# Challenges in creating herd immunity to SARS-CoV-2 infection by mass vaccination

###### tags: `articles`
[DOI](https://doi.org/10.1016/S0140-6736(20)32318-7)
[Main text](https://www.thelancet.com/action/showPdf?pii=S0140-6736%2820%2932318-7)
[Supplementary appendix](https://www.thelancet.com/cms/10.1016/S0140-6736(20)32318-7/attachment/edac053b-b974-45c4-8b29-00c2cf954f07/mmc1.pdf)
# Main text
- For any vaccine, efficacy and duration of protection are key issues.
- Priority groups for vaccination typically start with front-line health-care staff, those working in essential services, those with health conditions that predispose to severe morbidity from infection, and then moving down the age groups from old to young in accordance with case fatality rates.
- [Report 33 - Modelling the allocation and impact of a COVID-19 vaccine](https://www.imperial.ac.uk/mrc-global-infectious-disease-analysis/covid-19/report-33-vaccine/)
- Priority of mass vaccination:
- To minimise net mortality per year.
- To maximise the average years of life gained.
- An additional complexity: <br>The extra burden imposed by COVID-19 on health provision for patients who need urgent treatment for other conditions, which have implications for net mortality.
- 

# Supplementary appendix
## Demographic calculations on years of life gained by vaccinated different age groups
- Two vaccination strategies
- The strategy $1$: vaccinating people aged $75+$.
- The expected life years saved by vaccinating is $250,597.6$.
- The proportion of deaths averted is $74.57 \%$.
- The required vaccine doses is $2 \times 4,915,318$.
- The strategy $2$: vaccinating only people aged $55-74$.
- The expected life years saved by vaccinating is $207,239.8$.
- The proportion of deaths averted is $21.75 \%$.
- The required vaccine doses is $2 \times 12,926,518$.
## Mathematical model of vaccine impact
$\left\{
\begin{array}{**lr**}
\frac{dS}{dt}= −𝛽SI + 𝛾_1𝑅 + 𝛾_2𝑉 − 𝜂S & \\
\frac{dI}{dt}=𝛽SI − 𝛾I & \\
\frac{dR}{dt}=𝛾I − 𝛾_1𝑅 − 𝜂𝑅 & \\
\frac{dV}{dt}=𝜂(S + 𝑅) − 𝛾_2V & \\
\end{array}
\right.$
- Parameters:
- $𝛽 > 0$, the rate of infectious contact between infected and susceptible individuals.
- $𝛾 > 0$, the rate of recovery from infection.
- $𝛾_1 > 0$, the rate of losing immunity from natural infection.
- $𝛾_2 > 0$, the rate of losing vaccine induced immunity.
- $𝜂 > 0$, the per capita rate at which people in the susceptible and recovered classes acquire vaccine-induced immunity.<br>We define $𝜂 = 𝛼𝜀$
- $𝛼 > 0$ is the rate at which susceptible and recovered individuals are being vaccinated
- $𝜀 ∈ (0,1]$ is the vaccine efficacy.
- In this model $𝑅_0 = 𝛽/𝛾$.
- A disease-free equilibrium
- $𝑉^∗ = 𝜂/(𝜂 + 𝛾_2)$.
- $S^∗ = 𝛾_2/(𝜂 + 𝛾_2)$.
- The system is guaranteed to reach the disease-free equilibrium if the rate of vaccination $𝛼 \gt 𝛼^∗$, where $𝛼^∗ = \frac{𝛾_2}{𝜀} (𝑅_0 − 1)$.
- Want: $𝑉^∗ = 1 − 1/𝑅_0$ $\Big( (1-v)R_0=1\Rightarrow v=1 − 1/𝑅_0\Big)$<br>And we know $𝑉^∗ = 𝜂/(𝜂 + 𝛾_2)=𝛼𝜀/(𝛼𝜀 + 𝛾_2)$<br>Solve: $1 − 1/𝑅_0=𝛼𝜀/(𝛼𝜀 + 𝛾_2)$<br>$\Rightarrow (R_0-1)(𝛼𝜀 + 𝛾_2)=R_0𝛼𝜀$$\Rightarrow (R_0-1) 𝛾_2=𝛼𝜀$$\Rightarrow 𝛼 = \frac{𝛾_2}{𝜀} (𝑅_0 − 1)$
- The number of vaccinations between any two times $𝑡_1$ and $𝑡_2$ is given by the integral $\int_{t_1}^{t_2} 𝛼^∗(S(𝑡) + 𝑅(𝑡))𝑑𝑡$
- At $𝑡 = 0$, if we assume that the proportion of currently infected people is small, then $S(𝑡) + 𝑅(𝑡) ≈ 1 − 𝑉(𝑡)$ for $𝑡 > 0$.
- $\frac{𝑑𝑉}{𝑑𝑡} = 𝜂(1 − 𝑉) − 𝛾_2V$<br>$\Rightarrow 𝑉(𝑡) =\frac{𝜂(1 − 𝑒^{−(𝜂+𝛾_2)𝑡})}{𝜂 + 𝛾_2}$
- In an initial period of duration $T$, the number of vaccinations as a proportion of the population $𝑝_𝑐(𝑇) = \int_0^T 𝛼^∗(1 − 𝑉(𝑡))𝑑𝑡$$= \frac{𝛼^∗ \Big ({𝛾_2}^2 + 𝛼^∗𝜀\big (1 + 𝛾_2 − 𝑒^{−(𝜂+𝛾_2)𝑡}\big )\Big)}{(𝛼^∗𝜀 + 𝛾_2)^2} \ \ \ \Big|_0^T$$=\frac{R_0-1}{𝜀{R_0}^2}\Big[𝛾_2TR_0+(R_0-1)(1-e^{-R_0𝛾_2T}) \Big]$
- At the disease-free equilibrium, the number of people targeted for vaccination is constant, so the critical proportion to vaccinate in one year is ${𝑝_𝑐}^* = ∫_{t_1}^{t_1+1} 𝛼^∗ \frac{1}{𝑅_0} 𝑑𝑡$ $= \frac{𝛾_2}{𝜀} (1 − 1/𝑅_0)$
- When vaccinating at the critical rate, the time taken to reach $𝑥\%$ of the herd immunity level of coverage required to block transmission, $𝑇_𝑥$, is $𝑇_𝑥 = \frac{−𝐷 \ ln \big[1 − \frac{𝑥}{100}\big ]}{𝑅_0}$