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REVIEW: PREVALENCE & INCIDENCE of MYOPIA

(@) Peter R. Greene, Ph.D.

(#) Edward S. Vigneau, B.S.

(&)Judith Greene, M.P.H.

(@) (&) B.G.K.T. Consulting Engineers, BioEngineering, Huntington, New York, 11743, corresponding author at prgreeneBGKT@gmail.com+1 631 935 56 66

(#) Pre-medical, S.U.N.Y. , 11735

Key words - prevalence, incidence, exponential equations, myopia, time constant, onset age

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ABSTRACT

Purpose. Herein we relate prevalence-time data, incidence rate data, age of onset , system plateau level, and system time constant, using exponential equations, as they apply to progressive myopia, potentially useful over a wide age range.

Methods. Cross-sectional refraction data is analyzed, N=9 studies, total number of subjects N = (345, 981, 7.6K, 39, 421K, 383, 2K, 12K, 255) = 444.6K , age range 5 to 39 years. Basic exponential equations allow calculation of the prevalence vs. time function Pr(t) [%] and the incidence rate function In(t) [%/yr] , system time constant t0 [yrs.], onset age t1 [yrs.], and saturation plateau level <S> [%] .

Results. The myopia prevalence as a function of time Pr(t) [yrs] and myopia incidence as a function of time In(t) [%/yr] are continuously generated and compared with prevalence/incidence data from various reports investigating student populations. For a general medical condition, typical values for time constant t0 may range from one week to 5 years, depending on the health condition. Typical plateau levels may range from 35% to 95% . Data from recent demographic studies of myopia, N=9, are calculated for prevalence Pr(t) with accuracy +/-14% , and incidence In(t) within +/-2.6 %/yr., onset t1 = 1.5 yr., time constant t0 = 4.5 yr. By comparison, linear regression can predict myopia prevalence Pr(t) within +/- 11% , and estimates a constant myopia incidence rate In(t) = 4.7 % per year , [ 95% CI: 2.1 to 7.3 % /yr.].

Conclusions. The initial incidence rate at onset age In(t1) and system time constant t0 are inversely related. For myopia, onset age, time constant, and saturation plateau level are fundamental system parameters derived from age specific prevalence and incidence data.

Key words - prevalence, incidence, exponential equations, myopia, time constant, onset age

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INTRODUCTION

Various high-percentage high-incidence medical conditions, acute or chronic, start at a

particular age of onset t1 [yrs.], accumulate or progress rapidly, with a system time constant t0 [yrs.], typically from one week to five years, and then level off at a saturation plateau level <S>,

ultimately affecting ten to ninety-five percent of the population. Saturation plateau level

describes the percentage of the population at which the incidence rate levels off.

Medical conditions may be communicable or non-contagious, endemic or epidemic,

inherited or acquired. Equally important to determining the prevalence of myopia in a

population Pr(t) [%] is the incidence rate function In(t) [%/yr.]. For myopia, an incidence rate of

sixteen percent per year, means one can expect sixteen new myopes per year from each class of

one-hundred students, at that age bracket. Practical ophthalmic examples of prevalence and

incidence equations include retinopathy of prematurity ( R.O.P.), juvenile onset myopia, with

onset age ranging from t1 = 0 to 12 yrs., adult onset glaucoma [17] with onset age t1 = 30 years,

macular degeneration, cataract development, or presbyopia, with onset around age 40.

Population demographic and epidemiologic surveys often include prevalence measurements, but

not incidence data, incidence but not prevalence data, or prevalence without age-specific data. A

system time constant t0 is rarely cited. In this report, we relate these fundamental parameters with basic

equations, as applied to the progression of myopia, of epidemic proportions in some populations, (e.g.

Pan, Ramamurthy, Saw [11]). Some reports do include prevalence and incidence rates, at age-specific

time intervals, allowing for an improved understanding of the problem of myopia, [1] [2] [3] [4]

[5] [6] [7] [8] [9] [11] [12] [13] [14] [16]. Eight of the nine populations used in our analyses were

mainly from Southeast Asia, reporting that the prevalence in late teens and early 20’s is high.3

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M E T H O D S

Myopia prevalence as a function of time is modeled with the following exponential equation :

Eq. (1) Pr(t) = 0.90 [ 1 - exp(-(t-t1)/t0) ]. The derivative d[Pr(t)]/dt is given by

Eq. (2) In(t) = 0.90 [ (1/t0) exp(-(t-t1)/t0) ], where t0 [yrs.] is the system time constant, t1 [yrs.] is

the onset age, and <S> = 0.90 is the saturation plateau level. For myopia, the saturation plateau level

<S> = 0.90 indicates that by age 25 to 30 years old, 90% percent of these populations will develop

myopia. The general format for the exponential prevalence equation, including the saturation plateau level, is:Eq. (2a) Pr(t) = <S> [ 1 – exp ( -(t-t1) / t0 ) ], derived in the the Results section, Eqs. 4 & 5.

The specific value <S> = 0.90 (90 percent) is determined by iterating in increments of 5% from

<S> = 85% to 100%, as calculated from the data in Table I . Using a more highly-educated subset of

the Tay, et al. [14] data , the plateau level would increase to <S> = 0.95 . Values for the time

constant t0 are calculated incrementally in steps of 0.25 years, from 3.5 to 5.5 years, with a

relative minimum <std.err.> realized at t0 = 4.5 years. Likewise, values for t1 from 1.0 to 2.5 yrs.,

in increments of 0.25 yrs., yield a minimum error at t1 = 1.5 yrs.

In words, the age of onset parameter t1 represents an age at which no prevalence of myopia

exists. This corresponds to an intercept on the time axis. The time constant parameter 1/t0 represents

the initial rate of the prevalence function. Minimizing the sum of the residuals, i.e. the <std.err.>, is the

conventional “goodness of fit” criteria used here which ultimately yields an optimal t0 value of 4.5

years for these data. Data are presented as a function of years of education, compiled from N=9

studies, Table I & II , Fig. 1, 2, & 3 . Least squares regression, including correlation coefficient r, +/-

<std.err.> accuracy and regression equation, are calculated for comparison. Data are from the studies

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of Goh & Lam [5], Lin et al.[8][9], Saw et al.[12], Fan et al.[1] [2], Greene[3], Tay et al.[14], and Lam

& Goh [7], Table I.

Statistics.

Least squares regression calculates an average incidence rate In(t) = 4.7 % per year.

Confidence limits on the incidence rate are In = [ 95% CI: 2.1 to 7.3 %/yr ]. This means one can

expect 2.1 to 7.3 new cases of myopia per year, on average, from each class of 100 , Fig. 3. The

exponential equations predict much higher initial incidence rates In(t1), i.e. 15% to 20% per year,

consistent with the incidence rate data, as shown in Fig. 2. Table II displays the exponential

prevalence function Pr(t) and incidence function In(t). The myopia prevalence vs. time function Pr(t)

[yrs] and myopia incidence vs. time function In(t) [%/yr] are continuously generated and

compared with student data, Fig. 1 and 2.

R E S U L T S

Myopia incidence rates [%/yr] are 5 to 10 times more rapid, during the juvenile and teen years,

compared to the college and university years, Table I and II . Figure (1) presents prevalence as a

function of years of education Pr(t), using the data from Table I, and theory from Eq. (1). Figure (2)

shows incidence as a function of time In(t), using data from Table I , and theory from Eq. (2).

For comparison with the exponential model, Fig. 1 & 2, Fig. 3 least squares regression statistics

yield the following: Eq. (3) prev(t) = 0.047(t) + 0.178 (x 100 %) where t = years of education,

correlation r = 0.88, N=9, <std.err.> accuracy = +/-10.6%, constant incidence rate <I> = 4.7 % per

year, [95% CI: 2.1 to 7.3 %/yr].

Eq. (1) and (2) are derived setting the number of new incident subjects proportional to the

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residual fraction of normal subjects, yielding : Eq. (4) In(t) = k x [ 1 - Pr(t) ]. From this, the

inverse relation between incidence and time constant also can be derived, Eq. (5) In(t1) = <S> / t0.

Table 1 presents 9 demographic studies, including myopia prevalence [%], incidence rate [% /

yr.], years of education.

T A B L E 1 - Population Surveys

# Rpt. #Subj. Ages Incid. Prev. Loca- Educ. [yr] [sex] [yr.] [%/yr] [%] tion [yrs.] -------------------- ---------------------- ------------------------ 1. Lin1 345 20<t<25 1.0% 92% Taiw. Medi. (1996) [m&f] 16-20

2. Saw 981 5<t<7 16% 40% Sing. Elem. (2005) [m&f] 1-3

3. Fan1 7.6K 5<t<16 14% 37% HongK. Elem. (2004) [m&f] 6-7

4. Gre 39 20<t<23 0.6% 89% USA Engr. (1986) [m] 15

5. Tay 421K 22<t<32 2.4% 43% Sing. Misc. (1992) [m] >12

6. Lam 383 16<t<17 4.0% 70% HongK. EL-HI (1991) [m&f] 10

7. Goh 2K 19<t<39 n.a. 87% HongK. Univ. (1994) [m&f] 16

8. Lin2 12K 16<t<18 n.a. 84% Taiw. EL-HI (2004) [m&f] 12

9. Fan2 255 6<t<11 8.2% 44% HongK. Elem. (2004) [m&f] 4-5 -------------------- ---------------------- --------------------- [ M ] = myopia prev. [%] <I> = myopia incidence [%/yr]

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T A B L E II - Prevalence & Incidence as a function of time t0 = 4.5 yrs., t1 = 1.5 yrs.

Time t (yrs.) Pr(t) [%] In(t) [%/yr]

--------------------- -------------------------------- ----------------------------------

1.5 0.0 17.9

2.5 17.9 14.4

3.5 32.3 11.5

4.4 43.8 9.2

5.5 53.0 7.4

6.5 60.4 5.9

7.5 66.3 4.7

8.5 71.0 3.8

9.5 74.8 3.0

10.5 77.8 2.4

11.5 80.2 2.0

12.5 82.2 1.6

13.5 83.7 1.3

14.5 85.0 1.0

15.5 86.0 0.8

16.5 86.8 0.6

17.5 87.4 0.5

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D I S C U S S I O N

Basic exponential equations can continuously predict the Prevalence Pr(t) [%] and Incidence

In(t) [%/yr] functions for myopia. The incidence function is derived from the prevalence function, the

prevalence is the integral of the incidence. Thus, these are coupled equations. The initial incidence rate

is inversely related to the system time constant, Eq. 5. For instance, an initial incidence In(t1) = 33 new

cases per year, per 100, implies a time constant t0 = 3 yrs. for plateau <S> = 1 . For our analysis of

myopia development, the time constant is t0 = 4.5 years, which is comparable to the individual myopia

time constant 3 < t0 < 5 yrs., reported by Medina & Fariza,[4].

Least squares regression, while useful for prevalence data, Eq. (3-1), Fig. 3, underestimates the

incidence rate. Nevertheless, this approximation is reasonable, at a constant incidence level Incidence =

4.7 %/yr. [95%CI: 2.1 to 7.3 %/yr.]. The exponential equations are quite accurate, able to predict the

incidence rate data within +/-2.6 %/yr, over the entire age range, Figure (2).

In terms of selection criteria, these data sets pertain to studies where the age specific prevalence

and incidence numbers for progressive myopia are available, and were thus appropriate to our analysis.

As of 2012-2013, these were the only detailed studies we could find. The net accumulated prevalence of

myopia in the late teens and early 20’s is very high in these studies. Undoubtedly, that is why these

studies were done.

Pan, Ramamurthy, & Saw [11] report 12 worldwide age-specific prevalence studies, examining

juvenile myopia from China, India, Malaysia, Nepal, United States, S. Africa, and Chile. More recent

prevalence studies include Kim et al. [6], Sun et al. [13], who report myopia prevalence Pr = 95.5%,

N=5083, for college students, and Yip et al. [16], and Lee et al. [15] who report 83.3% myopia

prevalence, N=2805 among university students.

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Usually, incidence data In(t)[ %/yr] and prevalence Pr(t) [%] are not available simultaneously.

Similar exponential equations for prevalence as a function of time or age Pr(t) are derived by Michael &

Bundy [10], with the time constant parameter TAU = t0 notation, as used here. Basic exponential

equations for prevalence Pr(t) and incidence In(t) yield four key system parameters: onset age t1

[yr], time constant t0 [yr], saturation plateau level <S> [%] , and initial incidence rate In(t1) [%/yr].

These then, in turn, allow direct comparisons between one data set and another, both cross-sectional and

longitudinal.

CONCLUSIONS

In terms of relevance and clinical implications, in this report we investigate the two simplest

mathematical models available - linear regression and exponential regression. Each has advantages, as

applied to ophthalmic prevalence and incidence studies, allowing modeling of progressive myopia,

glaucoma, presbyopia, cataract, or a more general medical condition, each with a specific onset age

range and system time constant. These equations are accurate over the course of 2 to 3 decades. The

correlation co-efficient r and r.m.s. residual error statistic +/- <std.err.> , also known as “scatter”,

can determine which is better, for a particular situation .

The initial incidence rate at onset age In(t1) and system time constant t0 are inversely related,

Equation 5. For myopia, onset age, time constant, and saturation plateau level are fundamental system

parameters derived from age specific prevalence data. In turn, these characteristic system parameters

may be compared with the results of future myopia intervention studies’ outcomes.

The literature abounds with various mathematical models of prevalence for contagious or

acquired diseases, far too many to cite, some requiring multiply linked differential equations (SEIR),

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including linear & exponential models, compared here, power law, binomial, quadratic, Markov,

random, oscillating, etc. For progressive myopia, the variables of interest, in addition to time, might

include age, discussed here, ethnicity, location, discussed here, inherited factors, and the near-point

environment .

N O M E N C L A T U R E

Pr(t) = prevalence as a function of time [ % ]

In(t) = incidence rate as a function of time [ %/ year ]

t0 = time constant [years] , typically 1 wk. to 5 yrs.

t1 = onset age [years], typically 1 yr. to 30 yrs.

<S> = saturation plateau level [%], typically 30-95% for myopia

95%CI = 95% confidence interval

K – 12 = kindergarten through 12-th grade

r = correlation coefficient, typically -1 to 1

exp(-t/t0 ) = exponential function , 0 < e-t/t0 < 1

<std.err.> = accuracy of regression, typically +/- 2% to +/- 14%

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REFERENCES

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[2] Fan DS, Cheung EY, Lai RY, Kwok AK, Lam DS. Myopia progression among preschool Chinese children in Hong Kong. Ann Acad Med Singapore. 2004 Jan;33(1):39-43.

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[14] Michael E, Bundy DA. Herd immunity to filarial infection is a function of   vector   biting rate. Proc Biol Sci. 1998 May 22;265(1399):855-60.

[15] Pan CW, Ramamurthy D, Saw SM.Worldwide prevalence and risk factors for myopia.Ophthalmic Physiol Opt. 2012 Jan;32(1):3-16.

[16] Saw SM, Tong L, Chua WH, Chia KS, Koh D, Tan DT, Katz J. Incidence and progression of myopia in Singaporean school children. Invest Ophthalmol Vis Sci. 2005 Jan;46(1):51-7.

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Acknowledgements - Guidelines from the Helsinki Declaration, for the use of human subjects in ophthalmic and vision research, were followed. Declaration of interest : The authors report no conflicts of interest. The authors alone are responsible for the content and writing of the paper. Patient confidentiality was maintained using retrospective data with subject I.D. deleted.

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FIGURE LEGENDS

Figure (1) - The myopia prevalence function Pr(t) [%] vs. years of education, N=9 studies. Pr(t) = 0.90 *[ 1 - exp(-(t-t1)/to ) ] , with to = 4.5 yrs. , t1 = 1.5 yrs. Medical students (N=345) are from Hong Kong, engineers (N=39) are from the U. S.

Figure (2) - Myopia incidence function In(t) [%/yr.] i.e. # of new cases per year, per class of 100, N = 7 studies. In(t) = 0.90 [ (1/to) exp(-(t-t1)/to) ] +/-2.6%/yr., to = 4.5 yrs., t1 = 1.5 yrs.

Figure (3) - Linear regression, myopia prevalence as a function of years of education, N=9 demographic studies. Pre-medical N=345 students, H.K., engineers N=39 students, U.S. On average, myopia incidence = 4.7 %/yr., confidence limits [95%CI: 2.1 to 7.3 %/yr.] .

FIGURES

Figure 1

Figure 2

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Figure 3

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