## 4 Data Analysis
Both M1 and M2 estimate a similar positive slope of viral load before ART initiation, specifically 0.43 and 0.42, respectively. Additionally, the models estimate similar percentages for the proportion of individuals who started ART treatment at time zero (time of diagnosis), with M1 estimating 31% and M2 estimating 32%. We also assess the performance of our model by comparing the observed viral load values with the predicted values for each individual in the HCCD. Individual random effects, including the change point, are predicted using the conditional mean, which is obtained by averaging the parameters from extra iterations of the imputation step after convergence. To further contextualize our model-based estimates, we also implemented the approach denoted as \(log1 plus^{*}\), adapted from the rule-based algorithm developed by [1], which uses viral load surveillance data to infer ART initiation timing. Specifically, \(log1 plus^{*}\) detects ART initiation when there is a decline of more than one \(\log_{10}\) unit in viral load between two consecutive measurements occurring within a defined time window (e.g., three months). Additionally, ART initiation is inferred when a subject transitions from being detectable to being undetectable (i.e., left-censored measurement). While the original method was primarily used to identify ART initiation in a subset of individuals affected by the treatment, \(log1 plus^{*}\) generalizes this logic to the entire sample. Furthermore, instead of \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Method** & **Parameter** & \(\mathbf{\alpha}\) & \(\mathbf{\beta_{1}}\) & \(\mathbf{\beta_{2}}\) & \(\mathbf{\beta_{3}}\) & \(\mathbf{\beta_{4}}\) & \(\mathbf{\pi}\) & \(\mathbf{\lambda}\) \\ \hline M1 & est & 0.43 & 10.51 & 3.92 & 1.69 & & 0.31 & 1.11 \\  & se & 0.02 & 0.04 & 0.02 & 0.02 & & 0.02 & 0.04 \\ M2 & est & 0.42 & 10.52 & 3.89 & 2.72 & \(-\)0.10 & 0.32 & 1.09 \\  & se & 0.02 & 0.03 & 0.02 & 0.02 & 0.07 & 0.02 & 0.04 \\ \hline \hline \end{tabular} Estimates of the variance components: M1: \(\sigma_{A}^{2}=0.49\), \((\sigma_{\rm 811}^{2},\sigma_{\rm 822}^{2},\sigma_{\rm 833}^{2})=(4.61,0 64,1.92)\), \(\sigma_{e}^{2}=0.10\), M2: \(\sigma_{A}^{2}=0.34\), \((\sigma_{\rm 811}^{2},\sigma_{\rm 822}^{2},\sigma_{\rm 833}^{2},\sigma_{\rm 844}^{2})=(4 2.38,0.80,0.70,0.50)\), \(\sigma_{e}^{2}=0.11\),  
\end{table} 
Table 1: Estimates in fitting HCCD data to the StEM algorithm with one-compartment (M1) and two-compartment (M2) specification for the post-ART trajectory. Figure 1: Trace plots for all parameters in M1. imputing ART initiation at the midpoint between the two relevant viral load measurements, \(log1 plus^{*}\) assigns the initiation time to the earlier measurement in the pair. This adjustment is biologically motivated, reflecting the expectation that viral load suppression begins shortly after ART initiation. Figure 3 showcases the findings for nine selected individuals, chosen to represent typical patterns. For each individual, we present the predicted trajectory from M1 and M2, in addition to the observed viral loads. The predicted change point and the corresponding viral load at the time of ART treatment are indicated on the plot with different symbols for the two models. Predicted ART initiation by the \(log1 plus^{*}\) is also highlighted in the plot. It is important to note that the trajectories produced by the two models exhibit slightly different trends, with some trajectories displaying censoring (IDs 4 to 9) and others without censoring (IDs 1, 2, and 3). Our model fits the fully observed data quite well and predicts a reasonable ART initiation time. For example, the change point is estimated right before the decline of the viral load for ID 1, while for ID 3, the change point is estimated beyond the last observed viral load, where the observed viral loads are ever-increasing, indicating no ART initiation. For ID 2, the ART initiation time occurs at about the time when the viral load starts to decline.