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Multi state Markov model with no classification errors with application to vertical transmission

Disease progression occur through a number of states making the use of transition models applicable in estimating the transition probabilities from different states. Transition probabilities are often expressed in terms of transition intensities(rates). Multi-state models based on Markov process are a well established method of estimating rates of transition between stages of the disease. One of the well known multistate Markov models is the birth-death model. In this paper we obtain transition probabilities for a four-state model in a process of a vertical transmission of HIV. We have considered a case of no errors of classification. A system of Kolmogorov forward equations are formed from the Chapman Kolmogorov equation and used to obtain the expressions for the transition probabilities which are in form of transition intensities. Using estimates for the transition intensities we are able to estimate the transition probabilities over a period of 50 years. We apply our data to the four state Healthy, Infected, Aids and Death model. It is observed that the probability of remaining in a state or transiting from one state to another generally decreases with time.