MULTI-STATE MODELS OF HIV/AIDS BY HOMOGENEOUS SEMI-MARKOV PROCESS

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1 American Journal of Biostatistics 4 (2): 21-28, 2014 ISSN: Z.G. Dessie, Tis open access article is distributed under a Creative Commons Attribution (CC-BY) 3.0 license doi: /ajbssp Publised Online 4(2) 2014 (ttp:// MULTI-STATE MODELS OF HIV/AIDS BY HOMOGENEOUS SEMI-MARKOV PROCESS Zelalem Getaun Dessie Department of Statistics, College of Science, Bair Dar University, Bair Dar, Etiopia Received ; Revised ; Accepted ABSTRACT Multi-state stocastic models are useful tools for studying complex dynamics suc as cronic diseases. Te purpose of tis study is to determine factors associated wit te progression between different stages of te disease and to model te progression of HIV/AIDS disease of an individual patient under ART follow-up using semi-markov processes. A sample of 1456 patients as been taken from a ospital record at Amara Referral Hospitals, Amara Region, Etiopia, wo ave been under ART follow up from June 2006 to August Te states of disease progression adopted in te multi-state model were defined based on of te following CD4 cell counts: 500(SI); 200 to 499(SII); <200(SIII); and Deat (D). Te first tree states are named as good. Female patients were 1.6 times more likely to move from state 2 to state 1 tan tose of male patients (adjusted HR = 1.60, CI = ). Patients, wo is not drug addicted, were 2.49 times more likely to move from state 3 to state 2 tan tose of drug addicted(adjusted HR = 2.67, CI = ). Patients wit tuberculosis were 2.67 times more likely to move from state 3 to state 4 tan tose wit no tuberculosis (adjusted HR = 2.67, CI = ). On te oter and, te probability of staying in same state until a given number of mont decreases wit increasing time. Multi-state modeling is a powerful approac for studying cronic diseases and estimating factors associated wit transitions between eac stage of progression. Te major predictors of te intensity of transitions between different states of HIV/AIDS patients were gender, age, drug addicted and TB status. Te dynamic nature of te AIDS progression is confirmed wit particular findings tat tere is more likely to be in worse state tan better one unless interventions are made. Keywords: Disease Progression, Markov Cain, HIV/AIDS 1. INTRODUCTION Te rate of spread of te HIV/AIDS epidemic as reaced a socking level. Te expansion of te epidemic as now become a burning issue globally and tis is particularly so more important in developing countries. Te disease being one witout any cure is still account for economic, social and ealt crises in many developing countries. On a global scale, te HIV epidemic as stabilized, altoug wit unacceptably ig levels of new HIV infections and AIDS deats. An estimated 34 million people worldwide were living wit HIV in 2011 among wic 23.5 million are living in sub- Saaran Africa. Etiopia is one of te countries ardest 21 it by HIV/AIDS epidemic. An estimated of 790,000 were living wit HIV in 2011(UNAIDS, 2013). Most studies of AIDS ave analyzed te factors associated wit terapeutic failure, suc as deat, opportunistic disease, detect Viral Load (VL) or low levels of immunity, using semi parametric Cox models (Pradeep et al., 2010; Muralidar et al., 2010; Keiser et al., 2010; Koete et al., 2010; Neuaus et al., 2010). However, it is also important to consider cronicity and te relatively slow progression of te disease and to assess te multiple immune states resulting from te effect of te virus on te uman immune system. Te use of multi-state Markov models to analyze te factors associated wit transitions between different states of

2 cronicity as been suggested for cronic diseases and te cost-effectiveness of various terapeutic regimes (Si et al., 2007; Pan et al., 2007; Gil et al., 2007). Recent studies ave sown tat te predicted probability of patients tat canging teir status given is/er current status allows te measurements of medical scientific progresses due to te advances in te treatment of te HIV/AIDS (D'Amico et al., 2009). Since te discovery of HIV/AIDS, numerous matematical models Blasi and Manca (2004) ave been developed to describe infectious disease transmission dynamics, understand te mecanisms of epidemic propagation, assess te impact of interventions on public ealt or forecast te future of epidemics. Among recent papers in biomedicine, Masala et al. (2014; Gosu and Dessie, 2013; Giuseppe et al., 2007) analyzed HIV/AIDS dynamic evolution as defined by CD4 levels from a macroscopic point of view by means of omogeneous semi-markov stocastic processes. Numerical analyses of te omogeneous semi-markov process are dealt by Corradi et al. (2004; Janssen and Manca, 2001). Oter more readings include (Davidov and Zelen, 2000; Viladent and Van Ackere, 2007; Satten and Sternberg, 1999; Baryarama et al., 2005). In tis study, te autor proposes omogeneous Markov process to study te evolution of HIV/AIDS. Te omogeneous approac considered is a piecewise Markov process, wit te transition intensity functions being step functions. Te autor is interested in determining factors associated wit te progression between different stages of te disease. Several researcers ave studied Markov processes wit covariates and a procedure to obtain te parameters in a model wit covariates as been reported (Maciulis et al., 2009; Gentlemann et al., 1985; Matieu et al., 2007; Pѐrez-Ocόn et al., 2001). Te autor also present te results of modelling of te waiting time of HIV/AIDS so as to predict te future waiting time of a patient. Tis is: Te conditional probability of staying in te starting state until mont t. 2. MATERIALS AND METHODS 2.1. Etics Statement Tis investigation was conducted according to te principles expressed in te Declaration of Bair Dar University, Etiopia. It was approved by te researc etics committee at te University of Bair Dar and all participants wo agreed to participate in tis study signed a consent form Data and Model Descriptions Te target population for tis study was patients under te follow up of ART at Amara Referral Hospitals, Amara Region, Etiopia from June 2006 to August Multistage sampling was used to select study subjects. First, all te Referral Hospitals in te Amara Region were stratified in to Western and Eastern area. Te calculated sample size was proportionally allocated to Eastern (n = 482) and Western (n = 874) areas respectively. Te sampling frame consists of HIV/AIDS patients wo ave visited te ospitals since te initiation of ART. Te study may consider all HIV infected patients under ART wose age is >15 years regardless of teir treatment category during te study period in Amara Referral Hospitals, Amara Region, Etiopia. Te states of disease progression adopted in te multi-state model were defined based on te following CD4 cell counts as in Oliveira et al. (2013; Giuseppe et al., 2007): 500 cells/mm 3 (SI); 350 to 499 cells/mm 3 (SII); <350 cells/mm 3 (SIII); and deat(d). Te deat state is considered to be an absorbing state-meaning tat once a patient is in te deat state se/e will never be in te oters states and rater stays tere forever. Te time elapse between state transitions was determined using te difference (in monts) between te dates of CD4 tests. In Fig. 1 te grap model is pictured. It sows all te immunological states a HIV infected patient can go into. All te states apart from tan Deat are interrelated and also improvements are considered. It is also possible tat an examination will sow tat te patient s state as not canged Modelling Semi-Markov Processes Te Transition-Specific Semi-Markov Model A semi-markov process is a stocastic process, {X(t): t 0}, were an embedded Markov cain governs te state to state transitions of te process wile a separate probabilistic mecanism determines te time spent in eac state. It is assumed tat te transition probabilities depend on te current state and te time spent in eac state depends upon te current and next state. Several autors ave discussed te use of Markov processes tecniques in ealt. Te autor cites only some of tem, i.e., (Jackson et al., 2003; D Amico et al., 2005; Foucer et al., 2005; Barbu and Limnios, 2009).

3 Te azard function of te semi-markovian process corresponds to te probability of jumping towards state j, given tat te process occupies state i for duration x Equation 6: Fig. 1. Communication between te states of te process i j Pij fij ( x) Gij ( x) = wit i, j E Si. ( x) λii ( x) = λij ( x) j= i (6) In tis part, Homogeneous Semi-Markov model are described by using te notation of Foucer et al. (2005). On a complete probability space te autor define two random variables. X n :Ω S be te stocastic process wit state space S = {S 1,S 2 S m } and T n :Ω R be te time of te n t transition, wit Ω domain of te process and R set real numbers. Te kernel Q = [Q ij ] associated wit te process and te transition matrix P ij of te embedded Markov cain is defined as follows Equation 1 and 2: Q ( t) = P[ X = j, T T t X = i] (1) ij n+ 1 n+ 1 n n Pij = lim Qij ( t) (2) t And it is necessary to introduce te probability tat te process may leave state i in a time t as Equation 3: m H ( t) = P[ T T t X = i] = Q ( t) (3) i n+ 1 n n ij j= 1 Foucer et al. (2005) defines te density probability function, of te waiting time in state i before passing to state j as follow Equation 4: P( x < T 1 1, ) (, ) lim n Tn x X n j X n i fij xθ + < + + = = ij + 0 = (4) were, θ ij is te parameter vector of te density probability function f ij (,). Te distribution and te value of parameters can vary between transitions. As usual in survival analysis, we deduce from f ij (x) te corresponding survival function and azard function, respectively S ij (x) and g ij (x) Equation 5: ij P( x < T T < x + T n+ 1 n n+ 1 T x, X = j, X = i) G ( x) lim = n n+ 1 n (5) It is assumed tat process spends some time in a given state and random time as distribution G ij (x), Weibull distribution: -Te azard function is defined as Equation 7: ij 1 θij 1 Gij ( x) = θij x, x 0 σ ij θ (7) Foucer et al. (2005) also defines te azard function wit covariates as follow:- T 1 2 ij ij 0, ij βij ij ij ij ij ij G ( x, z ) = G ( x)exp( z ), were z = ( z, z,..., z ) te vector of n ij covariates, specific to te transition i j. In te same paper, tey proved tat te Parameter estimations and likeliood metods Equation 8: m L = { P f ( T T )} r= 1 X X X X, r r Z X { S ( T T )} X, r r Z X r 1, r r 1, r, 1,, X r r 1 r 1., 1,, X r r 1 1 δ, r δ n ij, r (8) were, δ,r is equal to 1 if te transition r is observed for te individual and 0 if censored Semi-Markov for Predicting te Probability of Waiting Time After solving te evolution equations of te semi- Markov model, it is appropriate to give some concrete applications of tese processes as models of evolution of te probability of waiting time of some system. In tis part, Numerical solution of te Homogeneous Semi-Markov model is described by using te notation of Giuseppe et al. (2007). For any omogeneous semi-markov process {X(t), t 0}, te transition probabilities are given by (9) for wic te solutions sould be obtained using te progression (10): ϕ ij ( t) = P[ X ( t) = j X (0) = i] (9)

4 m t ϕij ( t) = (1 Hi( t)) δij + Qil ( τ ) ϕlj ( t τ ) dτ l= 1 0 (10) Here δ ij represents te Kronecker delta. An approximate solution of (10) can be obtained using te general numerical integration formula given in Corradi et al. (2004). In te same paper, tey proved tat te numerical solution of te process converges to te discrete time HSMP described as an infinite countable linear system Equation 11: m k ϕij ( k) = dij ( k) + vij ( τ ) ϕij (( k τ ) ) l= 1 τ = 1 (11) were, stands for te step measure of te approximation and: 0 if i j dij ( k) = 1 H i ( k) if i = j 0 0 if k vij ( k) = Qij ( k) Qij (( k 1) ) if i = j In matrix form, Equation 12 becomes: k Φ ( k) V ( τ ) Φ (( k τ ) ) = D ( k) τ = 1 (12) (13) Te fact tat te matrix Φ (k) is stocastic is already proved in Corradi et al. (2004; Janssen and Manca, 2001). For solving te evolution Equation 13, Corradi et al. (2004) proposed te following algoritm wit suggested matrix form Equation 14: V m T P G T Q T Φ T D T T T T Φ = D (14) Te variables involved are te following: = number of states of SMP = number of periods to be examined for te transient analysis of SMP = matrix of order m of te embedded Markov cain in SMP = square lower-triangular block matrix of order T+1 wose blocks are of order m = It represents te kernel of SMP = block vector of order T+1 te block of wic are square matrices of order m = block vector of order T+1 te block of wic are te diagonal square matrix of order m 24 V T S T = square lower-triangular block matrix of order T+1wose blocks are of order m = block vector of order T+1 te block of wic are te diagonal square matrix of order m. Te diagonal element of eac block t are sii = Qij ( t) m j= 1 Given an epoc T is fixed, matrices G T and P, te algoritm solves te linear system (14) for te unknown matrix Φ T by means of a purely iterative procedure. Te algoritm is: (0) Read te inputs: m, T, P, G T (1) Construct: Q T, V T, D T V = I ; Q = 0; S = 0; D = I (0) (0) (0) (0) for t = 1 to T Q( t) = P * G( t) for i = 1 to m sii( t) = Q ( t) 1 end for * i V( t) = Q( t) Qt 1 D( t) = D(0) S( t) end for (2) Given Φ (0) =D (0), solve for Φ T for t = 1 to T Φ = D ( t) ( t) for s = 1 to t Φ = Φ + V Φ ( t) ( t) ( s) ( t s) end for end for (3) Print te results, Φ T, Q T In te above ( ) represent te usual row column matrix product ( ) stands for element by element product and (1) is te m-component s sum vector. In tis study, a computer program for solving te evolution equations is developed in te R statistical software version RESULTS AND DISCUSSION 3.1. Descriptive Statistics Te study analyzed data obtained from 1456 AIDS patients during seven years of follow-up. Female represent about 58% of individuals and 42.5% of

5 transitions concerns patients over 40 years old. Te proportion of a patients wo is not drug addicted accounted a larger proportion in te sample (about 52.0%) compared to tose of addicted patients (42.5%). Te information presented above is summarized in Table Multivariate Analysis According to univariate strategies, 10 factors out of 72 possible (6 covariates * 12 transitions), were selected. Finally, te multivariate model uses te 8 regression parameters given in Table 2. Women tend to move quickly from state 2 to state 1. More precisely, Female patients were 1.6 times more likely to move from state 2 to state 1 tan tose of male patients (adjusted HR = 1.60, CI = ). Similarly, Te azard of moving from state 3 to state 2 for female patients is about 23% iger tan for male patients (adjusted HR = 1.23, CI = ). Patients, wo is not drug addicted, were 2.49 times more likely to move from state 3 to state 2 tan tose of drug addicted (adjusted HR = 2.67, CI = ). Patients wit tuberculosis were 2.67 times more likely to move from state 3 to state 4 tan tose wit no tuberculosis (adjusted HR = 2.67, CI = ). Similarly, patients wit tuberculosis were 2.28 times more likely to move from state 2 to state 4 tan tose wit no tuberculosis (adjusted HR = 2.28, CI = ). Te azard of moving from state 3 to state 4 for a patients less tan 40 years of old were about 2% lower tan tose of a patient greater tan 40 years old (adjusted HR = 0.98, CI = ). Table 2 reveals tis in detail Results of Semi-Markov for Predicting te Probability of Waiting Time Table 3 and Fig. 2 sows tat te result of semi- Markov for predicting te probability of waiting time. Te conditional probability tat a patient stays in state one, two and tree for at least 30 monts are 0.167, and respectively. It is increasing wit increasing seriousness of te disease. Witin te good states, it is more likely for a patient to stay in a worse state tan in a better one. Of course, te deat state is an absorbing state, i.e., once a patient enters te deat state e/se stays in same state forever. It is also interesting to find out tat te conditional probability of staying in same state until a given number of mont decreases wit increasing time. So tere is a possibility of canging from one state to anoter wic is a non-zero probability. Table 3 and Fig. 2 reveal tis in detail. 25 Fig. 2. Te probability tat a patient stays in same state of disease for at least t monts Table 1. Caracteristics of 1456 patients diagnosed wit AIDS under Antiretroviral Treatment (ART) Variables Frequency (%) Sex [n (%)] Female 853(58.6) Male 603(41.4) Drug Addicted [n (%)] No 757(52.0) Yes 619(42.5) Age [n(%)] <40 922(63.3) (36.7) Baseline Weigt [n (%)] <60 kg 925(63.5) 60 kg 531(36.5) TB Status [n (%)] Negative 1085(74.5) Positive 371(25.5) Baseline Immunological State [n (%)] SI: CD (23.9) SII: (200<CD4<500) 549(37.7) SIII: (CD4 350) 559(38.4) Transition between States [n (%)] (10.4) (3.6) (8) (1.9) (14.1) (8.7) (11.5) (3.4) (6.7) (15.5) (10.8) (5.4)

6 Table 2. Multiple multi-state model in patients diagnosed wit AIDS under ART, Amara Referral Hospitals, Amara Region, Etiopia Adjusted HR Adjusted HR Variables Transition (95% CI for its HR) p-value Sex Female Vs male* (1.02, 2.49) 0.040** (1.01, 1.48) 0.047** (0.34, 0.85) 0.008** TB status Positive Vs negative* (1.52, 4.68) 0.001** (1.29, 3.99) 0.004** Drug addicted No Vs Yes* (1.48, 4.19) 0.001** (0.97, 0.98) 0.019** Age 40 Vs >40* (0.97, 0.99) 0.000** (*) Reference Category; (**) Hazard Ratio (HR) is a significant at α = 0.05 Table 3. Te probability tat a patient stays in same state of disease for at least T monts States T = 1 T = 5 T = 10 T = 15 T = 20 T = 25 T = 30 T = 35 T = 40 T = 50 T = 60 T = 70 State State State Discussion Te Markov process framework developed by Kay (1986), continued by Gentlemann et al. (1985) and applied by Anderson et al. (1991; Foucer et al., 2005; D'Amico et al., 2009; Giuseppe et al., 2007; Gosu and Dessie, 2013), for studying te effect of disease indicators on survival as been used in tis study for constructing survival probabilities for different states of disease progression on HIV/AIDS, introducing nonomogeneity in time. Te present paper is also to estimate transitions between states based on CD4 counts and to predict te probability of waiting time in te starting state until mont t. Te use of te multi-state model, as compared to te Cox regression, improved substantially te understanding of variation in risk factors related to te evolution of tis cronic. Foucer et al. (2005) used multi-state models to estimate transitions between states based on VL and CD4 counts. Te most significant difference between studies tat use multi-state models is te definition of te different states. Altoug CD4 count, VL and te incidence of opportunistic infections are used to assess AIDS progression from a clinical point of view, few studies ave used tese multiple variables. Te ig cost of te VL assays is a major obstacle to te adoption of tis test, terefore restricting its availability compared to CD4 cell counts test. As a result, we found tat gender, age, drug 26 addicted and TB status were predictors of te intensity of transitions between different states. Tis result is similar to te result obtained in previous study (Oliveira et al., 2013; Foucer et al., 2005). Te conditional probability of waiting time in te starting state until mont t decreases wit increasing of time; wile wit increasing CD4 count, te probability of waiting time in te starting state until mont t decreases. Specifically, AIDS patient wo is in te tird state of te disease may ave a igest waiting time until mont t compared to te patients wo is in te first and second stage of te disease wit a given time. Tis result is similar to te result obtained in previous studies (Gosu and Dessie, 2013; Giuseppe et al., 2007; Corradi et al., 2004). 4. CONCLUSION Te semi-markov process model is applied to capture te AIDS dynamic progression of a patient. Te model considers te randomness of te time tat a patient spends in a given state of te disease. Te following can be concluded from tis study. Wen estimating survival in patients wit cronic diseases, suc as AIDS, tat involve te transition between different states, it is fundamentally important to consider progression between te different degrees of cronic. AIDS multi-state modeling is capable of

7 identifying te factors associated wit eac transition between te degrees of cronic defined by CD4 count ranges. Te multi-state model analysis sowed tat te major predictors of te intensity of transitions between different states of HIV/AIDS patients were gender, age, drug addicted and TB status. Witin te good states, it is more likely for a patient to stay in a worse state tan in a better one at any time of te process. Te deat state is an absorbing state, i.e., once a patient is deat state e/se stays in same state forever. Te conditional probability of staying in same state until a given number of mont decreases wit increasing time. So tere is a possibility of canging from one state to anoter wit non-zero probability. Te dynamic nature of te AIDS progression is confirmed wit particular findings tat tere is more likely to be in worse state tan better one unless interventions are made. It is recommendable to keep up te ongoing ART treatment services in most effective ways wit te careful considerations of recent disease status of patients. 5. ACKNOWLEDGEMENT My sincere appreciation and tanks go to my staffs of Statistics Department at Bair Dar University for teir unreserved knowledge saring and cooperation. My appreciation also goes to Felege-Hiwot Referral Hospital, Etiopia ART members and clinic staff for teir cooperation during te time of data collection. Last but not te least; I would like to acknowledge my friend Addisu Abebe (PD in Economics) for teir kind advice and material support. 6. REFERENCES Anderson, K.M., P.M. Odell, P.W. Wilson and W.B. Kannel, Cardiovascular disease risk profiles. Am. Heart J., 121: PMID: Barbu, V.S. and N. Limnios, Semi-Markov Cains and Hidden Semi-Markov Models toward Applications: Teir Use in Reliability and DNA Analysis. 1st Edn., Springer Science and Business Media, New York, ISBN-10: , pp: 240. Baryarama, F., J.Y.T. Mugisa and L.S. Luboobi, An HIV/AIDS model wit variable force of infection and its application to te epidemic in Uganda. Am. J. Applied Sci., 2: DOI: /ajassp Blasi, J. and R. Manca, Numerical treatment of omogeneous and non-omogeneous semi-markov reliability models. Commun. Stat. Teory Met., 33: DOI: /STA Corradi, G., J. Janssen and R. Manca, Numerical treatment of omogeneous semi-markov processes in transient case-a straigtforward approac. Metodol. Comput. Applied Probability, 6: DOI: /B:MCAP D Amico, G., J. Janssen and R. Manca, Homogeneous semi-markov reliability models for credit risk management. Dec. Econo. Finance, 28: DOI: /s D'Amico, G., J. Janssen, D. Giuseppe and R. Monica, HIV progression troug two different temporal scales according to non-omogeneous semi-markov models. Proceedings of te 13t International Conference on Applied Stocastic Models and Data Analysis (MDA 09), pp: Davidov, O. and M. Zelen, Designing cancer prevention trials: A stocastic model approac. Stat. Med., 2: DOI: / ( )19:15<1983::AID- SIM534>3.0.CO;2-E Foucer, Y., E. Matieu, P. Saint-Pierre, J. Durand and J. Daures, A semi-markov model based on generalized Weibull distribution wit an illustration for HIV disease. Biometrical J., 47: DOI: /bimj Gentlemann, R.C., J.F. Lawless, J.C. Lindsey and P. Yan, Multi-state Markov models for analysing incomplete disease istory data wit illustrations for HIV disease. Stat. Med., 13: DOI: /sim Gil, J., A. Alioum, C. Ketzoian, J.C. Desport and M. Druet-Cabanac et al., Disease progression and survival in ALS: First multi state model approac. Amyotrop. Lateral Scler., 8: DOI: / Giuseppe, D., G. D Amico, A. Girolamo, J. Janssen and S. Iacobelli et al., A stocastic model for te HIV/AIDS dynamic evolution. Mat. Problems Eng., 2007: DOI: /2007/65636 Gosu, A.T. and Z.G. Dessie, Modelling progression of HIV/AIDS disease stages using semi- Markov processes. J. Data Sci., 11:

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