Semi-Markov Modelling of HIV/AIDS Disease Progression

Background: HIV/AIDS epidemic continues to be the main challenge in the world. According to United Nations Program on HIV/AIDS (UNAIDS) and the World Health Organization (WHO) reports of 2013, 35 million people were living with HIV worldwide, with 2.1 million new infections and with 1.5 million deaths occurred each year. Among these, 24.7 million lived in sub-Saharan Africa with 1.5 million new infections and 1.1 million AIDS deaths. Method : The main objective of this study is finding factors affecting HIV/AIDS disease progression. This study was conducted to investigate the effect of factors on HIV/AIDS disease progression. Patient follow-up data is obtained at Yirgalim General Hospital. A sample of 370 Patient data from a follow-up cohort is obtained at Yirgalim General Hospital. Multivariate generalized hazard regression model was employed to investigate the disease progression using both time independent and time dependent covariates. Result : The study revealed that the risk of transition differs by patient's body mass index. Increase in the body mass index reduces the risk of transiting into the next worst states. The effects of sex, weight, age and body mass index of patients are significantly associated with AIDS disease progression. The risk of transition differs by patient's body mass index. Increase in the body mass index reduces the risk of transiting into the next worst states. The effect of sex, weight, age and body mass index of patients are significantly associated with AIDS disease progression. The results further revealed that the semi-Markov model with Weibull waiting time distribution has smaller log likelihood and AIC values compared to a semi-Markov model with exponential waiting time distribution. Conclusion : T ransition probabilities are highly dependent on the choice of waiting times. We recommend that while choosing waiting time distributions for semi-Marko models one should consider appropriate distributions as waiting time distribution effect have a significant change on the estimated model parameters. In addition, this study recommends that concerned bodies should look at deferent contributing factors of AIDS diseases progression in addition to the ART services administered for slowing the current level of high diseased population in the country.

Inference is difficult when the process is only observed at discrete time points, with no information about the times or types of events between observation times. Studies show that mortality among HIV-infected individuals depends on their CD4 cell counts and other associated risk factors. CD4 cell counts and risk factors accelerate or decelerate the state transitions of the AIDS disease [4]. Several studies are mainly concerned with estimation of transition and survival probabilities using Markov models without considering effects of covariates on the disease progression [5, 6, and 7]. The probability of dying increases in the worse transition states. The probability of being in healthy state after he/she started the treatment is higher as compared with any other working state [4]. Some factors may increase/decrease the hazard rate of transition between disease states. When patients get older and infected with TB, AIDS disease transition rates to death state increases [8]. In semi-Markov Models, we are able to choose a parametric distribution with more freedom than in traditional Markov Chain. Parametric semi-Markov models are powerful for studying chronic diseases and estimating factors associated with transitions between different stages of disease progression. Literature shows there exists a growing interest to apply parametric semi-Markov models for disease progression [9].
The development of R packages such as the smm package [10] for disease modelling is motivated by the interest to include time-varying characteristics of individuals to transition rates through a parametric approach. Epifani et al. [11] used Bayesian estimation for a parametric Markov renewal model applied to seismic data. Their work focused on the development of methodology for Bayesian inference on a parametric Semi-Markov process, from the elicitation of the prior distribution to the computation of posterior summaries. The aim of this study is to find factors affecting HIV/AIDS disease progression using parametric semi-Markov model with interval censoring.
The method used tin this analysis is formalized in 2. In 3 we describe our main result. In 4 we discuss our results and describe our findings of this study by comparing with the methods presented in 2. In 5, we conclude with a discussion.

Methods
Data for this study were obtained from Yirgalim General Hospital. Yirgalim General Hospital is located 300 km south of Addis Ababa in Yirgalim town of Sidama zone, the Southern Nation's Nationalities Region. The Hospital was inaugurated in 1968. The ART clinic started follow-up on HIV patient in 2000. We adopted a simple random sampling procedure to select sample of HIV patients from the lists of patients who follows ART between September 2008 and August 2015. Accordingly, the following sample size determination formula [12] is used: SI: CD4 T cells count  500 x 6 10 T cells/L SII: 350 x 6 10 T cells/L  CD4 T cells count < 500 x The random variable   and for j i  . The Semi-Markov kernel is given by , namely t, represents duration time of the process [9].
Equation 7 represents the probability of a patient making its next transition to state j , given that he/she entered state i at time t and The probability density function of the waiting time in state i before passing to state j is From density function, we can derive the cumulative probability function, of waiting time in state $\textit{i}$ as defined by: In semi-Markov modelling one can choose any eligible waiting (sojourn) time distribution.
Thus, we considered both exponential distribution and Weibull distribution as the waiting time distribution. For exponential distribution, the hazard function is constant. The hazard function of the waiting time is given by: Further, the Weibull distribution generalizes the exponential distribution by using two parameters, which is more flexible and well adapted to various shapes. Its hazard function is defined by: is the shape parameter of the Weibull probability distribution.
The hazard function of the Semi-Markov process, which represents the probability of transition towards state j between time t and t t   , given that the process is in state i for duration t can be derived as follows: (11) The parametric model allows to incorporate covariates in the distribution of sojourn times using a proportional-hazards regression model [16].

Results and Discussions
The aim of this study is to find factors affecting HIV/AIDS disease progression using a parametric semi-Markov model with interval-censored data. We use the R package semi-Markov to analyses the data and estimate the parameters of the semi-Markov model. Table 1 shows comparison of waiting time distribution for fitting progression of the disease. Based on the result the log likelihood of exponential waiting time distribution (-2 * log-likelihood = 5159.791 and AIC=5167.791) is higher than the log likelihood of Weibull sojourn time distribution (-2 * log-likelihood = 5056.362 and AIC=5064.362). Therefore, the Weibull waiting time distributions are preferred for HIV/AIDS disease progression. Moreover, differences in the transition estimates from state to state in both waiting time distributions are obtained. Exponential waiting time distribution generates wider confidence intervals as compared with Weibull waiting time distributions. The Weibull waiting time distribution is preferable as compared with the exponential waiting times distribution for fitting our data. According to the univariate analysis covariates sex, age, weight and body mass index are found to be significant. However, TB co-infection, religion, educational status, place of residence, occupational status and opportunistic infections are not associated with increasing or decreasing the risk of developing the progression of the HIV/AIDS disease. Table 2   In Table 3, we considered multivariable generalized hazard regression model to investigate disease progression to the next higher healthier state when covariates, sex, age, weight, and BMI of patients are simultaneously included. p<0.001) for are significantly different from 0. This depicts that the effect of gender is significant for a patient to progress to the next higher disease state.

Results in
The estimate of coefficient associated to the transition from state one to state two (  =

Discussions
This study was intended to assess the effects of covariates on the

Conclusions
This study aims to assess the effects of covariates on AIDS disease progression using 370 follow-up data obtained from Yirgalim General Hospital. Univariate regression analysis using generalized hazard regression revealed that effects of sex, age, weight and body mass index are significantly associated with AIDS progression. Other factors such as the effects of