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ISSN: 2155-6180
Journal of Biometrics & Biostatistics
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Comparison of Sojourn Time Distributions in Modeling HIV/AIDS Disease Progression

Ayele Taye Goshu and Tilahun Ferede Asena*

School of Mathematical and Statistical Sciences, Hawassa University, Hawassa, Ethiopia

*Corresponding Author:
Asena TF
School of Mathematical and Statistical Sciences Hawassa University
Ethiopia
Tel: +251 46 220 5311
E-mail: [email protected]

Received date: April 01, 2017; Accepted date: July 20, 2017; Published date: July 25, 2017

Citation: Goshu AT, Asena TF (2017) Comparison of Sojourn Time Distributions in Modeling HIV/AIDS Disease Progression. J Biom Biostat 8: 358. doi: 10.4172/2155-6180.1000358

Copyright: © 2017 Goshu AT, et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

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Abstract

In this paper we apply semi-Markov models to the HIV/AIDS disease progression and compare two sojourn time distributions, namely, exponential and Weibull probability distributions. We obtained data from 370 HIV/AIDS patients who are under a clinical follow up from September 2008 to August 2015. The results of the study reveal that within the "good" states, the transition probability of moving from a given state to the next worse state has a parabolic pattern - increases with time until it gets optimum and then declines over time. In the Weibull case, as compared to exponential, the conditional probability of staying in a good state before it moves to other good state is growing faster at the beginning, reaches peak, and then declines faster over long period of time. The conditional probabilities from any good states to death event are all linearly increasing over time. Similar behavior of disease progression is found for exponential and Weibull waiting time distributions. The probability of staying in same good state of the disease declines over time, keeping higher value for healthier state. Moreover, the Weibull distribution under the semi-Markov model leads to dynamic probabilities with higher rate of decline and smaller deviations. Weibull distribution is flexible in modeling and so it can preferably be used as waiting time distribution under semi-Markov modelling and monitoring HIV/AIDS disease progression over time.

Keywords

Disease; HIV/AIDS; Progression; Semi-Markov model; Transition probability

Introduction

Disease modeling and mapping is becoming important in biomedical sciences for investigation of the future status of particular patients disease results. Markov model can describe the natural course of disease progressions. These types of models are used particularly in medical applications where stages or levels of diseases are represented by the states in the model. Both semi-Markov and hidden Markov models are the most exciting tools for modeling a wealth of biological and biomedical data. In pathology individuals suffering from a particular disease passes through a series of disease stages either to the worst state or to a better states. Numerical analyses of the homogeneous semi-Markov process are dealt by Corradi et al. [1] and Janssen and Monica [2]. Other more readings include D’Amico et al. [3], Goshu and Dessie [4].

The movement to the pervious state and moving forward to the next state or progression to adjacent state of a disease can be characterized by some well-defined states representing the various stages of the disease. Many disease have a long preclinical phase during which the disease progresses and eventually become clinically recognized. It is then important to know what factors accelerates the progression and when this progression does occurs. In disease modeling, disease progression modeling has gained an at most important because slowing or preventing disease progression may be more effective strategy for reducing morbidity than trying to prevent onset. HIV/AIDS disease is a continuum of progressive damage to the immune system from the time of infection to the manifestation of severe immunologic damage by opportunistic infections, neoplasm’s, wasting, or low CD4 lymphocyte count that define AIDS.

This study deals with comparison of sojourn times for semi- Markov models of longitudinal data, which refers to data on individuals measured repeatedly at different times. Semi Markov models for disease progression are dealt with numerous researchers in the past two decades. Among them early papers on disease progressions for Semi Markov studies are more recent one [5-10]. Comprehensive introductions in semi-Markov are Janssen and Manca [11] while Levy [12] is the first to introduce the semi-Markovian process. A Semi- Markov model for disease progression has been studied in recent decades [13,14].

The report shows that 19 million of the 35 million people living with HIV today do not know that they have the virus. The report highlights efforts to increase access to ART are working. In 2013, an additional 2.3 million people gained access to the life-saving medicines. This brings the global number of people accessing ART to nearly 13 million by the end of 2013. Based on past scale-up, UNAIDS projects that as of July 2014 as many as 13,950,296 people were accessing ART. By ending the epidemic by 2030, the world would avert 18 million new HIV infections and 11.2 million AIDS-related deaths between 2013 and 2030. Infection by the human immunodeficiency virus (HIV) gradually evolves to the acquired immune deficiency syndrome (AIDS), and AIDS evolves to death if not handled carefully.

One may consider the progression of HIV infection to AIDS and then to death as a stochastic process. By splitting the progression into various states of the disease based on the immunological indicators namely CD4+ count including death as one state. In this study five states will be considered namely, state one (CD4 count>500); state two (350<CD4 count<500); state three (200<CD4 count<350); state four (CD4 count<200); and state five (Death) based on the seriousness of the Sickness.

According to the study of Goshu and Dessie [4] on Progression of HIV/AIDS Disease Stages Using Semi-Markov Processes at Felege Hiwot referral hospital, Ethiopia, the probability of being in a better state is non-zero, but less than the probability of being in worse state. This suggests that patient transition from one state to another depends on whether how long he/she has spent in that state. Their study also recommends that the survival probability of an HIV/AIDS patient depends on his/her current state of the disease in such a way that the lower CD4 counts the higher is the risk to be in worse health state or death state.

Accurate and detailed models of HIV/AIDS disease progression are crucially important for reliable early diagnosis and the determination of effective treatments. For the purposes of patient staging developing progression models are more effective [15]. Models of disease progression are among the core tools of modern medicine for early disease diagnosis, treatment determination and for explaining symptoms to patients. Disease Progression Modeling is the modeling of the progression of a target disease with computational methods, which is an important technique that can help with the early detection and management of chronic diseases. By characterizing the entire disease progression trajectory, disease progression modeling also facilitates disease prognosis improvement, drug development, and clinical trial design. Examples of staging systems that are based on disease progression can be found in diseases such as cancer, HIV, TB, Alzheimer, daiabetes, and chronic obstructive pulmonary diseases. Every disease can be uniquely characterized by a progression of symptom and pathology, whose understanding is vital for accurate diagnosis and treatment planning. The purpose of this study is to model the disease progression of HIV/AIDS patients using exponential and Weibull waiting time distributions based on data obtained from Ethiopia.

General Methodology

Markov models for disease progression

The Markov assumption is restrictive and not necessary realistic. In recent days semi-Markov models are highly utilized to model the progression of HIV/AIDS Disease stages [4,13-15]. It is known that the environment of the semi-Markov process is much richer than the Markov chain and complete application of the semi-Markov process is the right choice to model the progression of diseases states under the homogenous time scheme [7]. Time homogenous semi-Markov models assumes that the trajectory of the process depends only on the amount of time spent in the current state, allowing the sojourn times in each state to have an arbitrary distribution.

The semi-Markov model

The semi-Markov models are studied in detail [2,7,16,17]. Numerical analyses of the homogeneous semi-Markov process are dealt [1,2]. Other more readings include in DAmico et al. [3]. See Goshu and Dessie [4] for application of the models to HIV/AIDS disease progression.

In semi-Markov model we have two important quantities, the first is the state of the transition denoted by Xn and the second is the time of the transition. Defining these two important variables we are able to consider the randomness of the state along with the randomness of the time elapsed in each state. Thus we define the two random variables simultaneous as:

equation (1)

Xn represents the state at the nth transition of the Markov process with state space S={S1, S2, ..., Sm}, and Tn represents the time of the nth transition.

The kernel Q=[Qij] and the transition probabilities Pij embedded in the Markov process are:

equation (2)

equation (3)

Furthermore, it is necessary to introduce the probability that the process will leave state i in time t which is given as:

equation (4)

The distribution of waiting time in each state i, given that the state j is subsequently occupied is:

equation

This can be computed as:

equation (5)

equation

The transition probabilities are given by (6) for which the solutions should be obtained using the progression or evolution equation (7).

equation (6)

equation (7)

Corradi et al. [1] proved numerical solution for (7) that converges to the discrete time HSMP described as an infinite countable linear system as:

equation (8)

where h represents the discretization step, and

equation

equation

The evolution equation in (8) can be written in the matrix form as:

equation (9)

equation (10)

Algorithm and R codes

For solving the evolution equation (10), Corradi et al. [1] proposed the following algorithm. Given an epoch T is fixed, matrices GT and P, the algorithm numerically solves the linear systems for the unknown matrix ΦT. The variables involved are the following:

m=number of states of the process.

T=number of periods to be examined for the transient analysis.

P=matrix of order m of the embedded Markov process.

GT=square lower-triangular block matrix of order T+1 whose blocks are of order m.

QT=represents the kernel of the Markov process.

Φ =block vector of order T+1 the block of which are square matrices of order m.

DT=block vector of order T+1 the block of which are the diagonal square matrix of order m.

VT=square lower-triangular block matrix of order T+1 whose blocks are of order m.

ST=block vector of order T+1 the block of which are the diagonal square matrix of order m. The diagonal element of each block at time t is given by equation.

The algorithm is:

(a) Read m, T, P, G

(b) Construct QT, VT, DT

equation

equation

equation

equation

equation

equation

equation

end for

end for

(c) Return the results ΦT, QT

Goshu and Dessie [4] implemented this algorithm by writing R codes. Same R codes are used in this study for the computations.

Given the solutions of the evolution equation equation for transition probabilities, the reliability function is computed as:

equation (11)

Sojourn time distributions

We suppose that the sojourn or waiting time in a given state is random and have a distribution. It is assumed that process spends some time in a given state and random time has distribution G(t). Two sojourn distributions are considered here for modelling progression of the disease. The first one is the exponential distributionequation whereequation is expected time that the process spends in state i before it enters state j from i with scale parameter σij>0. The second distribution we considered is the Weibull distribution equation, and with scale parameter σij>0 and shape parameter vij>0. When the shape is equal to 1, the Weibull becomes exponential distribution.

Data

The Data for this study were obtained from Yirgalem General Hospital, which is located 300 km South of Addis Ababa in the Yirgalem town of Sidama zone, the Southern Nation’s Nationalities and Peoples regional state. There are HIV patients enrolled for ART follow up in the hospital since 2000. We adopted a simple random sampling procedure to collect our data from the lists of patients who are under follow up of ART from 2008 to 2015. The following sample size determination formula Cochran [17] is used:

equation (12)

Where Zα/2 is the upper equation points of standard normal distribution with α=0.05 significance level, which is Zα/2=1.96. The term p represents proportion of death among HIV/AIDS patients. It was obtained from the previous comparable study done by Mathieu et al. [7] on data taken from Felege-Hiwot Referral Hospital which is p=0.134. The degree of precision d selected for this study was taken to be 0.03. With total number of N=1570 HIV/AIDS patients at the Yirgalem General Hospital, the sample size for this study was estimated to be 375 patients.

Referring the CDC immunological classification [17] of HIV/AIDS infected patients, we have five states, where the first four states are the good states and the last state is bad state or death state. Thus, based on the seriousness of the cases we have the following states:

SI: CD4 count>500 × 106 cells/L

SII: 350 × 106 cells/L < CD4 count ≤ 500 × 106 cells/L

SIII: 200 × 106 cells/L < CD4 count ≤ 350 × 106 cells/L

SIV: CD4 count ≤ 200 × 106 cells/L.

D: Death.

The death state D is considered an absorbing state, while the other four "good" states all communicate each other.

Results and Discussion

HIV/AIDS disease progression

We study the progression of HIV/AIDS disease and health risk factors and its relationship with the well-beings of HIV/AIDS patients in order to provide theoretical support for decision-making on health intervention, prognostics and prevention using flexible waiting time distributions. For comparison purpose we use exponential distribution and a two parameter Weibull distribution as awaiting time distribution of the semi Markova stochastic process. Semi-Markov models explicitly define distributions of waiting times, giving an extension of continuous time and homogeneous Markov models based implicitly on exponential distributions. To see the difference in waiting time distribution of diseases progressions we produced results in both distributions as well. Finally, we compare the two models on HIV/AIDS disease progression and estimate the transition probabilities using these waiting time distributions of stochastic process.

The data analyzed in this study were collected at Yirgalem General Hospital during September 2008 to August 2015 follow up period in every six months at known and fixed time points but the transition between levels of the state space could occur at any time. Frequencies and estimated transition probabilities of between the states are summarized from the data and displayed in Table 1. Each patient was followed throughout the study period on the change of status of the disease under ART follow up. Among these patients, 69 (18.4%) were dead.

State I II III IV D
I 171 (0.438462) 169 (0.433333) 32 (0.082051) 7 (0.017949) 11 (0.028205)
II 240 (0.495868) 76 (0.157025) 132 (0.272727) 22 (0.045455) 14 (0.028926)
III 59 (0.166667) 183 (0.516949) 52 (0.146893) 41 (0.115819) 19 (0.053672)
IV 12 (0.079470) 30 (0.198675) 77 (0.509934) 7 (0.046358) 25 (0.165563)

Table 1: The Transition Probability Matrix Computed from the Progression Data.

The model we consider is a continuous time Markov model. It helps us to compute transition probabilities and mean waiting times over time. The solution for the transition probabilities equation at time t is obtained using the defined algorithm given m=5 states, T=204 months, transition probability matrix P as given in Table 1. For the Weibull sojourn time distribution, the shape parameter is taken be 1.5. The scale parameters are estimated from the data.

All analysis is based on the algorithm defined. The results are plotted in Figures 1-7.

biometrics-biostatistics-probability-patient-sojourn

Figure 1: Conditional probability that a patient will be in state j after t months given that s/he is initially in state i. The plots in left panel are for exponential sojourn time and those in right panel for Weibull sojourn time distributions.

biometrics-biostatistics-weibull-sojourn-time

Figure 2: The Conditional probability that a patient will be in state death state after t months given that she/he is initially in state i∈{SI,SII,SIII,SIV}. The plot in left panel is for exponential sojourn time and in right panel for Weibull sojourn time distributions.

biometrics-biostatistics-weibull-sojourn-time

Figure 3: Conditional Probability of being in next state j after month t given the starting state i. The plot in left panel is for exponential sojourn time and in right panel for Weibull sojourn time distributions.

biometrics-biostatistics-statej-starting-state

Figure 4: Conditional Probability of being in next statej after t months given a starting state i. The plot in left panel is for exponential sojourn time and in right panel for Weibull sojourn time distributions.

biometrics-biostatistics-probability-exponential-sojourn

Figure 5: Conditional probability of being in next state j after t months given a starting state i. The plot in left panel is for exponential sojourn time and in right panel for Weibull sojourn time distributions.

biometrics-biostatistics-probability-sojourn-time

Figure 6: Conditional probability of being in next state j after t months given the starting state i. The plot in left panel is for exponential sojourn time and in right panel for Weibull sojourn time distributions.

biometrics-biostatistics-weibull-sojourn-time

Figure 7: The conditional probability that a patient stays in a good state of the disease for at least t months. The plot in left panel is for exponential sojourn time and in right panel for Weibull sojourn time distributions.

The plots in Figure 1 display the several conditional probabilities that a patient will be in state j at time t given that she/he is initially in state i - within the good states. These progressions are computed for SI to SII, SII to SIII and SIII to SIV of a specific HIV/AIDS patient.

Figure 1a is plotted for the exponential waiting time distribution. The parabolic curves of the probability get peaks at in the timeprobability points (48, 0.314) for state I to II, (36, 0.161) for state II to III, and (12,0.054) in state III to IV. The peaks may indicate there is time when a patient will be at highest risk of being at worse state. Moreover, the transition probability from SII to SIII is the lowest as compared to the others. It is interesting to find out that, within the good states, the transition probability from a given state to the next worse state increases with time gets optimum at a time and then decreases with increasing time considering exponential waiting time distribution.

Figure 1b is plotted for the Weibull waiting time. The progressions from state SI to SII, SII to SIII and SIII to SIV of a specific HIV/AIDS disease states to the next consecutive disease states resulting a similar parabolic curve with optimal/peak points (48, 0.314), (72, 0.160) and (36, 0.054) in the time-probability axis as exponential waiting time distribution. Interestingly, we observed a marked increment in progression from SI to SII, SII to SIII and SIII to SIV of a specific HIV/AIDS patient considering Weibull waiting time distribution compared to exponential waiting time distribution. This fact suggests a relationship between clinical outcomes and the duration of waiting times which could explain the increase or decrease in transition probabilities of disease stages of patients in the cohort. In addition for Weibull as compared to exponential model the conditional probability of staying in a good state before it moves to other good state is growing faster at the beginning, reaches peaks, and then declines faster over long period of time.

Figure 2 displays the conditional probabilities that an HIV/AIDS patient who is currently in a given state SI, SII, SIII, SIV will be in the death state after t months. These transitions are plotted from SI to D, SII to D, and SIII to D and SIV to D. These figures express the probability of a patient will be in the death state where ever he/she is. Based on the exponential waiting time distribution the probability of dying after 204 months is 0.396 for a patient who is in the first stage, 0.404 for one who is in the second stage, 0.4298 for one who is in the third stage and 0.5057 for one who is in the fourth stage of the disease. The probability of dying will increase by nine percent (9 percent) for a specific HIV/ AIDS patient in state three compared to an HIV/AIDS patient in state one. Similarly the probability of dying will increase by twenty percent (20 percent) for a specific HIV/AIDS patient in state four compared to an HIV/AIDS patient in state one.

For Weibull distribution model, the probability of dying after 204 months is 0.213 for a patient who is in the first stage, 0.224 for one who is in the second stage, 0.256 for one who is in the third stage and 0.353 for one who is in the fourth stage of the disease. Each plot is an increasing parabolic curve over time with no optimal/peak point. The probability of dying will increase by five Percent (5 percent) for a specific HIV/ AIDS patient in state three compared to an HIV/AIDS patient in state one. Similarly the probability of dying will increase by sixty six percent (66 percent) for a specific HIV/AIDS patient in state four compared to an HIV/AIDS patient in state one. This can be interpreted as the probability that an HIV/AIDS patient with any one of the good states will be in death state is increasing with time. Moreover, a patient who is in the fourth state has the highest probability of dying after any given t months, while that of one who is in first state is the lowest probability throughout the time. In comparison with the Weibull and exponential waiting time, the exponential waiting time gives higher probabilities of an HIV/AIDS patient with any one of the good states will be in death state.

This result also shows that over long time, Weibull leads to smaller deviation of the probabilities than the exponential case. Special medical intervention may be required to slow down the rate of decline. For instance, at time 204 months, the differences between the probabilities are smaller for the Weibull as compared to exponential distribution.

Figures 3-7 display conditional probabilities of patient making changes in disease states given his/her current status using exponential and Weibull waiting time distribution. The results show that the probabilities of being in the some state for both waiting time distribution across the study period decrease over the time.

But the probabilities highly decreases as one considers exponential waiting time distributions compared with Weibull waiting time distributions demonstrating that low probabilities (small rates) tend to be those with low waiting time rates and, consequently, slow progression. Conversely, it demonstrates that high probabilities (high rates) tend to be those with high waiting time rates and have early progression.

Thus physicians can monitor patient’s disease status at each state and take corrective actions as needed as early as possible if one considers Weibull waiting time distributions. Therefore considering Weibull waiting time distribution is much preferable than the exponential waiting time distributions in such a way that help with the early detection and management of diseases.

We computed the probability of staying in same state. It is interesting to find out that the conditional probability of staying in same state decreases with increasing time for both waiting time distributions (Figure 7). This result explains that an HIV/AIDS patient in a specific good state of the disease will stay in that state with a nonzero probability. The patient is more likely to be in a good state than worse state, for example, the probabilities to be in state I, II, III and IV at time 24 months are about 0.545, 0.433, 0.321 and 0.187 respectively under exponential model.

This result also indicates that an HIV/AIDS patient in a specific state of the disease the probability of being in same state decreases over time. With the good or alive states, the results show that probability of being in a better state is non-zero, but less than the probability of being in worst states.

Discussion

This study is intended to model the progression of HIV/AIDS so as to predict the future clinical state and survival probability of a patient in order to provide theoretical support for decision-making in health intervention, prognostics and prevention using flexible waiting time distributions. Accordingly, different probability plots are produced from the semi-Markov model based on the data on CD4+ counts of the patients under ART follow up during September 2008 to August 2015.

The results show that high deviations are observed when exponential waiting time distribution used in modelling HIV/ AIDS disease progression as compared with Weibull waiting time distribution having lower deviations. The observed high deviation in exponential waiting time distribution results higher growth rate and lower deviations in Weibull waiting time distribution results in low growth rate.

The study revealed that within the good states, the transition probability from a given state to the next worse state increases with time gets optimum at a time and then decreases with increasing time while considering exponential waiting time distribution. These are consistent with findings of Goshu and Dessie [4]. Furthermore, for a Weibull waiting time distribution, the conditional probability of staying in a good state before it moves to other good state is growing faster at the beginning, reaches peaks, and then declines faster over long period of time.

We found similar findings as in Goshu and Dessie [4] that the probability that an HIV/AIDS patient in any one of the good states will be in death state is increasing over time. Moreover, a patient who is in the fourth state has the highest probability of dying after any given t months, while that of one who is in first state is the lowest probability throughout the time. This is because of the fact that the fourth state is the highest in sickness stage and a patient at this stage is more likely to move to the absorbing state. Such patients need effective medical care to help them in order to divert the disease progression to the healthier states.

The results also show that high deviations are observed when exponential waiting time distribution used in modelling HIV/ AIDS disease progression as compared with Weibull waiting time distribution having lower deviations. The observed high deviation in exponential waiting time distribution results higher growth rate and lower deviations in Weibull waiting time distribution results in low growth rate.

Conclusions

This study is intended to model the progression of HIV/AIDS so as to predict the future clinical state and survival probability of a patient. The semi-Markov model is used to model the progression using varying waiting time distributions. Thus, the following conclusions are drawn from this study.

Within the good states, the transition probability from a given state to the next worse state increases with time gets optimum at a time and then decreases with increasing time while considering exponential and Weibull waiting time distribution. But the probability gets narrower for exponential waiting time distribution as compared with Weibull distribution. This indicates patients will suffer some time in their follow period and thus it is recommended that physicians can help patients with the early detection and management of their diseases symptoms and intervene their follow ups and alter their medications systems. We observed a marked increment in progression from the good state to the next consecutive worst state of a specific HIV/AIDS patient considering Weibull waiting time distribution compared to exponential waiting time distribution.

The Weibull case as compared to the exponential model provides the conditional probability of staying in a good state before it moves to other good state is growing faster at the beginning, reaches peaks, and then declines faster over long period of time. The probabilities of being in the some state for both waiting time distribution across the study period decrease over the time. In comparison staying in the some good state declines faster for Weibull waiting time distribution than in exponential waiting time.

The findings indicate that probability of staying in same good state of the disease declines over time, keeping higher value for healthier state. The survival probabilities are all decreasing for both waiting time distributions with increasing time. Moreover, the Weibull distribution under the semi-Markov model leads to dynamic probabilities with higher rate of decline and smaller deviations. Weibull distribution is flexible in modelling and so it can preferably be used as waiting time distribution under semi-Markov modelling and monitoring HIV/ AIDS disease progression over time.

Acknowledgments

The authors would like to thank Zelalem Getahun Dessie for permitting us to use the R function developed for the semi-Markov algorithm. Our special appreciation goes to Hawassa University and Arba Minch University for financial supports.

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