## 2 ZIE-Based Nonlinear Random Change Point Model  
In longitudinal data analysis, understanding how a specific outcome changes over time is crucial. Random effects change point models offer a valuable framework for examining changes in time trajectories by incorporating individual change points. These changes are typically induced by external events, causing deviations from the original data pattern. Our objective is to determine when individuals initiate HIV treatment using their HIV viral load measurements over time. Without ART, the viral load fluctuates significantly after HIV infection until it stabilizes at a set point. If untreated, the viral load eventually increases, leading to AIDS [11]. Starting ART, however, results in substantial decreases in HIV viral load. To simplify our modeling approach, we assume the HIV diagnosis occurs after the viral set point and focus on modeling the changes in viral load dynamics post-ART initiation. Traditional random effects change point models often assume that the longitudinal outcome can be described by a segmented linear mixed-effects model. However, as mentioned in the introduction, linear assumptions may not be suitable for many real-world applications. While linear models might adequately fit the observed data, they may not be appropriate for data subject to censoring, which is common with HIV viral load measurements, especially after ART initiation. Extensive research has been conducted to understand the dynamics of viral load following ART initiation and to evaluate the effectiveness of these drugs in treating HIV. Building upon biological and clinical knowledge, ref. [12] proposed a virological model to approximate the patterns observed in viral load data. This model is represented by the equation:  
\[V(t)=P_{1}e^{-\gamma t}+P_{2},\]  
where \(V(t)\) represents the total virus at time \(t\), and \(P_{1}\) and \(P_{2}\) are baseline values. The parameter \(\gamma\) corresponds to the viral decay rate and can be interpreted as the turnover rate of productively infected cells and long-lived or latently infected cells in an ideal therapy setting. Refs. [13; 14] provide detailed discussions of this model. For our problem, we can consider the following random change point model to describe the viral loads \(y_{ij}\) for individual \(i\) at \(j^{th}\) visit with time \(t_{ij}\) after HIV diagnosis:  
\[y_{ij}=a_{1i}(t_{ij}-\tau_{i})^{-}+\log_{10}(b_{1i}e^{-b_{2i}(t_{ij}-\tau_{i}) ^{+}}+b_{3i})+e_{ij}. \tag{1}\]  
Here, \(y_{ij}\) is a log 10-transformed viral load, \(\tau_{i}\) is the (single) change point which induces the change of viral load trajectory, and \(e_{ij}\) is the error term. We use \(\log_{10}\) transformation in line with standard practice in HIV clinical research and surveillance, where viral load thresholds and treatment response metrics (e.g., a 1-log reduction) are conventionally interpreted on the \(\log_{10}\) scale. While natural logarithms are common in general modeling, the base-10 scale facilitates direct clinical relevance and comparability with prior studies. The functions \(x^{-}\) and \(x^{+}\) correspond to \(\min(x,0)\) and \(\max(x,0)\), respectively. The quantity \(a_{1i}\) represents the subject-specific regression coefficient that captures the viral load slope before the change point, while \(b_{1i}\), \(b_{2i}\), and \(b_{3i}\) represent subject-specific mixed effects governing the viral trajectory after the change point. We define these as  
\[a_{1i}=\alpha_{1}+\alpha_{1i},\quad b_{1i}=\beta_{1}+\beta_{1i},\quad b_{2i}= \beta_{2}+\beta_{2i},\quad b_{3i}=\beta_{3}+\beta_{3i}.\]  
Here, \(\alpha_{1}\), \(\beta_{1}\), \(\beta_{2}\), and \(\beta_{3}\) represent the population parameters (fixed effects), while \(\alpha_{1i}\), \(\beta_{1i}\), \(\beta_{2i}\), and \(\beta_{3i}\) denote the random effects, typically assumed to follow normal distribution with a mean of zero. It is worth noting that the pre-change point segment and the post-change point segment meet at the intercept \(\log_{10}(b_{1i}+b_{3i})\) when \(t_{ij}=\tau_{i}\).