Template

Oral cholera vaccine

Assumption - Vaccination is implemented in a all-or-nothing fashion - Two-dose efficacy is \(VE\) - The first-dose efficacy is \(ve_1\)

What the efficacy of the second vaccine dose, \(ve_2\), has to be if the want the same number of people is averted,

set.seed(42) # to make it reproducible
N = 1000
VE = 0.8 # vaccine efficacy under two dose regimen
vacc_cov = 0.6 # vaccine coverage
ve1 = 0.4 # efficacy of the first dose

vacc_protected_VE = VE*vacc_cov*N
vacc_protected_ve1 = ve1*vacc_cov*N
vacc_protected_ve_ve2 = vacc_protected_ve1 + (VE-ve1)*vacc_cov*N

Implementing the first and the seocond dose separately

Suppose that a population size is \(N\) and the vaccine coverage, \(0<=v<=1\). We introduce a parameter, \(\pi\), to represent the proportion of the first-dose recipients who receive the second dose. If \(\pi=1\), the number of vaccine recipients is \(Nv\) and the number of two-dose recipients is also \(Nv\) and the number of vaccinees who only received a single dose is zero. More generally, however, if \(\pi < 1\), the number of complete two-dose recipients is \(Nv\pi\) and those who have only received one dose is \(2Nv(1-\pi)\).

set.seed(42) # to make it reproducible
N = 1000
ve2 = 0.8 # vaccine efficacy under two dose regimen
ve1 = 0.4 # efficacy of the first dose
vacc_cov = 0.6 # vaccine coverage
pi = 0.95 # proportion of the first-dose recipients who again received the second dose
vacc_protected_ve2 = ve2*vacc_cov*N
vacc_protected_ve1 = ve1*vacc_cov*N
vacc_protected_ve1_ve2 = 
  pi*(vacc_protected_ve1 + (ve2-ve1)*vacc_cov*N) + 2*(1-pi)*ve1*vacc_cov*N

vacc_protected_ve1

[1] 240

vacc_protected_ve1_ve2

[1] 480

These are estimates and algebraic relationship wouldn’t hold. $$ \[\begin{align} c_2 &= v_1 \pi \\ c_{1+} &= v_1 \pi + v_1(1-\pi) + v_2(1-\pi v_1/v_2) \end{align}\] $$ \(c_2\) and \(c_{1+}\) represent coverage for complete two-dose regiment and at least one dose, respectively. \(\pi\) represents the proportion of the vaccinees who received the first dose and went on to receive the second dose.

Estimates from Pezzolli et al. (2020)

cov_first = 90.3 # v_1
cov_second = 88.2# v_2
cov_two <- 69.9 #c_2
cov_one_plus <- 84.6 #c_1+

Boundary conditions for the pi for the vr1plus must not be bigger than 1

# create a function that calculates 
ocv_round_cov_calc = function(vc1=0.95, vc2=0.95, vr1plus=NULL, vr2=NULL, pi=0.7){
  
  if (pi < (vc1+vc2-1)/vc1) {
    stop(paste0("pi must be larger than (vc1+vc2-1)/vc1, ", (vc1+vc2-1)/vc1))
  } else if (pi > (vc2/vc1)) {
    stop(paste0("pi must be smaller than vc2/vc1, ", vc2/vc1))
  }
  vr2 = vc1*pi  
  # vr1plus = vc1*pi + vc1*(1-pi) + vc2*(1-pi*vc1/vc2)
  vr1plus = vr2 + vc1*(1-pi) + vc2*(1-pi*vc1/vc2) 
  vr1 = vr1plus - vr2
  # TODO there are several limiting conditions
  # vaccine coverage is 
  return (list(vr1plus=vr1plus, vr2=vr2, vr1=vr1))
}
ocv_round_cov_calc(vc1=0.56, vc2=0.46, vr1plus=NULL, vr2=NULL, pi=0.80)

$vr1plus
[1] 0.572

$vr2
[1] 0.448

$vr1
[1] 0.124

When the first dose is distributed, apply the When the second dose is distributed, identify the number of two-dose and one-dose recipients and compare that number with the previous round

Summarize results