Distribution of Incubation Period of COVID-19 in the

Canadian Context: Modeling and Computational

Study
Subhendu Paul1,* and Emmanuel Lorin1,2
1 Schoolof Mathematics and Statistics, Carleton University, Ottawa, K1S 5B6, Canada
2 Centrede Recherches Mathématiques, Université de Montréal, Montréal, H3T 1J4, Canada
* subhendu.paul@carleton.ca

ABSTRACT

We propose a novel model based on a set of coupled delay differential equations with fourteen delays in order to accurately
estimate the incubation period of COVID-19, employing publicly available data of confirmed corona cases. In this goal, we
separate the total cases into fourteen groups for the corresponding fourteen incubation periods. The estimated mean incubation
period we obtain is 6.74 days (95% Confidence Interval(CI): 6.35 to 7.13), and the 90th percentile is 11.64 days (95% CI: 11.22
to 12.17), corresponding to a good agreement with statistical supported studies. This model provides an almost zero-cost
approach to estimate the incubation period.

Introduction
The outbreak of coronavirus disease 2019 (COVID-19), first appeared in Wuhan (China) and spread around the world1 , is
creating dramatic and daily changes with profound impacts worldwide. People with underlying medical condition, respiratory
disease, diabetes, cancer etc., and older people are vulnerable to severe complications and death from coronavirus, although we
are discovering new features of COVID-19 every day. In the absence of vaccination and proper medication, we can merely
obey some non-pharmaceutical containments, lockdown, social distancing, hand hygiene, face masking, mobile app to trace
corona-positive individuals, to help prevent the spread of the this infectious disease. To minimize transmission of the virus
through human-to-human interaction, quarantine of individuals with exposure to infectious pathogen is an effective strategy for
containing contagious diseases. To govern a quarantine period for asymptomatic as well as presymptomatic individuals, it is
essential to fully estimate the incubation period of COVID-19 disease.
The incubation period of an infectious disease is the time between when a person is infected by a virus and when the first
symptoms of the disease are noticed. Estimates of the incubation period for COVID-19 range from 2 to 14 days, according to
the investigation so far. Precise knowledge of the incubation period is crucial to control infectious disease like COVID-19; a
long incubation period means a high risk of further spreading the disease. The distribution of the incubation period can be
used to estimate the basic reproduction number R0 , a key factor of epidemics, in order to measure the potential for disease
transmission. It is indeed difficult to obtain a good estimate of the incubation period on the basis of limited data.
There are several statistical studies2–12 , using a single measure11 , estimating the incubation period of the current pandemic.
In addition to those statistical approaches, there are numerous analytical and computational studies based on mathematical
models, involving Ordinary Differential Equations (ODE)13–17 as well as Delay Differential Equations (DDE)18–23 , to calculate
the basic reproduction number R0 and understand the underlying dynamics of the epidemic. Researchers usually consider a
single delay models, occasionally two delays.
To the best of our knowledge, we are proposing for the first time a mathematical model, comprising fourteen delays, to
estimate the incubation period utilizing publicly available data of the total number of corona-positive cases. This approach
is free from any special type of samples in order to produce the distribution of the incubation period. It is then almost cost
free, as it only involves a small scale computations. After a single calculation employing this method, we can generate the
current distribution as well as previous distributions of the incubation period. We can also observe the change in the incubation
period. In the statistical based approach, it is usually difficult to consider a large incubation period if the sample size is small.
However, in this approach, we can go well beyond 14 days, the incubation period we have set for the current work. In this
context, we demonstrate the incubation period of the COVID-19 epidemic in Canada employing publicly available data of
confirmed corona-positive cases24 . As of November 7, 2020, the World Health Organization (WHO) had confirmed a total of
251,338 cases of COVID-19 in Canada, including 10,381 deaths1 .
Figure 1. Estimation of the total number of confirmed coronavirus cases (T ) compared to the available data24 . Here blue
circles indicates the results obtained from model, and red line is the publicly available data24 .

There are several studies on incubation period mainly based on Chinese patients that can only provide a rough estimation
for rest of the world. The incubation period may depends on age25 (median-age / country), hard immunity, public health
system, corona testing capacities, daily corona cases, etc. For a better estimation of the incubation period for a particular region,
we need to study local patients. Data collection is a bottleneck in studying the incubation period. However, one can easily
estimate the incubation period using the approach we propose and publicly available data of confirmed cases.

Results and Discussion

After estimating the model parameters with sufficiently small values of the error functions, we compare in Figure 1 total corona
cases calculated with our model and the available data24 . This shows excellent agreement between the model results and the
data. In Figure 2, the confirmed cases 211,735 of 276 days are divided into fourteen groups. The ith compartment Ti , defined
in Equation (3), is the confirmed cases of 276 days corresponding to the incubation period of i day(s) for i = 1, 2, · · · , 14. In
addition Ti is the frequency of the incubation period of i day(s), and using the bar chat, we obtain a mean incubation period
of 6.89 days, a median of the incubation period of 6 days, 90th percentile of 11 days, 95th percentile of 12 days and 99th
percentile of 13.5 days. The bar chat shows that mode of the incubation period is of 6 days, and there is a second peak for
the incubation period at 10 days. However, the second peak is strongly dominated by the first. From the bar chat presented in
Figure 2, we can also obtain the probability densities of incubation period of the first k days during the epidemic, thanks to
(k)
the total confirmed cases of the first k days starting on January 22, 2020. The probability densities pi of the first k days and
corresponding incubation period i days for i = 1, 2, · · · , 14 can be defined as
(k)
(k) Ti
pi = (k)
, (1)
∑14
i=1 Ti

(k) (100) (200) (276)

where Ti s are defined in Equation (3). The probability densities pi , pi and pi for i = 1, 2, · · · , 14 are presented in
Figure 3. The density curves for the first 100, 200 and 276 days are similar, and the densities obey a remarkable configuration:
(100) (200) (276) (100) (200) (276)
for 2 ≤ i ≤ 5, pi > pi > pi , for 7 ≤ i ≤ 9 and 11 ≤ i ≤ 14 , pi < pi < pi ; although the second peak, for i
(200) (100) (276)
= 10, does not follow the decreasing-increasing convention. For the incubation period of 10 days p10 > p10 > p10 , an

2/10
Figure 2. The cumulative data of confirmed corona cases as of October 23, 2020 is splitted into several incubation periods.

Figure 3. Probability densities of incubation period, presented in Equation (1). The ’first 100 days’ indicates that the density
of incubation period based on the cumulative data of the first 100 days during the epidemic starting from January 22, 2020 and
similar for other two.

3/10
Figure 4. Probability density function of the lognormal distribution of the incubation period with µ = 1.79 and σ = 0.52. The
result based on the total confirmed corona cases of 276 days. The blue circles indicate the densities obtained from the model
calculation.

Figure 5. Probability density function of the lognormal distribution of the incubation period with µ = 1.83 and σ = 0.53. The
result based on the confirmed corona cases of a particular day, October 23, 2020. The blue circles indicate the densities
obtained from the model calculation.

4/10
 Table 1. A list of several studies along with sample size, mean and lognormal parameters µ, σ .

Author Data size Mean (days) µ σ

Backer et al.2 88 6.4 1.796 0.349
Lauer et al.3 181 5.5 1.621 0.418
Li et al.4 10 5.2 1.425 0.669
Bi et al.5 183 4.8 1.570 0.650
Jiang et al.6 40 4.9 1.530 0.464
Linton et al.7 158 5.6 1.611 0.472
Zhang et al.8 49 5.2 1.540 0.470
Ma et al.9 587 7.4 1.857 0.547
Leung et al.10 61 7.2 1.780 0.680
McAloon et al.11 Meta 5.8 1.63 0.50
Jing et al.12 1084 8.29
Math Model (raw data) 211,735 6.89
Math Model (lognormal) 211,735 6.7 1.788 0.520
Math Model (raw data) 2,587 7.17
Math Model (lognormal) 2,587 7.0 1.827 0.528

oscillatory behaviour is observed. The descending pattern for 2 ≤ i ≤ 5 and the ascending order for 7 ≤ i ≤ 9 and 11 ≤ i ≤ 14
indicate that the mean incubation period is rising. Now, we fit the frequency data for the first 276 days, presented in the bar chat
Figure 2, with the lognormal distribution function and obtain the lognormal distribution parameters µ = 1.79 and σ = 0.52.
Figure 4 shows the lognormal distribution function of the incubation period of the first 276 days and population size 211,735
(276)
with pi for i = 1, 2, · · · , 14. The estimated incubation period, obtained using lognormal distribution, has a mean of 6.74
(95% CI: 6.35 to 7.13), and the 90th percentile is 11.64 days (95% CI : 11.22 to 12.17). In addition, we focus on the distribution
of the incubation period for a single day, October 23, 2020 which is the 276th day of the epidemic, with 2258 confirmed cases.
(k)
The probability density p̂i for a single day can be calculated as

(k) (k−1)
(k) Ti − Ti
p̂i = (k) (k−1)
, (2)
∑14
i=1 Ti − ∑14
i=1 Ti

(k)
where Ti is defined in Equation (3). The estimated incubation period, obtained from frequency table of 276th day and
population size of 2258, has a mean of 7.14 days, a median of 7 days, the 90th percentile of 11 days, 95th percentile of 12.5
days and 99th percentile of 14 days. We generate the lognormal distribution function from the 276th day’s frequency data
and obtain the lognormal distribution parameters µ = 1.83 and σ = 0.53. Figure 5 shows the lognormal distribution of 276th
(276)
day along with p̂i . The estimated incubation period of 276th day, population size of 2258 and obtained using a lognormal
distribution, has a mean of 6.98 days (95% CI: 6.41 to 7.55) and the 90th percentile of 12.29 days. A list of several studies
along with the present calculation “Math.-Model” are presented in Table 1; we present the mean incubation period for the raw
data as well as the lognormal distribution function. The list shows that our calculated data are closed the value reported by Ma
et. al.9 using a larger sample size of 587. The calculated mean incubation period using two different ways, the raw data as
well as the lognormal distribution are indeed closed, indicating that the raw data calculated using our mathematical model,
are statistically significant for a lognormal distribution (statistical p value less than 0.001). It follows from the “Math.-Model”
calculation, presented in Table 1, that the mean incubation period of 276th day, population size 2258, is greater than the mean
incubation period of 276 days, population size 221,735 which demonstrates that the mean incubation period of COVID-19 is
slightly increasing with time.

Methods
In this section, we introduce a compartment based infectious disease model including a total of seventeen partitions, Lockdown,
Susceptible, Infected and fourteen compartments of Total confirmed cases (LSIT). The model is constructed as a set of coupled
delay differential equations involving several variables and parameters.

5/10
The Model
Modeling the spread of epidemics is an essential tool for projecting its outcome. By estimating important epidemiological
parameters using the available database, we can make forecasts of different intervention scenarios. In the context of compartment
based model, where the population of a region is distributed into several population groups, such as susceptible, infected, total
cases etc., is a simple but useful tool to demonstrate the panorama of an epidemics.
In this article, we introduce a infectious disease model, extending the standard SIR model, including the phenomenon
lockdown, a non-pharmaceutical way to prevent the spread of the epidemics. The schematic diagram of the model is presented
in Figure 6 with several compartments and various model parameters. The following are the underlying principles of the present
model.

• The total population is constant (neglecting the migrations, births and unrelated deaths) and initially every individual is
assumed susceptible to contract the disease.

• The disease is spread through the direct (face-to-face meeting) or indirect (through air current, common used or delivery
items like door handles, grocery products) contact of susceptible individuals with the infective individuals.

• The quarantined area or the compartment for corona cases contains only members of the infected population who are
tested corona-positive.

• The virus always kills some percent of the people it infects; the survivors percent represents the recovered group.

• There is a non-pharmaceutical policy (stay at home), commonly known as lockdown, to stop the spread of the disease.

Based on the above principles, we consider several compartments:

• Susceptible (S): the group of individuals who can be infected.

• Infected (I): the group of people who are spreading the contiguous disease.

• Total cases (T ): the group of individuals who tested corona-positive (Active cases + Recovered + Deaths).

• Lockdown (insusceptible) (L): the group of persons who are keeping themselves safe.

The goal of the present model is to estimate the distribution of the incubation period of COVID-19. In this goal, we split the
compartment T into J subcomponents T1 , · · · , TJ , where
J J
(k)
T (t) = ∑ Ti (t) or T (k) = ∑ Ti . (3)
i=1 i=1

(k)
In (3) k represents the time index and Ti represents the total corona-positive cases corresponding the incubation period τi ,
presented in Figure 6. The time-dependent model is the following set of coupled delay differential equations:


dS SI
= −β (t) − α(t)S + ν(t)L ,




 dt N
S(t − τi )I(t − τi )

 dI SI
= β (t) − ∑Ji=1 δi (t)β (t)


 ,
dt N N (4)
 dTi S(t − τi )I(t − τi )

 = δi (t)β (t) ,


 dt N
 dL


= α(t)S − ν(t)L ,

dt
where α(t), β (t), δi (t), for i = 1, · · · , J and ν(t) are real positive parameters respectively modeling the rate of lockdown, the
rate of infection, the rate of tested corona-positive corresponding the incubation period τi and the rate of ignoring lockdown,
respectively. It follows from (4), that for any t

L(t) + S(t) + I(t) + T (t) = N , (5)

where N (constant) is the total population size.

6/10
 Figure 6. Schematic diagram of the compartmental based epidemic model, presented in Equation 2.

We solve (4) using matlab inner-embedded program dde23 with particular sets of model parameters. To solve the initial
value problem (4), in the interval [t0 ,t1 ], we consider L(t0 ), S(t0 ), I(t0 ) and T (t0 ) as follows:

 L(t0 ) = 0 ,

 S(t0 ) = N − L(t0 ) − I(t0 ) − T (t0 ) ,
(6)

 I(t0 ) = q ,
T (t0 ) = Te(t0 ) ,


where Te(t0 ) is the available data at time t0 , and q is the initial value adjusting parameters. Initially, there is no lockdown
individual so that we can consider L(t0 ) = 0.

Parameter estimation of the model

We focus on the exponential growth phase of the COVID-19 epidemic in Canada; one can use the approach to estimate the
incubation period distribution for any region affected by the infectious disease. The time resolved (daily updated) database24
provides the number of total corona-positive cases. The optimal values of p(t) = (q, α(t), β (t), δ1 (t), · · · , δJ (t), ν)T , that is the
set of initial values and model parameters, is obtained by minimizing the root mean square error function E(p(t)), defined as
s
1 M
E(p(t)) =
M ∑ (T (k) (p(t)) − Te(k) )2 , (7)
k=1

where Te(k) is the available data of total corona-positive cases on the particular kth day, and T (k) is the calculated results
obtained from System (4). The integer M, used in (9), is the size of the data set. Due to the complexity of the error function, the
minimization using the matlab function fminsearch requires a very large number of iterations.

Numerical experiment
In this section, we propose a detailed description of the computational procedure for the proposed model. On 23 January 2020,
a 56-year old man admitted to Toronto hospital emergency department in Toronto with a new onset of fever and nonproductive
cough, and returning from Wuhan, China, the day prior26, 27 . It is believed this is the first confirmed case of 2019-nCoV in
Canada, and according to the government report, the novel coronavirus arrived on the Canadian coast on January 25, 2020, first
reported case. The above information suggests that the start date of the current pandemic in Canada is possibly xsto be January
22, 2020. Additionally, some research studies reported that the estimation of the incubation period of COVID-19 is from 2 to
14 days1, 28 . As a consequence, in the present study we consider 14 delays, τ1 = 1 day, τ2 = 2 days, · · · , τ14 = 14 days. Here we
consider a calculation of 276 days, from January 22, 2020 to October 23, 2020. We decompose the time domain of 276 days
into two parts : the time domain splitter is in the interval where the first wave is slowed down and the “second wave” begins, i.e.
the splitter is in the interphase of two different scenarios. In this goal, we can choose the parameters p(t) as
p(t) = p(1) from January 22, 2020 to July 19, 2020 ,

(8)
p(t) = p(2) from July 20, 2020 to October 23, 2020 ,

7/10
 Table 2. The estimated values of the model parameters for two different time domains defined in Equation 6.

Parameters Estimated value Parameters Estimated value

α (1) 0.01376 α (2) 0.00102
β (1) 0.44952 β (2) 0.73348
(1) (2)
δ1 0.01331 δ1 0.02180
(1) (2)
δ2 0.07014 δ2 0.02454
(1) (2)
δ3 0.07104 δ3 0.04129
(1) (2)
δ4 0.09705 δ4 0.05931
(1) (2)
δ5 0.15866 δ5 0.11374
(1) (2)
δ6 0.13964 δ6 0.14658
(1) (2)
δ7 0.06694 δ7 0.13907
(1) (2)
δ8 0.08066 δ8 0.10781
(1) (2)
δ9 0.08007 δ9 0.08989
(1) (2)
δ10 0.12072 δ10 0.06755
(1) (2)
δ11 0.04584 δ11 0.05151
(1) (2)
δ12 0.02497 δ12 0.03834
(1) (2)
δ13 0.00901 δ13 0.03833
(1) (2)
δ14 0.00427 δ14 0.02056
ν (1) 0.00114 ν (2) 0.00085

(1) (1) (2) (2)

where p(1) = (q, α (1) , β (1) , δ1 , · · · , δ14 , ν (1) )T and p(2) = (α (2) , β (2) , δ1 , · · · , δ14 , ν (2) )T are some constants. The ca-
pability of an optimization package depends on the initial values of the parameters: for q, α, β , ν we consider any positive
random number less than unity, where as a choice of δ = (δ1 , · · · , δ14 )T is tricky. For this purpose, we consider a vector of
14 positive random numbers δ such that δ1 < · · · < δ4 > δ5 > δ6 > · · · > δ14 and ∑14 i=1 δi = 0.9. We observe, from numerous
numerical experiments, the renormalization factor 0.9 works perfectly for the computation.
For a complete calculation, we run the matlab code twice. Firstly, we run the code for the period January 22, 2020 to July
(1) (1)
19, 2020 to obtain the estimated value pest of p(1) , presented in Table 2, and the value of the error function E(pest ) = 54.92.
(1)
Then, we run the code for the entire period from January 22, 2020 to October 23, 2020, but using the estimated value pest for
(2)
the interval January 22, 2020 to July 19, 2020, and obtain the estimated value pest of p(2) for the rest of the period, presented in
(1) (2)
Table 2, and the value of the error function E(pest , pest ) = 48.38, defined as
v
1 u M1 M
u
(1) (2) (1) (2)
E(pest , pest ) = t ∑ (T (k) (pest ) − Te(k) )2 + ∑ (T (k) (pest ) − Te(k) )2 , (9)
M k=1 k=M +1 1

where M1 corresponds the date July 19, 2020.

Conclusion
In this paper, we have derived a mathematical model based on a set of coupled delay differential equations, which was used to
estimate the incubation period with good agreement with statistical works. Using the proposed model and publicly available
data of confirmed cases, one could accurately estimate the incubation period in any region. We obtain the distribution of the
incubation period from the population, so that it is better than any sample-dependent result. We have considered fourteen
delays, but it is possible to consider an arbitrary number of delays. After estimating the model parameters, one can estimate the
incubation period of confirmed cases over a long period, over a small time interval, and even over a single day. The present
approach can be used with a large-scale computation to estimate the recovery period of COVID-19.

8/10
References
1. Organization, W. H. Rolling updates on coronavirus disease (covid-19).[cited 2020 april 14] available at: https://www.
who. int/emergencies/diseases/novel-coronavirus-2019/events-as-they-happen. (2020).
2. Backer, J. A., Klinkenberg, D. & Wallinga, J. Incubation period of 2019 novel coronavirus (2019-ncov) infections among
travellers from Wuhan, China, 20–28 january 2020. Eurosurveillance 25, 2000062 (2020).
3. Lauer, S. A. et al. The incubation period of coronavirus disease 2019 (covid-19) from publicly reported confirmed cases:
estimation and application. Annals internal medicine 172, 577–582 (2020).
4. Li, Q. et al. Early transmission dynamics in Wuhan, China, of novel coronavirus–infected pneumonia. New Engl. J.
Medicine (2020).
5. Bi, Q. et al. Epidemiology and transmission of covid-19 in shenzhen China: Analysis of 391 cases and 1,286 of their close
contacts. MedRxiv (2020).
6. Jiang, X. et al. Is a 14-day quarantine period optimal for effectively controlling coronavirus disease 2019 (covid-19)?
medRxiv (2020).
7. Linton, N. M. et al. Incubation period and other epidemiological characteristics of 2019 novel coronavirus infections with
right truncation: a statistical analysis of publicly available case data. J. clinical medicine 9, 538 (2020).
8. Zhang, J. et al. Evolving epidemiology of novel coronavirus diseases 2019 and possible interruption of local transmission
outside Hubei province in China: a descriptive and modeling study. MedRxiv (2020).
9. Ma, S. et al. Epidemiological parameters of coronavirus disease 2019: a pooled analysis of publicly reported individual
data of 1155 cases from seven countries. Medrxiv (2020).
10. Leung, C. The difference in the incubation period of 2019 novel coronavirus (sars-cov-2) infection between travelers to
Hubei and nontravelers: The need for a longer quarantine period. Infect. Control. & Hosp. Epidemiol. 41, 594–596 (2020).
11. McAloon, C. et al. Incubation period of covid-19: a rapid systematic review and meta-analysis of observational research.
BMJ open 10, e039652 (2020).
12. Qin, J. et al. Estimation of incubation period distribution of covid-19 using disease onset forward time: a novel cross-
sectional and forward follow-up study. medRxiv (2020).
13. Paul, S. & Lorin, E. Lockdown: a non-pharmaceutical policy to prevent the spread of covid-19. mathematical modeling
and computation. Res. Gate preprint DOI: 10.13140/RG.2.2.13857.22882 (2020).
14. Bouchnita, A. & Jebrane, A. A hybrid multi-scale model of covid-19 transmission dynamics to assess the potential of
non-pharmaceutical interventions. Chaos, Solitons & Fractals 109941 (2020).
15. Ng, K. Y. & Gui, M. M. Covid-19: Development of a robust mathematical model and simulation package with consideration
for ageing population and time delay for control action and resusceptibility. Phys. D: Nonlinear Phenom. 132599 (2020).
16. Eshragh, A., Alizamir, S., Howley, P. & Stojanovski, E. Modeling the dynamics of the covid-19 population in australia: A
probabilistic analysis. arXiv preprint arXiv:2005.12455 (2020).
17. Kucharski, A. J. et al. Early dynamics of transmission and control of covid-19: a mathematical modelling study. The lancet
infectious diseases (2020).
18. Qaddura, Y. & Mavinga, N. Analysis of a vector-borne diseases model with a two-lag delay differential equation. The
North Carol. J. Math. Stat. 4, 12–28 (2018).
19. Vyasarayani, C. P. & Chatterjee, A. New approximations, and policy implications, from a delayed dynamic model of a fast
pandemic. arXiv preprint arXiv:2004.03878 (2020).
20. Yin, Z., Yu, Y. & Lu, Z. Stability analysis of an age-structured seirs model with time delay. Mathematics 8, 455 (2020).
21. Menendez, J. Elementary time-delay dynamics of covid-19 disease. medRxiv (2020).
22. Chen, Y., Cheng, J., Jiang, Y. & Liu, K. A time delay dynamic system with external source for the local outbreak of
2019-ncov. Appl. Analysis 1–12 (2020).
23. Dell’Anna, L. Solvable delay model for epidemic spreading: the case of covid-19 in italy. arXiv preprint arXiv:2003.13571
(2020).
24. https://resources-covid19canada.hub.arcgis.com/datasets/case-accumulation/data. .
25. Tan, W. Y. T., Wong, L. Y., Leo, Y. S. & Toh, M. P. H. S. Does incubation period of covid-19 vary with age? a study of
epidemiologically linked cases in singapore. Epidemiol. & Infect. 148 (2020).

9/10
26. Marchand-Senécal, X. et al. Diagnosis and management of first case of covid-19 in Canada: Lessons applied from sars.
Clin. Infect. Dis. (2020).
27. Silverstein, W. K., Stroud, L., Cleghorn, G. E. & Leis, J. A. First imported case of 2019 novel coronavirus in Canada,
presenting as mild pneumonia. The Lancet 395, 734 (2020).
28. https://www.worldometers.info/coronavirus/coronavirus-incubation-period/. .

Acknowledgements
This research is supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) and Mathematics of
Information Technology and Complex Systems (MITACS) Ref. IT#19228.

Author contributions statement

S.P. has derived the model, has developed the matlab code, has analyzed the calculated results, and has prepared all figures and
tables. S.P. and E.L. have drafted the original article. Both authors have contributed to the editing of the article. Both authors
have read and approved the final article.

Competing interests
The authors declare that they have no competing interests.

Additional information
Correspondence and requests for materials should be addressed to S.P.

10/10