AD A058 91i1 FLORIDA STA TE UNIV TALLAHASSEE DEPT Off STATISTICS Pfl IL’SON THE NEGATIVE BINOMIAL CONVERGENCE IN A CLASS Off M—DIMENSIOt4A——EtC(U)JUl. 78 H LACATO . N A LANGBERG AFOSR—7k’2581

UNCLASSIFIED FSU~ STAT ISTICS~ Mle6le AFOSR—TR—78—1232 NI.

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_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ - ~~U No. ‘lLiELi J7. AUT HOR(s ) ‘. ~T~~A C’ ‘~)R ‘~~~A MT NUMBER(,)

~~~~~~~~ 76— ~lr’9 a—’H. Lacayo and Naftah A . r. r~ ’.~ rn ‘ -

• 9. PERFORMING O R G A N I Z A T I O N NAME AND A DDRESS ~~. P R O C R A M ELEM E NT.P ROJEC T , T A Su(

The Florida State Univer sityDepartment of Statistic s “ 61102F 2304/A5Tallahassee , Florida 32306

II . CONTROLLING OFFICE NAME AND A D D R E S S ~~. _ PCRT DA T E

Air Force Office of Scien tific Research /NM I ,~~~~~~~~tB~~R

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Boilin g AFB , Washington , OC 2033 1014 . MONITORIN G AGENCY NAME & A DDRESS ,t di f fe r en t irom ‘.~~ntro i I in . t ~)‘h c e , IS. SECURITY CLAS S. (of this report)

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IL SUPPL EMENTARY NOTES

19. K EY WORDS (Continue on r e v e r s e s ide ii necessary and iden t l ly by block number)

Exponential distributions , convergence in law , stochastic processes,rn-dimensiona l simple epidemic s , negative binomial distributions.

— 20. A B S T R A ~~~~~Con I9nue on revers, s i de i i necessary and ide n t i ty by b lock number)

We consider a population which is exposed to m infections , and consisti n i t i~

’ll y of N susceptibles. At each point In time at most onesusceptible becomes Infective , and only from one cause. This rn-dimensiona lsimple epidemic i s a stochastic process , (X ..~(t),..,, S (t), with

r~~J~ ~~~~~~~~~~~~~~~~~~~~ N ,1 t (i~\_g..~1,

~~~~~~~~components counting the number of infectives from the respective cau1~es attime t. We show that if the transition rates of cause 1 through m at

time t pF~jci i ven by (continued on r~ v~ r~~t ~~~~~~~~ DD

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ai~N ,~(t)(

~ —

~~~~~ •~~~~~ :~ j~~~ ~~~~~~~j ) j j , ~ ~ • . . , ct > 0 and

if Lim X~ .(0) b . ~ ~1. ~~~. . . .} . :~~ n t ’( 1 (t) ; . . ., X , (t) ) converges a~N.+co j ,1 1

N - .~~ to a random vector t .rtth in d .~ne.ndent negative bincmia]. components.

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~~~SRTR 78 12 32I

~ ~N ThE NEGATIVE LINOMIAL CONVERGENCEIN A CLASS OF ~1-DIME?~TSIONAL SThPLE EPIDEMICS

-

by ?fz -~Ll “ ~~

~~~~J~~~~~~~a~~~~~~~ Nafta1 i A /Langberg~~

FSU Statistics Report H464AFOSR Technical Report No. 84AFOSRN Technical Report No. 9

~LL L i ~.._IJuly, 1978

The Florida State UniversityDepartment of Statistics

.~~~~~Tallahassee Florida 32306 ;• , j, ~~~

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Research sponsored by the Air Force Office of Scientific Research, APSC,USAF, under Grants AFOSR 74-25811) and AFOSR 76-3109.

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I AI~ POØCE 0TPI~~ OF SCIENTIFIC R~SZ~~C~ (AFSC)NOTICE OP TPt~~MITTAL TO DDCThic tocb.iioal re~~rt ~~~~ ; h€~~ reviewed and is

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~~~~~~- - — —~~~~~~~~~~~ -— - ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ . -- ——.~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ -~~~-

On the Negative Binomial Convergencein a Class of rn-Dimensional Simple Epidemics

by

H. Lacayo and Naftali A. Langberg

ABSTRACT

We consider a population which is exposed to in infections, andconsist initially of N susceptibles. At each point in time at most onesusceptible becomes infective, and only from one cause. This rn-dimensional

simple epidemic is a stochastic process , (XN 11t), ~~ m(t~)~ ‘~‘it~ ~orIponents

counting the number of infectives from the respective causes at time t.

We show that if the transition rates of cause 1 through in a~ tine t are

given by a1 XN j (z [1 - ~~

. I O ~, 1 1(t) - XN j())’ i = 1~ ...~~ m , a1, . .. , a~ > 0

and if Lia XN ~ (J) b. c{l , 2, . . . } , thea (X. 1(t), . . . , X,1 1,(t)) converges asN+. 1 I i

N.. to a random vector with independent negative binomial components.

Key Words: Exponential distributions, convergence in law, stochasticprocesses, a-dimensional simple epidemics, negative binomial

distribut ions.

_ _ _ _ _ _

_ _ _ _ _ _ _ _ _ _ _ _ - —— - ~~~~ -.---- --.- . -.~~~ - . .- —- - — - -

S 1

1. Introduction and Stmaary

Introduction.

We consider a population which is exposed to in infections and

consists initially of N susceptibles and bN 1 infectives from cause

I ,

i, i = 1, . . ., in. At each point in time at most one susceptible becomes

an infective, and only from one cause. Once an individual enters the

infective state he remains there and cannot be a carrier of any other

infection. This rn-dimensional simple epidemic is a stochastic process,

(XN 1(t), .., X 14 (c)), ~iit1. :~mponer.ts :~u~cin; the nu~’ther of infectives

fron the respectiv: causes at time t. We show that if the transition

rates of cause 1 through m at time t are given by:

— ~~~.

~~~(X 1~~~ (t) — bN,j)], i = 1, • . . , m , a1, . . ., a~ > 0, (1.1)

and if

Lim bN i — b. c {1, 2, ...}, i 1, . . , m (1.2)1

then for every posittve real numbers t and B, and in nonnegative integers

. . . ,

Lirn P{XN •(t) � k. + b., i - 1, ..., in) = 11 P{X.(t) � k. + b.} (1.3)N-pco ,1 1 1 1 1 1

and

Lia E(X N i (t))8 • E(X~(t))8. 1. • 1, •. . , in , (1.4)

where X1(t), . .• , X~(t) are negative binomial random variables with

— -~~----- - ---~~- - - ~~ — —- ------- - - -—-—-- -.--—-. - .— -- -.-- --—-- -- -- ~ .— - - - ----.- -- -—-- -—-~ -— -. , .- — S ~~~~—~~S--~-----—..~

t .

2

-cu trespective parameters b

1 and e , i - 1, . . ., in. The only previous workin the area of rn-dimensional simple epidemics is the one by Billard, Lacayoand Langberg (1978), who proved (1.3) and (1.4) with the additionalassumption that • 02 ... In that case the interinfection

random times are independent of the infection causes. Generally, as

will be pointed out in Section 2 this property does not hold. The proof of

(1.3) in the cited reference depends on the special structure of the

interinfection times. To obtain (1.3) and (1.4) we have approached theproblem from an alternate viewpoint.

!~~mary.

In Section 2 we present a rigorous definition of an rn-dimensionalsimple epidemic, and describe the ones that are the subject of ouranalysis. Statements (1.3) and (1.4) are proven in Section 3. In this

section we present an rn-dimensional simple epidemic in a population con-

sisting initially of infinitely many susceptibles. This theoretical

simple epidemic is instrumental in the proofs of (1.3) and (1.4).

_____ — ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ :~~‘.

-:.~ —.5— - - -

. 5 ‘3 S - —a

3

2. The in-dimensional Simple Epidemic

An a-dimensional simple epidemic in a population consisting initially

of N susceptibles and bN ~. infectives from cause i, i - 1, . . . , in can be

I

described by N bivariate random vectors (TM 1 I ~~~~ 1) , . . ., (T ~ F ,j.N,1~ N,1~TN1 , . . ., ~~~~ are the random interinfection times, and

~N,l’ • • •

~~ ~N,N

are discrete random variables with values in {l, ..., a} designating therespective infection causes. Let SN ~ be equal to ~

TN , k 1, . . ., N.‘ jal

The finite dimensional joint distributions of (XN 1(t)~ .. . , x. (t)) are~ ,m

determined by (TN1 , ~N,l~ ’

.~~~~~~~ ~ N,N ~N,N~’

through the following setequalities:

XNj (t) bN i + k) - U ( SN r � t < S~~~+p ~1I(~N,q

1) - k), t > 0 (2.1)

k 0, ..., N, and i — 1, . .. , in. (S~~0 = 0, SN N+j =

We assume throughout that (TN,l, ~N,l~’ ~ N,N ~~~~~~~ determined

by (2.2) and (2.3).

— j — 1, . .. , N) —

m r . N in k-I.C 11 u1~

1114(bN ~

+ j L][ 11 C ~~ al(bN ~ ~ I(~ i))]1, (2.2)

i—i j—0 ‘ k—I i—i ‘ ~~ ~N

where r. — ~

I(t~ a i), i — 1, ..., m.1 rpl N

- I IijtjP{TNk I~Nl, “ ‘ ~N,k-l

) tkl k — 1, ...~~ N} — e j~l (2.3)

where t , ..., t are positive real numbers, and p is equal toN - + _i_ I aj(b

i + I I(~q i)), for j — 1, . . ., N.i—l cpl

In the particular case discussed in Billard, Lacayo and Langberg (1978),

condition (2.3) reduces to independent exponential interinf.ction times

— -~~~~~

- --- 5 ,- - - — -- - - - --~~~~~~~~~~~~~~~~~ —— —

~~~~~~~~~~ _ _

3

4

with rates equal to 0N - k + 1(1 bN j + k - 1), k = 1, . . . , N, independent1—1

of F~ , . . . , . In addition equation (2.2) simplifies toN,N N inC TI 1T~ (bN j + j))I~ ii C I bN 1

+ k - i)i~~. Finally we note that ifi—i j •0 ‘ k—i i—i(2.2) and (2.3) hold, then the in-dimensional simple epidemic satisfies

(1.1). Conversly, if (1.1) is satisfied, then equations (2.2) and (2.3)

follow by the inherited Markovian structure of the rn-dimensional simple

epidemic.

_ _ _ _ _ _ _ _ _ _ -- ~~~--

- Srn -- --

5

3. ~Iain Results

Let (X1(t) , ~~~~~~~

X~(t)) be an in-dimensional simple epidemic in a

population consisting initially of b~ infectives from cause i,

i • 1, 2, ..., in , and infinitely many susceptibles. This epidemic can be

described by a sequence of bivariate random vectors, (T1, F~L) , (T2 , ~2~

•

T~, T2, ..., are the interinfection times, and

~~ ~2 ., are random variables with values in U, ..., i n ), which

kdesignate the respective infection causes. Let Sk be equal to Ik = 1, 2 The finite dimensional joint distributions of (X1(t),

X~(t)) are determined by (Tk,

~k~’ k = 1, 2, . .. , through the following

set equalities:

CX~(t) - b1 + k) = U (Sr

� t < S~41, ~ I(~q = i) = k) (3.1)rk q—1

for every positive real number t, positive integer k, and i — 1, . . ., in.

We assume throughout that the sequence CT1, ~~ ) , (T2 , F~ ), . .. , is decer-

mined by (3.2) and (3.3).

k — 1, . . . , n)

in r i j— lC 11 ll~ (b~ + J )~1[ ~ C I cz~ (b. + •

~~~~ i (~ = 1))’l (3.2)

i i j0 j”l izil 1 q=1 q

where r~ ~ 1(1 a i), i — 1, . . . , in , and n — 1, 2 q-1 q

- 1 ~~~P{Tkkl, ~~~~~~~ ~k-l

> tk, k — 1, . . ., n} = e j.l (3.3)

in i-iwhere t1, . . . , ~~ are positive real numbers , p~ - ~

a~ (b~ + ~ I(~ — 1))

‘ i.i q.l q

j a l , ...,n, a n d n — l , 2

L _ _ _ . _ _

______________________________ — -S — — — 5— S--S — -~~ 5’F ~~~~~~~~~~~~~~~~~~~~~-:

~~~~=-=-~--- — -- —

j1 ,

6

We will show that for every two positive real numbers t and B

(XN 1(t), . . . , ~~ ~~~~

converges in law to (X1(t) , . . ., X Ct)) as

N.., and that lAm E(XN ~(t))8

= E( X~

(t)) 8 for i = 1, . . .~~ in. First

we introduce two lemmas.

Leiinna 3.1. Let U1, U2, . . ., be an i.i.d. sequence of exponential random

variables with mean 1. Further let ~‘b+r-l’ ‘ tb+r_l ,b+~_l ) be the

order statistic of a sample of size b + r - 1 from Ul~ Then ~ (b + j - l)~~U

j—land tb+r..1 r are identically distributed.

Proof. It suffices to note that the spacings; tr+b_ l,r_j+l - Tr+j_l ,r_ jI

1, . . ., r, (tr+b..l,O = 0) are independent exponential random variables

with means ~qua1 respectively to (b + j - 1)~~, j = 1, ..., r.

inLenina 3.2. Let t and B be positive real numbers; let c = Sup

~ bN 1

andN i=l ‘

a — max a.. If (1.2) holds then; (i) P(SN r ~ t) � (1 - e t)1’ forl�i�in 1

r • 1, . . . , N, and (ii) Sup E(XN j(t)) c co~ for i = 1, .. .

~~ m.

Proof. We note that (ii) is a corollary of (i). To prove (i) let

U1, 02, ...~~ be as in Lenina 3.1. Since ~t5N,k ~ tJ � ~~~~~~ i~

1U;j

� act),j—1

Ci) follows from Lenina 3.1.

Theorem 3.3. Let B and t be positive real numbers. If (1.2) holds then;

Ci) (XN 1(t), ..., X~ m (t)) converges in law as N + to (X1(t), . . ., X1(t)) ,

and (ij Lim E(XN ~(t))8

— E(X~(t))B for i — 1, ..., in.

- ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ e r . ~~~~—~~.-e-—~.-—--——---—------ — - - . -

.e 5

_ _ _ _ _ _ _ _ _ _ _ _ _ _ _

7

Proof. We note that (ii) is a corollary of Ci) and Lemma 3.2 Cii).

To prove Ci) let k1, . . km be m nonnegative integers, A be the set

. . ., r~)~r1 ‘~ k~, i = 1, ... , in) and let R = ~ r~. Since for N

sufficiently large, P{X~~~

(t) � + = ~~~. .. ~~~~~~~~~~ =

ZPSN R � t < S~~1~+1 JiI(~N,q

i) = 1~~, 1 , . . . , in)I(~ ~ !-T)}, (i) follows

from Lemma (3.2) (1) , the doninatel convergence theorem, and some simple

calculations.

For ref erence purposes the following two wz~1l known results are presented.

Lemma 3.4. Let U1, 02 1 . .., be as in Leitma 3.1, and let p1, p2, . . . , be

a sequence of positive real numbers. Further we denote by ~k

the density

fu~ction of ~~~~~~~ k = 1, 2 Then, (~~) u;~i

fr+i(t)

P{Z p1U. � t <

~ p 10} and (jj~ fr(t) (li ii.) ~ e~ ’i~ II (p~

-

~ j=1 ~ j-4 ~ j=1provided p1. u~. are distinct.

i�j

We are ready to state and prove the main result of the paper.

Theorem 3.5. Let t and 8 be positive real numbers, and k1, ~~~~~~~ k~ be

nonnegative integers. Further let A be the set Cr1, . . . , r~Ir± � k1,

i — 1, . .. , mi. If conditions (1.1) and (1.2) are satisfied, then the

following hold.

m b +r - l k(i) Lim P(X.~ 1(t) � k~ + b~1 i • 1, . . ., m) = ~ (

b ~

)e0ib1~i - e~~~

t) i

__ A i l I

(ii) lAm E(XN . (t))8 is the 8th moment of a negative binomial random variable,] .

with parameters b~ and e 0it, i • 1, . . ., m.

- -‘i—- - - •‘~— - -

!~~~ ~~~~~~~~~~~~~~~~~~

8

Proof. By Theorem 3.3 it suffices to show that X1(t), ~~~ X~(t) are inde-peñdent, and have the respective negative binomial marginal distributions.Let k1, ~~~~~~~

km be nonnegative integers, ‘iith sum equal to k. Further, letB denote the set

£. c U, ..., i n) ,

JjI(tq = 1) = k1, i • 1, ...,

We note that P{X.(t) = k. ~. b., i = 1, ..., in) equals

I P{Sk � t < S~+1kq £q~ q = 1, ...~~ ki P{~q = 1q’ q = 1, . . ., Ic).

From Lemma 3.4 and equation (3.2) it follows that P{X.(t) = + b~. i — 1, ..., a)equals

in b. + k. - 1 -a.b.t k. k+l -e.t k+11 b ~

e 1 1 a.’k.! ~ e Ii (8. - O.) 1, wherei—i i B j=l i—ii*j

j-1 me . ~ ~ a.I(& = i), ~ = 1, • . ., k + 1.~ q~l i=1 q

By reusing Leunna 3.4 we obtain that P{X~ (t) = k1 + b1, i 1, . .. , k} equalsin b. • k. - 1 -a.b.t k. k.’.l k+1~ b -

‘1 e 1 3. a. 11c.!

~ I O~~U. � ti ii O~~.1=1 1 P~ jii~~ ~ j2~

To complete the proof it suffices to show that

in k k+l k+1 inII a3

1k.! ~ PC I � t} ii = II (1 - e 1 ~ I

(34)i—l 1 ~ j=2 ~ j=2 ~ 1=1

Let F denote the left side of equation (3.4). Further let

B. [Ci , £~~~I ...~

tq1IL2~ ~~ 1k ~ ... rn },

q~~I(Lq a r) -

Jc~, r — 1, ..., in , r ~ i, I 1(1. — I) a Ic1

— 13.qa2 q

_____ -~~~~

- - -

-

9

i n k in k÷l k+lWe observe that F(t) equals II aj~

1k~! 1 1 PC I O ’U. � t} Ti e 1.i—i i—i B

~ ja2 ~ j2 ~

By a simple calculation and Lemma 3.4, it follows that

in k in -at k+l k+1II a.~k.! Ic 1 1 P{ Z o t l u . � t n o~~., where 0 . =cit i—l 1 1 i—l B~ j=3 J~~ ~ ~~3~•••::J , i —Li

in j-iI ‘CL = r), 1 = 1, . . ., in , J = 3, ..., k + 1. Consequently Equation

r-1 q—2 q

::�i

(3.4) is established by an induction argument on k applied to each of thek+2 k+l

sums I I ~~ ~U. � t} II ~ :‘.., ~ = i, ..., in.B. j=3 -‘‘ j—3’~~’1

Corollary 3.6. Let X.~(t) the total number of infectives at time t. If

(1.1) and (1.2) are satisfied , then the following hold.

(i) X.~(t) converges in law as N + to a sum of in independent negative-a .t

binomial random variables with respective parameters b , e , i = 1, . . ., in , and

(ii) Lim E(X (t))8 equals to the 8th moment of the limiting random variable___ N

given in Ci).

En t’-’e particular case discussed in Billard, Lacayo and Langberg (1978),

X~(t) converges in law to a negative binomial random variable with parameters

~ b~ and e~~t. This fact played a major role in the proof of (1.3) for

i—I.the symmetric case as presented in the cited reference. Finally we note

that it is of interest to investigate the asymptotic behavior of rn-dimensional

simple epidemics, that do not have the f4arkovian structure.

—- -— --- -

~~~~~~~~~~~

.__ —5- .——- _55- - _ - --- ——5— -, -~~~~~~ - —5---— ----5----

5- -S..

~~—r--’

-

.

10

REFERENCE

(1) Billard, L., Lacayo, H. and Langberg, Naftali A. (1978). The Syimnetricrn-dimensional Simple Epidemic. PSU Statistics Report M46l.

_ _

— -5-——--55-- - -

- —-~~~~~~- - -• - - -~~~-- - — -~~~

- UNCLASSIFIEDSECURI~TY CLASSIFICATION OF ThIS PAGE

REPORT DOCUMENTATION PAGE1. REPORT NUMBE R 2. GOVT ACCESSION NO. 3. RECIPII!rJT S CATALOG NUMBER

FSU No. 1464AFOSR No. 84AFOSRN No. 9 _________________________

4. TITLE -

5. TYPE OF REPORT 8 PERI OD COVEREDOn the negative Binomial Convergence Technical Reportin a Class of ~-Diniensiona1 Simple 6. PERFORMING ORG. REPORT NIJHBF.REpidemics

- FSLJ Statistics Report M4647. AUThOR(s) 8. CONTRACT OR GRANT NUMBER(s)

H. Lacayo AFOSR 74-2531DNaftali A. L~ngberg AFOSR 76-3109

9. PEqFORTIING ORGANIZATION NA~1E 8 ADDRESS 10. PR OGRA:.l ELEMENT, PROJECT, TASK AREA 8The Florida State University 1~ORK UNIT NUMBERSDepartment of StatisticsTallahassee, Florida 32306

11. CONTROLLING OFFICE NAME 8 ADDRESS 12. REPORT DATEThe U.S. Air Force July, 1978Air Force Office of Scientific Research 13. NUMBER OF PAGESBolling Air Force Base, D.C. 20332

1014. FIONITORING AGENCY NAME 8 ADDRESS (if 15. SECURITY CLASS (of this report)

different from Controlling Office)Unclassified

l5a. )ECLASSIFTCATION/DO!i’NGRADING SCHEDULE

16. DISTRIBUTION STATEMENT (of this report)

Approved for public release; distribution unlimited .17. DISTRIBUTION STATE-lENT (of the abstract entered in Block 20, if different from reportj

18. SUPPLE.IENTARY NOTES

19. KEY WORDS

Exponential distributions, convergence in law, stochastic processes, rn-dimensional simpleepidemics, negative binomial distributions

20. ABSTRACTWe consider a population which is exposed to m infections, and consist initial ly of N

susceptibles. At each point in time at most one susceptible becomes infective, and only fromone cause. This in-dimensional simple epidemic is a stochastic process, CX N ,l (t) , ...

~~

with components counting the number of infectives from the respective causes at time t .We show that if the transition rates of cause 1 through m at time t are given by

aiXN j (t)[l - ~~~ ~~(X~~~(t) - XN ~(O))i,

i - 1, . . ., m , a1, ..., a~ > 0 and if Lint XN j (O) -

bi t Cl , 2, ...), then (XN 1 (t), . . ., X.~~ (t)) converges as N + ~ to a random vector with

independent negative binomial components.

L - ~~~~ - — - --— - -~~~~~ ~~~~~~~- - - .~~~~~~~~~