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A multitype stochastic population model: An extended version
MATHEMATICAL
A
Multitype
487
BIOSCIENCES
Stochastic Population
Model:
An
Extended
Version S. C. PORT The Rand Corporation,
Communicated
Santa Monica,
California
by Richard Bellman
ABSTRACT A previous article [l] introduced and analyzed a simple multitype population model. That model was special insofar as the input process was assumed Poisson. This article analyzes a modified model that has for its input process an arbitrary renewal process. This modification allows far greater flexibility in fitting the model to some real situation than does the case of Poisson inputs.
INTRODUCTION
A previous article [I] introduced a multitype stochastic population model, motivated it with examples, and analyzed some of its mathematical consequences. The purpose here is to do the same with a modified model that includes the original one as a special case. This new model is the same as the previous one except that the input process is an arbitrary renewal process instead of a Poisson process. This allows far greater flexibility in fitting the model to a real situation. The new model is as follows. A particle may be in one of r different states. Particles enter the system in bunches that arrive according to a renewal process q(t) having waiting times {IV,}, which possesses a density and finite variance. The total number of particles in the ith bunch is X,. The processes r(t) and {Xi} are independent. The composition of the ith bunch is {x,~, . . . , x+}; where conditional on Xi, the r-vector is multinomial (Xi; CI~,. . . , ct,). After arrival into the lxil, . . . 3 x,} system, the particles all act independently of each other and transform Mathematical
Copyright
Biosciences
4 (1969), 487-498
0 1969 by American Elsevier Publishing Company,
Inc.
488 their PJt)
S. C. PORT
states according to a semi-Markov process having transition and final probabilities lim,_ ocIPJt) = Tl,. We will also assume
law that
m s
]P&t) -
IIj] dt < co.
0
As before, let M,(t) and Yj(t) denote, respectively, the number particles that enter the system in state j by time t and the number particles in state j at time 1. For RI and 2, negative, set E[exp(&MLl))
exp(&
of of
C(t)>1= @& &>,
and for 0 < s < 1, set E(ss”)
= G(s),
and set pr(t) = crjPjj(t), tcj[l - Pjj(t)] = pz(t) and p3(t) = & Then all of the information about the Yj(t), M,(t) follows following integral equation that the ai(jll, 2,) satisfy.
@,(a,, 1,) = P(w, >
r) +
s
cQJt). from the
G{1 + [exp(& + 1,) - ~IPI(~- s>
+ bp&) -“ll~,(t - s> + lexp(&)- llkdt - 4)L(L
&>dP(W, < s). (1)
This equation may be derived by the usual renewal-type argument and is left to the reader. We note that if the Wi are exponentially distributed with mean I/,u, then the equation becomes
@dL &> = and direct substitution
exp(-put)
[
1+
P
s
1
&Ps expG.4ds ,
0
shows that t (2)
Here
[exp@,+ A) - llpl(d + (exp&> - l)p2b) + (exp&> - lk&)~.
G(s) = GU +
Setting R, = iB1, & = i& in (2) and comparing Mathemafical
Biosciences
4 (1969), 487-498
with Eq. (4) of our previous
MULTITYPE POPULATION
489
MODEL
Thus in the case of Poisson
article, we see that these are the same. the integral
Eq. (1) becomes
the basic equation
arrival,
of [l].
We will state and prove our general results for processes in continuous time. Naturally, analogous results hold in discrete time. Unfortunately, Eq. (1) cannot be solved in the general case. However, we can use it to derive the behavior of the moments of Y,(t) and M,(r). Only the first two moments will be of interest to us. Set ,~r = EX,, pz = Z?X,(X, - l), y = (EW,)-l. Under our assumptions, Erl(t) -
ty + Z = [Var WI -
(Z5WJz]y2/2,
THEOREM 1. Under the assumptions qf the model converges in probability to (~LY~,LQ, ~,LQII~). Moreover, EYjW
= yp1njt
[M,(t)/t,
t+co. Yi(t)/t]
+ pu,(yJ + r&Z) + o(l)
(3)
where J=
J[ia,zJJt) i=l
-
rI
J dt,
0
Var Yj(O =
PpfyK3 + kJCy +
EM,(t)
p,ynJt
+ 40,
= l/rUlajt + prajZ + o(l),
(4) (5)
and
THEOREM 2. Cov[M,(t),
Under the assumption
Yj(Ol= P&“JJ
Let us now compare when WI is exponential;
of the model,
+ (p&?,+ p1kq,ylt + o(t).
our general results with those for the special case that is, q(t) is Poisson with the same parameter
values. Then the first term in expansion of the means is the same and the second term differs by the factor involving I. Likewise, in the asymptotic expansion of the variance, we see that there is an additional term involving I. The exponential waiting has the gamma density, f,,,(x)
time is a special case of the gamma.
= &
x”-repas,
If WI
a > 0, 2) > 0,
then EW2 = ‘(’ + ‘1
EW,=?,
1 a
a2
2
Mathematical Biosciences 4 (1969), 487498 33
S. C. PORT
490 and thus Z = Q(V ~- + 1) cc2
v(1 -
2d=1’__ CC2
cc2
v)
cc2
ThusZ>Oifvl.
Thecasev=I
is the exponential distribution. An interesting special case occurs when the arrival times are constant. Here it is better to formulate the model in discrete time, and it is convenient to take one time unit as the time between successive group arrivals. Then if Y,(n) is the number of particles in statej at time n, we see that
where Xni denotes at time i. Thus
the number
of particles
E[exp(iOYj(n))]
= fi
in state j at time rz that arrive
E[exp(SX,J].
i=l
Now
We therefore
see that
Var Y,(n) = n[(Var X,)IIS + Z~rfI~(l Thus, E
computing
the characteristic
if%&(n) - EYkn))
exp
[
(Var Yj(n))1/2
1
function,
+ o(n).
we have
=fJ l-;$$n;+u(d)l [
= Hence we have established
Il,)]
1
3
-E + 0(t)17’-exp(-
5).
the following.
THEOREM 3. For the case of constant group arrivals the number of partials Y,(n) in state j at time n normalized by its mean and variance is asymptotically normally distributed. Mathematical
Biosciences 4 (1969), 487-498
MULTITYPE
POPULATION
PROOF OF THEOREM
491
MODEL
1
We first consider
Y&t). Setting ;1, = 0 in (1) and setting
E[exp(~WNI = Q,,@>, we see that t @,,(A> = 1 -
G[l +
F(t) +
(exp(4
-
l)p(t -
s)]@~_,(A)~F(s)
s 0
where P(0 = PI(t) + Pdf) = li %P,j(Q, i=l
and F(t) = P(W, < t). A differentiation
on 4 yields +
+
@‘I =
s
G'(...)e”p(t - s)@,,_,S(s)
s
G(- . .)a;_, dF(s).
+
(7)
0
0
Let ,ur = EX, and ,u~ = EX,(X,
-
1). Setting ii = 0 in (7) yields t
t EY,(t) = ,ul p(t s
s) S(s)
+
I
EY,(t -
s) S(s).
(8)
0
0
This integral equation is readily solvable by Laplace transforms. We will illustrate the technique here and henceforth leave such details to the reader. Denote Laplace transforms by a circumflex (^) and let the argument of the transform function be 4. Then (8) yields
Hence
Thus t
EYj(O =
PI
5 1~0
-
s)W’,
+ * . . + W,& ds)
j=l
0
t =
Pl
s
P(t
-
s) dWs>,
0
where here and in the following
N(t) = Eq(t) Mathematical
is the renewal Biosciences
function.
4 (1969), 487-498
S.C.
492
; J
Under
our assumptions
on the waiting
n(s) ds, where n(s) is the renewal
0
times it is known
density,
that
PORT
N(t) =
and that
00 I 0
In(s) - (EW,)-‘l
Also
f
a,
I =
I
[n(s) - (EW,)y] ds =
tls < cc.
EW(-$y2 = Var 1
0
- tpgnj =
: J
[p(t -
/Ll
0 =
s)n(s) -
*
ds
t
t
s
[p(t -
p1
yrIjl
rIj]n(s)ds + pin,
s) -
0
; inCs)Plnj J
Now
2(EW,)2
From (9) we see that +
Set y = (ECZ/,)-‘. EYj(f)
W, - (EW,)2
I
[PI(S)
-
~1 ds.
0
yl =
plnjz
t-•
O(l),
+
co.
0
Also t
1
I
~ [p(t - s) - l-I&I(s) ds = 0
I
[p(t -
s) -
rI,][n(s)
-
y] ds
”
0
The first integral on the right is clearly o(1) as I + co, and the second is r.I + o(l), where Z J =
I
[P(s) -
rtj] ds.
0
Hence /Al
j
[p(t -
s) - rI,]n(s> ds = p1yJ +
o(l).
0
Thus EYj(t) We must Mathematical
-
now analyze
tplyI’j
= pl?J
the second
Biosciences 4 (1969), 487-498
+ ~1njZ + o(l). moment
(10)
EYj(tj2.Differentiating
MULTITYPE
POPULATION
(7) on il and setting
493
MODEL
I = 0 yields the equation
t EYj(t)2 = 2
s 0
t.
,u&t
s)EY,(t
-
s) dF(s)
+ p2
t
t
+ ,a1 p(t s
Solving,
-
s) dF(s)
+
s 0
EY;(t
-
J 0
p2(t -
s) dF(s)
s) dF(s).
we find that” f EY;(t)
= 2,ul
+
p(t -
s 0
,u2
s)EYj(t
p”(t -
s
-
s)n(s) ds
s)n(s) ds + EYj(t).
(11)
0
To establish
our results we make good use of the following. Let a(t) andb(t) be nonnegatioefinctions
LEMMA 1.
such that a(t) - a,,
b(t) - b, are in L,(O, co), and set 00 J, =
s
[u(t) - a] dt,
0 m
J, =
s
[b(t) -
b] dt.
0
Suppose
M(t) > 0, M(t) r co,
t+
co, awl M(t) -
th4, + J,,,
t -+ co.
Then
t b(t -
s
s)M(t -
s)a(s) ds -
0
Proqf.
aoboMot’ ~ = aoboJ,,lrt + M,b,J,t 2
+ o(t).
(12)
Moaobot2 ~. 2
(13)
We can write
t
s
b(t -
s)M(t -
s)a(s) ds -
0
=
s ”
b(t -
aoboMot2 ~ 2
s)[M(t
-
s) -
(t -
.s)M,]a(s)
ds
t + MO (t - s)u(s)b(t s
0
-
s) ds -
Mathematical Biosciences 4 (1969), 487-498
494
S. C. PORT
Now
b(t)pf(t)
- tMo] - bJ,,,
t*
a(t) -
00 ;
t--too.
4
Thus
s
b(t -
s) - (t - s)M,]a(s) ds N a,bJ,t,
s)[M(t -
t -+ co.
(14)
0
On the other hand, 1
t
MO (t f
s) ds = MO sb(s)a(t - s) ds
s)a(s)b(t -
I
0
0
s s
= MO sb(s)[a(t -
s) -
a”]
0
t
+ M,a,
Now integrating
by parts,
s(b(s) -
bo)
+
M,a,b,t”
2
.
we find that t
t
s
s[b(s) -
b,] ds = t
;
s
0
=
0
0
t.I, + O(f) = o(t).
tJ, -
(16)
Also t I
t sb(s)[a(t
-
s) -
a,] ds =
f
0
s[b(s) -
bo][u(t -
s) -
a,]
0
+ b.
s[u(t -
s) -
uo].
0
The first term on the right is clearly o(t), and the second term is
b,
s
s[a(t -
s) -
a,] ds = b,
0
J’
(t -
s)[u(s)
-
oo] ds
0
t =
tb,
I
t
[u(s) -
0 =
Mathematical
(15)
Biosciences
tb,J,, + o(t).
4 (1969), 478-489
a,] -
b,
s
0
s[u(s) -
a,] ds
MULTITYPE
POPULATION
495
MODEL
Thus
s
sb(s)[a(t - s) -
a,] ds
= tb,J, + o(t).
(17)
0
Substituting
(16) and (17) into (15) yields t
MO
Moaobot2
s
(t - s)a(s)b(t - s) ds = -
2
+ M,b,J,t
+ o(t).
(18)
0
Using this and (14), we find from (13) that (12) holds. We may now proceed to analyze EY:(t). Taking b(t) = p(t), M(t) EY,(t), and a(t) = n(t) in the lemma and using (lo), we find that
=
t 2p,
s
From
p(t -
s)EY,(t -
s)n(s) ds
(11) we then find that EYf(t) = jJU:y2n;t2 + [2p,ynj(pIyJ + (1uJCY + /JIynj)t
+ plnjz)
+ 2/-&n;z1t
+ oC.0.
On the other hand,
Var Yj(t = EY;(t)
-
= Ppfy~3
[EYj(t)12 + p2qy
+ plynjlt
+ o(t).
(19)
Turning our attention now to M,(t), we may proceed in a similar manner. Setting A, = 0 in (1) and letting Qt(A) = E[exp(&I4,(t))], we find that @,,(I) = 1 - F(t) + Proceeding
s
G[l + (exp(l)
-
l)aj]@,t-,(n)
dF(s).
as before, we find that EM,(t)
= p,a,N(t) Mathematical
Biosciences
4 (1969), 487498
S. C. PORT
496 and
I
EM,(t)” = 2,~~ EM,(t - s)n(s) ds + ,u2u;N(t)+
EM,(t)
0
= 2&x;
s
N(t -
s>n(s) ds + p&v(t)
+ pu,cqv(t).
0
In Lemma
I set b(t) = 1, M(t) = N(t), and a(s) = n(s).
Then
t
s N(t
-
s)n(s) ds = y$
+ 2yzt + o(t).
0
Thus EM;(t)
= 2,134
y”t” + 2yIt [2
since [EM,(f)]”
1
+ ,u&yt
= ,u;~+V(t)~ = &c;{[N(t)
-
= 2p;“;yl
+ p;tC2jy2tz + o(1).
Var M,(t)
= [2~$4yZ
+ ,qyt
+ o(t),
yt]” + 2yt[N(t)
-
yt] + yBt2]
Thus + p2c& + pug]t
+ o(t).
(20)
We may now easily establish the law of large numbers for Y,(t) and M,(t). Indeed, by (19) we know that Var Yi(t) = O(t), and thus P
Likewise
>F
This completes
M,(t) - ~#q”j
[I
t
-Y,(t)=0 E2t2
Varp.(t) > & Q A= s2t2 1
I
the proof of Theorem
OF THEOREM
[ Biosciences
1.
n
2
Computing
Mathematical
<
1
[I. I
t
(20) shows that p
PROOF
[Iy-ynjpl I
a%,, an, an, 1++o’
4 (lY6Y), 487-498
1 0 [ t.1.
MULTITYPE
POPULATION
497
MODEL
we find that t
+
Pupl(~) + Pu(Pl(S) + P,(s)wj(t - s) + ~z(s))EMj(t - s)>n(s) ds.
+ ,444 Consider becomes
the third
term
on the right.
(21)
Since p1 + pz = CQ, this term
t ,uuj
s
EYi(t -
s)n(s) ds.
0
In Lemma 1 set b(t) = 1, a(s) = n(s), and M(t) = EYi(t). of Eq. (3)
Then in view
t EYj(t -
,uuj 0
y2/A,rI .t2 s)n(s) ds - 3 2 = Qj?/lu$J
Similarly, in the fourth and thus this term is
+ njl]tCrjp,2yIIjlt
term on the right, pi(s)
+ o(t)
(22)
+ p3(s) = ~~==,M,P~~(s),
t p1 P,,(s)EM,(t s
-
s)n(s) ds.
0
In Lemma 1, set b(t) = PJt), view of Eq. (5) we see that
EM,(t)
= M(t),
and a(t) = n(t).
Then in
t ,ul
I
P&)EM,(t
- s)n(s) ds -
&I
jy%jtz 2
0
= 2yIIj/&t
+ o(t).
(23)
Now EYj(t)EMj(t)
-
t2&y2Crjnj = &tcjy[yJ
Thus from (21)-(23)
+ o(t).
we see that
Cov(Yj(t>2 Mj(t>) = This establishes
+ njZ]t + ,&IIjajlt
Theorem
[2/-4ujynj~
2.
+
(p(2) +
pl)QjnjY]t
+
O(t).
w Mathematical Biosciences 4 (1969), 487498
498
S. C. PORT
ACKNOWLEDGMENT This research is supported by the United States Air Force under Project RANDcontract no. F44620-67-C-0045-monitored by the Directorate of Operational Requirements and Development Plans, Deputy Chief of Staff, Research and Development, Hq USAF.
REFERENCE 1 S. Port, A multitype stochastic
Afuthematical
Biosciences
population
4 (1969), 487498
model, Math. Biosci. 2(1968), 129-137.
A
Multitype
487
BIOSCIENCES
Stochastic Population
Model:
An
Extended
Version S. C. PORT The Rand Corporation,
Communicated
Santa Monica,
California
by Richard Bellman
ABSTRACT A previous article [l] introduced and analyzed a simple multitype population model. That model was special insofar as the input process was assumed Poisson. This article analyzes a modified model that has for its input process an arbitrary renewal process. This modification allows far greater flexibility in fitting the model to some real situation than does the case of Poisson inputs.
INTRODUCTION
A previous article [I] introduced a multitype stochastic population model, motivated it with examples, and analyzed some of its mathematical consequences. The purpose here is to do the same with a modified model that includes the original one as a special case. This new model is the same as the previous one except that the input process is an arbitrary renewal process instead of a Poisson process. This allows far greater flexibility in fitting the model to a real situation. The new model is as follows. A particle may be in one of r different states. Particles enter the system in bunches that arrive according to a renewal process q(t) having waiting times {IV,}, which possesses a density and finite variance. The total number of particles in the ith bunch is X,. The processes r(t) and {Xi} are independent. The composition of the ith bunch is {x,~, . . . , x+}; where conditional on Xi, the r-vector is multinomial (Xi; CI~,. . . , ct,). After arrival into the lxil, . . . 3 x,} system, the particles all act independently of each other and transform Mathematical
Copyright
Biosciences
4 (1969), 487-498
0 1969 by American Elsevier Publishing Company,
Inc.
488 their PJt)
S. C. PORT
states according to a semi-Markov process having transition and final probabilities lim,_ ocIPJt) = Tl,. We will also assume
law that
m s
]P&t) -
IIj] dt < co.
0
As before, let M,(t) and Yj(t) denote, respectively, the number particles that enter the system in state j by time t and the number particles in state j at time 1. For RI and 2, negative, set E[exp(&MLl))
exp(&
of of
C(t)>1= @& &>,
and for 0 < s < 1, set E(ss”)
= G(s),
and set pr(t) = crjPjj(t), tcj[l - Pjj(t)] = pz(t) and p3(t) = & Then all of the information about the Yj(t), M,(t) follows following integral equation that the ai(jll, 2,) satisfy.
@,(a,, 1,) = P(w, >
r) +
s
cQJt). from the
G{1 + [exp(& + 1,) - ~IPI(~- s>
+ bp&) -“ll~,(t - s> + lexp(&)- llkdt - 4)L(L
&>dP(W, < s). (1)
This equation may be derived by the usual renewal-type argument and is left to the reader. We note that if the Wi are exponentially distributed with mean I/,u, then the equation becomes
@dL &> = and direct substitution
exp(-put)
[
1+
P
s
1
&Ps expG.4ds ,
0
shows that t (2)
Here
[exp@,+ A) - llpl(d + (exp&> - l)p2b) + (exp&> - lk&)~.
G(s) = GU +
Setting R, = iB1, & = i& in (2) and comparing Mathemafical
Biosciences
4 (1969), 487-498
with Eq. (4) of our previous
MULTITYPE POPULATION
489
MODEL
Thus in the case of Poisson
article, we see that these are the same. the integral
Eq. (1) becomes
the basic equation
arrival,
of [l].
We will state and prove our general results for processes in continuous time. Naturally, analogous results hold in discrete time. Unfortunately, Eq. (1) cannot be solved in the general case. However, we can use it to derive the behavior of the moments of Y,(t) and M,(r). Only the first two moments will be of interest to us. Set ,~r = EX,, pz = Z?X,(X, - l), y = (EW,)-l. Under our assumptions, Erl(t) -
ty + Z = [Var WI -
(Z5WJz]y2/2,
THEOREM 1. Under the assumptions qf the model converges in probability to (~LY~,LQ, ~,LQII~). Moreover, EYjW
= yp1njt
[M,(t)/t,
t+co. Yi(t)/t]
+ pu,(yJ + r&Z) + o(l)
(3)
where J=
J[ia,zJJt) i=l
-
rI
J dt,
0
Var Yj(O =
PpfyK3 + kJCy +
EM,(t)
p,ynJt
+ 40,
= l/rUlajt + prajZ + o(l),
(4) (5)
and
THEOREM 2. Cov[M,(t),
Under the assumption
Yj(Ol= P&“JJ
Let us now compare when WI is exponential;
of the model,
+ (p&?,+ p1kq,ylt + o(t).
our general results with those for the special case that is, q(t) is Poisson with the same parameter
values. Then the first term in expansion of the means is the same and the second term differs by the factor involving I. Likewise, in the asymptotic expansion of the variance, we see that there is an additional term involving I. The exponential waiting has the gamma density, f,,,(x)
time is a special case of the gamma.
= &
x”-repas,
If WI
a > 0, 2) > 0,
then EW2 = ‘(’ + ‘1
EW,=?,
1 a
a2
2
Mathematical Biosciences 4 (1969), 487498 33
S. C. PORT
490 and thus Z = Q(V ~- + 1) cc2
v(1 -
2d=1’__ CC2
cc2
v)
cc2
ThusZ>Oifv
Thecasev=I
is the exponential distribution. An interesting special case occurs when the arrival times are constant. Here it is better to formulate the model in discrete time, and it is convenient to take one time unit as the time between successive group arrivals. Then if Y,(n) is the number of particles in statej at time n, we see that
where Xni denotes at time i. Thus
the number
of particles
E[exp(iOYj(n))]
= fi
in state j at time rz that arrive
E[exp(SX,J].
i=l
Now
We therefore
see that
Var Y,(n) = n[(Var X,)IIS + Z~rfI~(l Thus, E
computing
the characteristic
if%&(n) - EYkn))
exp
[
(Var Yj(n))1/2
1
function,
+ o(n).
we have
=fJ l-;$$n;+u(d)l [
= Hence we have established
Il,)]
1
3
-E + 0(t)17’-exp(-
5).
the following.
THEOREM 3. For the case of constant group arrivals the number of partials Y,(n) in state j at time n normalized by its mean and variance is asymptotically normally distributed. Mathematical
Biosciences 4 (1969), 487-498
MULTITYPE
POPULATION
PROOF OF THEOREM
491
MODEL
1
We first consider
Y&t). Setting ;1, = 0 in (1) and setting
E[exp(~WNI = Q,,@>, we see that t @,,(A> = 1 -
G[l +
F(t) +
(exp(4
-
l)p(t -
s)]@~_,(A)~F(s)
s 0
where P(0 = PI(t) + Pdf) = li %P,j(Q, i=l
and F(t) = P(W, < t). A differentiation
on 4 yields +
+
@‘I =
s
G'(...)e”p(t - s)@,,_,S(s)
s
G(- . .)a;_, dF(s).
+
(7)
0
0
Let ,ur = EX, and ,u~ = EX,(X,
-
1). Setting ii = 0 in (7) yields t
t EY,(t) = ,ul p(t s
s) S(s)
+
I
EY,(t -
s) S(s).
(8)
0
0
This integral equation is readily solvable by Laplace transforms. We will illustrate the technique here and henceforth leave such details to the reader. Denote Laplace transforms by a circumflex (^) and let the argument of the transform function be 4. Then (8) yields
Hence
Thus t
EYj(O =
PI
5 1~0
-
s)W’,
+ * . . + W,& ds)
j=l
0
t =
Pl
s
P(t
-
s) dWs>,
0
where here and in the following
N(t) = Eq(t) Mathematical
is the renewal Biosciences
function.
4 (1969), 487-498
S.C.
492
; J
Under
our assumptions
on the waiting
n(s) ds, where n(s) is the renewal
0
times it is known
density,
that
PORT
N(t) =
and that
00 I 0
In(s) - (EW,)-‘l
Also
f
a,
I =
I
[n(s) - (EW,)y] ds =
tls < cc.
EW(-$y2 = Var 1
0
- tpgnj =
: J
[p(t -
/Ll
0 =
s)n(s) -
*
ds
t
t
s
[p(t -
p1
yrIjl
rIj]n(s)ds + pin,
s) -
0
; inCs)Plnj J
Now
2(EW,)2
From (9) we see that +
Set y = (ECZ/,)-‘. EYj(f)
W, - (EW,)2
I
[PI(S)
-
~1 ds.
0
yl =
plnjz
t-•
O(l),
+
co.
0
Also t
1
I
~ [p(t - s) - l-I&I(s) ds = 0
I
[p(t -
s) -
rI,][n(s)
-
y] ds
”
0
The first integral on the right is clearly o(1) as I + co, and the second is r.I + o(l), where Z J =
I
[P(s) -
rtj] ds.
0
Hence /Al
j
[p(t -
s) - rI,]n(s> ds = p1yJ +
o(l).
0
Thus EYj(t) We must Mathematical
-
now analyze
tplyI’j
= pl?J
the second
Biosciences 4 (1969), 487-498
+ ~1njZ + o(l). moment
(10)
EYj(tj2.Differentiating
MULTITYPE
POPULATION
(7) on il and setting
493
MODEL
I = 0 yields the equation
t EYj(t)2 = 2
s 0
t.
,u&t
s)EY,(t
-
s) dF(s)
+ p2
t
t
+ ,a1 p(t s
Solving,
-
s) dF(s)
+
s 0
EY;(t
-
J 0
p2(t -
s) dF(s)
s) dF(s).
we find that” f EY;(t)
= 2,ul
+
p(t -
s 0
,u2
s)EYj(t
p”(t -
s
-
s)n(s) ds
s)n(s) ds + EYj(t).
(11)
0
To establish
our results we make good use of the following. Let a(t) andb(t) be nonnegatioefinctions
LEMMA 1.
such that a(t) - a,,
b(t) - b, are in L,(O, co), and set 00 J, =
s
[u(t) - a] dt,
0 m
J, =
s
[b(t) -
b] dt.
0
Suppose
M(t) > 0, M(t) r co,
t+
co, awl M(t) -
th4, + J,,,
t -+ co.
Then
t b(t -
s
s)M(t -
s)a(s) ds -
0
Proqf.
aoboMot’ ~ = aoboJ,,lrt + M,b,J,t 2
+ o(t).
(12)
Moaobot2 ~. 2
(13)
We can write
t
s
b(t -
s)M(t -
s)a(s) ds -
0
=
s ”
b(t -
aoboMot2 ~ 2
s)[M(t
-
s) -
(t -
.s)M,]a(s)
ds
t + MO (t - s)u(s)b(t s
0
-
s) ds -
Mathematical Biosciences 4 (1969), 487-498
494
S. C. PORT
Now
b(t)pf(t)
- tMo] - bJ,,,
t*
a(t) -
00 ;
t--too.
4
Thus
s
b(t -
s) - (t - s)M,]a(s) ds N a,bJ,t,
s)[M(t -
t -+ co.
(14)
0
On the other hand, 1
t
MO (t f
s) ds = MO sb(s)a(t - s) ds
s)a(s)b(t -
I
0
0
s s
= MO sb(s)[a(t -
s) -
a”]
0
t
+ M,a,
Now integrating
by parts,
s(b(s) -
bo)
+
M,a,b,t”
2
.
we find that t
t
s
s[b(s) -
b,] ds = t
;
s
0
=
0
0
t.I, + O(f) = o(t).
tJ, -
(16)
Also t I
t sb(s)[a(t
-
s) -
a,] ds =
f
0
s[b(s) -
bo][u(t -
s) -
a,]
0
+ b.
s[u(t -
s) -
uo].
0
The first term on the right is clearly o(t), and the second term is
b,
s
s[a(t -
s) -
a,] ds = b,
0
J’
(t -
s)[u(s)
-
oo] ds
0
t =
tb,
I
t
[u(s) -
0 =
Mathematical
(15)
Biosciences
tb,J,, + o(t).
4 (1969), 478-489
a,] -
b,
s
0
s[u(s) -
a,] ds
MULTITYPE
POPULATION
495
MODEL
Thus
s
sb(s)[a(t - s) -
a,] ds
= tb,J, + o(t).
(17)
0
Substituting
(16) and (17) into (15) yields t
MO
Moaobot2
s
(t - s)a(s)b(t - s) ds = -
2
+ M,b,J,t
+ o(t).
(18)
0
Using this and (14), we find from (13) that (12) holds. We may now proceed to analyze EY:(t). Taking b(t) = p(t), M(t) EY,(t), and a(t) = n(t) in the lemma and using (lo), we find that
=
t 2p,
s
From
p(t -
s)EY,(t -
s)n(s) ds
(11) we then find that EYf(t) = jJU:y2n;t2 + [2p,ynj(pIyJ + (1uJCY + /JIynj)t
+ plnjz)
+ 2/-&n;z1t
+ oC.0.
On the other hand,
Var Yj(t = EY;(t)
-
= Ppfy~3
[EYj(t)12 + p2qy
+ plynjlt
+ o(t).
(19)
Turning our attention now to M,(t), we may proceed in a similar manner. Setting A, = 0 in (1) and letting Qt(A) = E[exp(&I4,(t))], we find that @,,(I) = 1 - F(t) + Proceeding
s
G[l + (exp(l)
-
l)aj]@,t-,(n)
dF(s).
as before, we find that EM,(t)
= p,a,N(t) Mathematical
Biosciences
4 (1969), 487498
S. C. PORT
496 and
I
EM,(t)” = 2,~~ EM,(t - s)n(s) ds + ,u2u;N(t)+
EM,(t)
0
= 2&x;
s
N(t -
s>n(s) ds + p&v(t)
+ pu,cqv(t).
0
In Lemma
I set b(t) = 1, M(t) = N(t), and a(s) = n(s).
Then
t
s N(t
-
s)n(s) ds = y$
+ 2yzt + o(t).
0
Thus EM;(t)
= 2,134
y”t” + 2yIt [2
since [EM,(f)]”
1
+ ,u&yt
= ,u;~+V(t)~ = &c;{[N(t)
-
= 2p;“;yl
+ p;tC2jy2tz + o(1).
Var M,(t)
= [2~$4yZ
+ ,qyt
+ o(t),
yt]” + 2yt[N(t)
-
yt] + yBt2]
Thus + p2c& + pug]t
+ o(t).
(20)
We may now easily establish the law of large numbers for Y,(t) and M,(t). Indeed, by (19) we know that Var Yi(t) = O(t), and thus P
Likewise
>F
This completes
M,(t) - ~#q”j
[I
t
-Y,(t)=0 E2t2
Varp.(t) > & Q A= s2t2 1
I
the proof of Theorem
OF THEOREM
[ Biosciences
1.
n
2
Computing
Mathematical
<
1
[I. I
t
(20) shows that p
PROOF
[Iy-ynjpl I
a%,, an, an, 1++o’
4 (lY6Y), 487-498
1 0 [ t.1.
MULTITYPE
POPULATION
497
MODEL
we find that t
+
Pupl(~) + Pu(Pl(S) + P,(s)wj(t - s) + ~z(s))EMj(t - s)>n(s) ds.
+ ,444 Consider becomes
the third
term
on the right.
(21)
Since p1 + pz = CQ, this term
t ,uuj
s
EYi(t -
s)n(s) ds.
0
In Lemma 1 set b(t) = 1, a(s) = n(s), and M(t) = EYi(t). of Eq. (3)
Then in view
t EYj(t -
,uuj 0
y2/A,rI .t2 s)n(s) ds - 3 2 = Qj?/lu$J
Similarly, in the fourth and thus this term is
+ njl]tCrjp,2yIIjlt
term on the right, pi(s)
+ o(t)
(22)
+ p3(s) = ~~==,M,P~~(s),
t p1 P,,(s)EM,(t s
-
s)n(s) ds.
0
In Lemma 1, set b(t) = PJt), view of Eq. (5) we see that
EM,(t)
= M(t),
and a(t) = n(t).
Then in
t ,ul
I
P&)EM,(t
- s)n(s) ds -
&I
jy%jtz 2
0
= 2yIIj/&t
+ o(t).
(23)
Now EYj(t)EMj(t)
-
t2&y2Crjnj = &tcjy[yJ
Thus from (21)-(23)
+ o(t).
we see that
Cov(Yj(t>2 Mj(t>) = This establishes
+ njZ]t + ,&IIjajlt
Theorem
[2/-4ujynj~
2.
+
(p(2) +
pl)QjnjY]t
+
O(t).
w Mathematical Biosciences 4 (1969), 487498
498
S. C. PORT
ACKNOWLEDGMENT This research is supported by the United States Air Force under Project RANDcontract no. F44620-67-C-0045-monitored by the Directorate of Operational Requirements and Development Plans, Deputy Chief of Staff, Research and Development, Hq USAF.
REFERENCE 1 S. Port, A multitype stochastic
Afuthematical
Biosciences
population
4 (1969), 487498
model, Math. Biosci. 2(1968), 129-137.