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Stochastic compartmental mooeld with branchingparticles and disasters: sojourn time and relatedcharacteristicsA. Vijayakumar a , B. Krishna Kumar a & B. Thilaka aa Department of Mathematics , Anna University , Madras, India , 600025Published online: 05 Jul 2007.

To cite this article: A. Vijayakumar , B. Krishna Kumar & B. Thilaka (2000) Stochastic compartmental mooeld withbranching particles and disasters: sojourn time and related characteristics, Communications in Statistics - Theory andMethods, 29:2, 291-318, DOI: 10.1080/03610920008832485

To link to this article: http://dx.doi.org/10.1080/03610920008832485

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GOMMUX. STATIST.-THEORY METH., 29(2), 291-3 18 (2000)

ABSTRACT

Compartmental modelling has applicationk in a variety of areas such

as drug kinetics in pharmacology, studies of metabolic systems, dynamics of

Copyright O 2000 by Marcel Dekker, Inc. www.dekker.com

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292 VIJAYAKLIMAR, KRISHNA KUMAR. AND THILAKA

chemica! reaction and ecology. The monographs hy And~rsnn (iU83), Giba!di

& P~erriei (1383) aiid Gatis et d., (1379j are concerned wish cornpartnwn-

tai .na!x:si. cf biological a ~ r i ~co',og:lcal systems. I?_ sr;stema;ic study of such

$ : : ! ? ! ! ! c s ! ! - . r:.-.. -z--

,4Ac,.., -?-.LlnmQ I--- 2]q- baepn msdG g s7.. c?-.<;rvs~: {?QQ?! :>-:! '.<:!:-.- , .d.=.T:

r-------- ---.. ...-- " -,, i i i i i i i i , . i"""", UllY J;JI\-GUCLI i l r l " ~ ) . - . - \

. . ~rp.ciiastic ..". nlodc;;ing and statisticel in r ience in conipartmelita; systems

have wit.nessed R ~emarkab!e deve!onment in the recent pzst as Sj1~31! i~ RPR-

shaw (1986). It is usually concerned with the study of movement of particles

within a system. Most oi these rnobe!s a s u m e that the partic!es do not rc-

produce. Recently, it has become quite relevant to study stochastic compart-

--- --.L---:- i L - - ..~ . . v

ILlrlit& y : : r r r l I l L i i e pzrticies in! each co-llcil,rtment repreduce s:m:iar

- - - L : - I - - T- - - I 1 L Z - I - - - - . , , . . . p n i i l L ; C b . l i i ~ ~ i i uivivgy; ~di H Y S ~ ~ I I ~ Y x i s e a l e ro mumcions occurring ir:

self-reproducing entities such as bacteria. viruses. toxins, hlood cells etc. The

.r'i;jjs tho bc -,-. ---.--.... the :ii-,i:-,l ,,-- --- ,,.I A > , -Ei- --I-& -::= .

--.. i i i i , t :~b i~~d! :,%j iilS biiil t l i i s ..II ~ p ~ i ~ t i i i ? i ~ : p ~ ~ v : : ? d r ~

some iniricaie exanipies. Tile uevelopment of such strains is a very undesirabie

phenomenon in the treatment of infectious diseases, particu!nr!y b e e a ~ s e swh

w s i s i n n t strains rzn mr2ltipiy ?Q rllch 277 extent thlt the perscns cannot be - - - - - . .. . . - - . . -. . . . .

' 2 1 I L 1 .- b l e i t w u e u e L u v e l y WILII diliibioiics.

Piifi (1968) and nenshaw (1973) stildied two-compartmenta~ sys-

terz!s witfi partic!es in each corrisartment ~ e r f ~ r r n i n z u a simn?e A- - birth and death

process. Gerald (1978) and Gerald & Matis j1979j have analysed an irre-

versibie id- compartmentai system with birth, death and immigration of

particles, Recently, Krishna Kumar and Parthasarathy jlY93a, 1993b) have

analysed a more general semi-Markov compartmental system wherein the par-

t , - .;,I,, ,,- ;,, .- each coinpartrnent reproduce simihr particles accol-riing to a ivlarkov

branching process.

Sojourn times play a vita! role in compartmental analysis as discussed

by Olson and Matis (1985) and they give added insight into the system kinet-

ics. Estimated sojourn t i n e moments are increasingly used nowadays in the

statistical analysis of kinetic data for one or more of the following reasons as

indicated in Matis et al., (1983):

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STOCHASTIC COMPARTMENTAL MODELS 293

(i) they are of inherent bio!ogical interest.

,*,,*<, 1 . . 3 . : I T . , , ,, , ,,- -,.,,,< ,.:.? ---.---. - A -.-- + ,--- * . , ; aLx;L; \ i s s a j -.b" .jZ.) t.zD.i & t i V d i i 01 t t e E:e G i : i ;^ST . .ruli

sojourn times for biolcgical interpsetability and their relative invariance over

different structurai assumptions of the modei. Simulation study of kinetic

models based on sojourn time moments are found to be very useful as explained : .. . , r T - % , ,: ~ - - . .- - ' ; .- . " ~ 7 uJ -" "/iatis -35 ;vR2e:i;rii: (?YE>/. R'iat:s i i ~ ~ 8 ; "as anajysed the - ~ o + h r . ~ u ~ ~ ~ ~ ~ i ~ ~ . ~ crrbi .St .q

- ? - ,"ariati!lrl ~ cf tilt. residtlilce (so~or?rii\ time distributions for a klark=via,n and a \ 0

Y-r;i - - 3 . : " ........ ,;21i iZE s;-stcm ..:. ;fh:-:;.llt 5 >13'? :-hit15 n':n~fi:-!ep ."..,""-" Y.-..Vl...l~ y Y . Y . " . V Y .

. . I'haj-P h a 4 J-WPTI -7 gfrat "& cf !-..erest irl she d ~ . ~ ~ l f i t \ ~ ? e r ~ t . I:!<! zr!a!y$io - - - - - - - - - - - - - -.

of models for the growth of - A - =TO ciih:mt Y - Y J V V U tn .,- Aic- Us"

asters (caiastruwi~es) ieading to death or iarge-scaie emigrations. Kapian et

aE., j1975), Hanson and Tuckweil (19781, Fakes e t al., (1979) and Trajstman

(1978) considered snch disaster mode!s associated with br~nching processes.

Disasters can occur in the form of epidemics, famine, forest fire, earthqua,ke ... * n . . . or :ixe lactors. Aisn in the trea.tmen:, ~ ? f in:ec:mus diseases? batcteria or viruses

which are exposed to antibiotics may survive with certain probability. A sim-

ilar phenomenon appears to occur when population of insects are exposed to

insecticides. Chen and Renshaw (1995) have studied a Markov branching pro-

cess regulated by emigration to avoid excessive growth of population level and

discussed the limiting distribution of propulation size. More recently, Renshaw

and Chen (1997) have analysed a birth-death process with mass annihilation

which hrings d o ~ n the population ]eve! to zero and an immigration compo-

nent reviving further growth of the population. However, this aspect has not

received due attention in compartmental modelling. Compartmental systems

incorporating both the branching of particles and disaster factor can be effec-

tively applied to modelling many biological and ecological sitnations.

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In this paper, we cons id~r a compartment.a! model with P. Markos. branch-

ing process under the infilience of disasters. Expressions are ihund for the mean

of tot&! 3o;n:irr: J - f i m ~ , the mean of the total number of deaths, *he rr;?,ean of total

ImI:i',er of birth3 znd the .,:leari vf tire i;ui:ibcr G p emigrant i;art;c';es durilig &?

. , . 3 . ': uiiiit: --..- ( 2 , t j . izieres-,ing re;ations &nzse are on;airie& Siiiiie

soecizl czses are also disc!:sser! i:: detail..

any of the compartments say i j either reproduces Ic partides of in?i!ar type

with probability a t ' h + o(h) for k = 0 , 2 , 3 , . . . , or does not reproduce with

probability 1 - ay) h i o(h) where

with a t ) 2 O for all k. In addition, we assume that disasters occur in the

it"ompartment (i = 1,2 , . . . , N) as a renewal process with distribution

function Gi(t) and the corresponding probability density function gi(t). If

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STOCHASTIC COMPARTMENTAL MODELS 295

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296 VIJA'PAKUMAR, KRISHNA KUMAR, AND TWILAKA

, .\ i , e t s)?'itj = E([lii(tj) J y be the mean sojourn time of a!! p.rtic!es in

compartment j originating from an initial particle in compartment i and , --.

_S:'(t) < rn for a!! i , j = 1 , 2 , . . . , N. On differentiating (2.1) with respect t o

s j and setting s = (0; 0; . . . O)j we obtain after some algebraic manipulations,

where bij is the Kronecker delta function and

is the mean number of offspring of a particle in ith compartment.

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STOCHASTIC COMPART.MENTAL MODELS

where I is the unit matrix of order N. il r . . 1 in our nor:;nr Dapei ,, s * - - r - w q

. . 0 .. i U l l l l \ i i A

- - - r,v,4ra,amizr CE ni.: ! , : ~ ~ 6 / i , . . : ~ r tnc p r~ccss w d e r

consideration, we have obtained the Laplace transform M ( z ) of M ( t ) , the

matrix of the mean number of particles alive a t time t , as

Thus from (2.4) and (2.5), we have

which on inversion yields Dow

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298 VIJAYAXUMAR, KRISHNA KUMAR, AND THILAKA

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STOCHASTIC COMPARTMENTAL MODELS 299

For tractable solutions, we consider the foiiowing special cases:

1 -. 1 3--*% - " - ~ - - - - . 3 - , : u3r71LLlle the &s&crs the :"'"cnmzaSmfrLt ocr:_rr as a ro issn~. '~-!

. . ,-. , - process with parar;;etc; pi &and the tirrAe diStributiors f e ~ ~ ~ U i 2 gp;.;:zl .-..+.ria

distribution F;(t) , so that (3.1) becomes

and for j = 2 , 3 , . . . , n'

On taking Laplace transform, we obtain

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YISAYAKUMAR, KRllSHNA K U M A R , AND T H I L A K A

Inversion of (3.3) yield

--. I...-,. V ( k ) ( . ' \ :- ' L - W I ~ C I C 1. ( L J ~b UX k- fold ~ o i i ~ o l u t i o n of Fiij with iiseif. These recursive

. . , .. nr L-. L . -1 re!&ions exp]icit ~ x p r e 3 s ~ 0 ~ ~ fGz sjij ( t ) , ̂I 5 j , = 1 , 2 , . . . , uji DacK

substieutiorl.

(b) If the disasters occur as a renewai process with inter-occurrence time

distribution Gi( t ) , and the residence times are exponentiaily distributed with

parameter vz for the ath compartment ji = i, 2 , . . . , iYj, a simiiar procedure

as above leads to

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STOCHASTIC COMPARTMENTAL MODELS

" A X N - @BMPARTMENTAL MAMMZELAP-Y SYSTEM * - - - - -

A rnaziiaikii-y compa~tmental system hss enc ccntrra! campartmezt acd

severa! peripheral compartments. Particies in the central compartment can

L - . iii,; 4ii t h ~ peripheral csm~ar',r;:ents or t:r: the szr:oun&ng ca~i.ir~nment but

7- - , - . - . * * - - - ----- - - - - - I L . . -rrr.r.** -.*inr.r.r.~ . \ ~ " . . - ~ ~ ~ r . ~ i F,-.l-,?--7Tf_ LlUL vlLe-vci>a. 1 U L L l l t Z l 1 I l " V C i L ~ C L L L L ~ L ~ G D c L i L & " & ‘ 5 pLl,p;,Lxu‘ ~ " A L , ~ u &

merits is inadmissible. Such models are often useful for analysing the kinetics

of distrihlitinn of a niaterid which is injected into the plasma and which enters

only the interstitia! space btit not the intra-ce!!u!a: space of the crgans. ?./Isre

:';&ails be h u n d in & Perrier (1975) and J.cq~.. (1985), Usinv o ihp

notation of eariier sectiocs; assuming compartment 3 as the central compart-

ment. we iiWe 2 f"r j; = 2 , 3 , . . . , N, P ~ , ~ + ~ 2 for i(i = i , 2 , . . . , ;%I)

and the other pi; 's are zero. Equation (2.2) now reduces to - 4 f *

gZ)(t) = I e-'ii'"[l - Gi[u)][1 - ~ ( u ) ] d u J 0

+ti I n i t e - a i " ~ ; ~ - ~ , ( u ) l ~ ~ ( u ) ~ / ' ) ( t - u)du " "

- Gi(u) j[ i - - u)dU,

i = 1, Z,. . . , IV (4.la)

and for j = 2,3, . . - , ?J

( 1 ) t

5'; ( t ) = r l & e-ail)u[l - F 1 ( u ) ] g l ( u ) S ~ ( t - u)du

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VIJAYAKUMAR, KRISHNA KUMAR, AND THILAKA

and for j = 2,3, . . . ,I%'

(bj For the case of exponentially distributed residence times with parameters

vi, i = 1,3, . . . , N and the d is~s ters nccurrino ---0 ac -- a renew?.! prncess with c.d.f.

Gi ( T L ] we get. correspondingly

and

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STOCHASTIC COMPARTMENTAL MODELS 303

5 . AN Ar-COMPARTMENTAL MARMOV SYSTEM

IT *--. Y Y I ; I ~ I i the residence tines in each compartment and the inter-disaster

time distribllcions are eYD03ectiai. th'i. is U; { t ) = i;,e--'i-t ~ i i r i f j t j - i?.p-":', - - > , - . , : " -,..

t > 0, 1 ; :, -, A 7 &L,,,-. L1,- - 7 . . . 3 A . , , 1 ------ ,,;.,, ,,ii 3;- ~ n - ~ ~ r : - e n i a l - - --- system -,y;:t$ bri?.rl&?;riz - " . % < , , irisucncc $21 <isasters becomes 2 I\lrarF>zv;az ziode. Tiie

integra! eqzatio~i (2.3) car? be cowerted mto the differentia! equation

where I is the unit matrix of order N, S ( % ) = (sjq)jt)), a square matrix of

so that Dow

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VIAYAKUMAR, KR!SHNA KUMAR, AND TfillLP.KP.

-A similar asymptotic b e h a ~ ~ i ~ u r has been studied by Olsor, & Matis (19853 arid

Parthasarathy & Sharafali (1989) in different contexts while studying the total

. - ~ ~ - - - tcorrrpar-tirrer!tal model wirn nirfn anG death process, ~ Y E : asscmi: th2t ~ S Q Z

and

where ?.. t z - - - [a!) + f i g i + v,], i = 1,,2

and (111 + ̂ /22) * J(711 + 1 2 2 1 2 - 4(1 i ihz - 7 1 2 1 2 1 )

6 1 , K2 = 2

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STOCHASTIC COMPARTMENTAL MODELS 305

6 . ---A T R T T T R A--- r \ m r \m r rnirsn IuaAei i ~ u ~ e s n a k w r e ? r o ~ . ~ n a

-. i ile effects of cumuiative processes frequently infiuence the quantitative

o e n o r t ~ of population dynamics. Such precesses hme ewked =uch interest, U Y y Y Y Y "

in biology and medicine while dealing with risk assessment using statistical . .

L - - L . - . - - - . . - . * k - -..T-+ . . ~t.t:~~ill~ue.~. lienee; it 1s oi interest to study L.ir rici-iLr~ of r~artjcies ivhii.h die

during the time (0 , t). Puri (1968) has studied the number of particles of single

type which die during the time (0, t) for a Markov branching process without

disasters and obtained some limit theorems. Here, we consider the cumulative

nurnber of particles which die during (0, t ) for the process dealt with in earlier

sections.

For the process under consideration, a particle of ith type can die either

d ~ e to disaster or ~ p o n its split. Define v z 23 - (+) \ " / zs,the tota! nurnber of particks

which die in the j th compartment during the time (0, t ) originatiag from a

particle in the i th compartment. Defining the joint probability generating

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VUAYAKUMAR. KRISHNA KUMAR. A N D THILAKA

Figure la. Mean sojourn time of ail pariicies in compartment i , initiated by - - - . . L : - I - :- L - - - L 1 x r - T:-- TL ------- L-- i-'"'L"'' :" '""'""3 ' "'""" '", ' '12". '12' " " 2 """"'> a:', r-- r---"--"--

and by considering the mutually exclusive possibilities, we have the integral

equation for G ( i ) ( s , t ) to be

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STOCHASTIC COMPARTMENTAL MODELS

Figure ii,. lviean sojourn time of all pa,itic:es in compartment 1, initiated Euj;

a particie in compartment 2 'is. Tirile. The parameters are:

t +st e- a!) [G(') (s, t - u)Ik [1 - G, (u ) ] [l - F, (u)]du

J U c-Q

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308 VIJAYAKUMAR, KRISHNA KUMAR, AND THILAK.4

Figwe 2s. Mean s g j x r r , time ~f a!! particles in cornpar tmat 1, iriitiateb by a partic!e ir? corr?prrtmer?t ! Vs. Time. The p s r ~ m t e r s are:

Differentiation of (6.1) with respect to s j and evaluation a t s = (1,1,. . . ,1)

..:,\lA" V l C l U 3

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STOCHASTIC CO.MPART.MENTP.L M O D E L

Figure 213. Mean_ sojo~urn time of a!! pa.rtic!es in compartment 1 initiated by a particie in compartment 2 Vs. Time. The parameters are:

+ b(') 1:' e-.:)' [l - Gi (u ) ] [1 - F, (u)] D:) ( t - u)du

The above equation can be written as the matrix renewal equation Dow

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is a diagona! matrix of order 1%' and K ( t ) is tfre square matrix rrf order N as

defined in section 2. Taking Laplace transforms of equation (6.3), we gei

,,, wnere R , " " : , ! i , (2) { ?v')

- W U W Y Lwl , rl l , l , al + hv2, - - - , al I p ~ q ~ ] G

and L is the square matrix of order N as in section 5. Thus, from equations

(5.2) and (6.5), me have

cl.,\n = V , - , A 3 - \ * I . (6.6)

A h , from equations (2.6) and j6.6), we have

which on inversion yields

From the above equation (6.7), we infer that the knowledge of one of these

means is sufficient to obtain the other two.

Special Cases:

(i) For the irreversible N- compartmental system discussed in section 3, we

assume that the inter-occurrence disaster times are exponentially distributed

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STOCHASTIC COMPARTMENTAL MODELS 311

and for 7 = 2 . 3 , . , , . !V

Similar anaiysis can be made for the N - compartmental mammillary system

described earlier.

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A . - ~ I L ViJAYAKUMAR, KRISHNA KUMAK, AND THILAKA

Let ~ ; ' ) ( t ) = E(Z,,(t)) be the mean of the total number of births in cornpart-

ment j d~ue to an in i t id partide in cnrr?partment i fnr 1, j = ! , 2 , . * . , N. Or,

differentiating (7.1) with respect to s, and evaluating at s = (1,1,. . . , I ) , we

obtain

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STOCHASTIC COMPARTMENTAL MODELS

in matrix notation? the above equstiorr can be -mitten as

- - -LK( t ) is the squwe matrix defined in section 2. On taking Lapiace trans- L L l l U

forms of the above equation (7.3) , we obiam

For the irreversibie 1%'-compartmental system discussed earlier, assum-

ing the inter-occurrence disaster times to be expanentiaiiy distributed with

parameters Pi(i = 1 , 2 , . . . , N) and the residence times to follow the general

distribiitioii Ivi'i(t) , = 1, 2 , . . . , N, :ve

and for j = 2 , 3 , . . . , N

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3 14 VIJAYAKUMAR, KRISHNA KUMAR, AND THlLAKA

Again for the irreversible iV-corn~artmental system, if the residence times are

exwnentia1:y Y distributed with parameters vi(i = 1 . 2 , . . . , W j and the disast,ers

occur as renewal pmcess, then

h s in the earlizr sectionsi we car? u'utain exyiicii; expressisns for the special cases

or' the rnaiiimiliary compartmental system and the Markovian compartmental

sy&riis.

P T n W A I h l T T A K T ) - Y l A= r'%x ST-? r m T 3. L v n r a u r u iii~iahfi QBY~. +irlvuisnfi~lONS

... , ,

Let Wiji t j denote the totai number of emigrations in compartment j

due to an initial particle in compartment z and

denote the joint probability generatifig - fcsct im cf the tsta! number of emigra-

t i ~ n s in the system due to an initial partide in the i th compartmeni;. By using

the totai probability arguments, we obtain the integral equation for ~ ( " ( s , t ) as

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STOCWAST!C COMPARTMENTAL. MODELS 315

Let f3jz'jt) = E(Wijjt)) be the mean of the eotai number of emi-

grant particles from compartment j due to an initial particle in compart-

-?_..I , . . - . .

, = 1. . . . , T \ On &gere2tlatin" 'C '', --.:'I. ----o--" --- IE .? 1 - I ! rull_l! lr.,u,.\,u t~ S j 2";

c- - -,

. 3 - ~ l : i . . . . , ~ ~ , mo'nta in

In matrix notation. the above equation ieads to

* t

E ( I ) = C3(t ) + lo K(t - u ) E ( u ) d u

a diagonal matrix of order N and Kjtj as defined in sectiiin 2, Takiiig Laplace

transforms of equation (8 .3)! we get,

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3i6 VIJAYAKUMAR, KP,!SHNA KUMAR, AND THlLAXA

The above relation implies that the mean number of partic!es in the system at

time t should be the same as the mean of the total number of births by time t ,

less the mean of the total number of deaths & emigrants together by time t .

ACKNOWLEDGEMENT

Anderson, D.N. (1983). Compartmental Modeling and Tracer Kznetics, Lecture

notes in biomathematics Vol. 50 (Berlin Springer).

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STOCHASTIC COMPARTMENTAL MODELS 3 14

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i<&, arltl TvIaiis; J ,H, (19T9j. :tan iL-- - - . -2s . - - - -" .'.,.Y,.. Pi- %mr,Qi -tTi,on- hc Luiiiuiautu vi o v i l r b b v u k y u s u u * u u

tal snGdejs zp~liert_ ta pc~iogical - system?, in Cnrn,~artrnentrtl A 7 d y , & 0";

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c;+ii h;~;,&,;oa- Prpqr "bF"L"J C?f "I'-"'b.":: ' --.A.

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L Limes I of particles iii seiiii-hiai.kov compaltmeii$,al, system with b r a n ~ h ; ~ g VS..~.

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318 VIJAYAKUMAR, KRISMNA KUMAR, AND THILAKA

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