Table 4 Summary of notation
Notation used in viral load and sensitivity models | |
 VL(t) | Viral load of an infected subject at time t post-exposure |
 \(t_w\) | The time at which the window period ends |
 \(t_p\) | The time at which the viral load peaks |
 \(t_s\) | The time at which the viral load reaches steady state |
 \(\lambda \) | Doubling time of the viral load during the window period |
 \(\tau \) | The life-time of the infection |
 \(C_0, C_w, a,b\) | Infection-specific calibration parameters |
 \(T^+(n)\) | The event that the test outcome is positive for pool size n, \(n \in {\mathbb {Z}}^+\) |
 \(N_I(n)\) | Number of infected specimens in a pool of size n, \(n \in {\mathbb {Z}}^+\) |
 Spec | Specificity of a test (constant for any pool size) |
 Sens(n) | Sensitivity of a pooled test, with pool size n, \(n \in {\mathbb {Z}}^+\) |
 Sens(n; i) | Conditional sensitivity of a pooled test, with pool size n, given that the pool |
| Â | Contains i infected specimens, \(i \in \{0,1,\ldots , n\}\), \(n \in {\mathbb {Z}}^+\) |
 \(\Phi (.)\) | The cumulative distribution function (CDF) of the standard normal distribution |
 z | A constant such that \(\Phi (z)=0.95\), i.e., \(z=1.6449\) |
 \(\chi \) | The number of nucleic acid copies per viral particle |
 \(x_{50},x_{95}\) | Viral load measurement at which the probability of testing positive is 50% and 95%, respectively |
 \({\widetilde{Sens}}(n;i)\) | Approximate conditional sensitivity of a pooled test, with pool size n, given that the pool contains i infected specimens, \(i \in \{0,1,\ldots , n\}\), \(n \in {\mathbb {Z}}^+\) |
 \(\beta \), \(\alpha \), \(\gamma \) | Calibration parameters for the approximation model |
 MSE | Mean squared error |
Notation used in the case study (prevalence estimation) | |
 s | Number of testing pools |
 n | Pool size |
 \(p_0\) | An initial estimate of p |
 \(c_f\) | Fixed testing cost per pool |
 \(c_v\) | Collection cost per specimen |
 B | Total testing budget |
 \(\overline{N}\) | The maximum pool size that can be used |
 \({\hat{p}}\) | The maximum likelihood estimator (MLE) of p |
 \(\sigma ^2(n,s;p)\) | The asymptotic variance of the MLE for a pool design (n, s), |
| Â | Given a prevalence rate of p |
 \(S_I(s)\) | Number of positive-testing pools among s pools |
 rBias | Relative bias of the MLE with respect to p |