Understanding unreported cases in the COVID-19 epidemic outbreak and the importance of major public health interventions
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Understanding unreported cases in the COVID-19
epidemic outbreak and the importance of major public
health interventions
Pierre Magal
University of Bordeaux, France
GdT Maths4covid19
Laboratoire Jacques-Louis Lions, Université Paris, April 21, 2020.
https://www.ljll.math.upmc.fr/fr/evenements/workshops-et-conferences
1/54
Co-authors
Quentin Griette, University of Bordeaux, France
Zhihua Liu, Beijing Normal University, China
Ousmane Seydi, Ecole Polethnique de Thies, Senegal
Glenn Webb, Vanderbilt University, USA
2/54Abstract
We develop a mathematical model to provide epidemic predictions for the
COVID-19 epidemic in China. We use reported case data from the Chinese
Center for Disease Control and Prevention and the Wuhan Municipal Health
Commission to parameterize the model. From the parameterized model
we identify the number of unreported cases. We then use the model to
project the epidemic forward with varying level of public health interventions.
The model predictions emphasize the importance of major public health
interventions in controlling COVID-19 epidemics.
3/54Outline
1 Introduction
2 Results
3 Numerical Simulations
4/54What are unreported cases?
Unreported cases are missed because authorities aren’t doing enough test-
ing, or ’preclinical cases’ in which people are incubating the virus but not
yet showing symptoms.
Research published1 traced COVID-19 infections which resulted from a busi-
ness meeting in Germany attended by someone infected but who showed
no symptoms at the time. Four people were ultimately infected from that
single contact.
1
Rothe, et al., Transmission of 2019-nCoV infection from an asymptomatic contact
in Germany. New England Journal of Medicine (2020). 5/54Why unreported cases are important?
A team in Japan2 reports that 13 evacuees from the Diamond Princess were
infected, of whom 4, or 31%, never developed symptoms.
A team in China 3 suggests that by 18 February, there were 37,400 people
with the virus in Wuhan whom authorities didn’t know about.
2
Nishiura et al. Serial interval of novel coronavirus (COVID-19) infections, Int. J.
Infect. Dis. (2020).
3
Wang et al. Evolving Epidemiology and Impact of Non-pharmaceutical Interventions
on the Outbreak of Coronavirus Disease 2019 in Wuhan, China, medRxiv (2020) 6/54Early models designed for the COVID-19
Wu et al. 4 used a susceptible-exposed-infectious-recovered
metapopulation model to simulate the epidemics across all major
cities in China.
Tang et al. 5 proposed an SEIR compartmental model based on the
clinical progression based on the clinical progression of the disease,
epidemiological status of the individuals, and the intervention
measures which did not consider unreported cases.
4
Wu, Joseph T., Kathy Leung, and Gabriel M. Leung, Nowcasting and forecasting
the potential domestic and international spread of the COVID-19 outbreak originating in
Wuhan, China: a modelling study, The Lancet, (2020).
5
Biao Tang, Xia Wang, Qian Li, Nicola Luigi Bragazzi, Sanyi Tang, Yanni Xiao,
Jianhong Wu, Estimation of the transmission risk of COVID-19 and its implication for
public health interventions, Journal of Clinical Medicine, (2020). 7/54Early results on identification the number of unreported
cases
Identifying the number of unreported cases was considered recently in
Magal and Webb6
Ducrot et al 7
In these works we consider an SIR model and we consider the Hong-Kong
seasonal influenza epidemic in New York City in 1968-1969.
6
P. Magal and G. Webb, The parameter identification problem for SIR epidemic
models: Identifying Unreported Cases, J. Math. Biol. (2018).
7
A. Ducrot, P. Magal, T. Nguyen, G. Webb. Identifying the Number of Unreported
Cases in SIR Epidemic Models. Mathematical Medicine and Biology, (2019) 8/54Outline
1 Introduction
2 Results
3 Numerical Simulations
9/54The model
Our model consists of the following system of ordinary differential equations
0
S (t) = −τ S(t)[I(t) + U (t)],
I 0 (t) = τ S(t)[I(t) + U (t)] − νI(t),
(2.1)
R0 (t) = ν1 I(t) − ηR(t),
0
U (t) = ν2 I(t) − ηU (t).
Here t ≥ t0 is time in days, t0 is the beginning date of the epidemic, S(t) is
the number of individuals susceptible to infection at time t, I(t) is the number
of asymptomatic infectious individuals at time t, R(t) is the number of reported
symptomatic infectious individuals (i.e. symptomatic infectious with sever symp-
toms) at time t, and U (t) is the number of unreported symptomatic infectious
individuals (i.e. symptomatic infectious with mild symptoms) at time t. This
system is supplemented by initial data
S(t0 ) = S0 > 0, I(t0 ) = I0 > 0, R(t0 ) ≥ 0 and U (t0 ) = U0 ≥ 0. (2.2)
10/54Why the exposed class can be neglected at first?
Exposed individuals are infected but not yet capable to transmit the
pathogen.
A team in China8 detected high viral loads in 17 people with COVID-19
soon after they became ill. Moreover, another infected individual never
developed symptoms but shed a similar amount of virus to those who did.
In Liu et al. 9 we compare the model (2.1) with exposure and the best fit
is obtained for an average exposed period of 6-12 hours.
8
Zou, L., SARS-CoV-2 viral load in upper respiratory specimens of infected patients.
New England Journal of Medicine, (2020).
9
Z. Liu, P. Magal, O. Seydi, and G. Webb, A COVID-19 epidemic model with latency
period, Infectious Disease Modelling (to appear) 11/54Compartments and flow chart of the model.
Asymptomatic Symptomatic
R
ν 1I ηR
τ S[I + U ]
S I Removed
ν2 I ηU
U
Figure: Compartments and flow chart of the model.
12/54Parameters of the model
Symbol Interpretation Method
t0 Time at which the epidemic started fitted
S0 Number of susceptible at time t0 fixed
I0 Number of asymptomatic infectious at time t0 fitted
U0 Number of unreported symptomatic infectious at time t0 fitted
τ Transmission rate fitted
1/ν Average time during which asymptomatic infectious are asymptomatic fixed
f Fraction of asymptomatic infectious that become reported symptomatic infectious fixed
ν1 = f ν Rate at which asymptomatic infectious become reported symptomatic fitted
ν2 = (1 − f ) ν Rate at which asymptomatic infectious become unreported symptomatic fitted
1/η Average time symptomatic infectious have symptoms fixed
Table: Parameters of the model.
13/54Comparison of Model (2.1) with the Data
For influenza disease outbreaks, the parameters τ , ν, ν1 , ν2 , η, as well as the
initial conditions S(t0 ), I(t0 ), and U (t0 ), are usually unknown. Our goal is to
identify them from specific time data of reported symptomatic infectious cases.
To identify the unreported asymptomatic infectious cases, we assume that the
cumulative reported symptomatic infectious cases at time t consist of a constant
fraction along time of the total number of symptomatic infectious cases up to time
t. In other words, we assume that the removal rate ν takes the following form: ν =
ν1 +ν2 , where ν1 is the removal rate of reported symptomatic infectious individuals,
and ν2 is the removal rate of unreported symptomatic infectious individuals due to
all other causes, such as mild symptom, or other reasons.
14/54Cumulative number of reported cases
The cumulative number of reported symptomatic infectious cases at time t,
denoted by CR(t), is
Zt
CR(t) = ν1 I(s)ds. (2.3)
t0
15/54Phenomenological model
We assume that CR(t) has the following special form:
CR(t) = χ1 exp (χ2 t) − χ3 . (2.4)
We evaluate χ1 , χ2 , χ3 using the reported cases data.
We obtain the model starting time of the epidemic t0 from 2.6
1
CR(t0 ) = 0 ⇔ χ1 exp (χ2 t0 ) − χ3 = 0 ⇒ t0 = (ln (χ3 ) − ln (χ1 )) .
χ2
16/54Estimation of the parameters
Name of the Parameter χ1 χ2 χ3 t0
From China 0.16 0.38 1.1 5.12
From Hubei 0.23 0.34 0.1 −2.45
From Wuhan 0.36 0.28 0.1 −4.52
Table: Estimation of the parameters χ1 , χ2 , χ3 and t0 by using the cumulative
reported cases Chinese CDC and Wuhan Municipal Health Commission.
Remark 2.1
The time t = 0 will correspond to 31 December. Thus, in Table 17,
the value t0 = 5.12 means that the starting time of the epidemic is 5
January, the value t0 = −2.45 means that the starting time of the
epidemic is 28 December, and t0 = −4.52 means that the starting time of
the epidemic is 26 December.
17/54Data
We use three sets of reported data to model the epidemic in Wuhan: First,
data from the Chinese CDC for mainland China, second, data from the
Wuhan Municipal Health Commission for Hubei Province, and third, data
from the Wuhan Municipal Health Commission for Wuhan Municipality.
These data vary but represent the epidemic transmission in Wuhan, from-
which almost all the cases originated from Hubei province.
18/54Fit of the exponential model (2.6) to the data
9.5 9000
9 8000
7000
8.5
6000
8
log(CR(t)+χ3)
5000
CR(t)
7.5
4000
7
3000
6.5
2000
6 1000
5.5 0
20 21 22 23 24 25 26 27 28 29 20 21 22 23 24 25 26 27 28 29
time in day time in day
9.5 9000
8000
9
7000
8.5
log(CR(t)+χ3)
6000
8 5000
CR(t)
7.5 4000
3000
7
2000
6.5
1000
6 0
23 24 25 26 27 28 29 30 31 23 24 25 26 27 28 29 30 31
time in day time in day
7.8 2400
7.6 2200
2000
7.4
1800
7.2
log(CR(t)+χ3)
1600
7
CR(t)
1400
6.8
1200
6.6
1000
6.4
800
6.2 600
6 400
25 26 27 28 29 30 31 25 26 27 28 29 30 31
time in day time in day
19/54Remark 2.2
As long as the number of reported cases follows (2.1), we can predict the
future values of CR(t). For χ1 = 0.16, χ2 = 0.38, and χ3 = 1.1, we
obtain
Jan. 30 Jan. 31 Feb. 1 Feb. 2 Feb. 3 Feb. 4 Feb. 5 Feb. 6
8510 12 390 18 050 26 290 38 290 55 770 81 240 118 320
The actual number of reported cases for China are 8163 confirmed for 30
January, 11 791 confirmed for Jan. 31, and 14 380 confirmed for Feb. 1.
Thus, the exponential formula (2.6) overestimates the number reported
after Jan. 30.
20/54Estimation of the parameters for the model (2.1)
Since the COVID-19 is a new virus for the population in Wuhan, we can
assume that all the population of Wuhan is susceptible. So we fix
S0 = 11.081 × 106
which corresponds to the total population of Wuhan (11 millions people).
From now on, we fix the fraction f of symptomatic infectious cases that are
reported. We assume that between 80% and 100% of infectious cases are
reported. Thus, f varies between 0.8 and 1.
We assume 1/ν, and the average time during which the patients are asymp-
tomatic infectious varies between 1 day and 7 days. We assume that 1/η,
the average time during which a patient is symptomatic infectious, varies
between 1 day and 7 days. We can compute
ν1 = f ν and ν2 = (1 − f ) ν. (2.5)
21/54Estimation of the parameters for the model (2.1)
Zt
CR(t) = χ1 exp (χ2 t) − χ3 = ν1 I(s)ds. (2.6)
t0
and by fixing S(t) = S0 in the I-equation of system (2.1), we obtain
χ1 χ2 exp (χ2 t0 ) χ3 χ2
I0 = = , (2.7)
fν fν
χ2 + ν η + χ2
τ= , (2.8)
S0 ν2 + η + χ2
and
(1 − f )ν fν
U0 = I0 and R0 = I0 . (2.9)
η + χ2 η + χ2
22/54Computation of the basic reproductive number R0
By using the approach described in Diekmann et al. 10 and by Van den
Driessche and Watmough 11 the basic reproductive number for model (2.1)
is given by
τ S0 ν2
R0 = 1+ .
ν η
By using ν2 = (1 − f ) ν and (2.8), we obtain
χ2 + ν η + χ2 (1 − f ) ν
R0 = 1+ . (2.10)
ν (1 − f ) ν + η + χ2 η
10
O. Diekmann, J. A. P. Heesterbeek and J. A. J. Metz, On the definition and the
computation of the basic reproduction ratio R0 in models for infectious diseases in
heterogeneous populations, Journal of Mathematical Biology 28(4) (1990), 365-382.
11
P. Van den Driessche and J. Watmough, Reproduction numbers and subthreshold
endemic equilibria for compartmental models of disease transmission, Mathematical
Biosciences 180 (2002) 29-48. 23/54Outline
1 Introduction
2 Results
3 Numerical Simulations
24/54Numerical Simulations
We can find multiple values of η, ν and f which provide a good fit for the
data. For application of our model, η, ν and f must vary in a reasonable
range. For the corona virus COVID-19 epidemic in Wuhan at its current
stage, the values of η, ν and f are not known. From preliminary information,
we use the values
f = 0.8, η = 1/7, ν = 1/7.
By using the formula (2.10) for the basic reproduction number, we obtain
from the data for mainland China, that R0 = 4.13 (respectively from the
data for Hubei province R0 = 3.82 and the data for Wuhan municipality
R0 = 3.35).
25/549000
8000
7000
6000
number of cases
5000
4000
3000
2000
1000
0
5 10 15 20 25 30
time in day
9000
8000
7000
6000
number of cases
5000
4000
3000
2000
1000
0
−5 0 5 10 15 20 25 30 35
time in day
2500
2000
number of cases
1500
1000
500
0
−5 0 5 10 15 20 25 30 35
time in day
26/54Turning point
Definition 3.1
We define the turning point tp as the time at which the function t → R(t)
(red solid curve) (i.e., the curve of the non-cumulative reported infectious
cases) reaches its maximum value.
27/54Absence of major public health interventions
For example, in the figure below, the turning point tp is day 54, which
corresponds to 23 February for Wuhan.
6
x 10
9
8
7
6
number of cases
5
4
3
2
1
0
0 10 20 30 40 50 60 70 80
time in day
Figure: In this figure, we plot the graphs of t → CR(t) (black solid line),
t → U (t) (blue solid line) and t → R(t) (red solid line).
28/54Strong confinement measures
In the following, we take into account the fact that very strong isolation
measures have been imposed for all China since 23 January. Specifically,
since 23 January, families in China are required to stay at home. In order to
take into account such a public intervention, we assume that the transmis-
sion of COVID-19 from infectious to susceptible individuals stopped after
25 January.
Remark 3.2
The private transportation was stopped in Wuhan after February 26.
Furthermore, around 5 millions people left Wuhan just before the February
23 2020 to escape the epidemic. So the rate transmission should be
divided by 2 after February 23 2020.
29/54Therefore, we consider the following model: for t ≥ t0 ,
S 0 (t) = −τ (t)S(t)[I(t) + U (t)],
I 0 (t) = τ (t)S(t)[I(t) + U (t)] − νI(t)
(3.1)
R0 (t) = ν1 I(t) − ηR(t)
0
U (t) = ν2 I(t) − ηU (t)
where (
4.44 × 10−08 , for t ∈ [t0 , 25],
τ (t) = (3.2)
0, for t > 25.
30/54Using the parameters χ1 , χ2 , χ3 obtain from the data for mainland China
7000
6000
5000
number of cases
4000
3000
2000
1000
0
0 10 20 30 40 50 60
time in day
31/54Using the parameters χ1 , χ2 , χ3 obtain from the data for Hubei province
4000
3500
3000
number of cases
2500
2000
1500
1000
500
0
−10 0 10 20 30 40 50 60
time in day
32/54Using the parameters χ1 , χ2 , χ3 obtain from the data for Wuhan city
1400
1200
1000
number of cases
800
600
400
200
0
−10 0 10 20 30 40 50 60
time in day
33/54Basic reproductive number
34/54Another model for τ (t)
The formula for τ (t) during the exponential decreasing phase was derived
by a fitting procedure. The formula for τ (t) is
(
τ (t) = τ0 , 0 ≤ t ≤ N,
(3.3)
τ (t) = τ0 exp (−µ (t − N )) , N < t.
The date N is the first day of the confinement and the value of µ is the
intensity of the confinement. The parameters N and µ are chosen so that
the cumulative reported cases in the numerical simulation of the epidemic
aligns with the cumulative reported case data during a period of time after
January 19. We choose N = 25 (January 25) for our simulations.
35/54τ(t )
4.× 10-8
3.× 10-8
2.× 10-8
1.× 10-8
0
0 10 20 30 40 50 60
Figure: Graph of τ (t) with N = 25 (January 25) and µ = 0.16. The transmission
rate is effectively 0.0 after day 53 (February 22).
36/54Predicting the epidemic in China
60 000
50 000
CR(t) CR(t)
CU(t) 50 000 CU(t)
40 000
R(t) R(t)
40 000
U(t)
μ = 0.16 U(t)
μ = 0.14
30 000
30 000
Data Data
20 000
20 000
10 000 10 000
0 0
20 30 40 50 60 20 30 40 50 60
A: Data January 19 - January 31 B: Data January 19 - February 7
60 000 60 000
CR(t) CR(t)
50 000 CU(t) 50 000 CU(t)
R(t) R(t)
40 000
U(t)
μ = 0.14 40 000
U(t)
μ = 0.139
30 000 30 000
Data Data
20 000 20 000
10 000 10 000
0 0
20 30 40 50 60 20 30 40 50 60
C: Data January 19 - February 14 D: Data January 19 - February 21
60 000 60 000
CR(t) CR(t)
50 000 CU(t) 50 000 CU(t)
R(t) R(t)
40 000
U(t)
μ = 0.139 40 000
U(t)
μ = 0.139
30 000 30 000
Data Data
20 000 20 000
10 000 10 000
0 0
20 30 40 50 60 20 30 40 50 60
E: Data January 19 - February 28 F: Data January 19 - March 6
37/54Predicting the Cumulative data for China with f = 0.8
CR(t)
CU(t)
60000 Cumulative Data
40000
20000
0
Jan 21 Jan 28 Feb 04 Feb 11 Feb 18 Feb 25 Mar 03 Mar 10 Mar 17
2020
38/54Predicting the weekly data for China
30000
I(t)
U(t)
R(t)
DR(t)
20000
10000
0
Jan 21 Jan 28 Feb 04 Feb 11 Feb 18 Feb 25 Mar 03 Mar 10 Mar 17
2020
39/54Daily number of cases
The daily number of reported cases from the model can be obtained by
computing the solution of the following equation:
DR0 (t) = ν f I(t) − DR(t), for t ≥ t0 and DR(t0 ) = DR0 . (3.4)
40/54Predicting the weekly data in China
4000
DR(t)
3500 Daily Data
3000
2500
2000
1500
1000
500
0
Jan 21 Jan 28 Feb 04 Feb 11 Feb 18 Feb 25 Mar 03 Mar 10 Mar 17
2020
41/54Multiple good fit simulations
We vary the time interval [d1 , d2 ] during which we use the data to obtain
χ1 and χ2 by using an exponential fit. In the simulations below we vary the
first day d1 , the last day d2 , N (date at which public intervention measures
became effective) such that all possible sets of parameters (d1 , d2 , N ) will
be considered. For each (d1 , d2 , N ) we evaluate µ to obtain the best fit
of the model to the data. We use the mean absolute deviation as the
distance to data to evaluate the best fit to the data. We obtain a large
number of best fit depending on (d1 , d2 , N, f ) and we plot the smallest
mean absolute deviation MADmin . Then we plot all the best fit with mean
absolute deviation between MADmin and MADmin + 5.
Remark 3.3
The number 5 chosen in MADmin + 5 is questionable. We use this value
for all the simulations since it gives sufficiently many runs that are fitting
very well the data and which gives later a sufficiently large deviation.
42/54Cumulative data for China until February 6 with f = 0.6
43/54Cumulative data for China until February 20 with f = 0.6
44/54Cumulative data for China until March 12 with f = 0.6
45/54Daily data for China until February 6 with f = 0.6
46/54Daily data for China until February 20 with f = 0.6
5000
4000
3000
2000
1000
0
Jan 29 Feb 12 Feb 26 Mar 11 Mar 25
2020
47/54Daily data for China until March 12 with f = 0.6
48/54Estimating the last day for COVID-19 outbreak in China
Level of risk 10% 5% 1%
Extinction date (f = 0.8) May 19 May 24 June 5
Extinction date (f = 0.6) May 25 May 31 June 12
Extinction date (f = 0.4) May 31 June 5 June 17
Extinction date (f = 0.2) June 7 June 12 June 24
Table: In this table we record the last day of epidemic.
49/54Cumulative data for France until Mars 30 with f = 0.4
50/54Cumulative data for France until April 20 with f = 0.4
51/54Daily data for France until Mars 30 with f = 0.4
52/54Daily data for France until April 20 with f = 0.4
53/54References
Z. Liu, P. Magal, O. Seydi, and G. Webb, Understanding unreported cases in
the COVID-19 epidemic outbreak in Wuhan, China, and the importance of major
public health interventions, MPDI Biology 2020, 9(3), 50.
Z. Liu, P. Magal, O. Seydi, and G. Webb, Predicting the cumulative number of
cases for the COVID-19 epidemic in China from early data, Mathematical
Biosciences and Engineering 17(4) 2020, 3040-3051. .
Z. Liu, P. Magal, O. Seydi, and G. Webb, A COVID-19 epidemic model with
latency period, Infectious Disease Modelling (to appear)
Z. Liu, P. Magal, O. Seydi, and G. Webb, A model to predict COVID-19
epidemics with applications to South Korea, Italy, and Spain, SIAM News (to
appear)
Z. Liu, P. Magal and G. Webb, Predicting the number of reported and
unreported cases for the COVID-19 epidemic in China, South Korea, Italy, France,
Germany and United Kingdom, medRxiv
Q. Griette, Z. Liu and P. Magal, Estimating the last day for COVID-19 outbreak
in mainland China, medRxiv.
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