EPA-670/2-74-067
August 1974
Environmental Protection Technology Series
OF THE KINETICS
OF VIRAL llCTtVAtlON
National Environmental Research Center
Office of Research and Development
U.S. Environmental Protection Agency
Cincinnati, Ohio 45268
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EPA-670/2-74-067
August 1974
A MATHEMATICAL ANALYSIS OF THE KINETICS
OF VIRAL INACTIVATION
BY
Robert M. Clark
Betty Lou Grupenhoff
George C. Kent
Water Supply Research Laboratory
Program Element No- 1CB047
NATIONAL ENVIRONMENTAL RESEARCH CENTER
OFFICE OF RESEARCH AND DEVELOPMENT
U.S. ENVIRONMENTAL PROTECTION AGENCY
CINCINNATI, OHIO 45268
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REVIEW NOTICE
This report has been reviewed by the National
Environmental Research Center, Cincinnati,
and approved for publication. Mention of
trade names or commercial products does not
constitute endorsement or recommendation
for use.
11
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FOREWORD
Man and his environment must be protected from the adverse
effects of pesticides, radiation, noise and other forms of
pollution, and the unwise management of solid waste. Efforts
to protect the environment require a focus that recognizes
the interplay between the components of our physical envi-
ronment—air, water, and land. The National Environmental
Research Centers provide this multidisciplinary focus through
programs engaged in
• studies on the effects of environmental contaminants on
man and the biosphere, and
• a search for ways to prevent contamination and to recycle
valuable resources.
This report describes a mathematical model which can be used
to characterize the response of viruses to a disinfecting
agent. Not only is the model itself presented, but a tech-
nique is described which can be used to estimate the model's
parameters. Both the model and the estimation technique are
being used to analyze experimental information resulting from
disinfection studies.
A. W. Breidenbach, Ph.D.
Director
National Environmental
Research Center, Cincinnati
111
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ACKNOWLEDGEMENTS
Dr. P. V. Scarpino, Professor of Environmental Engineering,
University of Cincinnati, provided guidance and assistance
throughout all phases of this work.
Mr. Arthur F. Hammonds of the Water Supply Research Laboratory,
NERC-Cincinnati, EPA, and Mr. Richard L. Manning, Office of
Water Programs, Office of Air and Water Programs, EPA, Wash-
ington, D. C. / assisted in the data processing work presented
in this paper.
Miss Jacqueline E. A. Kent assisted in the development of and
programming of several of the equations utilized in this analysis,
Ms. Catherine Hall, University of Cincinnati, assisted in the
preparation of this manuscript.
Miss Gruppenhoff is employed as an engineer by General Electric
Company in Cincinnati; Mr. Kent is employed by the Office of
Water Programs, Office of Air and Water Programs, EPA, Wash-
ington, D. C.
IV
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A MATHEMATICAL ANALYSIS OF THE KINETICS OF VIRAL INACTIVATION
INTRODUCTION
Pathogenic enteric viruses transmitted via the water route
present a potential hazard to public health because of their
resistance to natural or artificial disinfection mechanisms.
More than 100 different strains of enteric viruses, causing
such diseases as poliomyelitis, meningitis, jaundice, and
gastroenteritis, are excreted in human feces. The six major
groups of enteroviruses responsible for these diseases are
polioviruses, coxsackieviruses A and B, echoviruses, adeno-
viruses, infectious hepatitis, and reoviruses. Since these
viruses are able to survive in sewage, natural waters, and
water supplies, they may pose a health threat, particularly
as wastewater reuse becomes more common.^' ^
Of constant concern to public health officials is the ability
of viruses to pass through water treatment plants. The chlorine
levels must be adequate, not only for bacterial disinfection
but for viral inactivation as well. As a result of the need
for constant concern over proper disinfection levels, much
research effort has been devoted to the s,tudy of the basic
disinfection mechanisms.
Chick was probably the first investigator to attempt to under-
stand the laws of disinfe'ction by applying the principles of
first-order kinetics to bacteria and spore inactivation.3 Only
the experiments with anthrax spores conformed to first-order
kinetics, whereas bacteria apparently followed another pattern
of inactivation. Subsequent studies have obtained results
that confirmed first-order kinetic inactivation for bacteria.
Many research investigations have been directed toward the
study of the inactivation of viruses and enteric organisms.
As a result of these studies, the process of inactivation
has been found to be dependent on the time of contact between
the organisms and disinfecting agent, concentration of disin-
fecting agent, temperature, and pH. In addition, viruses may
form clumps of varying sizes and may cause aberrations due
to their existence in inactivation systems.4 one approach
to studying the interaction of these various factors is to
develop a kinetic model that will systematically account for
them. The development of such a model and its application
are discussed in this paper.
-------
MODEL DEVELOPMENT
One of the major features in this model is the consideration
of clumping or aggregation and its effect in explaining the
devitalization process and associated aberrations. For pur-
poses of this model, it is assumed that the virions exist
either as individual particles in a suspension or as aggregates
or clumps made up of two or more particles.5 Each individual
particle or aggregate will form a plaque-forming unit (PFU)
before the viral suspension is subjected to a disinfecting
agent. It is impossible to determine whether a PFU represents
a single infective unit. If the suspension contains single
particles as well as clumps of various sizes, the disinfection
process will continue until the last particle 'in the largest
clump is devitalized. When the clump is completely devitalized,
a PFU is destroyed, but it is obvious that a distribution of
different size clumps will lead to a non-uniform destruction
of PFU's thereby causing some unusual shapes in the disinfec-
tion curve.
In this discussion, it will be assumed that this distribution
of infective units represents the state of the suspension.
The percentage of aggregates or clumps of all sizes which
have been disinfected at any time represents the Nth state;
the percentage of undisinfected single particles represents
the N-lst state, etc. For illustrative purposes, let us
assume a suspension in which the maximum clump size is com-
posed of three viral particles and with clumps composed of
two particles as well as single particles. Following our
convention, state 1 is the percentage of undisinfected aggre-
gates with three virions; state 2, the percentage of undisin-
fected aggregates with two virions; state 3, the percentage of
undisinfected single particles; and state 4, the total per-
centage of aggregates (clumps of 1, 2, and 3 viral particles)
that have been devitalized at any point in time. Obviously,
under the action of a disinfectant, assuming ideal conditions,
state 4 would increase as the process continues until state 4
would be 100 percent.
We can impose a frequency distribution on the various states
in effect, assigning a percentage of the total plaque-forming
capability to each state. The initial condition of state 4(84)
must equal 0 percent at time equal to zero or before the disin-
fectant acts. The percentage of undisinfected singles plus
the percentages of clumps with two particles plus the percentage
of clumps with three particles would equal 100 percent when
time equals zero.
Associated with each state is a decay rate, k^, that represents
the probability of interaction of the destructive agent with
the undisinfected singles or aggregate. The process of devital-
ization is assumed to take place in the following manner: The
-------
clumps of three virions are reduced to two surviving virions,
and the clumps of two are reduced to one surviving virion all
the way along the chain of states until the clumps are no longer
infective and are registered as a decrease in total PFU.
The set of differential equations that describes the devitali-
zation process, where S^(i = 1 . . .4), the percent of plaque-
forming capability at each state is:
dsl
a-r = -
dS2
= k2S2 - k3S3
dt~ = k3S3
These are a set of linear first-order differential equations.
The parameters kj_ (i = 1 . . . 4) represents the devitalization
rate with k4 = 0, and s9 is the initial condition of state i
with sO = 0 at t = 0. X
The solution to Equation 1 is as follows :
0 e"klt e"k2t
S4 - klk2k3Sl [(-k!) (k3-ki) (k2-ki) + (-k2) (k3-k2)
(-k3)(k2-k3)(k!-k3) T (k3)(k2)
i
J
32 (_k2) (k3-k2) (-k3) (k2-k3) (k-3) (k2)
[e~k3t - 1]. (2)
The general closed form solution to a set of differential
equations as illustrated by Equation 1 is given by the fol-
lowing : 5
-------
I (Wi---1We''Cnt"s°' I3.
=l 'VV 'Vl- V • ' k?
where k>j. When j=k, (k.-k, ) = 1.
1 K
When j "Dressed as percent survival, the equation could be
written as percent survival = 100 - S^, where S^ is the last
or final state to be considered.
Figure 1 illustrates schematically the change taking place
during an experiment. Devitalized virus in an aggregate are
represented by a broken circle. In a devitalization chain,
the value for kj_, which indicates the rate of transition from
one state into the next, differs for each state.6 There are
also differences between chains. For example, k3 in the first
chain may be smaller than k3 in the second chain. This might
be attributed to different geometric configurations and re-
sulting interferences. We will assume, however, that k3 is an
average reaction rate for state 3 in all of the decay chains.
Equation 2 can be reformulated in the following manner :
(4)
where,
(k3-k1)
-k k s° k q°
KlK3bl K3b2
(k3-k2)
-k k q° k q°
1 2bl K2b2
_ _
3 ~ (k2-k3) (k1-k3) (k2~k3) b3
C0 = Sl + S2 + S3 (5)
We know that as t-*-°°, S.->-100 percent; therefore, Cn-*100 percent.
Equation 4 forms the basis for the mathematical model of the
kinetics of viral inactivation we wish to examine. However,
to use this equation, we must be able to estimate its param-
eters.
-------
•*- ( II
o
83 84 85
0
4
Figure 1. Schematic illustration of the physical change
taking place in a suspension of virions under
the influence of a devitalizing agent.
Devitalized virus in a clump is represented by
a broken circle. The values for X indicate the
rates of transition from one state into the
next in a devitalization chain. S? is the
initial percent of plaque-forming capability
in state i.
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ESTIMATION OF PARAMETERS
If we were to rewrite Equation 4 in terms of percent survival,
we would have the following:
100% - S° = -Cje-! - C^- - Cge- (6)
or
y - -C^l* - C2e-k2t - C3e-k3t (7)
where y = 100% - S,.
For simplicity/ we shall assume that our observations are
equidistant, as in Figure 2, and that the difference in the
successive abscissa values is h. With the use of our three-
term example, we find the value of the ith ordinate at
to + (i-l)h, where to is the value of yo at to, is then:
y± = -C^ expt-k-^Q + (i-l)h] - C2 exp[-k2tQ + (i-l)h]
-C3 exp[-k3tQ + (i-l)h] (8)
or, if we make the following substitutions:
-kih
e x = u-j^;
-k-,h
e 1 = u2;
-kih
e J- = u3;
-k-Lt^ + (i-l)h] = f1?
-C2 exp[-k2tQ + (i-l)h] = f2;
-Cx exp[-k3tQ + (i-l)h] = f3?
then, for five equidistant measurements, we have:
fi + f2 + f3 - y±
flul + f2U2 + f3U3
-------
100.0
TIME
Figure 2. Equally distant values for percent survival versus time,
-------
flUl + f2U2 + f3U3 =
flul + f2U2 + f3U3 =
flUl + f2U2 + £3U3 =
In general, the set of equations for N observations would be
as shown below:
flul
flul
fl + f2
+ f2U2 +
+ f2u2 +
+ h
f3U3
f3u3
- yo
= yl
= Y2
which would necessarily be satisfied identically. If the con-
stants U]_, u2, and U3 were known (or ^reassigned) , Equations 10
would comprise N linear equations in the three unknowns f^, f2,
and f3, and be solved exactly if N=3 or approximately by
least squares if N>3.
However, if the u's are also to be determined, at least six
equations are needed, and a difficulty occurs because the
equations are non-linear in the u's. This difficulty can be
minimized by the following method.
Let u, , Up, and u., be the roots of the algebraic equation:
3 2
u - a,u - a2u - a., = 0 (11)
so that the left-hand member of Equation 11 is identified with
the product (u-u^)(u-u2)(u-u3). To determine the coefficients
a]_, a2, 33, we multiply the first line in Equation 10 by ao,
and the second line by a2, and the third line by a^, and the
fourth line by -1, and add the results. If use is made of the
fact that each u satisfies Equation 11, the result is seen to
be of the form:
Y3 " aly2 ' a2yl " a3Y0 = °
-------
A set of N-4 additional equations of similar type is obtained
in the same way by starting instead successively with the second,
third . . . (N-3)th equations. In this way, we find that
Equations 10 and 11 imply the N-3 linear equations:8
Y2al + yla2 + yOa3 = y3
yN-2al
%-4a3 =
Since the ordinates y^ are known if N=6 , this set generally
can be solved directly for a^ , a2 , and 33, or it can be solved
approximately by the method of least squares if N>6.
In theory, after the a's are determined, the u's are found as
the roots of Equation 11 and may be real or complex. Equation 10
then becomes linear and the f's can be determined from the first
n of these equations or preferably by applying a least-squares
technique applied to the entire set.
We have examined the situations in which there are only three
terms to analyze in Equation 2. However, most often the situ-
ation will occur when there are n terms in the equation to be
solved. This would take the form as follows:
Sn = C0
(14)
Assuming that there are N points equally spaced at t=0 , 1, 2,
3 ... N-l, and following the logic described in this paper,
we get a set of equations similar to Equations 8:
fn =
flul H
flul H
h f2U2 H
h f^ H
h f 3u3 + . ,
h f 3u2 + . .
' • + f nun = S
'• + fnun= S
Again, following the logic described earlier, we have the fol
lowing N-n linear equations where the columns of data are
labeled 1 through n+1.
-------
(1)
ynal H
(2)
h yn-la2 H
(3)
h yn-2a3 +
h yn-2a3 +
(n) (n+1)
• • ' + Y0an = yn
' ' ' + ylan = yn+l
YN-2al + %-3a2 + YN-4a3 + ' ' ' + YN-n-lan = yN-l
(16)
After the a's have been determined by least squares, the values
for the c's can be found as roots of the, following equation:
un - a^11"1 - a2un~2 - ... - an_2u - an = 0 (17)
And once the u's have been found, the f's can be found from
Equation 15.
The application of. this approach presumes that the number of
terms that make up the model is known. Generally this number
is unknown, and a major part of the analysis becomes the esti-
mation of the optimum number of terms describing the disinfection
process. Even if the number of terms is known, the solution to
Equation 17 is often complex because of estimation errors in
determining the coefficients. To make this analysis usable,
we must be able to determine the number of terms (number of
states), that make up the inactivation process. The following
section describes a technique for estimating the number of
components that "best" describe the inactivation process.
OPTIMAL NUMBER OF TERMS
To determine the proper number of terms that will describe
the inactivation process, we would formulate the set of linear
equations shown in Equation 16. In this set, the column
labeled n+1 is the response or dependent variable, and the
columns 1 through n are the independent variables. Using
step-wise regression, we regress the independent variables
(1 through n) against the n+lst or dependent variable.^ As
each variable is forced into the equation, a value for its
coefficient is calculated. Each coefficient has an associated
sign. When the signed coefficient is substituted into Equa-
tion 17, it is possible that an equation with alternating
signs may result; for example, Equation 17 might look as
follows:
anun - a,un + a0un - ... + a ~u - a =0 (18)
U l z n-Z n
10
-------
According to Des Cartes' rule of signs:
The number of positive real roots of a
real albegraic equation either is equal
to the number Na of sign changes in the
sequence a0 , alf a2 , . . . an of co-
efficients where vanishing terms are
disregarded or it is less than Na by
a positive even integer.
Since the decay coefficients in Equation 14 are the positive
real roots in Equation 18, we can use Des Cartes' rule to
give us an indication as to the number of terms which optimally
describes the inactivation process. We will assume that when
the number of terms in the regression equation is one more
than the number of sign changes, the optimal number of terms
has been identified, and the variables in the regression equa-
tion are to be used in calculating kn. The approach will be
discussed beginning with the identification of the optimal
number of terms.
We can illustrate this approach by assuming a model of three
terms as follows:
y = 20.00e-°-10t + 30.00e-°-30t + SO^Oe'0' (19)
Table 1 (Page 12) contains values for Equation 19 which have
been generated at intervals of t=»0.50 to simulate a disinfection
curve. Table 2 illustrates the way in which these data are
organized to solve for the coefficients in Equation 17. As
shown in Equation 16, a matrix of data points is established
with n dependent variables. In this case, 27 independent
variables have been constructed. The value of yi - 100.00 is
the first value in the upper left-hand corner of the matrix,
and the value y2j = 5.7660 is the first value for the dependent
variable. The second value for the first independent value is
y-L = 83.7858, and the second value for y28 = 5.4274. This same
pattern is repeated throughout the matrix.
Table 2. MATRIX OF DATA FOR REGRESSION ANALYSIS
Var 1
100.00
83.786
Var 2
83.786
70.648
. . ., Var n ... Var 28
5.7660
5.4274
2.0359 1.9338 0.5203
1.9338 1.8370 0.4949
11
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Table 1. VALUES FOR EQUATION 19 AT INTERVALS OF t=0.5
t
0.00
0.50
1.00
1.50
2.00
2.50
3.00
3.50
4.00
4.50
5.00
5.50
6.00
6.50
7.00
7.50
8.00
8.50
9.00
9.50
10.00
10.50
11.00
11.50
12.00
12.50
e-o. loot
20. 0000
19.0245
18.0967
17.2141
16. 3746
15.5760
14.8163
14.0937
13.4064
12. 7525
12.1306
11.5390
10.9762
10.4409
9.9317
9. 4473
8. 9865
8. 5483
8.1313
7.7348
7. 3575
6. 9987
6. 6574
6. 3327
6. 0238
5. 7301
e-0. 3001
30.0000
25.8212
22. 2245
19. 1288
16.4643
14.1710
12.1970
10.4981
9. 0358
7. 7772
6. 6939
5. 7615
4. 9589
4. 2682
3. 6736
3.1619
2.7215
2. 3424
2.0161
1.7353
1.4936
1. 2855
1.1064
0. 9523
0.8197
0. 7055
e-o. soot
SO. 0000
38. 9400
30. 3265
23.6183
18. 3939
14. 3252
11.1565
8. 6887
6. 7667
5. 2699
4. 1042
3. 1963
2. 4893
1.9387
1. 5098
I. 1758
0.9157
0.7132
0. 5554
0.4325
0. 3368
0. 2623
0. 2043
0.1591
0. 1239
0. 0965
y
100. 0000
83. 7858
70. 6478
59.9613
51.2329
44. 0722
38. 1699
33.2806
29. 2090
25. 7997
22.9287
20. 4969
18.4245
16. 6478
15.1152
13.7852
12. 6239
11.6039
10. 7030
9. 9027
9.1881
8. 5467
7. 9682
7.4443
6. 9675
6. 5321
t
13.00
13.50
14.00
14.50
15.00
15.50
16.00
16.50
17.00
17.50
18.00
18.50
19.00
19.50
20.00
20. 50
21.00
21.50
22.00
22.50
23.00
23.50
24.00
24.50
25.00
25.50
.-0. lOOt
5. 4506
5. 1848
4.9319
4. 6914
4. 4626
4.2449
4. 0379
3.8410
3. 6536
3. 4754
3. 3059
3. 1447
2.9913
2. 8454
2. 7067
2. 5747
2.4491
2. 3296
2.2160
2. 1079
2. 0051
1.9073
1.8143
1.7258
1.6417
1.5616
e-0. 300t
0. 6072
0. 5226
0. 4498
0. 3872
0. 3332
0. 2868
0. 2468
0.2125
0. 1829
0.1574
0.1354
0.1166
0.1003
0. 0863
0. 0743
0. 0640
0. 0550
0. 0474
0. 0408
0.0351
0. 0202
0. 0260
0. 0223
0.0192
0.0165
0.0142
-0. SOOt
0.0751
0.0585
0.0455
0.0355
0. 0276
0.0215
0.0167
0.0130
0.0101
0. 0079
0. 0061
0. 0048
0. 0037
o. 0029
0. 0022
0.0017
0.0013
0.0010
0. 0008
0. 0006
0.0005
0. 0003
0. 0003
0. 0002
0. 0001
0.0001
y
6.1330
S. 7660
5. 4274
5.1141
4. 8235
4. 5533
4. 3016
4. 0665
3. 8467
3. 6408
3. 4476
3. 2661
3. 0954
2. 9347
2.7833
2. 6404
2. 5055
2. 3781
2. 257?
2. 1437
2. 0359
1.9338
1.8370
1. 7453
1. 6584
1.5760
t
26.00
26.50
27.00
27.50
28.00
28.50
29.00
29.50
30.00
30.50
31.00
31.50
32. 00
32. 50
33. 00
33.50
34.00
34.50
35.00
35.50
36.00
36.50
37.00
-0. lOOt
1.4854
1.4130
1.3441
1.3785
1.2162
1. 1568
1. 1004
1.0467
0. 9957
0.9471
0. 9009
0. 8570
0.8152
0. 7754
0..7376
0.7016
0. 6674
0. 6349
0. 6039
0. 5744
0. 5464
0. 5198
0. 4944
e-0. 300t
0.0122
0.0105
0.0091
0. 0078
0. 0067
0. 0058
0. 0049
0. 0043
0. 0037
0.0031
0. 0027
0.0023
0. 0020
0.0017
0.0015
0.0012
O.OOU
0. 0009
0. 0008
0. 0007
0. 0006
0. 0005
0. 0004
e-o. soot
( : •!•>'
6.
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Table 3 contains the results of the application of the step-
wise regression program to the matrix of data in Table 2.
The equations resulting from each step are as follows:
x28 - 0.9418X27 =
x28 - 1.1223X27 + 0.1402X23 =
p Q ^ *7 o "3 1
x - 1.1420x + 0.1565X J - 0.000089x =
x28 + 0.0318X27 - 0.9398X23 + 0.07968x8 - 0.01124X1 =
Equation 22 combines the maximum number of sign changes with
the minimum .number of variables in the equation and is, there-
fore, selected as the equation governing the number of terms
in the disinfection equation. This matches identically with
the three terms used in the simulated data. After the best
estimate has been made of the number of terms which makes up
the data, the next step in the analysis is to estimate the
decay coefficients in the equation. This step is described
in the following section.
Table 3. RESULTS OF REGRESSION ANALYSIS
USING DATA FROM TABLE 2
0
0
0
0
(20)
(21)
(22)
(23)
Step
1
2
3
4
Var
27
23
27
1
23
27
1
8
23
27
Coefficient
0.94179753
-0.14022224
1.12226027
0.00008941
-0.15649791
1.14201651
0.01124259
-0.07967716
0.93985016
-0.03183525
ESTIMATION OF PARAMETERS
Decay Rates
Based on the data in Table 1, the parameters for Equation 19
can be estimated using the techniques outlined in Appendix A.
The first decay coefficient to be calculated will be that
associated with variable ,x23 and the calculation is as follows
13
-------
[(x24 - rlX23) - (x2 - r^1)]
(23 _ 1} =
24 23 28 27
24 - r,x2J) - (x^b - r,xz/)]
L—r _ . _ ± _
(23 - 27)
28 27 24 23 2
24
Substituting the average values for x , x , x , x , x , and
x1, into Equation 24 yields the following values for r:
r, = 0.85 (25)
where
-k]h (26)
rl - e
From Equation 26, we can calculate k, as follows:
(27)
kx = - [ln(0.85)]/0.50
kx = 0.32
The decay coefficient associated with variable x is calculated
as follows:
In(r2) = ln(x2) - ln(2xX - x2) (28)
Ind^) = - 0.24
Substituting into Equation 27 for k2 we get the following:
k2 = (0.24)/0.50 (29)
k2 = 0.48
27
The decay coefficient associated with variable x is calculated
as follows:
In(r3) = In(x28/x27) (30)
In(r3) = ln(0.95)
Substituting into Equation 27 for k3 we get the following:
k3 = (0.05)/0.50
k3 = 0.10
14
-------
Coefficients
Once the decay rates in Equation 19 have been estimated, the
values for the coefficients are relatively easy to obtain.
Values for each exponential term can be caluclated at the
appropriate time interval and these values regressed against
the values of y in Table 1. Stepwise regression can then be
used to estimate the coefficients (Appendix D).
EXAMPLE INACTIVATION PROBLEM
To illustrate the utilization of this technique, it will be
applied to experimental data collected from a series of
electromicroscopy investigations conducted by Gordon Sharp
at the University of North Carolina.*-® Sharp prepared
electron micrographs of dilute preparations of T7 virus
that had been subjected to a devitalizing agent.
Figure 3 shows the inactivation curve, and Table 4 contains
the distribution of T7 coliphage particles resulting from
these experiments. Column 1 of Table 4 lists the group size
of the aggregates, that is, the number of particles in each
clump of virus. Column 2 lists the number of groups in the
suspension, and Column 3 lists the number of particles in
each group. Column 4 lists the percent of plaque-forming
capability that each group represents in the suspension.
For example, there are 770 groups in the suspension, but
610/770 or 79.1 percent of them are groups of single viral
particles, and 116/770 or approximately 15.1 percent of them
are groups of two viral particles, etc.
Table 4. T7 VIRUS DATA
Group Number of Number of Plaque-forming
size groups particles capability (%)
1 610 610 79.22
2 116 232 15.06
3 24 72 3.12
4 12 48 1.56
5 6 30 0.78
61 6 0.13
18 1 18 0.13
Total 770 1,016 100.00
15
-------
INACTIVATION OF COLIPHAGE
BY ULTRAVIOLET RAYS
50 100
IRRADIATION
TIME (SECONDS) AT 33 jjW/cm'
Figure 3. Inactivation of coliphage
16
-------
Table 5 contains the data from Figure 3, at intervals of
5 seconds, arranged in 10 columns of data. Table 6 contains
the coefficients associated with each set of variables as
they enter the stepwise regression equation. It is obvious
from the alternating signs that six variables will describe
the inactivation process. The decay coefficients per minute
calculated from the techniques outlined in Appendix A are
as follows:
/
r! = 1-56 (32)
r2 = 2.12
r3 = 2.32
r4 = 2.68
r5 = 2.83
rc = 4.68
D
Each of these values represents a k^ in Equation 14, and each
value of e^it can be generated at various intervals of t by
the program in Appendix B. If all six values of e'^it repre-
sented by Equation 32 are regressed against the values for y
as obtained from the graph in Figure 3 then Table 7 contains
the values for their coefficients. Using the program in
Appendix C, the values for each predicted Sj_ (percentage of
plaque-forming capability), can be calculated. The predicted
and actual values are shown in Table 8.
When the regression is performed, the values shown in Table 7
result. At the fourth step of the regression, the corrected
R2 begins to decrease which is an indicator that the regression
should be terminated at that point, and step 3 is, therefore,
used as the last step in the regression analysis. Equation 5
and the program contained in Appendix C, where k^ = 1.56,
k2 = 2.83, and k3 = 4.68, yields the following values for Sj_:
83 = 73.97%, 82 = 15.51%, and Si = 11.30%. Physically, this
means that there are ,73.97% singles, 15.51% doubles, and the
rest of the particles amount to approximately 11.30%. The
comparison between the results obtained from the model and
the electron micrographs is shown in Table 8. The agreement
seems reasonable.
17
-------
00
Table 5. DATA FROM DISINFECTION CURVE
Var 1 Var 2 Var 3 Var 4 Var 5 Var 6 Var 7 Var 8 Var 9 Var 10
100.00000 79.00000 58.00000 45.00000 36.50000 28.50000 25.00000 19.80000 17.50000 15.00000
79.00000 58.00000 45.00000 36.00000 28.50000 25.00000 19.80000 17.50000 15.00000 13.00000
000.16500 00.16000 00.15500 00.15000 00.14000 00.13500 00.13000 00.12000 00.11000 00.10500
000.16000 00.15500 00.15000 00.14000 00.13500 00.13000 00.12000 00.11000 00.10500 00.10000
-------
Table 6. RESULTS OF REGRESSION ANALYSIS
USING DATA FROM TABLE 5
Step .
1
2
3
4
5
6
7
Var
9
6
9
6
8
9
5
6
8
9
4
5
6
8
9
1
4
5
6
8
9
1
4
5
6
7
8
9
Coefficient
0.84449432
0.16137199
0.57866191
0.22635578
-0.22446799
0.73339338
-0.14587300
0.36096394
-0.35799450
0.95901767
0.12430537
-0.26913616
0.29413744
-0.36433149
1.01961917
-0.03634866
0.29008852
-0.30158431
0.22952950
-0.50623862
1.13438561
-0.03743081
0.32109643
-0.43943162
0.20853501
0.25010067
-0.67970749
1.22741919
19
-------
Table 7. RESULTS OF REGRESSION
USING DR. SHARP'S INACTIVATION DATA
Step
1
2
3
4
Var
2
2
6
1
2
6
1
2
4
6
Coefficient
96.77
89.08
6.64
58.89
4.12
37.77
61.68
-10.82
15.24
34.70
Corrected R2
as a percent t value
99.501 117.26
99.549 32.05
2.88
99.933 19.67
0.92
20.82
99.932 6.69
-0.23
0.32
3.55
Table 8. COMPARISON BETWEEN MODEL OUTPUT
AND ELECTRON MICROGRAPHS
Group size
Percent
plaque-forming
capability
(counted)
Percent
plaque-forming
capability
(predicted)
3
1
2
+4+5+6+18
79.22
15.06
5.82
73.97
15.51
11.30
SUMMARY AND CONCLUSIONS
The kinetics of viral inactivation have been examined from a
rational point of view. A mathematical model based on the
radionuclide chain decay concept was formulated and a solution
technique developed that allows for estimations of the optimal
number of terms in the equation and for estimating the equation's
parameters. With the use of data derived from electron micros-
copy, the model was tested and achieved reasonable results.
20
-------
Based on this discussion, it is obvious that the postulated
mathematical model and its solution techniques are superior
to others that have been formulated. The approach outlined
in this report not only determines the number of aggregate
groups in the suspension, but the values for decay coefficients
as well. There are some deficiencies in this approach, however,
and it is important that these be considered. The approach
suggested here is statistical in nature and is, therefore,
subject to experimental error in the various estimations made.
More importantly, the estimates of aggregate size and concen-
tration are blind. That is, if this approach estimates three
terms as optimal, there is no way to provide information on
the make-up of these aggregate groupings. They might be clumps
of single, double, and triple particles, or clumps of 20, 21,
and 25 particles. The values for the decay coefficients may
give some insight as to clump size, but these insights are
hardly sufficient. This technique must be coupled with a
physical assay approach incorporating electron microscopy.
A research project that combines the elements of mathematical
analysis with electron microscopy is currently underway.
21
-------
APPENDIX A
In this appendix, the mathematical justification for the
techniques used in the section entitled "Estimation of
Parameters" is developed. Table 9 contains the first three
values for the individual terms which make up the variables
1, 2, 23, 24, 27, and 28, as shown in Table 2. The first
variable to enter the stepwise regression equation is x27,
as shown in Equation 20. Looking at variables 27 and 28 in
Table 9, it is obvious that the term labeled f$ dominates
variable 27 and is most highly correlated with variable 28,
while the terms f^ and f.2 in variable 27 are relatively
insignificant. The next variable to enter the stepwise
regression equation is variable 23, and it can be seen that
terms ±2 and f$ in variable 23 are significant but that term
f^ is insignificant, and finally in variable 1, terms fi, f2/
and f3, are all significant. It can be concluded from this
that a variable enters the regression equation when one of
the terms which comprise it is significant enough to alter
the rate of change of the entering variable. Therefore, we
would expect that variables would enter the regression equation
with alternating signs associated with their coefficients,
since the entrance of each variable into the equation signifies
a significant change in the functions slope. Moreover, we
would expect that the variables entering the equation with
alternating signs represents the maximum change in the slope
of the function with respect to the other variables in the
regression equation. Using Equation 19 as an example, we
would, therefore, attempt to find a u^ such that bn in the
following equation is a maximum relative to its adjacent
variables:
fl(uf1 - u.uj) + *,(„»« - »±u» + f3(uf 1 - u.u") = bn 133)
or in a more simple form, we would attempt to find a u-^ such
that b23 is a maximum relative to bi and b27 in the following
set of equations:
x2 - U..X1 = b1 C34)
x24 - u.x23 = b23
x28 - u.x27 = b2?
22
-------
Table 9. FIRST THREE VALUES FOR SELECTED VARIABLES
Var
1
2
23
24
27
28
Time
0
0.5
1.0
0.5
1.0
1.5
11.0
11.5
12.0
11.5
12.0
12.5
13.0
13.5
14.0
13.5
14.0
14.5
Cie-0.500t.
50.0000
38.9400
30.3265
38.9400
30.3265
23.6183
0.2043
0.1591
0.1239
0.1591
0.1239
0.0965
0.0751
0.0585
0.0455
0.0585
0.0455
0.0355
C26-0.300tt
30.0000
25.8212
22.2245
25.8212
22.2245
19.1288
1.1064
0.9523
0.8197
0.9523
0.8197
0.7055
0.6072
0.5226
0.4498
0.5226
0.4498
0.3872
C3e-»-"°^
20.0000
19.0245
18.0967
19.0245
18.0967
17.2141
6.6574
6.3327
6.0238
6.3327
6.0238
5.7301
5.4506
5.1848
4.9139
5.1848
4.9139
4.6914
y
100.0000
83.7858
70.6478
83.7858
70.6478
59.9613
7.9682
7.4443
6.9675
7.4443
6.9675
6.5321
6.1330
5.7660
5.4274
5.7660
5.4274
5.1141
**!•
tf2-
23
-------
Therefore, u^ can be calculated from the following equation:
(x2 -
- (x
24
23
f
(x
24
23, , 28
) - (x
22.
(1 - 23)
(23 - 27)
(35)
23
since we know that bn is a maximum round the point x
Table 10 confirms that this is in fact the case for variable 23.
Table 10. bn FOR SELECTED VALUES OF Uj_
x2 - u x1
«"• LI n.Ch
-2.2800
-1.4613
-0.7931
x24
0
0
0
.5839
.5583
.5000
28
x
0.
0.
0.
27
- U2x
4857
4449
4569
1 27
For variables x and x , this computation is impossible since
there are no variables which can be used to make a computation
similar to Equation 2.
However, for the decay coefficient associated with x^-, the
following equation can be developed using the properties of
infinite series:
un(x2 - u
un+1(x1 - x2) as n
(36)
Therefore,
= ln(x2) - ln(2x1 - x2)
(37)
27
For the decay coefficient associated with x , the following
relationship can be developed:
n 28 n+1 27 n
u.x ~ui x -* 0 as n -> «>
(38)
Therefore,
ln(u±) = ln(x28) - ln(x27)
(39)
24
-------
APPENDIX B
PROGRAM FOR GENERATING VALUES OP e~kit
AT GIVEN INTERVALS OF t
DlMtNSION A(15),b< 15), C( 15)
MR = 1
MU = 15
5 READ(MR, 10 ) T , • M
10 FORMATIF5.2, 12)
IF IN)70, 70,15
15 REAO(MR,20) (A( J), J = 1,N)
READ(MR,20) (C( J) ,J=1,N)
20 FGRMATI 10F8.0)
E=2. 71828
S = 0.0
Y=0.0
WRITl(MW,25)
25 FORMAK • l'//26X, '-XT' )
V»RITE(M^,30)
30 FORMAT(25X, 'E TABLE')
35 FORMAT ( 25X, ' ---------- ')
VvRITE(MW,40) (A( J) , J=1,N)
40 FORMAT«//10X, 3(3X, '-« ,F5.3, 'T1 ) )
fcRITE(MW,45)
A5 FORMAT(5X,'T',6X,'E«, 2 ( 9X , • E • ) , 1 2X , • Y • )
t>0 DO 60 I = 1,N
t>5 B( I )=C( I )*E»»(-A( I )»S)
60 Y = Y4-B(I)
WRITEIMW.65) S , ( B ( K ) ,K= 1 , N ) , Y
05 FORMAT(/3X,F5.2,2X,11F10.4)
Y = 0.0
IF (fid)-. 0001) 5,50,50
10 STOP
EMD
25
-------
APPENDIX C
PROGRAM FOR CALCULATION
DIMENSION A(16),B(16),SUO),TEMP<10),STOR 009
STOR(1)=STOR(1)«A(J) 010
30 CONTINUE Oil
U1=TEMP(1)/STOR(1) 012
IF( 1-1)45,35,45 013
35 S(1)=B(1)«U1 014
WRITE (3,40) S(l) 015
40 FORMAT (•!'////' S(l> = «,F12.2///>
GO TO 41 017
45 TEMPI2I-1.0 017A
STCR(2)=1.0 01TB
DO 65 J = 1,N 018
IFU-J)SO,65,50 019
iO STOR(2)=STOR(2)»A(J) 020
TEMP(2) = TEMP(2)«(A(JI-AI I ) ) 021'
65 CONTINUE 022
U2=STOR(2)»SU)/TEMP(2) 023
IF(I-2)80,70t80 024
?0 S(2)=(B(2)-U2)«Ul 025
WRITE (3,75) S(2) 026
75 FORMAT (' S(2) = SF12.2///) 027
GO TO 41 02B
00 TEMP(3)=1.0 028A
STOR(3)=1.00 028B
DO 100 J=2,N 029
IF(I-JI85,100,85 030
85 STOR(3)=STCR(3)»A(J) 031
TEMP(3)=TEMP(3)*(A(JI-AII» 032
100 CONTINUE 033
U3=STOR(3)»S(2)/TEKP(3) 034
IF(1-3)115,105,115 035
105 S(3)=(B(3)-U2-U3)»U1 v 036
WRITE (3,110) S(3) 037
110 FORMAT (' SO) = ',F12.2///J 038
GO TO 41 039
115 TEMP(4) = UO 039A
STOR(4)=1.0 ' , 039B
DO 135 J=3,N 040
IF( I-JU20, 135,120 041
120 STCR(4)»STCR(4)»A(J) 042
TeMP(4)=TffMP(4)*(A( J)-A( I I ) 043
135 CONTINUE 044
U4=STOR(4)»S(3)/TEMP(4) 045
IK 1-4)150,140,150 046
140 S(4)=(B(4)-U2-U3-U4)«U1 047
WRITE (3,145) S(4) 048
145 FORMAT (' S(4) = SF12.2///) 049
GO TO 41 050
150 TfMP(5)=1.0 050A
STCR(5)=1.0 050B
26
-------
DO 170 J = 4,N
IHI-J>155,170,155 052
155 STOR(5)=STOR(5)»A(J) ot,3
TEMP<5»=TEMP(5)«(A(J)-AU)) 054
170 CONTINUE Ot,5
U5=STOR(5)«S(4)/TEMP<5) 056
IF(1-5)185,175,185 057
175 S(5)=(B(5>-U2-U3-U4-U5)«U1 058
WHITE (3,180) SI5) 059
180 FORMAT (• S(5) = «,F12.2///) 060
GO TO 41 061
185 TEMP(6)=i.O 061A
STOft(6)=1.0 061B
00 205 J=5,N 062
IF(I-J)190,205,190 063
190 STOK(6)=STOR<6J»A(J) 064
TFMP(6> = TeMP(6)«U( J)-A< I » 065
205 CONTINUE 066
U6=STOR(6)«S(5)/TFMP(6I 067
IF(I-6)220,210,220 068
210 S(6)=(8I6)-U2-U3-U4-U5-U6)»U1 069
WRITE 13,215) S(6) 070
215 FLRWAT I' S(6) = ',F12.2///) 071
GO TO 41 0/2
220 TEMP(7)=1.0 072A
. STOR(7)=1.0 072B
DO 240 J=6,N 073
IF(I-J)225,240,225 074
225 STOR(7)=STOR(7)«A(J) 075
TEMP(7)=TEfP(7)*(A(J)-A(I)) 076
240 CONTINUE 077
U7=STOR(7)«S(6)/TEMP(7) 078
IF( 1-7)255,245,255 079
245 S(7)=(B(7)-U2-U3-U4-U5-U6-U7)»U1 000
HR1TE (3,250) 5(7) 081
250 FORMAT (• S<7) = SF12.2///) 082
GO TO 41 083
255 TEMP(8)=1.0 083A
STOR(8)=1.0 083B
00 275 J=7,N 084
IF(I-J)260,275,260 085
260 STCR(8)aSTOR<6)»A(J) 086
TEMP(8)=TEMP(8)»(A( Jl-AI I ) ) 087
275 CONTINUE 088
U8=STOR(8)«S(7)/TEMP(8) 089
IF(I-8)290,280,290 090
280 S(8)=(B(8)-U2-U3-U4-U5-U6-U7-U8)«Ul 091
WRITE (3,285) S(8) 092
285 FORMAT (' S(8) = «,F12.2///) 093
GO TO 41 oq«
290 TEMP(9)=1.0 094A
STOR(9)=1.0 094B
DO 300 J=8,N °95
IKI-J)295,300,295 °96
295 STOR(9)=STOR(9)»A(J) °97
TEMP(9)=TEMP(9)«(A(J)-A(I)) 098
300 CONTINUE °"
U9=STOR(9)«S(8)/TEMP(9) 100
IF(I-9)315,305,315 l°l
305 S(9)=(B(9)-U2-U3-U4-U5-U6-U7-U8-U9)«U1 102
WRITE (3,310) S(9) l03
27
-------
'HO FORMAT (' S(9) = ',F12.2///) 104
GC TO 41 105
315 TEMP(10>=1.0 105A
STOR(10)*1.0 10!>8
DO 325 J=9,N 106
IFU-J)320,325,320 107
320 STORI10)=STOR(10)»A(J) 100
TfcMP(10)=TEMP«lO)*(A(J)-AU)) 1°9
325 CONTINUE HO
U10 = STOR(10)»S(9)/TEMP(10) HI
IF( 1-10)41,330,41 112
330 S(10)»(B(10)-U2-U3-U4-U5-U6-U7-U8-U9-UIO)»UI 113
hRITE (3,335) S(IO) 114
335 FORMAT (' S(10) = ',F12.2///1 US
41 CONTINUE 116
GO TO 2 H6A
42 STOP 117
END 118
28
-------
APPENDIX D
STEPWISE REGRESSION PROGRAM
c 1130 STEPWISE MULTIPLE DEGRESSION PROGRAM, 3/14/66 0010
C PHASES 1 AND 2 CAN. BE OVERLAID TO CONStRVE CORE. THE STtPS TO 0020
C READY PHASES \ AND 2 FOR OVERLAY ARE 0030
C 1. SET UP A COMMON AKEA CONSISTING OF R U , X6AR, S IGMA.F IN, 0040
C FQUT, DBS, NVAR, NOBS, NINDV, IRES, IFA. 0050
C 2. SET SIGMA ANO DATA EQUIVALENT IN PHASE 2. 0060
C 3. REPEAT PHASE 1 DEFINE FILE STATEMENT IN PHASE 2, 0070
C 4. REMOVE STATEMENT 101-3 FROM PHASE 1 AND INSERT IT 0080
C BEHIND DIMENSION COMMENTS CARD IN PHASE 2. 0090
C PHASE 1. TRANSFORM ORIGINAL DATA, COMPUTE AND PRINT MEANSt 0100
C STANDARD OEV I AT lOiSiS , ANQ SIMPLE CORRELATION CUEFFIC IENTS. 0110
C DIMENSIONS 0120
IMPLICIT REAL*8(Ai-H,C-Z)
DIMENSION DATA(30),CONST(12), ITRANI 30 I , JTRAN< 30 ) ,KTRAN< 30 > ,LTRAN( 3 0130
10) 0140
DIMENSION RIJ(30,30),XBAR(30>, SIGMA I 30 ) , AIO( 18 ) 0150
DIMENSION SIGB(30) ,8(30) , 10(30) 0160
C EQUIVALENCES 0170
EQUIVALENCE < S IGMAC 1 ) .DATA (1) ) 0180
C DEFINE DATA FILE 0190
DEFINE, FILE 10 ( 1000, 60, U, IFA )
C STATEMENT LABEL 101 IS NOT REFERENCED. IT MARKS THE FIRST 0210
C EXECUTABLE STATEMENT OF THE SOURCE PROGRAM. 0220
C ICCM IS FIXED DECIMAL REPRESENTATION OF ALPHABETIC COMKA. 0230
101 ICCM=27456 0240
C INITIALIZE DATA FILE 0250
IFA=l 0260
C READ 1.0. °270
READ (U It ENO=999) ( AID ( I ) , 1*1 , 18 )
1 FORMAT! 18A4)
C REAP CONTROL CARD 0300
REAOU,2)NVIN,NVAR,NOBS,NTRAN,NCCNS,FIN,FOUT,IRES 0310
2 FORMAT(2I2, 14, 2 12* 2F6. 3, 1 1 )
IF(FIN-FOUT)1020,690,690 331
690 IF(NTRAN)100Q|730,700 0340
C ' READ TRANSFORMATION CARDS 035°
700 READ(l,71HITRANm,,JTRANm,KTRANm,LTRANm,l = l,NTRAN> 0360
71 FQRMAT(36I2) 037°
IF(NCONS) 1000, 730, 720 038°
C READ CONSTANT CARD °39?
720 READ(1,72MCCNST
-------
C X(JI=X(K) 0610
760 DATA(JJ)=DATA(KK) 0620
GO TO 850 0630
C X(J)=-X(K) 0640
HO DATA!JJ)=-OATA!KK) 0650
GO TO 850 0660
C X(J)=LOG X(K) 0670
780 DATA(JJ)=DLOG(DATA(KK)) 0680
GO TO 850 0690
C X(J)=1/X(K) • 0700
790 DATA(JJ)=1.0/CATA(KK) 0710
GO TO 850 0720
C X( J)=X!K)+X!L) 0730
800 DATA!JJ)=OATA(KK)+DATA(LL) 0740
GC TO 850 0750
C X(JI=X(K)*X(l) 0760
810 DATA!JJ)=DATA(KK)»DATA(LU 0770
GO TO 850 0780
C X(JJ=X(K)/X(L) 0790
820 DATA!JJ)=DATA(KK)/DATA!LL) 0800
GO TO 850 0810
C XI J)=X(K)+C 0820
(330 DATA( JJ )=DATA ! KK)+CCNST
-------
...M,
C feRF°RK STEPWISE CALCULATIONS AND PRINT RESULTS. 1230
C INITIALIZE
DO 190 I=1,NVAR
SIGB(I)=0.0
190 Bii)=o.o
NENT=0
OF-OBS-l.O
NSTEP=-i
C TRANSFORM SIGMA VECTOR FROM STANDARD DEVIATIONS TO SQUARE 1320
C ROOTS OF SUMS CF SQUARES. 1330
DC 310 I=1,NVAR 1340
310 SIGMAl I > = SIGMA( I ) « ( CBS-1 .0 ) •«. 5 1350
C BEGIN STEP NUMBER NSTEP. 1360
i!00 NSTEP=NSTEP+1 1370
STDEE=(
-------
RSQP = RSC » 100.
WRITE(3,59) RSCP
59 FORMAT!' PERCENT VARIATION EXPLAINED R-SQ = ' , F 15. 3 ) 1773
CRSQ = !.-(( l.-RSQ)«(CBS-l.))/(OBS-DEPV-1.)
CRSGP = CRSG » 100.
WRITE(3,84) CRSGP
84 FORMAT!1 CORRECTED R-SG AS A PERCENT*',F20.3)
IDFN=OBS-DF-2.0 1800
IDFD=DF+1.0 1810
F=(SIGMA(NVAR)«*2-(STDEE»*2)*!DF+1.0))/((OBS-DF-2.0)*STOEE«»2) 1820
WRITE(3,66)IDFN,IDFD,F 1830
66 FORMAT!' GOODNESS OF FIT OR OVERALL F,F(',I 3,',',I 3,') = ',F8.3) 1840
WRITE(3,60)BSUEC 1850
60 FORMAT!* CONSTANT TFRM=•,I8X,F16.8) 1-860
WRITE(3,61) 1870
61 FORMAT!'OVAR COEFF STD OEV T VALUE' 1880
1) 1881
WRITE<3,62> 1890
62 FORMAT!• COEFF*) 1900
DO 430 1=1,NIN 1910
J=IO(I) 1920
T = L« D/SIGB! I ) 1930
WRITE(3,63)ID( I ),B(I),SIGB(I),T 1940
63 FORMAT!' • , I 3 , F18.8,F20.8,F 18.8) 1950
430 CONTINUE
C COMPUTE F LEVEL FOR MINIMUM VARIANCE CONTRIBUTION VARIABLE 1960
C IN REGRESSION EQUATION. 1970
FLEVL = VMIN«DF/RU(NVAR,NVAR) I960
IF(FOUT + t-LEVL)460,460,450 1990
C INITIALIZE FOR REMOVAL OF VARIABLE K FROM EQUATION. 2COO
4-iO K=NMIN 2010
NENT=0 2020
DF=OF+2.0 2030
GO TO 500 2040
C COMPUTE F LEVEL FOR MAXIMUM VARIANCE CONTRIBUTION VARIABLE 2050
C NOT IN EQUATION. 2060
460 FLEVL=VMAX«DF/(RIJ(NVAR,NVAR)-VMAX) 2070
IF(FLEVL-FIN)600,600,470 2080
C INITIALIZE FOR ENTRY OF VARIABLE K INTO EQUATION. 2090
470 K=NMAX 2100
NENT=K 2110
GO TO 500 2120
C OUTPUT FOR VARIABLE DELETED 2130
480 WRITE(3,64)NSTEP,K 2140
64 FORMAT!'OSTEP NUMBER ', 12 , 10X,'DELETE VARIABLE ',12) 2150
GO TO 425 2160
C UPDATE MATRIX 2170
bOO DC 540 I=1,NVAR 2180
IF! I-K)510,540,510 2190
510 DO 530 J=1,NVAR • 2200
IF(J-K)520,530,520 2210
520 RIJ(I,J)«RIJ(I,J)-RIJ(I,K)*RIJ(K,J)/RIJ 2310
32
-------
5BO CONTINUE
RIJ(K,K)=1.0/RIJIK,K)
GO TO 200
600 IF!IRES)610,640,610
PRINT RESIDUALS
blO IFA=1
lriRITE(3,67)
67 FORMAT!'0 OBS ACTUAL
WRITE(3,69)
69 FORMAT! •
DO 630 K=1,NG6S
READ!10'I FA){DATA!I),I = 1,NVAR)
EST=BSUBO
DO '620 1 = 1,NIN
J=ID(I)
620 EST = EST + B( I)»DATA(J)
RESID = DATAdMVAR)-EST
XNORD = RESID/STDEE
IF»DA6SlXNORO)-3.191,92,92
ESTIMATE
RESIDUAL
NORMAL')
DEVIATE')
91
92
30
94
31
93
68
630
C
640
999
C
C
1000
C
C
C
1010
C
C
1020
IF! DABS (XNORD 1-2. 193,94,94
WRITE(3,30)K,DATA(NVAR) , EST , RES ID, XNORD
FORMAT!' ' ,I4,4F12.2,« ««')
GO TO 630
WRITE!3,31)K,DATA(NVAR),EST,RESID,XNORD
FORMAT!' ' ,I4,4F12.2, ' »')
GO TO 630
WRITE(3,68)K,DATA«NVAR) , EST , RES ID, XNORD
FORMAT!' »,I4,4F12.2)
CONTINUE
NORMAL END OF JOB
GO TO 101
CALL EXIT
ERROR. NIN, NENT, VMIN, NCONS, OR NTRANS IS NEGATI
FOR CONTROL CARD ERROR.
STOP1
ERROR DEGREES CF FREEDOM =0. EITHER ADD MORE DATA
VE. CHECK
OBSERVATIONS
OR
DELETE ONE OR MCRE INDEPENDENT VARIABLES. SAMPLE SIZE MUST EXCEED
NUMBER OF INDEPENDENT VARIABLES BY AT LEAST 2.
STOP2
ERROR. F LEVEL FOR INCOMING VARIABLE IS LESS THAN
OUTGOING VARIABLE.
STOP4
END
F LEVEL FCR
2320
2330
2340
2350
2360
2370
2380
2390
2391
2392
2400
2410
2420
2430
2440
2450
2460
2461
2470
2471
2480
2481
2482
2483
2484
2485
2486
2487
2490
2500
2501
2520
2530
2550
2560
2570
2580
2590
2600
2610
33
-------
REFERENCES
1. Berg, G., "Virus Transmission by the Water Vehicle,"
I. Viruses, Health Lab Science, 3:86 (1966).
2. Berg, G., "Virus Transmission by the Water Vehicle,"
III. Removal of Viruses by Water Treatment Procedures,
Health Lab Science, 3:170 (1966).
3. Chick, H., "An Investigation of the Laws of Disinfection,"
Journal of Hygiene, 8:92 (1908).
4. Berg, G.; Clark, R. M.; Berman, D.; and Chang, S. L. ,
"Aberrations in Survival Curves," Transmission of Viruses
by the Water Route, Interscience Publishers, a division
of John Wiley and Sons, New York, New York, pp. 235-240
(1967) .
5. Clark, R. M., and Niehaus, J. F., "A Mathematical Model
for Viral Devitalization," Transmission of Viruses by
the Water Route, Interscience Publishers,a division of
John Wiley and Sons, New York, New York, pp. 241-245 (1967).
6. Clark, R. M., "A Mathematical Model of the Kinetics of
Viral Devitalization," Mathematical Biosciences 2, pp. 413-423
(1968) .
7. Willers, A., FR, Practical Analysis, Dover, New York (1948).
8. Hildebrand, F. B., Introduction to Numerical Analysis,
McGraw-Hill, New York, New York (1956).
9. Draper, N., and Smith, H., Applied Regression Analysis,
John Wiley and Sons, New York, New York (1967).
10. Sharp, G. D., "Electron Microscopy and Viral Particle
Function," Transmission of Viruses by the Water Route,
Interscience Publishers, a division of John Wiley and
Sons, New York, New York, pp. 193-217 (19671.
34
-------
TECHNICAL REPORT DATA
(Pease read Instructions on the reverse before completing)
. REPORT NO.
EPA-670/2-74-067
2.
3. RECIPIENT'S ACCESSION'NO.
4. TITLE AND SUBTITLE
A MATHEMATICAL ANALYSIS OF THE KINETICS
OF VIRAL INACTIVATION
5. REPORT DATE
August 1974; Issuing Date
6. PERFORMING ORGANIZATION CODE
'. AUTHOR(S) ~~ ~~~~'
Robert M. Clark, Betty Lou Grupenhoff,
and George C. Kent
8. PERFORMING ORGANIZATION REPORT NO.
9. PERFORMING ORG \NIZATION NAME AND ADDRESS
National Environmental Research Center
Office of Research and Development
U.S. Environmental Protection Agency
Cincinnati, Ohio 45268
1O. PROGRAM ELEMENT NO. T.CB047 '
ROAP 21AQE; Task 10
11. CONTRACT/GRANT NO.
12. SPONSORING AGENCY NAME AND ADDRESS
Same as above
13. TYPE OF REPORT AND PERIOD COVERED
14. SPONSORING AGENCY CODE
15. SUPPLEMENTARY NOTES
16. ABSTRACT •
Pathogenic enteric viruses transmitted via the water route present
a potential hazard to public health because of their resistance to
natural or artificial disinfection mechanisms. Of constant concern
to public health officials is the ability of viruses to pass through
water treatment plants. Therefore, many research investigations have
been directed toward the study of the inactivation of viruses and
enteric organisms. This report describes a mathematical model which
can be used to characterize the response of viruses to a disinfecting
agent. Not only is the model presented, but a technique is described
which can be used to estimate the model's parameters. Both the model
and the estimation technique are being used to analyze experimental
information resulting from disinfection studies.
17.
KEY WORDS AND DOCUMENT ANALYSIS
DESCRIPTORS
b.lDENTIFIERS/OPEN ENDED TERMS
c. COS AT I Field/Group
Computation, Computers, *Mathe-
matical models, *Viruses, Linear
regression, Disinfection, *Electron
microscopy
Exponential decay,
*Inactivation
12A
13B
8. DISTRIBUTION STATEMENT
19. SECURITY CLASS (ThisReport)'
UNCLASSIFIED
21. NO. OF PAGES
39
RELEASE TO PUBLIC
20. SECURITY CLASS (This page)
UNCLASSIFIED
22. PRICE
EPA Form 2220-1 (9-73)
VS. GOVERNMENT PRINTING OFFICE: 1974- 657-049/1025
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