Limiting theorems on ‘case’ reporting

Applied Mathematics Letters

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8CIISNCII ~DIRECT

I

Applied Mathematics Letters 17 (2004) 855-859

ELSEVIER

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L i m i t i n g T h e o r e m s on 'Case' R e p o r t i n g A . S. R . SRINIVASA RAO Centre for Ecological Sciences, Indian Institute of Science Bangalore, 560 012, India arni. srs. rao©ces, iisc. ernet, in

(Received May 2003; accepted June 2003) A b s t r a c t - - T h e relation between reported disease cases and actual cases (hypothetical) is defined. Certain situations where such sequences of relation converge or diverge is studied. (~) 2004 Elsevier Ltd. All rights reserved. K e y w o r d s - - M o n o t o n e convergence, Poisson process, Series.

1. A D J U S T M E N T

OF

UNDER-REPORTING

Usually, reporting of disease cases in epidemiology is not complete. Here, in this work, these cases are treated as points on the positive real line and focus their mathematicM properties. DEFINITION 1. A function ¢(y) that determines the closeness of reported cases R to total (actual) c a s e s T is defined as a efficiency function of cases. DEFINITION 2. Let R 1 , R 2 , R 3 , . . . ,RN and T1,T2,T3,... ,TN be the numbers of reported and total cases in the time intervals [~0, ~tl), [£1, £2), [~2, ~3),.-. [~N--1, ~N), where ~ g can be current calendar time or previous to the current calendar time. I f yj (j = 1, 2, 3 , . . . N ) be the estimated percentage of cases that were not reported for the year j and is available, then define total cases Ti at time i as the rat/o of reported cases Ri (Ri > O) and the efficiency function ¢(Yi). This means R~

T~ - ¢(y~),

for all i < N, (where ¢(y~) = 1 - y~) and y~ # 1.

(1)

We modify Definition 2 by removing the condition on time i. DEFINITION 3. Define Ti as the ratio of R~ (R~ > O) and ¢(yi), for all i E R +, i.e., R~ T~ = ¢(y~),

for all i e R + (here R~ > 0 and ¢(y~) = 1 - y~) and ~ # 1.

(2)

Later in Section 2, we see that actually 0 < y < 1. If ¢(y~) is a constant in (1), then R o( T or R = CT, where C is proportionality constant (= ¢(y)). I thank the Department of Science and Technology, New Delhi for funding this research (SR/FTP/MS-12/2001). 0893-9659/04/$ - see front matter @ 2004 Elsevier Ltd. All rights reserved. doi: 10.1016/j.aml.2004.06.017

Typeset by ~4hd,~-~.X

856

A.S.R. SRINIVASARAO

REMARK 1. ¢(y) --~ 1 implies [l --~ T. An important generalization is proved by the following

theorem. THEOREM 1. Let R, T E R + - {0}. For every ~-neighbourhood No(T) of T, there exist a 6-neighbourhood N~(1) of 1, around 1, with the property that for all Cn(Y) E N6(1) (¢(y) ~ 1), it follows that R , E N~(T,,). To put this simple, we state the theorem below Cn(Y) • g~(1),

(¢(y) # 1)

~

R~ • N~(Tn).

PROOF. Let a > 0. ¢~(y) • N~(1) ~ I¢~(Y) - 11 < ~. By definition (3), T~ = P ~ / ¢ , ( y ) , which implies~

IRa - Tn] =

~)~y) , ~(¢,(y)

¢.~

for all R~ • (R~),

- 1) ,

for all R~ • (R~),

]¢.(y) - 11,

for all n . • (P~).

(3)

Since ~-neighbourhood around 1 has radius at the most 1 - Tr (for ~r > 0), it follows that for a given r,

Now, choosing ~ = c~¢,~(y)/R~, we have

IR~ - T~I =

R¢_~y) ICn(Y) - 11 < c~.

Hence, the proof.

|

REMARK 2. P(ITn-Rnl < a) -- 1 implies limCn(y) -~ 1. It is not difficult to prove Cn(Y) • N~(1)

(¢(y) ~ 1) ~, R. • N~(T~). o

SOME PRELIMINARY RESULTS FUNCTION ON EFFICIENCY

Let Yl ~ Y2 ~ Y3..., then,

l-y1 i l--y2 <-l-y3..., ¢(Yl) _< ¢(Y2) <_ ¢(Y3) _< . . . .

(5)

This implies that if the percentage of cases that were not reported over the period decreases, then the efficiency function will increase with the time i. Now, take Yl < Y2 < Y3 . . - , then =~l-yl

>l-y2>l-y~>...,

¢(yl) > ¢(y~) > ¢(y3) > . . . .

(6)

This means ¢ is strictly a monotonic decreasing function. LEMMA 1. ~f R is always under-reported and the efficiency function nondecreasing, then se-

quence ( Rn) converges. PROOF. Since R is always under-reported they will never be equal to T. This means sequence (Rn) is bounded above. That the efficiency function is nondeereasing implies, that the percentage of eases not reported is nonincreasing (5). This means R1 ~ R2 _~ R3 ~ . . . . So R is bounded and also monotone, hence, by using MCT sequence (R~) converges. Another proof for Theorem 1 is given below.

Limiting Theorems

857

PROOF. In general from (3), T = R e ( y ) -1 = R(1 _ y ) - l , where y ¢ 1. Let us assume t h a t y will never be extinct. A higher value of y indicates poor r e p o r t i n g a n d a lower value of y indicates good r e p o r t i n g of cases. So

o < u < 1.

(7)

This means l>y>y2>y3

>...>0.

This m e a n s sequence (y~) is b o u n d e d a n d m o n o t o n e . Hence, b y M C T , (yn) converges to a limit a (say). B u t (y2~) = y~.y~ = a 2 (algebraic limit theorem). Therefore, a = 0, a n d hence, sequence (y~) converges to zero. This implies the sequence (¢~(y)) -~ 1 as n --~ 0% so R --~ T or

R~ C N~(T~).

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REMARK 3. 0 < y < 1 ~ lyl < 1 a n d R cannot be zero. So,

Ti = Ri(1 - y i ) - i r

(8) k=0

which is a geometric series a n d it is known that if Yi = 1 a n d R~ ¢ 0, then the series diverges. Also, total cases reported can be written as Ri(1 - y~)(1 - y i ) - i as n ~ c~. Consider the series ~-]~=1 T~ and take u m = T1 + T2 + T 3 + . . . Tin. If urn converges to U (say), t h e n we can say that ~ T~ converges. Using (8), we can write um=

R j yk j = l k=O

= ~

(9)

Rj(1 - y j ) - i

j=l = R I ( 1 - yl) -1 + R2(1 - y~)-z + R3(1 - y3) -1 + . "

+ Rm(1 -- ym) -1.

NOW, consider vm = R1 + R2 + R3 + ... Rm. Since y ¢ 1 a n d 0 < y < 1, every term in (9) is greater than the corresponding terms in v m . Then, um > v,~. If Cj(y) and l:lj are same with increase in time j above, then um = m R ( 1 - y)-] > m R = vm,

for m E N .

(10)

for m C N.

(11)

Therefore, v~ > m,

and hence, u,~ > m,

Thus, the series ~ T , diverges. B y the comparison test of convergence, we can say that ~ R~ also diverges. If there is no condition on Cj (y) and Rj, then also every term in (9) is greater than the corresponding term because R is a positive quantity and 0 < y < 1. In addition, u m < Um+l < urn+2 •. •, i.e., (Urn) is monotonic increasing sequence. We have proved a L e m m a 1 when Rj increases with the time j. Now, we will prove an important theorem if the sequence (Rn) decreases. THEOREM 2. If sequence (Rn) is decreasing and condition (5) is valid, then sequence (Tn) is

convergent. PROOF. Since (5) is given, the inequality

(1-

> (1 -y2)

> (1-

(12)

is true. Given t h a t the sequence (P~) is decreasing, so R1 > R2 > R3 > " - and R > 0 from Definition 3, so 0 is the infimum of (R~). Hence, m u l t i p l y i n g by R 1 , R 2 , R 3 . . . to the corresponding t e r m s in (12) will n o t change the inequality, so R1(1 - yl) -1 > R2(1 - y2) - 1 > R3(1 - y3) - 1 > ' " is also true a n d has same infimum as above, hence, b y M C T sequence (T~) is convergent. Hence, the proof.

(13)

II

858

A.S.R.

SRINWASA

RAO

COROLLARY 1. From Definition 3 and (7), we have Ri < Ti, for all i, hence, under the conditions of Theorem 2, sequence (Rn) is also convergent. If we assume reported cases, i.e., R will follow Poisson distribution (with parameter/3, say) over the time i and there is a decay in percentage of cases not reported, i.e., decay in y is an exponential rate, then the n th term of T,~ as defined in (3) is

Rn(1

- yn) -1 ~ e-fl/3R~ (1 -- yoe-cl'n)-I

(14)

Rn! -

R,~!

~(Yoe-d'n)k

k=O _ e-~/3Ro (1 + yoe - ~ ~ + y~e - ~ ~ ~ + ) R,~ ! "'"

e-t3/3R~ yoe-(~+d.n)/3R,~ y2e-(~+2.d.n)/3R~ -

R~

+

R~

+

nn~

+'"

(15)

Just a reminder, here R~,~3, d, y0 and n > 0. y0 is initial value of y before decay, which is a constant. Every term in (15) is positive and also these terms monotonically decrease (since (7)). All the terms following the initial term in (15) converge to the first term as y0 --* 1 and d ~ 0. Thus,

{ e-fl/3R'~yOe-(~+d' ' n)/3R~y2Oe-(~3+2"d' Rnl ' n)/3R'Rnl ~.Rn!

' . .} .e-j3/3R~< . R~'

e-~/3R~

(16)

(17)

R~=I In (17), R~ cannot begin from 0 as in general discrete mass functions, because it is a positive quantity. The right-hand side of (17) is slightly less than total probability of mass function, so (17) < 1. Hence, the sequence,

( yoe-(fl+m'd'n)~ R'~"~

(is)

] m=0,1,2,3... is bounded. Thus, it is convergent. Now, if we consider series (15), then

e-~]3R'~

~

P~-----Y-. <

yoe--(~+d'n)

e-~3/3R'~ R,~-------~. < ~- I,

R~=I < ~ YOe-(~+d'n)

<---~ 1,

Rn=l y2e-(~+2"d'n)/3R~ < R~!

Rn[

<'" 1,

R~=I

(19)

R~ <-~ 1, Y~e-(~+J'~n)/3R~ < N-~ z_, y~e-(¢+J'a'~)/3 ~

Thus, we see series (15) resembles series ~ 1 which is divergent. Hence, the sequence

(20) converges. Now, we can state the following theorem based on the results obtained.

Limiting Theorems

859

THEOREM 3. / f R n ~" Poisson mass function or Poisson process and y ~ exponential decay, then

the sequence (Tn) is divergent. PROOF. Given R~ ~ e-~/3R"/Rn! and y = (1 - yoe-d'n) -1. Here,/3 is Poisson parameter, yo is initial value of y, d is decay rate with time i. Now, using Definition 3 and from (14)-(20), then the result is proved. II Though the way in which reporting is introduced and discussed in this work is new, one can read in general about the principles of monotone convergence and divergence from the standard books [1,2]. 3.

ORDER

IN

REPORTING

Here one or multiple case reporting is considered. A case could be reported either once or more t h a n once. If it is more t h a n once, then it could be possible t h a t reported cases outnumber actual cases in t h a t particular time. DEFINITION 4. Total (actual) eases at the time i is defined as the sum of reported and error of reporting at the time i over positive reaI line, i.e., symbolically Ti = Ri ~=ei, for all i C R + and ei is error of reporting. As per this definition, a set A = {(Ri,T~), for all i C R } is an ordered set, because either Ri < Ti, Ri -= Ti, or Ti < Ri will hold true. We can also imagine Ri is around Ti with a radius e~. As the reporting improves or error of reporting reduces with increase in time i, then the radius e~ approaches zero. This results in decreasing the distance between R~ and T~ with increasing in time i. This encourages the following theorem. THEOREM 4. As a monotonic decreasing sequence (en) approaches to zero, then sequence (Rn)

converges to sequence (Tn). REFERENCES 1. W. Rudin, Principles of Mathematical Analysis, Tara-McGraw-Hill, Singapore, (1976). 2. E.G. Phillips, A Course of Analysis, Cambridge University Press, Cambridge, (1948).