An existence and uniqueness theorem for generalized birth and death processes

Annals of Nuclear Energy, Vol. 7, pp. 289 to 296 Pergamon Press Ltd 1980. Printed in Great Britain

AN EXISTENCE AND UNIQUENESS THEOREM FOR GENERALIZED BIRTH AND DEATH PROCESSES* V. CAPASSO Istituto di Analisi Matematica and Istituto di Matematica Applicata, Universita di Bari, 70121 Bari, Italy and S. L. PAVERI-FONTANA'~ Department of Engineering Science, State University of New York at Buffalo, Buffalo, N.Y. 14214, U.S.A. Abstract--A generalized version of the continuous-time birth-and-death stochastic model is formulated.

Employing semigroup methods, it is shown that the initial value problem subject to appropriate regularity requirements admits a unique solution for all positive times. Several examples from the biological and engineering sciences are given.

1. I N T R O D U C T I O N

Multitype branching and birth-and-death processes have been treated extensively in the literature in connection with problems originating from the biological, physical and engineering sciences. In this paper, a generalized multitype continuous-time age-independent birth-and-death branching model is studied. The theory of semigroups is employed to verify the existence and uniqueness of solutions for the pertinent abstract Cauchy problem. In the absence of sources and for the one compartment case, the process discussed here reduces to the classical Galton-Watson continuous-time branching process (see Harris, 1963). For the sourceless case, Athreya and Ney (1972) have studied the probability generating function counterpart of equations (5) and (6). More general models have been discussed by Mode (1971). Here our aim is to employ functional-analytic methods for the direct study of existence-anduniqueness questions for a fairly general--yet specific--class of continuous time multitype Markov processes. Alternatively, one could proceed by incorporating the model treated here in a more general framework, and then by employing results from the literature on Markov processes (see, for instance, Jagers, 1969; Ikeda et al., 1968, 1969; Savits, 1969). The semigroup method (Kato, 1966; Belleni-Morante, 1979) employed in this paper appears to be more straightforward. Previous applications of the semigroup method to some specific classes of birth-and-

death processes (Belleni-Morante, 1975; Capasso, 1977) are generalized here. Earlier applications of the semigroup approach to Markov processes may be found, e.g. Kato (1954), Reuter (1957) and BharuchaReid (1965). This paper is organized as follows. In Section 2 the problem is stated; in Section 3 the mathematical setting is established; in Section 4 the main existence and uniqueness theorem is proved. In the Appendix some examples from the applied sciences are presented. In the following by N we denote the set of non-negative integers I0, 1,2 . . . . I; by R the set of real numbers; by R+ the set [0, + ~ [ of nonnegative real numbers; by N N, with N E N - {01, the set of N-dimensional integer-valued vectors; by the infinitesum ~ n

n

289

~

...

NI:0

~

;

nN=0

for n = (nl . . . . . nN) ~ N N,

by r=0

we denote the fnite sum nl

nN

E - . . rN=O E; rl=O finally we set e 1 = (I, 0, 0 . . . . . 0), e 2 = (0, 1, 0 . . . . . 0), e 3 = ( 0 , 0 , 1 . . . . ,0), etc., with e k~N N for all k6{1,2 . . . . . N ] . 2. H Y P O T H E S E S

* This research was supported in part by the CNR (Italy) in the context of the Progr~m for Preventive Medicine. 5"On leave from Universith di Firenze, Italy.

= N

AND

DEFINITIONS

We study a stochastic system composed of N classes (compartments, or regions, or species, etc.),

290

V. CAPASSOand S. L. PAVERI-FONTANA

which, at any time t > 0, is characterized by the random vector fi(t)= (~dt), . . . . ~N(t)). Each component ~k(t) is a discrete random variable with values in N which denotes the number of elements in the kth class at time t. We suppose that the stochastic process (~(t))t~ R . i s a Markov process whose possible transitions are described as follows: 2.,,at + o(At) = Probability for an individual which is at time t in class k to be removed from class k during the time interval [t, t + At[, (l.1) Sk(t)At + o(At) = Probability that an individual is injected into class k from outside the system during the time interval It, t + At[, (1.2) n~(r) = Probability that the removal of an individual from class k results in the immediate simultaneous appearance of rl individuals in class 1, rE in class 2 . . . . . rN in class N. (1.3) Here k~(l . . . . . N}; r = (rl,r2 . . . . . rN) s N N ; o(At) denotes an infinitesimal of higher order than At. Observe that, for all k, nk(') is a probability function with state space NN; accordingly, we require: 0 < ~k(r) =< 1, for

k s {1,2 . . . . . N},

E~k(r) = 1,

k s [ l , 2 . . . . . N].

Remark 1 As illustrated by the examples given in the Appendix, the model discussed here is quite general. For instance, the model allows for migration processes among compartments (refer to parameters such as lk(e') with r ¢: k), for death events (refer to lk(O)), for immigration events (refer to sk), for simple or multiple birth events (refer to lk(n) with nl + n2 + . . . +nN > 2). On the other hand, there are limitations: for instance, the model does not allow for age dependent processes or for an infinite number of compartments. Finally observe that the processes to be studied are completely insensitive to the 'dummy' parameters lk(e k) ( k s {1, 2 . . . . . N}). It is usually convenient to set l,(e*) = rrk(e*) = 0 for all k. In the hypothesis that each individual in the system acts independently of each other, the significant transition probabilities of the process are Prob(h(t + At) = n - e k + rl~(t) = n)

= lk(r)nkAt + o(At), Prob(~(t + At) = n + ekl~(t) = n)

= sk(t)At + o(At), where n, r s N N. Then it is not difficult to show (Feller, 1957, ch. 7) that the evolution equations for the probability distribution at time t, P(n;t) = Prob{h(t) = n},

r ~ N N,

n S

N N,

are given by for

r

~ t P(n;t) = --[k=~ 2kn~+ s(t)]P(n;t)

Then if we let N

lk(r) = 2kr~dr),

+ ~ k=l

we have the constraints

~ tk(r)(nk + 1 -- rk) r=0 N

lk(r) => 0,

for

k ~ { 1 , 2 . . . . . N],

r ~ N N, (2.1)

× P(n+e k-r;t)+

~ Sk(t) k-1

0 <~/dr)=

2k< +~,

for

k s [ l , 2 . . . . . N}.

x P(n--ek;t),

n s N N,

(5.1)

r

(2.2)

where

Moreover..we recluire P(n;t)=0

O
+~,forks{1,2

. . . . . N},

t>0. (3)

if 3kE{1 . . . . . Nls.t. nk < 0 , (5.2)

and N

We shall also impose a regularity condition on the function s(.). In some parts of this paper, we shall add the requirement that, for all k, nk(') has a finite expected vector; that is, the requirement

~rilk(r)<+ao,

for

i , k ~ [ 1 , 2 . . . . . N}.

(4)

s(t) = ~ Sk(t).

(5.3)

k=O

System (5) must be supplemented by an initial condition P(n;0)=Po(n),

for

n ~ N N,

(6)

An existence and uniqueness theorem for birth and death processes such that

will be Po(n) e[O, 1],

for

n e N N,

Po(n) = 1,

useful

(7.1) (7.2)

introduce

the

positive cone

X+ = {f eXqYneNN:f(n) > 0].

(11)

Now we define the following operators from X to X :

n

N

A solution P(n;t) is sought for problem (5~(7), subject to the requirements P(n;t)E[0,1],

to

291

for

n ~ N N, t > 0 ,

[Af](n) = -

k-I

+

(8.1)

~ /k(r)(nk+ 1 - - r k ) f ( n + e k - r ) , k=l

P(n;t) < 1, for

t>0.

~ 2knkf(n)

(12)

r-O N

(8.2)

[R(t)f](n) = ~ sk(t)f(n - ek), t E R + ,

n

(13)

k=l

Now, consider the case in which where we set f(n) = 0 if 3k~{1 . . . . . N} s.t. nk < 0. Here [Af](n) denotes the nth component of the transformed vector A f The domain ~(A) of A is assumed to be

def

P~(t') = 1 - Z e ( n ; t ' ) >

0

n

for some time t" > 0. The customary interpretation is that at time t" there is a non-zero probability, P~(t"j, for the population in the system to be infinite. A Markov process is said to be 'honest' when such an explosive situation cannot occur. Specifically, then, the Markov process treated here is 'honest' if requirement (8.2) can be replaced by the more stringent constraint P(n;t) = 1,

for

t>0.

~(A)= { f ~X ~ [ A f ] ( n ) < z c } , while

~(R(t)) = X,

Vt~R+.

(15)

We also introduce the following auxiliary operators

(9)

N

[Uf](n) =

n

As we shall show, in agreement with results from the literature (Savits, 1969; Athreya and Ney, 1972), the non-explosive evolution of a process, equation (9), is connected to the validity of constraint (4).

('4)

~ ).kn~f(n),

(16)

k-1

[Kf](n) = ~ k=l

~ /k(r)(nk+ 1 - - r k )

f(n+e k-r).

r-0

(17) It can be easily shown that we can assume

3. THE MATHEMATICAL SETTING

~ ( n ) = @(g) = 2 ,

Our main purpose is to prove that for a suitable choice of the initial condition Po(n) which satisfies requirements (7), for a suitable choice of the 'source' functions Sk(t) which satisfy requirements (3) and for any choice of the parameters lk(r) which satisfy requirements (2), the initial value problem (5)-(6) admits a unique solution P(n;t) which satisfies requirements (8) for all times t ~ R + (but which is not necessarily 'honest'). We introduce the Banach space X of all N-indexed sequences of real numbers

(18)

where we have denoted by c~ the subset of X

~={fEX

~k~xnkf(n)

<~}.

(19)

Remark 2 Observe that definition (19) is motivated by the following reasoning. If (f(n)),~NN denotes a probability distribution of the process at some time t¢ R+, i.e, if f(n) = P(n ; t) ~ [0, 1], and

f = (f(n))n~NU ~ R(NN),

ZP(n;t)

such that

= 1,

n

[If It = ~ tf(n)l.

(10)

then the mean values

ii

Since we are dealing with probabilities, P(n; t), it

Q]k(t)> = El-r]k(t)] = y" nkf(n) n

A.~.E, 7/4-5

(;

(20)

292

V. CAPASSO and S. L. PAVERI-FONTANA

of the random variables Ilk(t), exist and are finite for all ke{1 . . . . . N}

ifffe~+

Proof Part (i) of this lemma follows directly from the definition (16) of H and from the explicit expression of (cd + H) 1

def

= 9~X+.

Moreover, let 9 0 be the linear manifold

VfEX,

9 o = {re X If has only a finite number of non zero components}.

V n e N N, V a > 0 : [ ( a l + H ) - l f ] ( n )

(21) =

It results to be dense in X and it can be shown that 9 0 c 9 c 9(A) c X,

(22)

from which we conclude that 9 itself is dense in X. The restriction of A to 90 will be denoted by Ao (i.e. A o = Al~o). By means of definition (12), the abstract version of system (5)-(7) can be written in the following form: d ~tu(t) = (A + R(t)-s(t)l)u(t),

2knk

f(n).

k=l

Part (ii) follows from the observation that (~l + H)- 1 e ~(X) implies (~l + H) e c£(X) and the equality H = (~I + H) - ~I e ~(X). Part (iii) is an obvious consequence of the definition ((13) of R(t). •

Lemma 2 I f assumptions (2) hold, then :

t > O, (23.1)

lim Ilu(t) - uoll = 0, t~0

~ +

(i) K [ 9 + ] c X+ (ii) Vf~ 9 : IlKfll< IlHfll; Vfeg+ : IIKfl[ = IIHfll.

+

where

Proof Uo = (P0(n)).~NN

(23.2) K.

is the initial state vector, u(t) denotes the state vector (P(n;t)),~N~ at time t, and d/dt denotes a strong derivative in X (see Kato, 1966, ch. 3). Before studying the initial value problem (23), we shall investigate some properties of operators H and K, along the lines suggested by Kato (1954). Employing standard notation let ~(X) be the space of bounded linear operators from X into itself, let oK(X) be the set of closed linear operators in X (Kato, 1966, ch. 3). We have

Lemma 1 I f assumptions (2) hold, then:

Lemma 2 directly follows from the definition (17) of • Now let

G, = - H + rK, r e [0,1],9(G,) = 9.

(24)

Then the following lemma holds as a consequence of Lemma 1 and Lemma 2 (see Kato, 1959, 1966, ch. 9; Beileni-Morante, 1979, ch. 13).

Lemma 3 I f assumptions (2) hold, then: (i) Vre [0,1],G r is the infinitesimal generator of a strongly continuous semigroup Zr(t) = exp (tGr), t e R+. (ii) V t e R +, Vre[0,1]: Zr(t)(~._~(X), IIZr(t)l < 1; Zr(t)l-X+] = X+. (iii) Vr,r'e[0.11, r < r ' , VfeX+, V n e N N, VteR+: fZ,(t)f](n) _-< [Zr,(t)f](n). (iv) VfeX,

VteR+ 9Z(t)f= lim Zr(t)f.

•

(i) V~ > 0 :(~I + H)-1 e .~(X), II(~I + H ) - 111 < -1,

(cd + H ) - I [ X + ] c X+.

Thanks to Lemma 3, we obtain the following theorem (see Kato, 1954).

Theorem 1

(ii) HE cg(X), H [ 9 + ] c X+.

I f assumptions (2) hold, then: (iii) VteR+ :R(t)e ~(X),

IIR(t)ll < s(t), R(t)[X+] c X+ ;

VteR+,

VfeX+ :

IIR(t)fl[ = s(t)llfll.

(i) (Z(t))teR+ is strongly continuous semigroup of linear operators such that V t e R + : Z(t)e,~(X), IIZ(t)ll _-< 1, Z(t)[X+] c X+.

An existence and uniqueness theorem for birth and death processes (ii) The limit in (iv) of Lemma 3 holds uniformly in each finite interval of time [0,i]. (iii) I f G is the infinitesimal 9enerator of Z(t), then

Let now be n # 0. Equation (28) gives lk(r)nk f * ( n -N a + ~ 2knk

N

f*(n) =

A0c -H+KcGcA. (iv) Z(t) is the smallest positive semigroup whose

9enerator is an extension of Ao.

293

2 Z k=l

r

+ r), (29)

ek

k=l

•

For the results obtained so far, we did not require that the coefficients lk(r) obey constraint (4). Now we have:

from which

Lemma 4

I & ( n ) l _-< Y, ~ k=l

I f equations (2) and (4) hold, then

N ~=

r

a +

Iq~*(n -

ek

+ r)l,

~ 2knk k=l

(30)

V t e R + , YfeX+ : nZ(t)fll = Ufll.

Proof In order to prove Lemma 4, we only need to prove the necessary and sufficient condition (ii) of theorem 3 in Kato (1956) according to which an at > 0 exists such that the characteristic equation

ASdf * = ctf *

where we have set Vn # 0 : ~b*(n) -

f_*fn) .

Y" nk k=l

(25)

admits no solution f * # 0. A* is the adjoint operator of A0 defined in X*, the conjugate space of X ; i.e. the space of all bounded linear forms on X defined by sequences IIf* II = {f*(n))neNN e R (NN)

If f * is a nontrivial solution of equation (28), q~* will be such that iim Nb*(n)l = 0,

which implies that

with norm

3 h e N N s.t./~ = IqS*(fi)l = maxl&(n)l. IIf*ll = sup

If*(n){

n

< oo.

n

From equation (30) we thus obtain It can be shown, following a method similar to that used in section 5 of Belleni-Morante (1975), that

N

N

k-1

k=l

i=1

#
N

[A*f*](n) = -

N

N

~ 2knd*(n)

,

(31)

k=l

k=l N

+ ~. ~ l k ( r ) n k f * ( n - - e k + r ) , k=l

(26)

which is impossible if m

N

k=l

i=1

t

(32) where we assume as domain of A* :

~(A~) = { f * e X * I A ~ f * e X * } .

(27)

Let now a > 0, f * • ~(A*), and consider the equation (st - A~)f* = 0.

(28)

If n = 0 ~r (0,0 . . . . . 0) E N N, in particular we obtain from equation (28): [ ( a I - A'~)f*](0) = 0

from which cif*(0) = 0 and then f*(0) = 0.

unless/1 = 0, which implies that also VneN N-

1,0j~:tk*(n)=0

and

f*(n)=0.

In equations (31) and (32) (r/)k = ~ lk(r)ri,

i, k~{1 . . . . . N /

(33)

r

To conclude, for any ct > ~t0 equation (28) admits only the trivial solution, as desired. •

294

V. CAPASSOand S. L. PAVERI-FONTANA 4. THE INITIAL VALUE PROBLEM

In this case we also have

Due to the results of Theorem 1 the initial-value problem

drw(t)I] = Ilexp(tG)u0 II

d

dtU(t) = Gu(t),

+

t > O,

f2

IJexp[(t - r)G] II IlR(r)w(r)lJ dr,

(34)

lim u(t) = Uo e ~(G),

Ilu0 I[ +

s(r) rlw(r) ll dr,

t > 0,

~-~0 +

subject to equation (2) and to definitions (10)-(19), admits a unique solution on R+. This has the form

u(t) = exp (tG)uo e ~(G), t > O.

VteR+,VfeS+:lexp(tG)fll

< Ilfll.

IIw(t)H < exp

(35)

It is useful to recall that VtER+ :exp(tG)[X+] c X+,

from which, applying Gronwall's lemma,

(36) (37)

If restriction (4) is satisfied, then an equality sign should replace the inequality in (37). Now we can state the following:

s(z)dr

),

u011, t > 0.

(40)

Going back to the original problem (23) we can observe that if u(t) is a solution of system

d

~ u ( t ) = (G + R(t) - s(t)l)u(t),

t > O,

/

lim u(t) = Uo e ~(G), t~O

(41)

+

then the function

Theorem 2 If the coefficients lk(r) obey (2), Uo e ~ ( G ) ~ X+ and HUol[ = 1, then u(t) = e x p ( t G ) u o e ~ ( G ) c a X + and [In(t)[/ < I at any t e R + . Moreover if the coefficients lk(r) satisfy also the additional requirement (4), then Ilu(t)[[ = 1 at any t e R + . In the following we will suppose that the 'sot~rces' Sk:R+ --*R+,

are continuously differentiable functions; in this case it can be easily shown that the function g : t e R + F-~g(t) = R ( t ) f e X is (strongly) continuous with its first (strong) derivative, for a n y f e X. Hence the following problem

dt W(t) = (G + R(t) )w(t),

t>0,

is a solution of system (38). Vice versa, if w(t) is a solution of system (38), then

u:t~R+ ~ u(t) = e x p ( -

ke{1 . . . . . N},

d

u:teR+ ]-*u(t)=exp(-/oS(Z)dr)w(t)e~(G)

/

fo'S(r)dr)w(t)~ ~(G)

is a solution of system (41). Hence system (41) itself admits a unique solution u(t) such that Vte R + : u(t) • ~(G). In particular if Uoe@(G)mX+, we have V t e R + : u(t)e ~(G) ca X+. Furthermore we have IIw(t)ll --- exp

(f:) s(r)dr

Ilu(t)ll, t > 0,

and, equivalently (38)

lim w(t) = Uo e ~(G),

Ilu(t)ll : exp -

t~O*

is equivalent to the Morante, 1979, ch. 5)

w(t) = exp (tG)uo +

integral equation

(Belleni-

f2

exp((t - r)G) R(r)w(r) dr, (39)

which, with the standard technique of successive approximations, can be shown to admit a unique solution such that if Uo • ~(G) c~ X+, then Vt e R+ :

w(t)e ~(G) ca X +.

s(z)dz IPw(t)Pl 5 P/uoll, t ~ 0, (42)

the inequality being due to equation (40). On the other hand, due to (iii) of Theorem 1, G is the restriction of A to ~(G); hence, if u(t) is the solution of system (41), one has

Vt e R+ : Gu(t) = Au(t). The treatment can be repeated for the case in which the coefficients lk(r) satisfy the additional requirement (4). In this case, inequality (42) becomes an equality.

An existence and uniqueness theorem for birth and death processes We can then state the following Theorem 3 I f requirements (2) are satisfied, Uo ~ ~(G) ~ X+ with [Iu0 I] = 1, and S k 6 ~ l ( R + , R + ) f o r k ~ {1 . . . . . n}, then a unique solution u(t) of system (41) exists on R+. Moreover Yt~R+:u(t)e~(G)c~X+,

[[u(t) I < l l u 0 [ =

1. (43)

In addition, if equation (4) is also obeyed, then in equation (43) the inequality can be replaced by an equality. • Theorem 3 is the main result of this paper. In terms of the original problem (5)-(6), or of its abstract version (23), Theorem 3 permits us to claim t h a ~ f o r the existence and uniqueness of solutions satisfying requirements (8)~it is sufficient to require the following: (a) the initial condition Po obeys equations (7) and it belongs to ~(G) c~ X+ ; (b) for all k~ {1,2 . . . . . N}, sk(t ) and dsk/dt are continuous functions of t on R+, and sk(t) obeys inequality (3); (c) the parameters lk(r) obey equations (2). Moreover, if the additional condition that all the rtk'S admit a finite expected vector, equation (4), is satisfied, then the Markov processes are 'honest' (equation (9)). Some comments are appropriate: Remark 3 The physical interpretation of assumption (a) is hindered by the fact that we do not know explicitly the form of ~(G). However, we do know that 9 + c ~(G). On account of definition (18) we are then in a position to claim t h a t ~ f o r the existence-and-uniqueness of solutions of our original problem (5)-(6)--a sufficient condition is that the initial probability Po satisfies equations (7) and has finite expected vector (provided, of course, conditions (b) and (c) are met). Remark 4 The hypothesis that the 'sources' Sk(t) are continuously differentiable functions on R+ can be weakened by requiring Sk(t) and dSk(t)/dt to be piecewise continuous functions on R + . I n fact, let tz, t2 E R+ be the first and second point, respectively, at which s(t) is not continuous with its first derivative. Our treatment can then be applied to the interval [0,tl]. Now, if u(tl) is the solution of system (41) at q , we can consider the new initial value problem d ~ u ( t ) = (G + R ( t ) - s(t)l)u(t), tl < t < t2,

295

(strongly) continuous solution of the integral version of system (41); this function satisfies system (41) for all the values of t6 R+ where s(t) and ds(t)/dt are continuous. • Remark 5 Requirements (2) on the parameters lk(r) are natural consequences of the definition. We have found also that the additional requirement ~rnk(r)

<+oc

(for all k c [ 1 . . . . . N ])

constitutes a sufficient condition for the process not to 'blow-up' in finite time (equation (9)). Note that the link between non-explosive processes and 'reproduction' processes involving a finite mean number of progenies per individual (or related conditions) has been discussed, for instance, by Harris (1963), Savits (1969) and Athreya and Ney (1973). • 5. CONCLUSIONS

In this paper, we have analyzed some mathematical features of a generalized birth-and-death branching process, employing a semigroup-theoretic procedure. As discussed in Section 4, we have established existence-and-uniqueness properties for the process described in Section 2, provided the following sufficient conditions are satisfied: (i) s:R+ --* RU+ ; (ii) s and its first derivative are piecewise continuous on R+; (iii) the initial distribution Po obeys requirements (8); (iv)

~, nPo(n) < + Q c , ;

(v) the coefficients lk(r) obey requirements (2). The additional requirement

~ rlk(r)

< + ~ (for allk)

has been shown to guarantee that the Markov processes are 'honest'. As a matter of fact, requirement (iv) has been established in a somewhat milder form. We observe that the usual assignment Po(n0)= 1 for some no s N N, Po(n) = 0 for n ~ no, is consistent with requirements (iii) and (iv). As a final comment, we would like to remark that a comparison with the literature suggests that it should be possible to relax conditions (ii) and (iv).

lira u(t) = u(tO ~ ~(G), t~r¢

and apply the above arguments to this system on the interval Its, t2]. In this way we construct on R+ a

REFERENCES Athreya K. B. and Ney P. E. (1972) Branching Processes, Ch. V, Sect. 7. Springer, Berlin.

296

V. CAPASSOand S. L. PAVERI-FONTANA

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APPENDIX

The object of this appendix is to give some indication on the variety of scientific areas in which the model discussed in the paper can be applied. The list is by no means complete. Additional examples may be found, e.g., in Mode (1971).

A.I Stochastic compartmental models It is assumed that a given drug is distributed among N regions ('compartments') in a biological system; that in each region its concentration is spatially uniform; that the flow rates among compartments and the leaking rates are governed by constant parameters; finally, that all molecules act independently of each other. Now, let the r.v. gk(t) denote the number of molecules of the drug in the kth compartment; let lk(ei) = Lik > 0 denote the probability, per unit time, for a molecule in the kth compartment to transfer to the ith compartment (i # k); let Ik(O) >=0 denote its probability, per unit time, to leak out; finally, for all other values of r e N N, set Ik(r) = 0. Then the probability distribution P(n;t) = Prob{h(t) = n) obeys equation (5), provided that Sk(t) is the probability (per unit time) that a molecule is injected into the kth compartment (see, e.g., Matis and Hartley, 1971; Capasso and Paveri-Fontana, 1978).

The same mathematical framework can be used for the analysis of radioactive chain decays; of 'truncated stepping-stone' models for the birthless populations (Renshaw, 1972); and of multigrade manpower arrangements in a company (McClean, 1976). A.2 Growth models for yeast populations Yeast cells reproduce by budding. When a daughter cell splits off, a scar is left on the mother cell. The number of scars on a cell is called its parity. According to a model by Gani and Saunders (1976), it is assumed that 2 > 0 (/~ > 0) is the probability per unit time for a cell to reproduce (to die). If one lets the r.v. ~k(t)(ke {1,2. . . . . N - 1}) be the number of cells with parity k - 1 ; and r~N(t) be the number of cells with parity greater or equal than N - 1, then one finds that P(n;t) obeys equation (5), provided that one makes the identifications: lk(O)= ,u (for all k); lk(eI + e k + l ) = 2 (for k:# N); lv(e I +eN) = 2; with all other parameters equal to zero. External sources may also be present. A.3 The mixing of generations problem Under conditions which are analogous to the ones of the previous example, let the r.v. ~k+ ~(t) denote the number of individuals in the kth generation (k = 0,1 . . . . . N - 2); let ~N(t) denote the number of individuals in the generations of order greater or equal than N - 1 ; and let, once again, ;. and ~u be the birth and death parameters, respectively. Then P(n;t) obeys equation (5) provided that one makes the identifications lk(O) = //(for all k), lk(ek + e k+ 1) = 2 (for k ¢: N) and lN(2eN) = 2, with all other parameters equal to zero.

A.4 The fssion process A simple one-group of delayed neutrons, one-energy group, point-reactor scheme is presented here to model the distribution of neutrons and precursors in a nuclear fission reactor. More general schemes are possible which also lead to equations of the type (4). For related material consult, for instance, Williams (1974). Let, for neutrons, 2c, ~./, 2, be the probabilities (per unit time) of non-fissile capture, fissile capture and leakage, respectively. For precursors let 2d be the probability (per unit time) of decay (which occurs with the release of one neutron). Finally let q(n,k) be the probability of n neutrons and k precursors being released after a fission event (n,k,~N;

here

,-o~ 1Z ) q(n,k)= . ~ o

Then, if the r.v.'s hi(t), ~2(t)denote the neutron and precursors population respectively, one finds that P(nl,n2;t) obeys equation (5) provided that one makes the identifications: 11(0) = 2c + 2, + 2yq(0,0); ll(ne I + ke 2) = 2yq(n,k) (for n + k > 0); 12(e~) = )-a with all other parameters equal to zero. External sources may be present.