Stochastic theory of compartments

BULLETII~ OF

~ATHE~ATIC~ BIOLOGY VOLUME 36, 1974

STOCHASTIC T H E O R Y OF COMPARTMENTS

[ ] P~TER PURDU~ Department of Statistics, University of Kentucky

The stochastic model of a compartment developed by Thakur, Rescigno and Sehafer is discussed without using generating functions. The behavior of the mean and variance of the number of particles present as a function of time is also discussed. We also allow both the input and output to be time dependent.

1. Introduction.

In a recent paper Thakur et al. (1972) discuss the study of a compartmental system as a stochastic process. In this model the input process is allowed to be time dependent but the output process is stationary in time. Also for the special ease where the initial number of particles in the system is determined t h e y discuss the time behavior of the moments. The basic tool in this analysis is the use of generating functions. Our aim in this paper is twofold. We first of all allow for both the input and the o u t p u t processes to be time dependent. Secondly we discuss the moment behavior when only the probability distribution of the number of particles in the system at time zero is known. Also we show how the analysis can be carried out without the use of generating functions.

2. Mathematical Model. Let X(t) denote the number of particles of some molecular species which is present at time t. As time progresses particles m a y enter or leave the system and so X(t) varies with time. The rules governing this variation are given by the following: (i) The Input Process.

Particles enter the system according to a non305

306

PETER PURDUE

homogeneous Poisson Process with intensity function h(t). This means that the probability of k arrivals in the time interval (0, t] is given by, e -re(t) [m(t)]~ k! where

/c = 0, 1, 2,

"'"

rt re(t) = J0 ~(~) dT.

(ii) The Output Process. Particles leave the system according to a pure death process with death rate t~(t). This means that the probability of a partitle leaving the system in (t, t + h) is ~(t)h + O(h) where 0(h) is an infinitesimal of higher order than h. (iii) Independence Criterion. It is assumed that particles in the system live or die independently of one another. The description given above is equivalent to that of Thakur et al. with the difference that we allow ~(t) to be a function of time. Under these assumptions it is clear that X(t) is a continuous time Markov chain with state space {0, 1, 2 . . . . }. The process can also be looked upon as a migration-death protess--see Bailey (1964, p. 115). However, instead of using generating functions we choose to analyze the process here b y a direct random variable technique. We believe that the direct approach using random variables avoids the analytic details of the generating function approach and leads to a better understanding of the basic probabflistic structure of the problem. Let,

P~(t) = P[X(t) = i]. 3. The Distribution P~(t). We begin b y showing that X(t) can be expressed as the sum of two independent random variables. Then we proceed to discuss the random variables in detail. Let, Y(t) = Number of particles present at time 0 which are still present at time t. Z(t) = Number of particles which arrive in (0, t] and are still present at time t. Then because of assumption (iii) we have Y(t) and Z(t) are independent and x ( t ) = y(t) +

z(t).

(1)

We proceed now to an analysis of Y(t) and Z(t). Let N(t) denote the total number of particles which enter the system in (0, t]. Then,

P[Z(t) = k] = ~ P[Z(t) = k i N ( t ) = n]P[N(t) = n].

(2)

STOCHASTIC THEORY

OF COMPARTMENTS

307

But, since PIN(t) = n] is known, this becomes, P[Z(t) = k] = e -re(t)

P[g(t) = k I N ( t ) = n]

[m(t)]~ n!

(3)

Consider now a single particle which entered the system at sometime x < t. Then the probability that this particle is still present at time t is just exp { - f tx t~(T) dr). Consequently, P [a particle entering the system in (0, t) is still present at time t] /,t A(x) x

= JoTn-ffJe p { - ;

~(')

dT~ dx.

~

(4)

I f we let,

then this probability can be expressed as h(t)/m(t). Since, by assumption, particles behave independently we have immediately by (4) .P[Z(t) = / , ; I N ( t )

r

= n] = \ / ~ ] L ~ ) j

h(t)1

.1 - - ; ~ j

•

(5)

F r o m (3) and (5) it follows t h a t

= =

~=~

\n] Lm(t)j

[,_ h(t) l"-~ [m(t)]n m-~J

n!

e_m(t)[h(t)]~ ~j, 1 [1 - h(t)] n-k k! . ~ (n - ~)! ~-T~J [m(t)]~-'~

= e_h<~)[~,(t)]~ k!

So, Z(t) is a random variable which has a Poisson distribution with parameter h(t). Consequently, E[Z(t)] = Var [Z(t)] = /t(t).

(6)

Turning now to the random variable Y(t) we will show t h a t it can be expressed as a random sum of random variables. L e t B(t) be a random variable which is equal to 0 or 1 depending upon whether a particle which is present at time 0 is still present at time t. Then, x(o)

Y(t) = ~ B,(t).

(7)

and P[B~(t)= 1 ] = exp{--J~t~(T) dT} Bo(t ) = 0

t > O.

i=

1, . . . . X(0)

(8)

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PETERPURDUE

Our final result then is, x(o)

X(t) -= ~, B,(t) + Z(t)

(9)

~=0

where B,(t) are independent and identically distributed Bernoulli random variables with parameter exp ( - f ~ t~(r) dr} and Z(t) is independent of the B~(t) and is Poisson with parameter h(t). In the special case where t~(z) = t~ for all r > 0 this corresponds to the results obtained via generating functions by Thakur et al. (1972).

4. Moment Behavior. In this section we examine the mean and variance of X(t) as a function of time. Let. E[X(0)] = No

(lO)

Var [X(0)] -- ao2. We first of all consider the mean of

X(t). F r o m (9),

E[Xit)]= E[iXo)]E[Bit)]+E[Z(t)]=

oexp(-fi,(r) dr} + h(t).

ill)

I n the special case where X(0) = x o we of course have,

EXit)= xo exp { - ; tL(r) dr} + h(t). For the variance of X(t) we use the known result (Parzen, 1962, p. 56)

)~o ] = E[(Xo)] Var [B(t)] + Var Var LFX( [ ~ OB,(t)

(Xo)[E(B(t)}]2.

Consequently, Var [X(t)] =/~o exp { - ; /.~(r) d r ) +(a~-

t~o)exp { - 2 f l tL(r)dr} + h(t).

(12)

I f we consider the special case where t~(r) = t~ for T > 0, then i12) gives, Var [X(t)] = t~o e -"~ + (a~ - t~o) e -2"~ + f : e-"a-~)h(r) dr. This gives an extension of the results reported in Thakur special case where X(0) = x o.

et. al. (1972) for the

STOCHASTIC THEORY OF COMPARTMENTS

309

Going back to the general ease we next consider the ratio of the variance to the mean for X(t). Var [X(t)]

/~o exp [ - fot/~(r) dr] + (ao2 -- Fo) exp [ -- 2 fot/~(r) dr] + h(t)

E[X(t)]

,o exp [-fo ,(r) dr] + h(t) { exp [-2fo~t~(r) dr] } = 1 + (a] -- /to) ~ o e x p [ - f o t/t(r) dr] + h(t) "

Then this ratio is < 1 whenever go2 - / t o < 0, i.e. when/~o > ao2 and equality occurs if and only if/to = ao2. Hence the result: whenever/to > ao2 then O<

Var [X(t)] El. E[X(t)]

From this we obtain the result that wherever/to > ao2 the relative standard deviation a~ (the ratio of a to the expectation) is less than or equal to [EX(t)] - 1/2. We remark that for the special ease ao2 = 0 this agrees with the results of Thakur et al. (1972). 5. Concluding Remarks. We have shown in this paper one approach to analyzing the mathematical model of a compartment when we allow the death rate to be a function of time. I f we want to consider a sequence of compartments where the output of one compartment is the input for the next we have to find the distribution function of the exit process. Toward this end we suggest that a compartment be viewed as a Queueing System and that known results on the exit process from a queue be used. This program is now under w a y and the results will be published at a later date.

LITERATURE Bailey, N. T . J . 1964. The Elements of Stochastic Processes with Applications to the Natural Sciences. New York: J o h n Wiley. Parzen, E. 1962. Stochastic Processes. San Francisco: Holden-Day. Thakur, A. I~., A. Rescigno and D. E. Schafer. 1972. "On t h e Stochastic Theory of Compartments: I. A Single Compartment System." Bull. Math. Biophysics, 34, 53-65. RECEIVED 4-26-73