1
2
Abstract
Dedicated to our friend and colleague,Stavros Busenburg.
DISTRIBUTED INDIVIDUAL RATES
FROM AGGREGATE POPULATION DATA
H.T. Banks
B. Fitzpatrick
Yue Zhang
Center for Research in Scientic Computation
North Carolina State University
Box 8205
Raleigh, NC 27695-8205
September15, 1994
We present a metho dology forestimationof the distribution of individu al ratesin
p opulationsfromobservationsofaggregateb ehavioroftheoverallp opulation. Discrete
andcontinuousprobabilitydistributionsarediscussed. Applicationtoanexamplefrom
vaccine protectionincludi ng computationalresults aregiven.
1
2
2
2
InvitedLecture,InternationalConference onDierentialEquationsandApplications toBiologyandto
Industry, HarveyMuddCollege,June1-4,1994,Claremont,CA
Research supp ortedinpartbytheNationalScienceFoundationundergrantNSF-UINT-9015007andin
partbytheAirForceOceofScienticResearchundergrantsAFOSRF49620-93-1-0198,F49620-93-1-0355,
P
0
=1
0 1 2
2
N
N
N
N
N
j M
j
j j j
j j
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j j
j
i
n
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2 Distributed Rate Models: Discrete Case
`
q R ;j ; ;:::;` N t
j;j ; ;:::;` t
N t f t;N t ; t ;q ; j ; ;:::;`;
N N
t N t t
t N t
t ;i ; ;:::;n
t t ;t ;t ;:::;t
j ; ;:::;`
q t
In this note we outline ideas and pro cedures that can b e used in obtaining individual rates
(growth, mortality,damage,susceptibility,fecundity,etc.) in p opulationsfrom aggregate or
total p opulation data counts. These ideas have b een shown to b e theoretically sound and
computationallyecientinseveralexamplesthatareimp ortantto researchersinthe
biolog-ical sciences. A sample of results on estimation of susceptibility and vaccination eciency
in a p opulation is presented in Section 4 b elow. While this application is rather simple, it
do es demonstrate the usefulness of the metho dology. We have tested morecomplex
exam-ples involving innite dimensional dynamicalsystems, e.g.,the partial dierentialequation
governed systems(Sinko-Streifer) in size-structured p opulation mo dels. Such problems
of-fer another levelof diculty - ecientapproximation of innitedimensional dynamics- as
well as that of estimating distributed rates. While we have successfully used the ideas
re-p ortedhereonsuchproblemswithexp erimentaldata(excellentresultsaredetailedin[BFZ])
from mosquitosh studies, b oth the theoretical and computational asp ects (involving
par-allelalgorithms on an Intel hyp ercub e) are to o involvedfor presentationin this short note.
Theoretical foundations for the ideas here and in [BFZ] are presented in detail in [BF][F1]
and [F2]. For abriefsurveyofother asp ects ofsuchproblemsand furtherreferences,see[B].
We assume that the total p opulation can b e divided into a nite numb er of sub classes
with eachsub classb eing distinguishedbyitsmemb ersp ossessing likerates. Theserates are
characterizedby vectorparameters =1 2 . Welet ( )denotethenumb er
in the p opulation class = 1 2 , at any time . This numb eris assumed to change
in timeaccordingto the parameter dep endent dynamics
_
( ) = ( ( ) ( ) ) =1 2 (2.1)
(0) =
where ( ) ( )is the total numb erinthe p opulation at time .
In the general situation we cannot distinguish the sub classes when observing the p
opu-lation (exceptions exist, of course, such as in the case of male/female sub classes and sexed
data). Thuswe assumethat we may observe ( )but not ( ) at various samplingtimes,
including the initial time = 0. That is, we are given observations ^
= 0 1 , for
( )at times =0 .
Ourgoalistousethisdataandtheknowndynamics(2.1)toestimatetheoveralldynamics
ofthe p opulationas wellas thoseof thesub classes. In particularwewould liketo determine
how the sub classes =1 2 , are distributed throughout the p opulation; i.e.,how the
characteristic parameters are distributed in the overall p opulation at any time . We
assumethatthe characteristicparameterforagivenindividualisintrinsicand xedintime.
j ( ) P X X P f g N N
f g N
N N
N
f g
N
f g
N N N
jN 0N j
f g N fNg N N N N N N N =1 =1
0 0 0 0 0
0
=1 =1
0
=1
0
0 0 0
0 0 =1 0 0 =1 0 =1 0 0 0 =1 0 2 0 0 =1
0 0 0
=0
0
0 0 0
0 0 =1 0 0 j ` j j ` j j j j j ` j j ` j j ` j j j j j ` j j ` j j ` j j j
i i i
n i i i j ` j j j i n i j
j j j
j j j
j j j
j j j
j
`
j
j j
j j j j
P t p t
N t
t
p t j t
P P N p P
p N t ;j ;:::;` t N t
P t t >
t t P N t P
N p ;j ; ;:::;`;
P p
N
:
P P t
q p t
t q
;i ; ; ;:::;n t t;P
P
J P t P
P p
N p f
J
f N
N t g t; t ;q N t ;
N t N t =p j ; ;:::;`
N N p f N
N t f t;N t ; t ;q
N
t t P p N t
P t N t p N t N t
( )= ( ) =
( )
( ) (2.2)
so that ( )istheprop ortion ofthep opulation inclass at agiventime . Notethatunder
these assumptions it suces to determine = (0) since = . Henceonce =
isknown, the dynamics(2.1)determine ( ) =1 , and ( )= ( ),
and thus, ( ),for all 0. Therefore,wehave
( )= ( ; )= ( ; ) (2.3)
where = =1 2 and
= =
(2.4)
In our discussions here, it is clear that (and ( )) can b e viewed as a probability
distributionon the parameterspace inthat ( )is theprobability that arandomly
selected individualin the p opulation at time will havecharacteristicparameter .
Given observations ^
= 0 1 2 , for ( ) = ( ), we may now formulate
the least squares estimationproblem intermsof . We seekto minimize
( )= ^
( ; ) (2.5)
over all discrete probability distributions = subject to (2.3) and (2.1) with
^
. Underreasonableregularityassumptionson in(2.1), itisrelatively
straight-forward to argue existence of solutions for the problem of minimizing in (2.5) for a given
setof observations ^
.
These ideas for a discrete numb er of characteristic sub classes can b e extended to the
moregeneralcaseofprobabilitydistributionsoveracontinuumofcharacteristicparameters.
Inpreparationforandto motivatethediscussions ofthecontinuouscase inthe nextsection,
we note that if is linear in , i.e. (2.1) has the form
_
( )= ( ( ) ) ( )
then we can equivalently dene evolution of our sub classes in terms of scaled p opulation
densities. To this end, dene the scaled density ~
( ) ( ) , = 1 2 , so that
~
(0)= since (0)= . Hence, if is linearin ,we may scale (2.1) to obtain
_ ~ ( ) = ( ~ ( ) ( ) ) (2.6) ~ (0) =
where ( ) = ( ; ) =
~
( ). One can then use this equivalentsystem in
mini-mizing (2.5). Of course, ( ) is still dened as in (2.2) where ( ) = ~
( ) with ~
( )
Z Z
Z
Z Z
Z Z
Z
M
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Q Q
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Q distribution
f g
N
2
2
N N
N
N
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!
1 2
0 0
0
~ ~
0 0 0 0
0
0
0
0 0
0
0
~ ~
0
0 0
0 0
q Q R
Q q ;q ;:::;q : Q Q
P Q t
q Q P Q Q Q
t q Q
dN q dP q P Q
dP dN N q
q N t q
N t q f t;N t q ; t P ;q
N q
t P N t q dP q
t f
N N t q
q t
N t q g t; t P ;q N t q
N q N q
q Q Q
dN t q N t q dP q
t P dN t q N t q dP q ;
dN q :
q N t q
Inageneralizationoftheconsiderations ofthe previoussectionthatwillincludethediscrete
case as asp ecial case,weassume that wehave ap opulation with p ossiblecharacteristicsin
acontinuumofvalues. More sp ecically,the feasibleor p ossible values of the characteristic
(vector) parameter lie in some given compact set . (In the previous section the
set consisted of preciselythe nite set ) For any Borel subset ~
of , we
dene [ ~
]astheprop ortionofthetotalp opulation at time =0p ossessingparameters
~
. Then is a probability distribution on and for any given Borel set ~
, the
total numb erof p opulation at time =0having parameters ~
isgivenby
(0; )= ( )= [
~
]
where and areinterpretedintheusualLeb esgue-Stieltjessenseand (0; )represents
the p opulation p ossessing parameter . Motivatedby(2.6), wedene ~
( ; )as
the solution to
_
~
( ; ) = ( ~
( ; ) ( ; ) ) (3.1)
~
(0; ) =
where
( ; )
~
( ; ) ( ) (3.2)
is the total numb er in the p opulation at time . Since we are assuming that is linear in
~
,this isequivalentto the continuumversion of(2.1). Thatis,if ( ; )isthe distribution
function for distributionof the parameter throughout the p opulation at time , and
_
( ; ) = ( ( ; ) ) ( ; ) (3.3)
(0; ) = ( )
we havethat the numb erof the p opulation p ossessing parameter in ~
is givenby
( ; )= ~
( ; ) ( )
and
( ; ) = ( ; )= ~
( ; ) ( )
= ( )
In this case the function ( ; ) is a function of b ounded variation with sp ecial
cases consisting of it b eing absolutely continuous (leading to a continuous distribution of
j j R R R X R R 8 < : X X 9 = ; ~ 0 0 ~ 0 0 0 0
0 0 1
0 =1 0 2 0 0 0 0 =1 + + 0
0 0 =1
Q Q Q i n n i i i k k Q k Q k K j
j q j j j
q j K K j K j N N N
jN 0N j
N N N
P Q
P P
2 P ! !
!
!N P
P
P
P 2 2
P
P
2P
f g
t
P t Q
N t q dP q
N t q dP q
N t q dP q
t P :
P
t
P
;i ; ;:::;n t P t ;t ;:::;t
J P t P
J
Q
Q Q Q Q
P ;P Q ; P ;P dP dP
Q
P A P A A Q P @A
f
P t P Q
J
Q
Q Q
Q
p p ; p ;K Z ;q Q
q Z
Q
P Q
P p
general case,one can approximatethe measureor distributionfunction by the discrete case
of Section 2 so that at least for computationaleorts, the situation inSection 2 is,in some
sense, the typicalor generic situation.
As in the discretecase wemay denethe probability distribution at any time
( )[ ~ ]= ~ ( ; ) ( ) ~ ( ; ) ( ) = ~ ( ; ) ( ) ( ; ) (3.4)
Once is known, one can use (3.4) along with (3.1) to determine the distribution of
parameters throughout the p opulation at any time .
As inthe discretecase, we wish to estimate the distribution or measure using
obser-vations ^
=0 1 , for the total p opulation ( ; ) at times . Thus, as
b efore, we denethe least squares estimationcriterion
( ) =
^
( ; ) (3.5)
where now is givenby (3.2) subject to (3.1) with = ^
. We seek to minimize over
( )denedas the setofallprobabilitymeasures(distributions)on the Borelsubsets of
. Wemaydeneametric on ( )underwhich,since iscompact, ( )is acomplete
metric space which is in fact compact. This is the Prohorov metric (see [BF],[Bill],[EK]
for a precise denition), and it can b e characterized by sequential convergence in that for
( ) ( ) 0 if and only if for all b ounded, continuous
real-valued functions on . Convergence in this metricis also equivalent to convergence
in distribution,i.e. [ ] [ ] for allBorelsets with [ ]=0.
Under reasonable regularity assumptions on in (3.1), it is not dicult to argue that
(3.1) has unique solutions and moreoverthat ( ; ) is continuous on ( ). Hence
one can guarantee existence ofsolutions to the minimization problemfor in (3.5).
The space ( ) is, exceptin the discrete case, an innite dimensional space and must
b e approximated for any computational eorts. The fundamental approximation theorem
for the set ( ) with the Prohorov metricmay b e stated in the following way. Let ^
b e a
countable densesubset of and dene
^
: 0 =1
^
(3.6)
where is the Diracdelta measureat and is the setof allp ositive integers. Then ^
is dense in ( ) inthe Prohorov metric.
Using this theorem,wemayapproximate any measure ( )by adiscretemeasure
= and for this discrete measure any computationaldiscussions for this section
that are p ertinentto solving the optimization problem for (3.5) reduceto those for solving
` 1 2
P
P
P N
N
N
N
0
j j
j j
n
j j
j
j
T T
j
`
j j
j
j v
v
`
j j
j
u j u
0
0
0
0 0 0 0
0 0
=1
0
0
0
0 0 0 0
0 0
0
0
=1 0
=1
4 Example: Susceptibility and Vaccination Eciency
J P
P
J P
J
P N p
N t N t P t P N t P
J P f N
N t P
P p col p ;p ;:::;p :
t;P p J P
J P p Ap p b c
p ; p
; j ; ;:::;`
X t j t X t
t X t X t
G t
; j ; ;:::;`
algorithm that can b e used to solve b oth the minimization problems of Sections 2 and 3.
To outline this algorithm, we describ e how one computes (
) for a given (current)value
or estimate
of a minimizing candidate. These steps then may b e used in one's favorite
optimization (in general, iterative) algorithm. Of course, derivatives of at
and/or
neighb oring p oint evaluations may also b e required. These are computedin the usual way,
giventhat one knows how to evaluate .
In the generalcase, given
, one uses (2.1) with = to solvethe initial value
problemfor ( )= ( ;
)andthen ( ;
)= ( ;
). Thiscanb eused directly
in (2.5) to compute (
). In the sp ecial case that is linear in and do es not dep end
explicitly on , the aforementioned steps lead to a solution ( ;
) that is linear in the
vector
= = ( )
It follows that (
) isalso linearin and hence (
) isa quadraticfunction
(
)= +
which must b e minimized subject to 0 = 1. This quadratic minimization
problemcan, of course,b e solvedusing one ofseveral p opularand widelyavailablesoftware
packages.
As we noted in the intro duction, the ideas outlined in Sections 2 and 3 can b e used with
readilyattainable p opulation data to estimatethe distribution of protection in vaccination
programs. We describ e here some sample results from application of the metho d to such
a problem discussed in some detail by Brunet, Struchiner, and Halloran in [BSH]. For our
discussions, we follow as closelyas p ossible the notation adopted in[BSH].
We assume that we have a p opulation with individually varying susceptibility to some
infection that is related to environmentalexp osure (the particular vector or mechanismfor
exp osure is not imp ortant to our discussions here). We are concerned with the ecacy
of a vaccination program with a vaccine which aords an individually varying protection
to memb ers of the p opulation. We hyp othesize that the vaccine oers dierent levels of
protection in a manner that pro duces sub classes in the vaccinated p opulation that can b e
characterized by susceptibility factors = 1 2 . (In light of the discussions in
Section 3, any probability distribution may b e approximated arbitrarily well by a discrete
probability distribution. Hence we may deal with a discrete numb er of sub classes.) Let
( ) denote the numb er in class that are still uninfected at time and let ( ) b e the
total numb erof vaccinated but not infected at time so that ( )= ( ).
The parameters whichrepresent aprop ortionality factor that measuressusceptibility
to environmental exp osure ( ), a rate of exp osure to infection, are b ounded ab ove by a
X X Z X P 8 > < > : f g 0 0 f0 g 0 f g N 2 0 0 0 0 0 0 0 =1 =1 0
0 0 0
0
0
0
0
0 0 0
1
0
=1
0 2
0 0 0
=1 0
0
0
j j v
v v i v
i
j j j
j j
j j j j
v v ` j j ` j j j v j
j j j
j
j j
t
v v
v i v i v v v
`
n
i
v i v i
j j
`
j
j j j j j
v
j
X X X t
t ; i ; ;:::;n
X t G t X t ;
X X ; j ; ;:::;`;
X X t X t =p
X t X t P X t p X t ;
X t X X X
X t G t X t ;
X X ;
X t X G s ds ; j ; ;:::;`:
X t X t P
X X t ; i ; ;:::;n X t X X X
;:::;
J P X X t P
P p p ; p q ; N t X t
t X t
G
P ;
K ;
P ` G
G t
t :
t : : <t :
: <t:
where we assume that (0) is known. We are also given observations ^
for ( ) at
times =1 2 . Following[BSH],weassumethat sub classp opulation dynamicscan
b e describ ed by
_
( ) = ( ) ( )
(0) = =1 2
(4.1)
where,of course, will b e unknown. Dening ~
( ) ( ) so that
( )= ( ; )= ( )=
~
( ) (4.2)
we havethat the numb er vaccinated at time =0, satises = (0) = ~
(0).
Moreover,the system(4.1) isequivalentto
_ ~ ( ) = ( ) ~ ( ) ~ (0) = (4.3)
whichhas solutions
~
( )= exp ( ) =1 2
(4.4)
This expression can b e used in (4.2) to obtain ( ) = ( ; ). That is, once we are
givendata ^
for ( ) =0 1 (so that ( )= (0) = = ^
) and
suscep-tibility values , we can use (4.4) and (4.2) in solving the problemof minimizing
( )= ^
( ; )
over all = such that 0 = 1. If we identify = ( )= ( )
and ( )= ( ), this is exactlythe problemdiscussedinSections 2 and 3.
We used the ideas discussed previously to test several numericalexamples: a numb erof
dierentexp osure functions wereused topro ducesimulateddatawithfourdierenttyp es
of continuous distributions : uniform on [0 1], mo died gaussian or b ellshap e, a left
skeweddistributionand a rightskeweddistribution. In actualfact, weused adiscretization
as in (3.6) with = 100 and the equally spaced in [0 1] to generate our \data" for
the continuous distribution and p opulation. We tested b oth smo oth and piecewise smo oth
exp osure functions. In allcases we obtained excellent p erformance fromthe algorithms. In
Figures1 and 2 we depict the \true"(the solid lines) versus the estimated(the circlelines)
corresp onding to =31and 71resp ectively. The continuous usedinthesecalculations
was givenby
( )=
0 0 195
100( 195) 195 205
Some remarks on estimation techniques for size-structured population
models Frontiers of Theoretical
Biology
Estimation of growth rate distributions in
size-structured populationmodels
A parallel algorithm forrate
distri-bution estimation in size-structured population models
On the distribution of vaccine
protection under heterogeneous response
Convergence of Probability Measures
MarkovProcesses: CharacterizationandConvergence
Modeling and estimation problems for structured heterogeneous
populations
Rate distribution modeling for structured heterogeneous
popula-tions
[B] H.T.Banks,
, CRSC-Tech Rpt.No. 92-11, Oct. 1992, NCSU;in
(S. Levin, ed.), Springer-Verlag Lec. Notes in Biomath, Vol 100, 1994, to
app ear.
[BF] H.T. Banks and B.G. Fitzpatrick,
, Quart. Appl.Math 49 (1991), 215-235 .
[BFZ] H.T.Banks, B.G. Fitzpatrick,and Yue Zhang,
, to app ear.
[BSH] R.C. Brunet, C.J. Struchiner, and M.E. Halloran,
,Math. Biosci. 116 (1993), 111-12 5.
[Bill] P. Billingsley, , Wiley,NewYork,1968.
[EK] S.N.EitherandT.G.Kurtz, ,
Wiley,New York,1986.
[F1] B.G. Fitzpatrick,
, J. Math. Anal.Appl. 172 (1993), 73-91.
[F2] B.G. Fitzpatrick,
, CRSC Tech Rpt No. 93-20, Nov. 1993, NCSU; Pro c. Int. Conf. Control of
