Background
Methods
Data
SIR model with vaccination
Notation | Definition |
---|---|
\(\beta\) | Transmission rate |
\(\gamma\) | Rate of recovery (days\(^{-1}\)) |
\(e_{ij}\) | Percentage effectiveness against infection of i doses of vaccine j |
\(V_{ij}\) | Daily number of individuals who have received dose i of vaccine j |
N | Number of individuals in the population |
\(\mu _i\) | Factor reducing infectiveness due to i doses |
Model fitting
(a) Prior distributions used for initialising model fits | |||||||||
---|---|---|---|---|---|---|---|---|---|
Name | \(\bf {\gamma}\) | Prior distributions | Initial values | ||||||
\(\bf {\beta}\) | \(\bf {I,I_{ij}}\) | \(\bf {e_{ij},\mu _{i}}\) | \(\bf {S,S_{ij},R,R_{ij}}\) | ||||||
Core Model | 1/7 | \(\text {Unif}[0,10]\) | \(\text {Unif}[0,100]\) | \(\text {Unif}[0,1]\) | Table 2b | ||||
Sensitivity to \(\gamma\) | 1/3 | \(\text {Unif}[0,10]\) | \(\text {Unif}[0,100]\) | \(\text {Unif}[0,1]\) | Table 2b | ||||
Sensitivity to \(\gamma\) | 1/11 | \(\text {Unif}[0,10]\) | \(\text {Unif}[0,100]\) | \(\text {Unif}[0,1]\) | Table 2b | ||||
Double Removed | 1/7 | \(\text {Unif}[0,10]\) | \(\text {Unif}[0,100]\) | \(\text {Unif}[0,1]\) | Table 2c |
(b) Initial values when assuming all infected from start of pandemic moved to removed class | |||||||||
---|---|---|---|---|---|---|---|---|---|
S | \(\bf {S_{1A}}\) | \(\bf {S_{2A}}\) | \(\bf {S_{1P}}\) | \(\bf {S_{2P}}\) | R | \(\bf {R_{1A}}\) | \(\bf {R_{2A}}\) | \(\bf {R_{1P}}\) | \(\bf {R_{2P}}\) |
1266979 | 478347 | 312594 | 74665 | 384481 | 70887 | 42715 | 20895 | 6738 | 26715 |
(c) Initial values when assuming infected individuals are underestimated by 50% | |||||||||
---|---|---|---|---|---|---|---|---|---|
S | \(\bf {S_{1A}}\) | \(\bf {S_{2A}}\) | \(\bf {S_{1P}}\) | \(\bf {S_{2P}}\) | R | \(\bf {R_{1A}}\) | \(\bf {R_{2A}}\) | \(\bf {R_{1P}}\) | \(\bf {R_{2P}}\) |
1196092 | 435632 | 291699 | 67927 | 357766 | 141774 | 85430 | 41790 | 13476 | 53430 |
Results
Parameter | Median (50%) | Lower (2.5%) | Upper (97.5%) |
---|---|---|---|
\(\beta\) | 0.4060 | 0.3893 | 0.4425 |
\(e_{1A}\) | 0.3851 | 0.3425 | 0.4260 |
\(e_{2A}\) | 0.6402 | 0.6140 | 0.6647 |
\(e_{1P}\) | 0.1954 | 0.1041 | 0.2809 |
\(e_{2P}\) | 0.8392 | 0.8212 | 0.8559 |
Initial I | 54.0151 | 48.4328 | 59.5435 |
Initial \(I_{1A}\) | 14.8932 | 10.3012 | 19.8382 |
Initial \(I_{2A}\) | 2.4775 | 0.9300 | 4.7359 |
Initial \(I_{1P}\) | 0.7469 | 0.1234 | 2.0681 |
Initial \(I_{2P}\) | 2.9717 | 1.3856 | 5.1267 |
References | Infection | Delay | Dose 1 Effectiveness | Dose 2 Effectiveness | |||
---|---|---|---|---|---|---|---|
Dose 1 | Dose 2 | Oxford-AstraZeneca | Pfizer-BioNTech | Oxford-AstraZeneca | Pfizer-BioNTech | ||
\(\begin{array}{c}\displaystyle \text {Core Model}\\ \displaystyle (\gamma =1/7)\end{array}\)
| Sym. & Asym. | 21 | 14 |
\(\begin{array}{c}\displaystyle 38.5\%\\ \displaystyle [34.3,42.6]\end{array}\)
|
\(\begin{array}{c}\displaystyle 19.5\%\\ \displaystyle [10.4,28.1]\end{array}\)
|
\(\begin{array}{c}\displaystyle 64.0\%\\ \displaystyle [61.4,66.5]\end{array}\)
|
\(\begin{array}{c}\displaystyle 83.9\%\\ \displaystyle [82.1,85.6]\end{array}\)
|
Bernal et al. [5] | Sym. | 21 | 14 |
\(\begin{array}{c}\displaystyle 30.0\%\\ \displaystyle (24.3,35.3)\end{array}\)
|
\(\begin{array}{c}\displaystyle 35.6\%\\ \displaystyle (22.7,46.4)\end{array}\)
|
\(\begin{array}{c}\displaystyle 67.0\%\\ \displaystyle (61.3,71.8)\end{array}\)
|
\(\begin{array}{c}\displaystyle 88.0\%\\ \displaystyle (85.3,90.1)\end{array}\)
|
Pouwels et al. [6] | Sym. & Asym. | 21 | 14 |
\(\begin{array}{c}\displaystyle 46\%\\ \displaystyle (35,55)\end{array}\)
|
\(\begin{array}{c}\displaystyle 57\%\\ \displaystyle (50,63)\end{array}\)
|
\(\begin{array}{c}\displaystyle 67\%\\ \displaystyle (62,71)\end{array}\)
|
\(\begin{array}{c}\displaystyle 80\%\\ \displaystyle (77,83)\end{array}\)
|
Sheikh et al. [7] | Sym. & Asym. | 28 | 14 |
\(\begin{array}{c}\displaystyle 33\%\\ \displaystyle (23,41)\end{array}\)
|
\(\begin{array}{c}\displaystyle 33\%\\ \displaystyle (15,47)\end{array}\)
|
\(\begin{array}{c}\displaystyle 61\%\\ \displaystyle (51,70)\end{array}\)
|
\(\begin{array}{c}\displaystyle 83\%\\ \displaystyle (78,87)\end{array}\)
|
\(\begin{array}{c}\displaystyle \text {Sensitivity to } \gamma \\ \displaystyle (\gamma =1/11)\end{array}\)
| Sym. & Asym. | 21 | 14 |
\(\begin{array}{c}\displaystyle 40.8\%\\ \displaystyle [36.2,45.1]\end{array}\)
|
\(\begin{array}{c}\displaystyle 14.4\%\\ \displaystyle [4.52,23.5]\end{array}\)
|
\(\begin{array}{c}\displaystyle 62.5\%\\ \displaystyle [59.6,65.1]\end{array}\)
|
\(\begin{array}{c}\displaystyle 83.7\%\\ \displaystyle [81.8,85.5]\end{array}\)
|
\(\begin{array}{c}\displaystyle \text {Sensitivity to } \gamma \\ \displaystyle (\gamma =1/3)\end{array}\)
| Sym. & Asym. | 21 | 14 |
\(\begin{array}{c}\displaystyle 35.3\%\\ \displaystyle [31.2,39.2]\end{array}\)
|
\(\begin{array}{c}\displaystyle 26.3\%\\ \displaystyle [18.1,34]\end{array}\)
|
\(\begin{array}{c}\displaystyle 66.2\%\\ \displaystyle [63.8,68.4]\end{array}\)
|
\(\begin{array}{c}\displaystyle 84.2\%\\ \displaystyle [82.5,85.8]\end{array}\)
|
\(\begin{array}{c}\displaystyle \text {Double Removed}\\ \displaystyle (\gamma =1/7)\end{array}\)
| Sym. & Asym. | 21 | 14 |
\(\begin{array}{c}\displaystyle 36.4\%\\ \displaystyle [31.9,40.6]\end{array}\)
|
\(\begin{array}{c}\displaystyle 18.3\%\\ \displaystyle [9.14,26.9]\end{array}\)
|
\(\begin{array}{c}\displaystyle 63.3\%\\ \displaystyle [60.7,65.8]\end{array}\)
|
\(\begin{array}{c}\displaystyle 83.7\%\\ \displaystyle [81.8,85.3]\end{array}\)
|
Dose 1 Efficacy | Dose 2 Efficacy | ||||||
Polack et al. [30] | Sym. | 12 | 7 | – |
\(\begin{array}{c}\displaystyle 52\%\\ \displaystyle [29.5,68.4]\end{array}\)
| – |
\(\begin{array}{c}\displaystyle 95\%\\ \displaystyle [90.3,97.6]\end{array}\)
|
Voysey et al. [31] | Sym. | 21 | 14 |
\(\begin{array}{c}\displaystyle 64.1\%\\ \displaystyle [50.5,73.9]\end{array}\)
| – |
\(\begin{array}{c}\displaystyle 70.4\%\\ \displaystyle [54.8,80.6] \\ \text {95.8\% credible interval}\end{array}\)
| – |