITH PROBABILITY  P=1/2000, BE INFECTED WITH COVID, WHERE INFECTIONS 

ON DIFFERENT DAYS ARE TAKEN TO OCCUR INDEPENDENTLY.

       LET   N=NUMBER OF DAYS UNTIL YOUR FIRST INFECTION

      THEN N POSSESSES A PROBABILITY DISTRIBUTION WHICH IS

    GEOMETRIC WITH PROBABLITY  P=1/2000

      IN PARTICULAR, THE MEAN OR EXPECTATION OF N IS

                                 E(N)= 2000

     

 AND THE STANDARD DEVIATION OF N IS, TO A CLOSE APPROXIMATION

                                

                                ST.DEV. (N)= 2000

IF, FOR EXAMPLE, YOU ARE WORKING IN RETAIL AND YOUR CUSTOMERS ARE NOT

COMPELLED TO WHERE MASKS, THEN THE DAILY PROBABILITY OF INFECTION P

 COULD BE MUCH HIGHER THAN IF YOU ARE WORKING FROM HOME

              

         IF FOR EXAMPLE, P=1/100, THEN YOUR EXPECTED NUMBER OF DAYS TO  FIRST

  INFECTION   IS E(N)=100 WITH APPROXIMATE STANDARD DEV IATION 100.

        IF YOU HAVE BEEN DOUBLY VACCINATED THEN YOU COULD CONSIDER DIVIDING 

YOUR SUBJECTIVE ASSESSMENT OF  P  BY  9.  IF P=1/900. THEN E(N)=900 WITH 

APPROXIMATE STANDARD DEVIATION 900.

        THIS PROVIDES A SIMPLE WAY OF THINKING ABOUT YOUR CHANCES OF INFECTION.

       A BASELINE POPULATION VALUE OF P  IS

       P*=  NUMBER OF DAILY INFECTIONS / POPULATION SIZE

       FOR EXAMPLE.,IF THE NUMERATOR AND DENOMINATOR ARE RESPECTIVELY

      3000 AND 6000000,  THEN P*=1/2000

       WHEN SUBJECTIVELY ASSESSING YOUR OWN P YOU COULD C ONSIDER HOW 

MUCH AT RISK YOU ARE RELATIVE TO A RANDOMLY SELECTED MEMBER OF THE 

POPULATION WITH P=P*.

PROBABILITIES FOR GEOMETRIC DISTIBUTION

       NOTE THAT     

      F(n)  = Prob (N>n)  can be calculated as the nth power of 1-P, for your subjective choice of P

        You can therefore easily calculate your probability of not being infected with COVID during the next n days, for any specified value of n 

  (Approximate) Quartiles:  The first or lower (approximate) quartile Q1 of N is the value of n such that F(n)=3/4.                                  

                         Therefore

                                      Q1= log (3/4) / log (1-P).

where log denotes the natural logarithm

                         Similarly, the second (approximate) quartile, or median of N is

                                     Q2= log (1/2)/log (1-P)

  

                          and the third or upper (approximate) quartile of  N is

  

                                     Q3= log (1/4)/log (1-P)