State probabilities and hazard functions

For v 1, in case of an Erlang distribution, the rate function at age 0 is h (0) = 0, after which the rate increases and the kinetic profile has a log-concave 

The Weibull distribution allows noninteger shape parameter values, and the kinetic profile is similar to that obtained by the Erlang distribution for p, 1. When 0 p 1, the kinetic profile presents a log-convex form and the hazard rate decreases monotonically. This may be the consequence of some saturated clearance mechanisms that have limited capacity to eliminate the molecules from the compartment. Whatever the value of p, all profiles have common ordinates, p(l/X) = exp(-l). 

Consider the irreversible two-compartment model with survival, distribution, and density functions Si (a), F (a), /i (a) and So (a), 72 (a), /2 (a) for ages a of molecules in compartments 1 and 2, respectively. We will assume that at the starting time, the molecules are present only in the first compartment. The state probability p (t) that a molecule is in compartment 1 at time t is S (a) with t = a the external time t is the same with the age of the molecule in the compartment 1, i.e., pi (t) = Si (t). The state probability p2 (/,) that a molecule survives in compartment 2 after time t depends on the length of the time interval a between entry and the 1 to 2 transition, and the interval I, a between this event and departure from the system. To evaluate this probability, consider the partition 0 = ai a.2 o.n 1 an = t and the n — 1 mutually exclusive events that the molecule leaves the compartment 1 between the time instants a, i and a,. By applying the total probability theorem (cf. Appendix D), p2 (t) is expressed as 

This result can be generalized in the case of a catenary irreversible m-compartment model [347] the state probability in the compartment i (i = 2, rn) at t is given by 

An elegant form of the previous expressions is obtained in the frequency domain. The convolution becomes the product of the Laplace transform of the survival and the density functions  

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Figure 9.3 State probabilities and hazard functions with A = 0.5h 1, and u = 1,2, 3 and p = 0.5,1,1.5 for Erlang and Weibull distributions, respectively.

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