Non-homogeneous Markov Model Research Articles

Estimating multisite precipitation by a stepwise NHMM-VAR model considering the spatiotemporal correlations of precipitation amounts

• This study developed a stepwise NHMM-VAR multisite downscaling model. • The NHMM-VAR model performs well not only in modeling the statistical properties but also in reproducing the spatiotemporal correlations of multisite precipitation amounts. • The temporal intermittency of precipitation amounts is considered in the NHMM-VAR model. The nonhomogeneous hidden Markov model (NHMM) is a popular statistical downscaling (SD) approach that can be used for multisite precipitation estimation. However, the NHMM assumes that once the days with the occurrence of precipitation have been determined, the precipitation amounts at each station on wet days can be conditionally independently generated from a parametric family of probability distribution functions without considering impacts from the past precipitation amounts and the neighboring stations. Such assumptions may lead to underestimation of the spatiotemporal correlations (i.e., temporal autocorrelation and spatial cross-correlations) of actual multisite precipitation amounts. Thus, this study developed a stepwise downscaling method called the nonhomogeneous hidden Markov model - vector autoregressive (NHMM-VAR), in which the NHMM is used to determine the probabilities for dry days and marginal distributions of multisite daily mean precipitation amounts on wet days based on the large-scale atmospheric covariates, then a VAR(1) model is used to reproduce the spatiotemporal correlations of actual multisite daily mean precipitation amounts. A comparison experiment between the NHMM and the NHMM-VAR models is conducted against outputs from eight global climate models (GCMs) of Phase 6 of the Coupled Model Intercomparison Project (CMIP6) over the Pearl River basin (PRB) of South China. The results show that the NHMM-VAR model performs well not only in modeling the statistical properties but also in reproducing the spatiotemporal correlations of actual multisite daily mean precipitation amounts, which indicates that the VAR model does add value to the NHMM downscaling method.

Cost-Utility Analysis of Pharmacogenetic Testing Based on CYP2C19 or CYP2D6 in Major Depressive Disorder: Assessing the Drivers of Different Cost-Effectiveness Levels from an Italian Societal Perspective.

Background and ObjectivesMajor depressive disorder (MDD) is a common and severe psychiatric disorder that has enormous economical and societal costs. As pharmacogenetics is one of the key tools of precision psychiatry, we analyze the cost–utility of test screening of CYP2C19 and CYP2D6 for patients suffering from major depressive disorder (MDD) and try to understand the main drivers that influence the cost–utility.MethodsWe developed two pharmacoeconomic nonhomogeneous Markov models to test the cost–utility, from an Italian societal perspective, of pharmacogenetic testing genetic to characterize the metabolizing profiles of cytochrome P450 (CYP) 2C19 and CYP2D6 in a hypothetical case study of patients suffering from major depressive disorder (MDD). The model considers different scenarios of adjustment of antidepressant treatment according to the patient’s metabolizing profile or treatment over a period of 18 weeks. The uncertainty of model parameters is tested through both a probabilistic sensitivity analysis and a one-way deterministic sensitivity analysis, and these results are used in a post-hoc analysis to understand the main drivers of three alternative cost-effectiveness levels (“poor,” “standard,” and “high”). These drivers are first evaluated from an exploratory multidimensional perspective and next from a predictive perspective as the probability that a patient belongs to a specific cost-effectiveness level is estimated on the basis of a restricted set of parameters used in the original pharmacoeconomic model.ResultsThe models for CYP2C19 and CYP2D6 indicate that screening has an incremental cost-effectiveness ratio of 60,000€ and 47,000€ per quality-adjusted life year (QALY), respectively. The probabilistic sensitivity analysis shows that the treatments are cost-effective for a 75,000€ willingness to pay (WTP) threshold in 58% and 63% of the Monte Carlo replications, respectively. The post-hoc analysis highlights the factors that allow us to clearly discriminates poor cost-effectiveness from high cost-effectiveness scenarios and demonstrates that it is possible to predict with reasonable accuracy the cost-effectiveness of a genetic test and the associated therapeutic pattern.ConclusionsOur findings suggest that screenings for both CYP2C19 and CYP2D6 enzymes for patients with MDD are cost-effective for a WTP threshold of 75,000€ per QALY, and provide relevant suggestions about the most important aspects to be further explored in clinical studies aimed at addressing the cost-effectiveness of genetic testing for patients diagnosed with MDD.

Exploring Patterns of Train Delay Evolution and Timetable Robustness

Uncertainty affects the overall performance of railway systems. It can be captured tracking the successive departure and arrival events of trains at stations in the scheduled railways. Markov chains has shown to be an effective means to model variabilities in the departure and arrival delays of trains. The empirical data of actual operations are used to develop the Markov matrices. Transitions between delay states comprise the internal dynamics of running and dwelling processes in probabilistic terms, presenting particular patterns of deterioration, recovery and state keeping in the prevailing timetable setting. All these features conform to the practice of railway operations as well as train dispatchers and drivers. Hence, delay-based Markov chains becomes a simple and effective tool to predict delays in the following events of trains and distribution of delays over a longer period of time using simple performance measures. We have developed the homogeneous and non-homogeneous Markov models and tested their performances using the actual data collected in the Netherlands railway network. The analysis and evaluation show that the railway corridor considered in this study performs well and the corresponding Markov chain models can be used effectively to predict train delays and to evaluate timetable robustness. The outcomes of the homogeneous and non-homogeneous model are evaluated with respect to the observed data resulting that the non-homogeneous model performs better than the homogeneous model.

Peak-to-peak fuzzy filtering of nonlinear discrete-time systems with markov communication protocol

This study deals with the peak-to-peak fuzzy filtering problem for a class of nonlinear discrete-time systems with analog fading channels and communication protocol, in which the nonlinear system is modeled by the Takagi-Sugeno fuzzy model. In analog fading channels, a homogeneous Markov chain is forwarded to model the random time-varying amplitude attenuation. Aiming at alleviating the utilization of energy consumption and preventing data collision/congestion in constraint networks, a Markov communication protocol is exploited to orchestrate the data transmission, in which only one sensor can get permission to release the measurement during each time interval. In virtue of the merging strategy, a new joint Markov chain is presented to incorporate the fading channels and the communication protocol. Differently, in order to eliminate the obstacle of design conservatism, a novel peak-to-peak filter design methodology is developed, whose asynchronization is described by a nonhomogeneous hidden Markov model. Under the aforementioned framework, the resulting system is stochastically stable with a desired peak-to-peak performance index. To this end, a practical example is addressed to indicate the validity and applicability of the presented peak-to-peak filter design strategy.

Increase of Future Summer Rainfall in the Middle and Lower Reach of the Yangtze River Basin Projected With a Nonhomogeneous Hidden Markov Model

AbstractAmong the statistical downscaling tools available for regional climate simulation, the nonhomogeneous hidden Markov model (NHMM) is a powerful and efficient algorithm. It consists of establishing statistical relations between predictand and predictors through a hidden layer of Markov process and a covariate nonhomogeneous term as external forcing. Here, it is used to simulate summer daily precipitation in the middle and lower reach of the Yangtze River basin (MLYRB), including future projection. Results show that NHMM well identifies large‐scale circulation features that are physically consistent with regional rainfall patterns. MLYRB exhibits a general wetness for 1.5°C and 2°C global warming targets, with smaller (larger) increase for the west (east). Such changes correspond to increases in the occurrence frequency of synoptic weather patterns with stronger and more westward Western Pacific Subtropical High and stronger westerly jet. This contributes to moistening MLYRB with increasing occurrence frequency for those wetter rainfall patterns.

Steerable Music Generation which Satisfies Long-Range Dependency Constraints

Although music is full of repetitive motifs and themes, artificially intelligent temporal sequence models have yet to demonstrate the ability to model or generate musical compositions that satisfy steerable, long-range constraints needed to evoke such repetitions. Markovian approaches inherently assume a strictly limited range of memory while neural approaches—despite recent advances in evoking long-range dependencies—remain largely unsteerable. More recent models have been published that attempt to evoke repetitive motifs by imposing unary constraints at intervals or by collating copies of musical segments. Although the results of these methods satisfy long-range dependencies, they come with significant—potentially prohibitive—sacrifices in the musical coherence of the generated composition or in the breadth of satisfying compositions which the model can create. We present R<small>EGUALR</small> non-homogeneous Markov models as a solution to the long-range dependency problem which uses R<small>ELATIONAL</small> automata to enforce binary constraints to compose music with repeating motifs. The solution we present preserves musical coherence (i.e., Markovian constraints) for the duration of the generated compositions and significantly increases the range of satisfying compositions that can be generated.

Maintenance Decision Optimization of Multi-State System Considering Imperfect Maintenance

In the maintenance decision-making problem of a multi-state system, it is more in line with the actual situation to consider imperfect maintenance. However, under the influence of imperfect maintenance, the failure mechanism and maintenance process of the multi-state system are more complicated. Further discussion is needed about how to deal with the relationship between the states of various components and the overall state of the system, and how to make the optimal maintenance strategy. To address this issue, this paper proposes a multi-state system maintenance decision optimization method considering imperfect maintenance. Firstly, an imperfect maintenance model is established based on quasi-renewal theory. Secondly, the mapping relationship between the state of the components and the system is decoupled combining universal generating function with the non-homogeneous continuous-time Markov model, the reliability evaluation of the multi-state system is realized. Finally, the maintenance decision optimizing model considering imperfect maintenance is constructed, the optimal maintenance strategy of the multi-state system is obtained by optimizing the number of imperfect maintenance actions to maximize the expected benefit of the whole life cycle.

Group-Based Random Access and Data Transmission Scheme for Massive MTC Networks

Massive machine-type communications (mMTC) is one of the three generic services for the fifth-generation (5G) wireless communications system. To utilize fully the high rate transmission feature of the 5G system to support massive MTC devices (MTCDs), we propose a group-based random access and data transmission scheme, where the data packets of MTCDs are first aggregated by the MTC gateways (MTCGs) and then forwarded to the base station. The access process of the MTC network is divided into two phases, namely the intra-group transmission phase and MTCG forwarding phase. The entire resources are also partitioned for the two phases. We employ the discrete-time nonhomogenous Markov model to characterize the joint queue-length evolution of multiple MTCGs, which cannot be analyzed directly due to the exponential complexity and time-nonhomogeneity. To facilitate the analysis, we approximately decompose the joint nonhomogenous queue-length evolution process into multiple independent nonhomogenous queue-length evolution processes with the same state transition probabilities. Then, we establish an equivalent single queue-length evolution based homogenous Markov chain by constructing a virtual queue and determine the corresponding stationary distribution by using the Gauss-Jordan elimination method. An optimization problem is formulated to maximize the average network throughput subject to the constraints on the resource partition for the two phases and the MTCG forwarding threshold. By developing a modified differential evolution algorithm, we provide an efficient solution to the formulated problem, which can be arbitrarily close to the optimal solution. Simulation results show that the proposed scheme can efficiently improve the network performance over the existing schemes.

A multi-state Markov model using notification data to estimate HIV incidence, number of undiagnosed individuals living with HIV, and delay between infection and diagnosis: Illustration in France, 2008-2018.

Thirty-five years since the discovery of the human immunodeficiency virus (HIV), the epidemic is still ongoing in France. To guide HIV prevention strategies and monitor their impact, it is essential to understand the dynamics of the HIV epidemic. The indicator for reporting the progress of new infections is the HIV incidence. Given that HIV is mainly transmitted by undiagnosed individuals and that earlier treatment leads to less HIV transmission, it is essential to know the number of infected people unaware of their HIV-positive status as well as the time between infection and diagnosis. Our approach is based on a non-homogeneous multi-state Markov model describing the progression of the HIV disease. We propose a penalized likelihood approach to estimate the HIV incidence curve as well as the diagnosis rates. The HIV incidence curve was approximated using cubic M-splines, while an approximation of the cross-validation criterion was used to estimate the smoothing parameter. In a simulation study, we evaluate the performance of the model for reconstructing the HIV incidence curve and diagnosis rates. The method is illustrated in the population of men who have sex with men using HIV surveillance data collected by the French Institute for Public Health Surveillance since 2004.

Exploring the predictability of cryptocurrencies via Bayesian hidden Markov models

In this paper, we consider a variety of multi-state hidden Markov models for predicting and explaining the Bitcoin, Ether and Ripple returns in the presence of state (regime) dynamics. In addition, we examine the effects of several financial, economic and cryptocurrency specific predictors on the cryptocurrency return series. Our results indicate that the non-homogeneous hidden Markov (NHHM) model with four states has the best one-step-ahead forecasting performance among all competing models for all three series. The dominance of the predictive densities over the single regime random walk model relies on the fact that the states capture alternating periods with distinct return characteristics. In particular, the four state NHHM model distinguishes bull, bear and calm regimes for the Bitcoin series, and periods with different profit and risk magnitudes for the Ether and Ripple series. Also, conditionally on the hidden states, it identifies predictors with different linear and non-linear effects on the cryptocurrency returns. These empirical findings provide important benefits for portfolio management and policy implementation.

Data Analysis of the Risks of Type 2 Diabetes Mellitus Complications before Death Using a Data-Driven Modelling Approach: Methodologies and Challenges in Prolonged Diseases

(1) Background: A disease prediction model derived from real-world data is an important tool for managing type 2 diabetes mellitus (T2D). However, an appropriate prediction model for the Asian T2D population has not yet been developed. Hence, this study described construction details of the T2D Holistic Care model via estimating the probability of diabetes-related complications and the time-to-occurrence from a population-based database. (2) Methods: The model was based on the database of a Taiwan pay-for-performance reimbursement scheme for T2D between November 2002 and July 2017. A nonhomogeneous Markov model was applied to simulate multistate (7 main complications and death) transition probability after considering the sequential and repeated difficulties. (3) Results: The Markov model was constructed based on clinical care information from 163,452 patients with T2D, with a mean follow-up time of 5.5 years. After simulating a cohort of 100,000 hypothetical patients over a 10-year time horizon based on selected patient characteristics at baseline, a good predicted complication and mortality rates with a small range of absolute error (0.3–3.2%) were validated in the original cohort. Better and optimal predictabilities were further confirmed compared to the UKPDS Outcomes model and applied the model to other Asian populations, respectively. (4) Contribution: The study provides well-elucidated evidence to apply real-world data to the estimation of the occurrence and time point of major diabetes-related complications over a patient’s lifetime. Further applications in health decision science are encouraged.

Online common change-point detection in a set of nonstationary categorical time series

Categorical sequences are widely used in various domains to describe the evolutionary state of the process under study. This article addresses the problem of behavioral change detection for multiple categorical time series. Relying on the sequential likelihood ratio test, an online change detection method is proposed based on the joint modeling of all the categorical sequences. To model the joint probability density, a nonhomogeneous Markov model is used. It allows modeling the transition dynamics over time and considering their dependence on some exogenous factors that may influence the behavior changes. An adaptive threshold is learned using Monte Carlo simulations to detect different changes and reduce false alarms. The performance of the proposed method is evaluated using two real-world and four synthetic datasets. It is compared with two state-of-the-art change detection methods, namely logistic regression and homogeneous Markov model. The experimentation using synthetic datasets highlights the proposed method’s effectiveness in terms of both the detection precision and the detection delay. The real-world data are issued from a water network and school-to-work transition. The analysis of the model estimated parameters allows us to characterize the detected changes in a real-world context.

Relationships between Large-Scale Climate Signals and Winter Precipitation Amounts and Patterns over Iran

Precipitation is one of the most complex weather phenomena; its successful spatiotemporal modeling provides substantial information for designing water management systems. This research aimed to determine the relationship of sixteen large-scale climate signals with weather types (WTs) as well as precipitation amounts over Iran during the winter season (January–March). Daily precipitation data covering the period 1991–2010 were collected from 130 weather stations across the country. In addition, four atmospheric variables affecting the precipitation process over Iran were derived from National Centers for Environmental Prediction/National Center for Atmospheric Research (NCEP/NCAR) reanalysis datasets as predictors. Several nonhomogeneous hidden Markov models (NHMMs) were employed to identify the prevailing WTs governing the spatial distributions of daily precipitation in winter over Iran. A hidden state (i.e., WTs) in NHMMs was dependent on an immediately previous state of WT in the hidden layer and the reanalyzed atmospheric variables at the current time in the predictor layer. Daily WTs over the recording period were decoded using Viterbi’s algorithm. Results showed that the NHMM including the atmospheric variable of mean sea level pressure (Mslp) in the predictor layer had the minimum value (= 0) of Bayesian information criterion difference (BICD) among the other variables and was treated as the best model. The chosen model distinguished the eight spatially coherent WTs for winter that were in accordance with the eight synoptic patterns influencing the study area. Based on the Spearman’s rank method, amounts of winter precipitation at many stations in Iran were significantly correlated with four climate signals, including the atlantic multidecadal oscillation (AMO), East Atlantic/West Russia (EA/WR) pattern, Southern Oscillation Index (SOI), and Trans Polar Index (TPI). However, these station-based relationships could not exhibit well the spatial dependencies of the teleconnections and regional precipitation. It was also found that the unconditional and conditional frequencies of WTs were under the influence of a greater number of large-scale climate signals than the precipitation amounts, were under the influence of a greater number of large-scale climate signals. Moreover, the inherent nature of WTs in keeping spatial dependencies allowed for a better understanding of regional teleconnective relationships. These findings confirmed the advantage of regional rather than local assessments of teleconnective relationships.

A novel non-homogeneous hidden Markov model for simulating and predicting monthly rainfall

Monthly-to-seasonal precipitation forecasts are important in water resource management. Hidden Markov models (HMM) are widely applied in precipitation simulation due to its simplicity and advancements in associated computing techniques. HMMs, however, lack the flexibility to accommodate external factors in the dynamics. In this study, we consider a novel semiparametric Bayesian non-homogeneous hidden Markov model (BNHMM) that can explicitly incorporate the effects of observed covariates, e.g., climate index, on the state transition probabilities as well as the observation emission distributions. The proposed approach is tested for three rainfall stations in the service area of Tampa Bay Water, a regional water supply agency in the Southeastern United States. The BNHMM is first examined to simulate historical monthly rainfall data from the period 1979 to 2016. It is then evaluated in a retrospective mode to generate 3-month-ahead precipitation forecasts. Results indicate that the proposed BNHMM can capture historical rainfall properties well, and it is a promising alternative in providing operational rainfall forecasts. Although large-scale atmospheric and oceanic teleconnections, e.g., ENSO, have a strong influence in regional rainfall at the seasonal time scale, especially for the winter and early spring; its influence on monthly scale rainfall is limited. Potential improvement in forecasting performance is also discussed.

Detect differentially methylated regions using non-homogeneous hidden Markov model for bisulfite sequencing data.

DNA methylation plays an important role in many biological processes and diseases. With the rise of the whole genome bisulfite sequencing technique, aberrant methylation patterns can now be detected by comparing paired normal and disease samples at the single nucleotide level. We develop a novel Bayesian method for detecting differentially methylated regions from paired bisulfite sequencing data, and implement it as a R package called BSDMR. Based on a non-homogeneous hidden Markov model, BSDMR provides a better modeling strategy for the spatial correlation between CpG sites and takes into consideration the relationship between methylation signals from normal and disease samples. Simulations show that BSDMR performs well even under low read depth and has a smaller false discovery rates than existing methods. We also apply BSDMR to the colon cancer data from Gene Expression Omnibus. The detected DMRs are well supported by existing biomedical literatures.

ENSO Dynamics, Trends, and Prediction Using Machine Learning

AbstractEl Niño–Southern Oscillation (ENSO) has global effects on the hydrological cycle, agriculture, ecosystems, health, and society. We present a novel nonhomogeneous hidden Markov model (NHMM) for studying the underlying dynamics of sea surface temperature anomalies (SSTA) over the region 15°N–15°S, 150°E–80°W from January 1856 to December 2019, using the monthly SSTA data from the Kaplan extended SST v2 product. This nonparametric machine learning scheme dynamically simulates and predicts the spatiotemporal evolution of ENSO patterns, including their asymmetry, long-term trends, persistence, and seasonal evolution. The model identifies five hidden states whose spatial SSTA patterns are similar to the so-called ENSO flavors in the literature. From the fitted NHMM, the model shows that there are systematic trends in the frequency and persistence of the regimes over the last 160 years that may be related to changes in the mean state of basin temperature and/or global warming. We evaluated the ability of NHMM to make out-of-sample probabilistic predictions of the spatial structure of temperature anomalies for the period 1995–2016 using a training period from January 1856 to December 1994. The results show that NHMMs can simulate the behavior of the Niño-3.4 and Niño-1.2 regions quite well. The NHMM results over this period are comparable or superior to the commonly available ENSO prediction models, with the additional advantage of directly providing insights as to the space patterns, seasonal, and longer-term trends of the SSTA in the equatorial Pacific region.

Bayesian quantile nonhomogeneous hidden Markov models.

Hidden Markov models are useful in simultaneously analyzing a longitudinal observation process and its dynamic transition. Existing hidden Markov models focus on mean regression for the longitudinal response. However, the tails of the response distribution are as important as the center in many substantive studies. We propose a quantile hidden Markov model to provide a systematic method to examine the entire conditional distribution of the response given the hidden state and potential covariates. Instead of considering homogeneous hidden Markov models, which assume that the probabilities of between-state transitions are independent of subject- and time-specific characteristics, we allow the transition probabilities to depend on exogenous covariates, thereby yielding nonhomogeneous Markov chains and making the proposed model more flexible than its homogeneous counterpart. We develop a Bayesian approach coupled with efficient Markov chain Monte Carlo methods for statistical inference. Simulations are conducted to assess the empirical performance of the proposed method. The proposed methodology is applied to a cocaine use study to provide new insights into the prevention of cocaine use.

Separation of Residential Space Cooling Usage From Smart Meter Data

For demand response (DR) programs that focus on the energy consumption of heating, ventilation, and air conditioning (HVAC), the knowledge on HVAC usage of individual customers is of great value for DR program implementation. This paper presents a novel methodology to extract space cooling usage from smart meter data. A multisequence, non-homogeneous Factorial Hidden Markov Model (MN-FHMM) is proposed to disaggregate the whole-house energy consumption into an HVAC component and a baseload (i.e., the sum of non-temperature-sensitive loads) component. Rather than holding a constant transition probability over the whole process, a time-varying transition probability model is developed to characterize the dynamic nature of the evolution of users' energy consumption. We also discuss how to use the disaggregation results to estimate the HVAC related DR potential for individual customers, which can further inform DR programs to target customers more cost-effectively. Data experiments on ground truth data validate both the accuracy and the robustness of the proposed model as well as the effectiveness of the targeting strategy based on the disaggregation results.

An adaptive probabilistic data-driven methodology for prognosis of the fatigue life of composite structures

Data driven probabilistic methodologies have found increasing use the last decade and provide a platform for the remaining useful life (RUL) prediction of composite structures utilizing health-monitoring data. Of particular interest is the RUL prediction of composite structures that either underperform or outperform due to unexpected phenomena that might occur during their service life. These composite structures are referred as outliers and the prediction of their RUL is a challenge. This study addresses this challenge by proposing a new data-driven model; the Adaptive Non-Homogenous Hidden Semi Markov Model (ANHHSMM), which is an extension of the NHHSMM. The ANHHSMM uses diagnostic measures, which are estimated based on the training and testing data, and it adapts the trained parameters of the NHHSMM. The training data set consists of acoustic emission data collected from open-hole carbon–epoxy specimens, subjected to fatigue loading, while the testing data set consists of acoustic emission data collected from specimens, subjected to fatigue and in-situ impact loading, which can be considered as an unseen event for the training process. ANHHSMM provides better predictions in comparison to the NHHSMM for all the cases, demonstrating its capability to adapt to unexpected phenomena and integrate unforeseen data to the prognostics course.

A Markov model for the effects of virological failure on HIV/AIDS progression in tuberculosis co-infected patients receiving antiretroviral therapy in a rural clinic in northern South Africa.

The goal of antiretroviral therapy (ART) is to suppress viral replication to undetectable levels. These low viral load (VL) levels may not be attained in some patients, a situation representing potential virological failure during the course of treatment. To present the results of a Markov model exploring how virological failure and active tuberculosis (TB) affect the progression of HIV in patients on ART. A continuous-time non-homogeneous Markov model was used to model the progression of HIV/AIDS in patients on combination ART (cART). We define seven states in our model. The first five states are based on VL levels and the other two are absorbing states: death and withdrawal from the study. The effects of TB co-infection, baseline VL, lactic acidosis and treatment failure on transition intensities were assessed. The model shows that VL-based transition intensities do not follow a constant rate; rather, there are two different trends in HIV/AIDS progression. The first trend is an increase in the prevalence of state 1 (undetectable VL levels) in the first 0.5 years of treatment. The second trend follows thereafter and shows a slow decrease. Within the first 0.5 years of therapeutic intervention, the undetectable VL state is therefore attainable from any VL state. However, when virological failure occurs, there is an increased risk of death. Developing active TB while on cART increases the risk of viral rebound from undetectable levels to VLs between 50 and 10000 copies/mL by ~1.03-fold. From a VL between 10000 and 100000 copies/mL, developing TB while on cART increases the rate of viral rebound by ~2.5-fold. However, if TB is detected and treated at enrolment, rates of viral rebound from undetectable levels are reduced. The model confirms that virological failure, coupled with developing active TB while on cART, increases mortality rates irrespective of patient CD4+ count status. It also suggests that while TB at the time of cART initiation does not increase the risk of viral rebound, development of active TB after cART initiation does increase this risk. These findings highlight the importance of strengthening VL monitoring, which should be performed every 2 months, especially in patients with TB, and addressing unsuppressed VLs appropriately if they are detected.