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

Updated 23 August 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).

The formula for the number of 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

x = t

The meaning of the variables in given at the end.

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

where R₀ is the native reproduction number of the virus

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 if V is brought to the left, V = (P - P/R₀ - FCt+n)/E

So, 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 5.0, then

Efficacy of prior native Covid19 to prevent reinfection by Delta Covid19 (F) is about 0.5

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.

So, Pfizer PNV for Delta in Australia is (1 - 1/5 - 0.5*44026/25,800,000) / 0.88

Which is

Reinfection prevention efficacy of prior native infection (F) is 0.5

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

Reinfection prevention efficacy of a prior native infection (F) is 0.5

So, AZ PNV for Delta in Australia is (1 - 1/5 - 0.5*44026/25,800,000) / 0.67

Which is

The part played by prior Covid19 infection (0.5*44026/25,800,000) is so small it can be ignored for practical purposes.

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

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).

The formula for the number of 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)**x = t

The meaning of the variables in given at the end.

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

where R₀ is the native reproduction number of the virus

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 if V is brought to the left, V = (P - P/R₀ - FCt+n)/E

So, 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 5.0, then

Efficacy of prior native Covid19 to prevent reinfection by Delta Covid19 (F) is about 0.5

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__So, Pfizer PNV for Delta in Australia is (1 - 1/5 - 0.5*44026/25,800,000) / 0.88

Which is

**0.91**(or 91 percent of the whole population of Australia, including children) would need to be vaccinated with Pfizer to end this wave (the epidemic).Reinfection prevention efficacy of prior native infection (F) is 0.5

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

Reinfection prevention efficacy of a prior native infection (F) is 0.5

So, AZ PNV for Delta in Australia is (1 - 1/5 - 0.5*44026/25,800,000) / 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).The part played by prior Covid19 infection (0.5*44026/25,800,000) is so small it can be ignored for practical purposes.

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 relevant, whereas**the ability of both vaccines to prevent severe disease and death is the relevant fact.**