## Proportion Needed to Vaccinate (PNV, NNV, to end an epidemic)

Updated 23 October 2021

This article examines the proportion needed to vaccinate to bring an epidemic (pandemic) to an end. It discusses the proportion of the population rather than the percentage or the number. If you don't like math, then skip ahead to the conclusion.

The usual formula to calculate the proportion needed to vaccinate assumes that prior disease and the vaccination are both 100% effective in preventing infection and therefore spread. It relies on R₀ (the native reproduction number without restrictions) and not effR (effective R) and not Rs, the effective R amongst the susceptible.

Subscripts are difficult to use on this website so upper case letters are variables, and lower case letters are subscripts.

The formula for the number of new cases (Nt+n+1) that can be expected on day t+n+1 is:

x = t+n

Nt+n+1 =

x = t

For the epidemic to end the sum at the right of Equation A must be multiplied by 1 (or less, but the threshold is 1)

where R₀ is the native reproduction number of the virus

N is the number of cases on a day.

Nt+n+1 is the number of (new) cases that is to be determined from the (infectious) cases over the previous n days. The initial day is day t and the last infecting day is day t+n.

The range and distribution of incubation periods for the infection are known (or estimated). Ax is the proportion of the new cases that must have been infected on day x. Day x ranges from day t to day t+n.

P is the number of the total population

E is the efficacy of the vaccine to prevent infection

V is the number fully vaccinated

F is the efficacy of past infection to prevent reinfection, and

C t+n is the cumulative number of cases of infection so far on day t+n, the day before day t+n+1.

If, R₀ * (P – EV -FC t+n)/P = 1 (the threshold for ending the epidemic)

then V = (P - P/R₀ - FCt+n)/E

And, V/P = (1 - 1/R₀ - (FCt+n)/P)/E

And since V/P is the proportion needed to vaccinate,

If the R₀ of the Delta Covid19 is assumed to be 5.5, and

the efficacy (F) of prior native Covid19 to prevent reinfection by Delta Covid19 is about 0.74

Vaccine efficacy (E) of Pfizer is: 0.93 against alpha strain, and 0.88 against delta strain.

N Engl J Med 2021; 385:585-594. August 12, 2021.

then, Pfizer PNV for Delta in Australia is (1 - 1/5.5 - 0.74*144000/25,800,000) / 0.88

Which is

The Vaccine efficacy (E) of AstraZenica (AZ) is: 0.75 alpha, 0.67 delta. (Same source as above).

So, AZ PNV for Delta in Australia is (1 - 1/5.5 - 0.5*144000/25,800,000) / 0.67

Which is

Currently Australia has 57% of the >16 population vaccinated with an mRNA vaccine (Pfizer, but Moderna could be expected to have the efficacy) and 43% vaccinated with AZ. The combined PNV is still over 1 (impossible) and a 100% vaccination rate could not end the epidemic with out further vaccination (perhaps boosters) or some form of restrictions.

Usually the part played by prior Covid19 infection is so small it can be ignored for practical purposes where NPIs have already been successful in reducing case numbers.

A very significant point is that

This article examines the proportion needed to vaccinate to bring an epidemic (pandemic) to an end. It discusses the proportion of the population rather than the percentage or the number. If you don't like math, then skip ahead to the conclusion.

The usual formula to calculate the proportion needed to vaccinate assumes that prior disease and the vaccination are both 100% effective in preventing infection and therefore spread. It relies on R₀ (the native reproduction number without restrictions) and not effR (effective R) and not Rs, the effective R amongst the susceptible.

Subscripts are difficult to use on this website so upper case letters are variables, and lower case letters are subscripts.

The formula for the number of new cases (Nt+n+1) that can be expected on day t+n+1 is:

x = t+n

Nt+n+1 =

**R₀ * (P – EV -FC t+n)/P*** Σ (Ax Nx) from**(Equation A of this**__epidemic model__)x = t

For the epidemic to end the sum at the right of Equation A must be multiplied by 1 (or less, but the threshold is 1)

**So,****R₀ * (P – EV - (FC t+n))/P must be 1 (or less)**.where R₀ is the native reproduction number of the virus

N is the number of cases on a day.

Nt+n+1 is the number of (new) cases that is to be determined from the (infectious) cases over the previous n days. The initial day is day t and the last infecting day is day t+n.

The range and distribution of incubation periods for the infection are known (or estimated). Ax is the proportion of the new cases that must have been infected on day x. Day x ranges from day t to day t+n.

P is the number of the total population

E is the efficacy of the vaccine to prevent infection

V is the number fully vaccinated

F is the efficacy of past infection to prevent reinfection, and

C t+n is the cumulative number of cases of infection so far on day t+n, the day before day t+n+1.

If, R₀ * (P – EV -FC t+n)/P = 1 (the threshold for ending the epidemic)

then V = (P - P/R₀ - FCt+n)/E

And, V/P = (1 - 1/R₀ - (FCt+n)/P)/E

And since V/P is the proportion needed to vaccinate,

**The proportion needed to vaccinate, PNV, is (1 - 1/R₀ - (FCt+n)/P)/E****CONCLUSION**If the R₀ of the Delta Covid19 is assumed to be 5.5, and

the efficacy (F) of prior native Covid19 to prevent reinfection by Delta Covid19 is about 0.74

Vaccine efficacy (E) of Pfizer is: 0.93 against alpha strain, and 0.88 against delta strain.

N Engl J Med 2021; 385:585-594. August 12, 2021.

__https://www.nejm.org/doi/full/10.1056/NEJMoa2108891__then, Pfizer PNV for Delta in Australia is (1 - 1/5.5 - 0.74*144000/25,800,000) / 0.88

Which is

**0.92**(or 92 percent of the whole population of Australia, including children) would need to be vaccinated with Pfizer to end this wave (the epidemic).The Vaccine efficacy (E) of AstraZenica (AZ) is: 0.75 alpha, 0.67 delta. (Same source as above).

So, AZ PNV for Delta in Australia is (1 - 1/5.5 - 0.5*144000/25,800,000) / 0.67

Which is

**over 1**(or more than 100 percent (impossible) of the whole population of Australia (including children), So 100% full vaccination with AstraZenica of the whole population of Australia, including children,**would not**end this wave (**not**end the epidemic).Currently Australia has 57% of the >16 population vaccinated with an mRNA vaccine (Pfizer, but Moderna could be expected to have the efficacy) and 43% vaccinated with AZ. The combined PNV is still over 1 (impossible) and a 100% vaccination rate could not end the epidemic with out further vaccination (perhaps boosters) or some form of restrictions.

Usually the part played by prior Covid19 infection is so small it can be ignored for practical purposes where NPIs have already been successful in reducing case numbers.

A very significant point is that

**both vaccines are safe by reasonable standards**and pose less risk that Covid19 disease itself and both vaccines markedly reduce hospitalisation and death. The inability of vaccination to finally halt the**spread**of Covid19 in Australia is not as relevant as**the ability of both vaccines to prevent severe disease and death.**A third dose of vaccine or some form of NPIs could halt the epidemic.