J Chron Dis 1975, Vol 28. pp. 317-310. Pergamon Press. Printed in Great Britain

A DYNAMIC MODEL OF PREVALENCE FOR AN EPIDEMIOLOGIC STUDY OF A CHRONIC DISEASE WITHIN A HEALTH MAINTENANCE ORGANIZATION* JOHN P. MULL~~LY

and ARTHUR OLEINICK

Health Services Research Center, Epidemiology Studies Section, Kaiser Foundation Portland, Oregon 97215 (Received 5

I.

Hospitals,

June 1974)

INTRODUCTION

health care systems or, in the current vernacular, health maintenance organizations, can serve as data resources for epidemiologic studies. The membership is relatively stable in successful plans. Moreover, plan administrative structure facilitates collection of important demographic data and provides an efficient means of recovering medical information. While data acquisition and quality control are obvious major advantages to such data systems, their full utilization has been retarded, to some extent, by a failure to comprehend clearly the relationship between the classical measure of prevalence and treated prevalence, a quantity which can be derived from administrative data almost as a byproduct. The classical measure of prevalence requires some type of sampling survey of individuals with medical ascertainment of the presence or absence of a particular condition. Such survey data is not necessarily any easier or cheaper to obtain in a prepaid system than in the more traditional medical research settings. In contrast to survey data the epidemiologic study discussed in this paper employs as its data source the computerized utilization records of a 5 per cent sample of Kaiser Health Plan members for the years 1967-1971. This data source combined with a population file allows for easy calculation of treated prevalence rates, i.e. the rates at which Plan members seek medical care for specific diseases. The extent to which treated prevalence rates approximate classical prevalence rates is shown to depend upon such system and disease characteristics as immigration rate (joining the Plan), emigration rate (leaving the Plan), diagnosis rate (the rate at which developed disease is labeled in the system), the remission rate, and the interval of time over which labeled individuals are counted. PREPAID

*Data used in this paper were drawn from research supported by the National Institute of Allergy and Infectious Diseases, Contract No. l-AI-32500. Presented to American December 1973).

Statistical

Association

Biometrics 317

Section, New York, New York, (27

318

JOHN P. MULL~~LY and ARTHUR OLEINICK

Several previous studies dependent upon data from prepaid systems have not explicitly considered this problem but all have dealt with it in some fashion. Shapiro’s [l] study of coronary heart disease in the Health Insurance Plan of New York population utilized the survey method, a review of the medical records of all eligible study members to ascertain index cases. Avnet’s [2] work utilized ‘attended illness incidence’ rates which really measure the medical care load of various morbidities rather than their prevalence. In Kessner’s [3] report advocating the use of tracer conditions to evaluate medical care, the estimated prevalence more closely approximates ‘attended illness incidence’ since all the centers are of recent origin. The Kaiser Health Plan of Oregon has collected extensive utilization and diagnostic data in a 5 per cent simple random sample of Health Plan subscribers since September 1966. The present report represents an attempt to develop a model which relates the routinely available data in the form of treated prevalence rates to classical prevalence rates. Development of a successful model for such a relationship would clarify the utility of prepaid health care data systems and should prove of use to both the student of disease and the health planner. The model is developed in terms of individuals but the calculations are given in terms of person years since the prevalence of interest is an annual prevalence. II.

We consider (the Portland definitions and prevalence for

THE

MATHEMATICAL

MODEL

the chronic disease of interest to be present in the source population Metropolitan Area) at a constant prevalence rate. The following assumptions describe the various components of the dynamic model of an epidemiologic study within the Health Plan,

1. The source population Schematic representation;

@++J (4

Definition of terms. SND=symbol for the non-disease state; SD=symbol for the disease state; (SND)=number of source individuals in the non-disease state; (SD)= number of source individuals in the disease state; v,=the proportion of source nondisease individuals who enter the disease state per unit time (incidence intensity parameter); va=the proportion of source disease individuals who enter the non-disease state per unit time (remission intensity parameter). Assumptions. (i) constant incidence and remission rates; (ii) negligible disease specific death rate, (iii) approximate stability of the source population size. System of d@erentiaI equations. d(SND) -= dt

-v1(SND)+v,(SD)

d(SD) -=vI(SND)--*(SD). dt At equilibrium : Source Prevalence Rate =

(SD) (SND)+(SD)

= &*

’

Epidemiologic

Study of a Chronic Disease

319

2. The health plan population Plan members are categorized by the providers of medical services who attach diagnostic labels to them. The two dimensions of disease and diagnostic label lead to four mutually exclusive categories, in exactly one of which each plan member is at any point in time. Diagnostic label + -

Di

DL

DC

DL

Disease + EEI (b)

In its usual operating mode, i.e., plan members presenting morbidities to the providers, the computerized data system does not provide information concerning the remission of labeled individuals. Hence, there is no direct connection between the DL and the DL categories in our model system. Schematic.

y, 0 v, Di _ 0 % DL

=X=

DL

v2

6

DE

8

(4 Definition of terms.

v,==incidence intensity parameter v,=remission intensity parameter 0=diagnostic intensity parameter=the proportion of diseased-non-labeled individuals who present to the clinic and are diagnosed per unit time. Assumptions.

(i) constant incidence and remission intensities, equal to those in the source population. (ii) constant diagnostic intensity. 3. Coupling of source and health plan populations The rate of growth of the HMO is the net result of the rate at which new members enter the Plan and the rate of exit from the Plan. Since a medical examination is not required for admission into the Plan, we don’t know the status of entrants with respect to the disease of interest. In the model Plan considered here, we assume that there is no entrance selection in favor of or against diseased individuals in the source population. To some extent the open enrollment periods and group enrollment in the Kaiser Portland Plan tend to minimize selection effects over a wide range of morbidities. We also assume that each Plan member has the same probability of leaving the Plan, regardless of his disease-diagnosis status. This non-selective admission assumption is modeled by partitioning the known total number of entrants into disease and nondisease categories according to the source-population prevalence rate. That is to say,

320

JOHNP. Mu~~oo~~and ARTHUR~LEINICK

if h represents the number of immigrants into the Plan per unit time, then the rate at which non-diseased individuals enter is taken to be h,=h . (prevalence rate of nondisease individuals in the source population), while the rate of entrance of diseased individuals is assumed to be h,=h * (prevalence rate of diseased individuals in the source population). The following scheme shows the HP disease-diagnosis dynamics coupled with the source population. Schematic diagram.

HP

Source

populatton

FIG. 1

Definition of terms. u=the proportion

of HP members leaving the Plan per unit time.

h,=the

number of non-disease source individuals entering the Plan per unit time.

h,=the

number of diseased individuals entering the Plan per unit time.

h=the total number of new Plan members per unit time, I=I,fh2. Assumptions. Vl (i) the source disease prevalence rate is at its equilibrium value v1+v2

(ii) L = *

. A, 1

2

1, -

yv2 .

v

h

1

(iii) constant immigration and termination rates. The immigration process is under control of administrative and source factors related to the demand for HMO membership. Although the ultimate size of the HP depends on the rate of input to the Plan, h, as we will see later the ultimate disease prevalence rate and cumulative treated prevalence rate are independent of h. It follows then, that our expressions for the ultimate or equilibrium values of the HP disease prevalence rate and the cumulative treated prevalence rate, although derived under the assumption of a constant h, apply in general, regardless of how h changes through time under administrative control. 4. The growth of the 5 per cent sample The following table shows the growth of the 5 per cent sample over the study period 1967-1971.

Epidemiologic

Person-yeors

321

Study of a Chronic Disease

of Coverage

1967

1968

1969

1970

1971

FIG. la

The change in the size of the 5 per cent sample AN is equal to the difference between the number entering h * At and the number exiting p . N . At during the time interval At. Using a continuous model for this growth dynamics, we have dN/dt=h-p . N. is easily solved to give N,=i+(N,

This first order differential equation

-;)e-V

where NO is the size at time t=o (start of the observation period). Note that as t gets large N, approaches the limiting value h/p. For small t, N, may be approximated by a linear function of t. N,-N,,+(h-p.

from which we have AN=N,+,-N,-h-p

NJ.

t

. N,.

The following table and plot present the yearly levels and increments. I

I

I

I

I

5000

I 6000

I

Females

: AN=l006-0209N

300-

xX)-

t300

2000

XMO N

40x

(Person-years

FIG. 2

of coverage)

7th

JOHN P. MULL~~LY and ARTHUR OLEINICK

322

Females

Males Total

N

AN 846 832 605 506

4831 5677 6509 7114

AN 370 418 309 271

N 2361 2731 3149 3458

N 2470 2946 3360 3656

AN 476 414 296 236

Fitting least squares regression lines to these data we obtain Total: AN=1669-0.16 N; Males: AN=652-0.106 N; Females: AN=1006-0.209

N

These estimates of the proportions exiting during a year will play an important role in the procedure by which the ultimate cumulative treated prevalcnces of the allergic diseases are estimated. This estimation procedure involves the assumption that the exit intensity u is approximately constant over time. This is not an unreasonable assumption since under normal operating conditions, the probability ofJermination from the Plan is a person characteristic. The estimate of the input rate is h= 1669 per year for the 5-yr study period. As pointed out above, our main results do not depend on this input rate being maintained over a long period of time. 111.

THE

SYSTEM

OF DIFFEKENTIAL AND ITS SOLUTION

EQUATIONS

Expressing the net rate of change for a disease-diagnostic category as the sum of positive and negative component rates, the following system of differential equations describe the dynamics of the model.

(9 (ii)

d (Dli) -=I,-(pfv,) dt d (DL) =h,i-v, dt ?=0.

(iv)

* @X+-v,

* (DL)

* @is)-(p+v,+B)

(Dt)+v,

. (DL)-(v,+u)

d (DL) -vz . (DL)-(vl+p) dt

* (Dt)

* (DL)

* (DL).

Before discussing the solution of this system of linear differential equations, let us investigate its equilibrium behavior. 1. Equilibrium relationships Denoting the equilibrium total input rate as h *, the nondisease and disease input rates are A,*=-

* A*, and h,*= - VI * A*, respectively. When the equilibrium v1fv1 v1+v2 state has been reached the total rate of input to the Plan is exactly balanced by the total exit rate. A constant plan size necessitates a dynamic balance among the disease-diagnosis categories. A mathematical statement of these equilibrium conditions are :

total input rate=hl*+hl*=total

exit rate=u * [(DL)+(DL)+(DL)+(DL)]

(1)

Epidemiologic Study of a Chronic Disease

323

the rate of change of each category equals zero d(BL) -=-Z=-=- d(DL) dt dt

d(DL) dt

d(DL)_O .

(2)

dt

Setting the derivatives equal to zero, and solving the resultant system of linear equations, subject to the constraint (I), we obtain the following equilibrium values:

h,* * (pi-v,+ 8)-t-h,* * v* (DL)== (pi-v,). (II-I U)i-p . VP” (DC)=

h2* . v,+h,*

(3)

* (Pi-VI)

(l.l+vJ * (p’-Q)+p

(4)

‘VP

8 . v:: . [h,* . vl+hl* . (p-lVI)] CDL)== cl * (piV,~i VJ . [(p-i-V11 - (p-tO)ip

.

(6)

’ V21

We now define three prevalence rates which are of interest to the epidemiologist studying disease in the HMO setting: (DL)t-(DL) Prevalence Rate of Disease=PD= Plan Size Prevalence Rate of Treated Disease=PTD=

(DL) Plan Size

(DL) +@I-). Prevalence Rate of Label=PL= (Cumulative Treated Prevalence Rate) ‘Ian Size) The prevalence rate directly obtainable from the utilization records of the HP members is the Prevalence Rate of Label. In order to determine this rate at some point in time, one would simply review the utilization records of active members as of the observation time and count the number of these individuals who had ever received medical service for the morbidity of interest. The prevalence rate of label is equivalent to the cumulative prevalence rate of treated disease. The Portland Kaiser Health Plan has been providing comprehensive medical care to its members since 1942. Computerization of the utilization records of a simple 5 per cent random sample of subscriber units was initiated during September of 1966. The study period for the Allergy Study is 1967-1971. Since computerized utilization data is not accessible prior to 1967, our calculated prevalence of label is restricted to those individuals who received medical service for allergic diseases during the study period 1967-1971. A procedure for estimating the true cumulative prevalence rate of treated disease within the Plan from the prevalence of individuals receiving service during the observation period will be developed in a later section. We now give explicit expressions for the three prevalence rates defined above when the system is at equilibrium. Substituting Plan Size=h:+h:/w into the definitions of PD, PTD, PL, and employing the equilibrium values of (DL), (DL) and (DL) given by equations (3-6), we obtain:

JOHN P. MLJLL~~LYand ARTHUR OLEINICK

324

pD,h”’

PTD=

* Vti-L* * (Pc-tVl) h* *(I.l+vl+vs)

(7)

0 *(p+vJ * [ha* *v1+h1* . (p+vJl h’ *(p+v,+vz) . r(J.l+vI) *Q.l+e>+cc *&I

pL=

(8)

8 . [he** v1+L* . (p+vJl I* . Kp+vJ * (P+w-P * %I

Notice that the equilibrium input rates, h,* and k2*, enter into these expressions as ratios to the total input rate h*. Under the assumption of non-selective entrance into the Plan, the proportions of non-diseased and diseased entrants are hz*/h*=vp/(vl+vs) and hI*/h*=vl/(vl+vz), resp:ctively. Substitution of these ratios into expressions (7-9) yields (10) PD=L VI+v, (v1+v2).

PL=

(11)

0 * Vl . (p+vJ

PTD=

M.-w

. (p+J)+p

* v,]

8 *Vl. (p+v,+ve) . (Vlt-V2)*[(p+vJ . (p+6)+p . &I

(12)

From expression (10) we see that the model HP equilibrium disease prevalence rate is exactly equal to the disease prevalence rate in the source population. That this should be true is intuitively clear in light of the assumptions of nonselective entrance and exit. At equilibrium we see from expressions (10) and (12) that the ratio of the cumulative treated prevalence and the disease prevalence rates is given by PL 8 - (p+v,+vz) . (13) -= PD (p+vJ * (P+@+P * vz Clarification of this equilibrium relationship is assisted by noting the nature of the dependence of the ratio (13) on the system and disease parameters. It is easily seen that PL/PD increases with the diagnostic intensity 0 and with the remission intensity vp, whereas it decreases with the termination intensity p and with the incidence intensity vl. The following tables give equilibrium values of PL/PD for a variety of model parameter values. It is to be noted that for a disease present at a high prevalence rate in the equilibrium model HMO for which the diagnostic intensity is low, the cumulative treated prevalence may actually be less than the point prevalence rate of the disease. TABLE l(a).

FOR ~~-0.001,

p-O.04

VZ

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

PD

1.00

0.167

0.091

0.063

0.048

0.038

0.032

0.028

0 0.10 0

EQUILIBRIUM VALUESOF $

0

0

0

0

0

0

0

0

0.714

0.775

0.830

0.883

0.933

0.964

1.024

1.050

0.20

0.833

0.917

0.996

1.073

1.147

1.199

1.287

1.334

0.40

0.909

1.009

1.106

1.202

1.295

1.365

1.477

1.542

0.60

0.938

1.044

1.149

1.252

1.354

1.431

1.554

1.627

0.80

0.952

1.063

1.171

1.279

1.385

1.467

1.595

1.673

1.00

0.962

1.074

1.185

1.295

1.404

1.489

1.621

1.702

Epidemio!ogic

TABLE l(b).

\‘_! ____~~_

Study

of a Chronic

EQUILIBRIUM VALUES OF ‘$i

Disease

325

FOR vi =0.0005,

0

0.005

0.01

0.015

0.02

I .oo

0.091

0.048

0.032

0.024

0.025

0.03

0.035

_~ PD

0.020 0.016 .~~_~~_~

~...

0

v-o.04

0.014 ~~~_

0

0

0

0

0

0

0

0

0

0.10

0.714

0.775

0.832

0.885

0.936

0.982

1.027

1.069

0.20

0.833

0.917

0.998

1.075

I.151

0.40

0.909

1.010

1.109

1.205

I .29?. 1.484

1.572

0.60

0.935

1.151

1.255

1.461

1.561

1.659

0.80

0.952

I.174

I .282

1.391

1.497

I.603

1.707

1.oo

0.962

I .045 1.064 1.075

1.300 I .359

1.222 I .393

1.188

1.299

1.410

1.520

1.629

1.736

1.359

2. Transient behavior of the model

Standard methods of solving the basic system of simultaneous linear differential equations [(i-iv)] may be applied to derive the following expressions for each of the disease-diagnostic categories.

pi)==Cl . h+P+v*+O)e,,,LC2

. (m+p+v,+e)em’,

I

Vl

Vl

(14)

(15)

(DL)=

~1. (b,+ir+vl) vZ +

b I YZ. (ba+tr+vJek, el+ v2

Kz * (m+~+vd Va

+K

em*’+(fQ +p +vJp2t VY

(16)

* (p+vJ V2

(DL)=y,

*ebll+yz - ebrr+K,+K,

+en’rt+K3- em’*

(17)

where the introduced constants are functions of the model parameters and/or the initial values of the categories (DL),, (DL),, (DL),, (DL),,: A=2t.tfv,+v*+Q B=(j.tfv,)

- (p+S)+p

- Va

JOHN P. MULLOOLY and ARTHUR OLEIWCK

326

c=

-v,

* h,--h,

* c/l+VI)

D==2p+v,+v,

E-P - (p+v,+v,)

A 1 ~ rile=--+'----

1

c1=(m,-m2)

K,=

Ks=

VI * 8 *

lL,+v,

* (DL),-m?

a;-(DL);(m,+~+v,SO)

1

c,

(m,“+D *m,+E) Va * 8 * Ca (me2+D * mz+E)

1 Y’=(b,-bp)

-

v8.(DL),--KS. m,-K3*ml-ba~

fb, * (K,+K~+W-_(P+VJ

(DL),

* CDL), 1

y*=(BL),-(K,-tK,'rK,)-yl.

It is to be noted that this solution requires the discriminants (A*-4B) and (D*-4E) to be positive.

Epidemiologic

Study of a Chronic Disease

327

Figures 3(a-m) illustrate the transient behavior of the model system. In these examples we follow the dynamics of the model HP from its inception through the attainment of an approximate equilibrium state. The unfolding of the incidence, remission, diagnosis and termination processes, as expressed through the prevalence rates PD, PTD, PL, is followed numerically for various values of the system and disease parameters. It is to be noted that the disease prevalence rate in the Plan (PD) is immediately established at its equilibrium value which equals the disease prevalence rate in the source population. Comparing examples 31 and 3m we see that the dynamics of the prevalence rates are independent of the rate at which new Plan members are enrolled.

0 022 0 02 0 016 1

p

0,015

-

e / 00,

/ a-.-.-.-.-*-.

-m-.-.-.

PD

_--

?! a

l_*_.-.L*-*-’

*,=o~ool.

v**O~IO,

PTD

(“,,y”o.ol),

c(‘O.04, (X=lOOO)

8=0.20,

I z

-

FIG. 3a

! e a 5 05 g

__/~~~~~ce.4po 0.010 -

--xPL

---

.-.-.-.-.

x/x’ 0005

0.001

-

/

-

:/*--

024

,,,1l,,,,l,,,,l,,,,,,,,,l,,,,1 IO

l_.-.-a-.

20

-.-•-•PTD

30

40 f,

v,=o~ooi, v,=O.IO, ( &=O.Ol)

-

50

Yr

,8=0.20, p=o.oe, FIG. 3b

(X=lOOO)

60

‘+

JOHNP. MULL~~LY and ARTHUROLEINICK

328

x’

; 0015 e 8 ;0, 001 e 0

x/

x’

x’

X/

. x

/ r -.-.-.-.-.-.-.-.-.

7l

-__

PTD

_._.-*-*-e--a-*

-•-•

PD

.’

0005 -

I, I/ I,I,,,,,,,,/,,,,,,,(,/(((((,( IO 20 30 40 50

0001 024

f. v,=OOOl, l$=o40,

(A=

I 60

Yr

0.01),c?=oa, p =o~o4,O.=looo) FIG. 3c

pL_xcex-x

9015 xz e %

0.01-

.---; ./

d E

/

&

:-,

-

x-x

PD

.-.-.~.~~-,-,-.

_w_

PTD

l_~_.-.-.-.-.~,~,

w

0,005 -

0.001 IIlll,,,ll(,,,~ 024 IO

,,,,,,,,,,,,,,,,

20

30

40

,

50

r, yr qaoo1,

v,=o~Io, (q&=0.01),

8=0.50, p=O-08, IX=lOOO) FIG. 3d

60

I

Epidemiologic

329

Study of a Chronic Disease

PL

X-

,I

xlX

x” x’ x’ x’ ,/ x’ PD . x

.-.-.-.-.B._

-_-

.-.-.-. -.-.-.-.

*_.A.-.-.-.

PTD

.’

Fro. 3e

0020

-

-x-x

xpLyedv-x x’ !! e

0015

-

g

0.012

0, 9 e P

0~010 :

yx -

0005

0001

l-.-.-.-.

l-

___

PD

+:L._.-._._._._._._.

-

-.-.-.-,-. PTD

-

024

III,I/I,,,II,,JIII,Illl~llllllllnll/ IO

20

30

40 t,

v =O.OOl,

Y =om.

(A=

50

yr

~ol~,8=0~90.

FIG. 3f

p=ooe,(x=o~locO)

60

70

330

JOHNP. MULWLY

0.06

-

0.05

-

0.04

-

0.03

-

002

1

0.01

-

024

.---.-

--.---

and AKTHUROLEINICK

PD _--.-_-.__-.___*___*

.---.---.

___

_x_xL-x.h-x-x-x -.-.-.-. PTD

x-x ._.-.-•-•

-

/

llIll1lllllllIlllllllllllll,1ll IO

I,,,, 20

30

40 t,

5 =O.OOI,

I,,,

60

yr

(&=0.048),8=0~2,

v,=o~,

50

p=o.08,

1 2

~X=IOOO)

FIG. 3g

0070.06

PL-X

-

0,05-

.--_*---.--

0.04

-

/

003-

*r-)

x_d---x-x~Dx r.---.---

y-x/ *-9--L.-*

.---.

-

---.---.

___ e

PTD

-*-.-.-.-.

8’

002

-

O-01

-

0

II,

I I,,

24

IO

I I I1 I I I I I ,,,I,, 20

v,=oo2,

,I

1 I

40 f.

yI =o~ool,

1

30

I1

1,1,,,,,

50

[

60

yr

( ~=0048).e=o~5.

r=O~4,~x=~ooo~ FIG. 3h

0.06

-

0,05004

.--_.___.--

_.__ /x

-

/:-

0.03-

_ .___

E

y -+--=-_&~~pp~

-x-x

_-_

._-.w.-.-.-.--•-•PTD

-

:’

0.02

-

0.01

II,, 02

II,,,1 4

IO

I,,, 20

I,

f,,

30 t,

U, =o.ooi,

~$0.02,

f&=0.048),

I ,,,I 40

I f I, 50

yr

8=0~5+=0~08,

FIG. 3i

(X=1000)

,, 60

VW

Epidemiologic

u) g!

0.07

-

0.06

-

pL,-x-x -x-

o.os-

.---;w&---.-

oo4_

r~._.-.-~-o-*-*PTD.-*-~

z 5

0,030.02

-

0.01

-

PD -.-._-.---.--

--.---.---.--

11,11,,,11,,l,1 024

I,,, IO

20

I

,,,I

40

30

f, v, =OOOl, v*=OO2. (&=0.048),

-

-x-x

X-X

e

P s! a

331

Study of a Chronic Disease

I,,,

_.

,I

50

_-_ -

I,,,,,,, 60

yr

8=0,9, ,u=oG4, (X=lOOO) FIG. 3j

0.06 f

0.05

-x-x--x--xPL _.--.-.-.-.PTD

.---.---Y.$- h, - (p+vl+v*+e)+hl * (!.lL+v1+v*) *

(21)

336

JOHN P. MUL~LY

and ARTHUR OLEINICK

The differential equations describing the scheme

0

cs

CL _

(d) are easily solved to give (22)

+[ (CS),-&]e-(V+f”’ (LS)=

Y *h P. @+I-9

-[

(LS)o--&]e-(U+f*).f+[

(LS),-i]e-V’.

(23)

Making the assumption that fi can be approximated by an appropriate constant fi*, the system of differential equations implicit in the scheme

are easily solved to give (LS)=(LS), (LS)=(LS),

ae-(P+fl*)‘j

(24)

*e-‘-“[I -e-f1*‘7.

(25)

Hence, the cumulative number of person-years for Plan members who receive medical service for the chronic disease during the 1 years from the start of observation is given by S=(IS)+(LS), s=

.e-CP+f** +[(LS),+(LS),+(LS), 1cp'.

which using the above results becomes

Feh _[ (L~)o-_L]

a

e-CP+f’)“--(L$),

P * (P-t?)

For a very long study, the cumulative number of Plan members service for the disease during the study period approaches S=p - h/p - (p+p), which, since the number of Plan members h/u, implies that the asymptotic cumulative treated prevalence given by PL=-= Y u+p

who receive medical a limiting value of approaches the limit rate in the HMO is

treated incidence rate termination rate+treated incidence rate’

(27)

Let us now investigate the behavior of S’, the cumulative number of Plan members treated, during an observation period of short duration. Expanding the exponential terms in expression (26) one sees that for small t > 0,

Epidemiologic

337

Study of a Chronic Disease

As a first attempt at estimation of the equilibrium cumulative treated prevalence in the Health Plan, we overlook the probable dependence of the contact-diagnosis rate during the study period on pre-study period treatment for the disease, and equate fi* and fi. In this case equation (28) simplifies to SNSOSIF which since S+,?=N,

* so-p

* S] ’ t,

(29)

the size of the 5 per cent sample, becomes

S-&-V * (No--so)-p * So] * t. (30) Setting At equal to one year, the yearly increments of the cumulative number of person-yr are given by AS,=fi * (N--S),_,-p t=l, 2, 3,4. As a first approximation scheme for estimatingf’ squares regression method.

. St_.,

(31)

and p we employ the ordinary least

3. An example using Kaiser health plan data Rather than attempting to estimate both the treated incidence and termination rates,f’ and p, from the limited amount of available data (4 yrly increments in the cumulative number of diagnosed person-yr of coverage, S) employing the simplified model (expression 30, 31), we use the loss rates based on the entire 5 per cent sample experience [p=O.106 (males), p=O.209 (females)] and estimate the single parameterp from the regression of (AS+p * S) on (N-S). The following tables display the data in a form convenient for this approach, Males, all combinations of asthma

Independent Variable (N-S,,) Dependent Variable (AS+o.l06S,)

2361 32.4

2698.6 14.03

3106 17.66

Females, all combinations of asthma

Independent Variable (N-S,) Dependent Variable (AS+O.209$,)

2470 17.8

2928.2 13.92

3332 19.65

3401.9 17.25

3660.6 23.34

___...-___

3614.2 25.44

3833.5 25.73

Males, all combinations of hayfever

Independent Variable (N-S,) Dependent Variable (As+0.106s0)

2361 46.0

2685 34.78

3073.1 37.65

3352.5 49.08

3584 67.50

Females, all combinations qf hayfever

Independent Variable (N-S,) Dependent Variable (AS+O.209S,J

2470 64.3

2881.7 48.24

3260.9 53.71

3523.9 58.71

3728.8 66.51

338

JOHNP.MULLOOLY~~~ARTHUROLEINICK

This estimation scheme yields the following estimates of treated incidence rates: Males : { Females : Males : Hayfever { Females : Asthma

$=0.0066 f= =0.0064 ?=0.0156 f’=O.O18 1

which upon substitution in expression (27) gives the following estimates for the equilibrium cumulative treated prevalence rates : Asthma Hayfever

Males : Females :

Pj&=O.O59 PL,,=O.O30

Males : Females:

P&,,=O.128 PL,,=O.O80

It is inyresting to compare these estimated equilibrium cumulative treated prevalence rates (PL,,) with the study period cumulative treated prevalence ratesin 1971 (Table 2). It is the lower loss rate experienced by the males which results in their extrapolated level being considerably higher than the ultimate level for females. This extrapolation procedure is made under the assumption of constant loss and incidence-diagnosis parameters. The incidence and remission of allergic rhinitis and asthma were studied in 6563 residents of Tecumseh, Michigan, who were surveyed on two occasions separated by an average interval of 4 yr [4]. The estimated incidences of probable asthma and probable allergic rhinitis over this time period were reported as 1.10 per cent (females), 1.02 per cent (males) and 1.67 per cent (females), 2.16 per cent (males), respectively. Remissions where considered to have taken place when disease which satisfied second survey criteria was estimated from age of onset to have been present at the first survey and then absent for two or more yr before the second survey. Remission was reported to have occurred in approximately 16 per cent (females), 24 per cent (males), of persons with asthma and 5 per cent (females), 10 per cent (males) of persons with allergic rhinitis. The values of the incidence intensity parameters corresponding to the reported rates in Tecumseh are v,=O.O028 (females), v,=O.O026 (males), for asthma; and v,=O.O042 (females), v,=O.O054 (males) for allergic rhinitis. Assuming that half of the remissions occurred in each of two yr before the second examination we estimate the remission intensity parameters to be: v,=O.O8 (females), va=O. 12 (males) for asthma; and v,=O.O25 (females), v,=O.O5 (males) for allergic rhinitis. The model equilibrium source prevalence rates, vl/(vl+vz), incidence and remission intensities are: PD=0.034 PD=0.144

corresponding

to these

(females), PD==0.021 (males), for asthma; and (females), PD=0.098 (males), for allergic rhinitis.

The actual prevalence rates calculated for people who had two examinations were reported as : 0.038 (females), 0.041 (males) for asthma; and 0.093 (females), 0.080 (males) for allergic rhinitis.

Epidemiologic

Study of a Chronic Disease

339

Considering the approximations made in calculating the remission intensities from the Tecumseh data, the agreement between the actual prevalence rates and the equilibrium source prevalence rates as specified by the model are seen to be fairly good. Finally, it is of interest to note that if estimates of v1 and vZ are available, expression (21) may be rearranged to give the basic diagnostic intensity parameter 0 as a function offi, vl, v2, and u. Recalling the model assumption of non-selective enrollment we may substitute h,/hz=v,/vz into expression (21) and obtain ef’

* (v,+v2> * (v*+vz+p). v*

* (v,+v,+p)--v2

(32)

*_P

As an illustration of the use of this relationship we suppose that the incidence and remission intensity parameters (vl, v,) calculated from the Tecumseh experience with probable asthma and probable allergic rhinitis characterize the dynamics of asthma and hayfever in the Portland Kaiser population. By substituting the Tecumseh v1 and vZ values, and the values of u andfi estimated from the cumulative treated prevalence into equation (32) we may calculate a value for 8. The following table gives the values of 8 as calculated from expression (32) using estimated values of u andff.

Asthma (all combinations) Parameter values Males

Females

Tecumseh:

v1=0.0026, v2=0.12

VI=0.0028, v2=0.08

From 5 per cent sample:

u=O.l06

u=O.209

Estimated from cumulative treated prevalence:

f* =0.0066

j-*=0.0064

Equation (32);

9=-0.94

Hayfever (all combinations)

0=0.51 [probable Allergic Rhinitis]

Parameter values Males Tecumseh : 5 per cent sample: Estimated from cumulative treated prevalence: Equation (32):

Females

VI=0.0054, vz=o.O051

v1=0.0042, vz=O.O25

p=O.106

u=O.209

f*=0.0156

f*=0.0181

e=oso

8=0.23

With respect to these calculations for females, the relative magnitudes of the 8s for asthma and hayfever are as one would expect; higher for the more serious disease, asthma. These values for the females have the following interpretation: (1) on the average during a year, 51 per cent of the females in the undiagnosed asthma disease state present to the system and are treated for the disease. (2) on the average during a year, 23 per cent of the females in the undiagnosed hayfever disease state present to the system and are treated for the disease. An inadmissible estimate of 8 is obtained for the males in the asthma category. These calculations are intended to illustrate the use of equation (32) when the parameter set is estimated from the same data source.

340

JOHN

P. MULL~~LY and ARTHUR OLEINICK

The absence of ragweed pollenosis in the Northwest is likely to substantially alter the rates in our area. V.

DISCUSSION

The present report provides the conceptual framework within which cumulative treated prevalence in a medical care system can be related to source prevalence. It is of interest to note that our expression for the equilibrium source population prevalence can be obtained as a special case of Chang’s Illness-Death process [5] (limiting distribution when death is not included). The model also contains a parameter 8 which is a composite of system accessibility and diagnostic accuracy. This parameter may be useful in evaluating various medical care systems. Estimation of the composite evaluative parameter, 8, requires incidence and remission rates. The present study employed data from the Tecumseh survey. The most appropriate estimates would be derived from survey of the Health Plan population. Such a survey should be carefully planned so as to maximize the yield of data on incidence and remission of several morbid conditions since 8 is likely to vary between various morbidities. To our knowledge no such survey with these objectives in mind has been performed. REFERENCES 1. Avent HH: Physician Service Patterns and Illness Rates, Group Health Insurance, Inc., 1967 2. Kessner DM, Bashshur RL, Kalk CE: Development of methodology for evaluation of neighborhood health centers, Health Service. Research Study, Institute of Medicine, National Academy of Sciences, February 1972 3. Shapiro S, Weinblatt E, Frank CW, Sager RV, Densen PM: The H.I.P. study of incidence and prognosis of coronary heart disease: Methodology. J Chron Dis 16: 2181, 1963 4. Broder I, Higgins MW, Mathews KP: The epidemiology of asthma and allergic rhinitis in a total community, Tecumseh, Michigan-3. Second survey of the community, J Allergy Clin Immuno 53: 127,1974--4. Natural history, J Allergy Clin Immune 54: 100, 1974 5. Chiang CL: Introduction to Stochastic Processes in Bioslatistics. New York: John Wiley, 1968