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Understanding unreported cases in the COVID-19epidemic outbreak and the importance of major public

health interventions

Pierre Magal

University of Bordeaux, France

GdT Maths4covid19Laboratoire Jacques-Louis Lions, Université Paris, April 21, 2020.

https://www.ljll.math.upmc.fr/fr/evenements/workshops-et-conferences

https://www.ljll.math.upmc.fr/fr/evenements/workshops-et-conferences

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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

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Abstract

We develop a mathematical model to provide epidemic predictions for theCOVID-19 epidemic in China. We use reported case data from the ChineseCenter for Disease Control and Prevention and the Wuhan Municipal HealthCommission to parameterize the model. From the parameterized modelwe identify the number of unreported cases. We then use the model toproject the epidemic forward with varying level of public health interventions.The model predictions emphasize the importance of major public healthinterventions in controlling COVID-19 epidemics.

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Outline

1 Introduction

2 Results

3 Numerical Simulations

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What 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 notyet showing symptoms.

Research published1 traced COVID-19 infections which resulted from a busi-ness meeting in Germany attended by someone infected but who showedno symptoms at the time. Four people were ultimately infected from thatsingle contact.

1Rothe, et al., Transmission of 2019-nCoV infection from an asymptomatic contactin Germany. New England Journal of Medicine (2020).

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Why unreported cases are important?

A team in Japan2 reports that 13 evacuees from the Diamond Princess wereinfected, of whom 4, or 31%, never developed symptoms.

A team in China 3 suggests that by 18 February, there were 37,400 peoplewith the virus in Wuhan whom authorities didn’t know about.

2Nishiura et al. Serial interval of novel coronavirus (COVID-19) infections, Int. J.Infect. Dis. (2020).

3Wang et al. Evolving Epidemiology and Impact of Non-pharmaceutical Interventionson the Outbreak of Coronavirus Disease 2019 in Wuhan, China, medRxiv (2020)

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Early models designed for the COVID-19

Wu et al. 4 used a susceptible-exposed-infectious-recoveredmetapopulation model to simulate the epidemics across all majorcities in China.Tang et al. 5 proposed an SEIR compartmental model based on theclinical progression based on the clinical progression of the disease,epidemiological status of the individuals, and the interventionmeasures which did not consider unreported cases.

4Wu, Joseph T., Kathy Leung, and Gabriel M. Leung, Nowcasting and forecastingthe potential domestic and international spread of the COVID-19 outbreak originating inWuhan, China: a modelling study, The Lancet, (2020).

5Biao 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 forpublic health interventions, Journal of Clinical Medicine, (2020).

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Early results on identification the number of unreportedcases

Identifying the number of unreported cases was considered recently inMagal and Webb6

Ducrot et al 7

In these works we consider an SIR model and we consider the Hong-Kongseasonal influenza epidemic in New York City in 1968-1969.

6P. Magal and G. Webb, The parameter identification problem for SIR epidemicmodels: Identifying Unreported Cases, J. Math. Biol. (2018).

7A. Ducrot, P. Magal, T. Nguyen, G. Webb. Identifying the Number of UnreportedCases in SIR Epidemic Models. Mathematical Medicine and Biology, (2019)

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Outline

1 Introduction

2 Results

3 Numerical Simulations

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The model

Our model consists of the following system of ordinary differential equationsS′(t) = −τS(t)[I(t) + U(t)],I ′(t) = τS(t)[I(t) + U(t)]− νI(t),R′(t) = ν1I(t)− ηR(t),U ′(t) = ν2I(t)− ηU(t).

(2.1)

Here t ≥ t0 is time in days, t0 is the beginning date of the epidemic, S(t) isthe number of individuals susceptible to infection at time t, I(t) is the numberof asymptomatic infectious individuals at time t, R(t) is the number of reportedsymptomatic infectious individuals (i.e. symptomatic infectious with sever symp-toms) at time t, and U(t) is the number of unreported symptomatic infectiousindividuals (i.e. symptomatic infectious with mild symptoms) at time t. Thissystem is supplemented by initial data

S(t0) = S0 > 0, I(t0) = I0 > 0, R(t0) ≥ 0 and U(t0) = U0 ≥ 0. (2.2)

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Why the exposed class can be neglected at first?

Exposed individuals are infected but not yet capable to transmit thepathogen.

A team in China8 detected high viral loads in 17 people with COVID-19soon after they became ill. Moreover, another infected individual neverdeveloped 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 fitis obtained for an average exposed period of 6-12 hours.

8Zou, L., SARS-CoV-2 viral load in upper respiratory specimens of infected patients.New England Journal of Medicine, (2020).

9Z. Liu, P. Magal, O. Seydi, and G. Webb, A COVID-19 epidemic model with latencyperiod, Infectious Disease Modelling (to appear)

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Compartments and flow chart of the model.

S I

R

U

SymptomaticAsymptomatic

τS[I + U ]

ν1I

ν2I

Removed

ηR

ηU

Figure: Compartments and flow chart of the model.

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Parameters of the model

Symbol Interpretation Method

t0 Time at which the epidemic started fittedS0 Number of susceptible at time t0 fixedI0 Number of asymptomatic infectious at time t0 fittedU0 Number of unreported symptomatic infectious at time t0 fittedτ Transmission rate fitted

1/ν Average time during which asymptomatic infectious are asymptomatic fixedf Fraction of asymptomatic infectious that become reported symptomatic infectious fixed

ν1 = f ν Rate at which asymptomatic infectious become reported symptomatic fittedν2 = (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.

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Comparison of Model (2.1) with the Data

For influenza disease outbreaks, the parameters τ , ν, ν1, ν2, η, as well as theinitial conditions S(t0), I(t0), and U(t0), are usually unknown. Our goal is toidentify them from specific time data of reported symptomatic infectious cases.To identify the unreported asymptomatic infectious cases, we assume that thecumulative reported symptomatic infectious cases at time t consist of a constantfraction along time of the total number of symptomatic infectious cases up to timet. 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 toall other causes, such as mild symptom, or other reasons.

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Cumulative number of reported cases

The cumulative number of reported symptomatic infectious cases at time t,denoted by CR(t), is

CR(t) = ν1t∫

t0

I(s)ds. (2.3)

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Phenomenological model

We assume that CR(t) has the following special form:

CR(t) = χ1 exp (χ2t)− 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

CR(t0) = 0⇔ χ1exp (χ2t0)− χ3 = 0 ⇒ t0 =1χ2

(ln (χ3)− ln (χ1)) .

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Estimation of the parameters

Name of the Parameter χ1 χ2 χ3 t0From China 0.16 0.38 1.1 5.12From Hubei 0.23 0.34 0.1 −2.45From Wuhan 0.36 0.28 0.1 −4.52

Table: Estimation of the parameters χ1, χ2, χ3 and t0 by using the cumulativereported cases Chinese CDC and Wuhan Municipal Health Commission.

Remark 2.1The 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 5January, the value t0 = −2.45 means that the starting time of theepidemic is 28 December, and t0 = −4.52 means that the starting time ofthe epidemic is 26 December.

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Data

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 theWuhan Municipal Health Commission for Hubei Province, and third, datafrom 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.

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Fit of the exponential model (2.6) to the data

20 21 22 23 24 25 26 27 28 295.5

6

6.5

7

7.5

8

8.5

9

9.5

time in day

log

(CR

(t)+

χ3)

20 21 22 23 24 25 26 27 28 290

1000

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7000

8000

9000

time in day

CR

(t)

23 24 25 26 27 28 29 30 316

6.5

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9

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log

(CR

(t)+

χ3)

23 24 25 26 27 28 29 30 310

1000

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7000

8000

9000

time in day

CR

(t)

25 26 27 28 29 30 316

6.2

6.4

6.6

6.8

7

7.2

7.4

7.6

7.8

time in day

log

(CR

(t)+

χ3)

25 26 27 28 29 30 31400

600

800

1000

1200

1400

1600

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2000

2200

2400

time in day

CR

(t)

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Remark 2.2As long as the number of reported cases follows (2.1), we can predict thefuture values of CR(t). For χ1 = 0.16, χ2 = 0.38, and χ3 = 1.1, weobtain

Jan. 30 Jan. 31 Feb. 1 Feb. 2 Feb. 3 Feb. 4 Feb. 5 Feb. 68510 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 30January, 11 791 confirmed for Jan. 31, and 14 380 confirmed for Feb. 1.Thus, the exponential formula (2.6) overestimates the number reportedafter Jan. 30.

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Estimation of the parameters for the model (2.1)

Since the COVID-19 is a new virus for the population in Wuhan, we canassume 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 arereported. We assume that between 80% and 100% of infectious cases arereported. 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, variesbetween 1 day and 7 days. We can compute

ν1 = fν and ν2 = (1− f) ν. (2.5)

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Estimation of the parameters for the model (2.1)

CR(t) = χ1 exp (χ2t)− χ3 = ν1t∫

t0

I(s)ds. (2.6)

and by fixing S(t) = S0 in the I-equation of system (2.1), we obtain

I0 =χ1χ2exp (χ2t0)

f ν= χ3χ2

f ν, (2.7)

τ = χ2 + νS0

η + χ2ν2 + η + χ2

, (2.8)

andU0 =

(1− f)νη + χ2

I0 and R0 =fν

η + χ2I0. (2.9)

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Computation of the basic reproductive number R0

By using the approach described in Diekmann et al. 10 and by Van denDriessche and Watmough 11 the basic reproductive number for model (2.1)is given by

R0 =τS0ν

(1 + ν2

η

).

By using ν2 = (1− f) ν and (2.8), we obtain

R0 =χ2 + νν

η + χ2(1− f) ν + η + χ2

(1 + (1− f) ν

η

). (2.10)

10O. Diekmann, J. A. P. Heesterbeek and J. A. J. Metz, On the definition and thecomputation of the basic reproduction ratio R0 in models for infectious diseases inheterogeneous populations, Journal of Mathematical Biology 28(4) (1990), 365-382.

11P. Van den Driessche and J. Watmough, Reproduction numbers and subthresholdendemic equilibria for compartmental models of disease transmission, MathematicalBiosciences 180 (2002) 29-48.

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Outline

1 Introduction

2 Results

3 Numerical Simulations

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Numerical Simulations

We can find multiple values of η, ν and f which provide a good fit for thedata. For application of our model, η, ν and f must vary in a reasonablerange. For the corona virus COVID-19 epidemic in Wuhan at its currentstage, 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 obtainfrom the data for mainland China, that R0 = 4.13 (respectively from thedata for Hubei province R0 = 3.82 and the data for Wuhan municipalityR0 = 3.35).

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5 10 15 20 25 300

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time in day

nu

mb

er

of

cases

−5 0 5 10 15 20 25 30 350

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of

cases

−5 0 5 10 15 20 25 30 350

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cases

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Turning point

Definition 3.1We 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 infectiouscases) reaches its maximum value.

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Absence of major public health interventionsFor example, in the figure below, the turning point tp is day 54, whichcorresponds to 23 February for Wuhan.

0 10 20 30 40 50 60 70 800

1

2

3

4

5

6

7

8

9x 10

6

time in day

nu

mb

er

of

cases

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).

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Strong confinement measures

In the following, we take into account the fact that very strong isolationmeasures have been imposed for all China since 23 January. Specifically,since 23 January, families in China are required to stay at home. In order totake into account such a public intervention, we assume that the transmis-sion of COVID-19 from infectious to susceptible individuals stopped after25 January.

Remark 3.2The private transportation was stopped in Wuhan after February 26.Furthermore, around 5 millions people left Wuhan just before the February23 2020 to escape the epidemic. So the rate transmission should bedivided by 2 after February 23 2020.

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Therefore, we consider the following model: for t ≥ t0,S′(t) = −τ(t)S(t)[I(t) + U(t)],I ′(t) = τ(t)S(t)[I(t) + U(t)]− νI(t)R′(t) = ν1I(t)− ηR(t)U ′(t) = ν2I(t)− ηU(t)

(3.1)

whereτ(t) =

{4.44× 10−08, for t ∈ [t0, 25],0, for t > 25. (3.2)

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Using the parameters χ1, χ2, χ3 obtain from the data for mainland China

0 10 20 30 40 50 600

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s

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Using the parameters χ1, χ2, χ3 obtain from the data for Hubei province

−10 0 10 20 30 40 50 600

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Using the parameters χ1, χ2, χ3 obtain from the data for Wuhan city

−10 0 10 20 30 40 50 600

200

400

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time in day

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Basic reproductive number

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Another model for τ(t)

The formula for τ(t) during the exponential decreasing phase was derivedby a fitting procedure. The formula for τ(t) is{

τ(t) = τ0, 0 ≤ t ≤ N,τ(t) = τ0 exp (−µ (t−N)) , N < t.

(3.3)

The date N is the first day of the confinement and the value of µ is theintensity of the confinement. The parameters N and µ are chosen so thatthe cumulative reported cases in the numerical simulation of the epidemicaligns with the cumulative reported case data during a period of time afterJanuary 19. We choose N = 25 (January 25) for our simulations.

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0 10 20 30 40 50 60

0

1.×10-82.×10-83.×10-84.×10-8

τ(t)

Figure: Graph of τ(t) with N = 25 (January 25) and µ = 0.16. The transmissionrate is effectively 0.0 after day 53 (February 22).

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Predicting the epidemic in China

20 30 40 50 600

10000

20000

30000

40000

50000

A: Data January 19 - January 31

μ = 0.16Data

CR(t)CU(t)R(t)U(t)

20 30 40 50 600

10000

20000

30000

40000

50000

60000

B: Data January 19 - February 7

μ = 0.14Data

CR(t)CU(t)R(t)U(t)

20 30 40 50 600

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30000

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C: Data January 19 - February 14

μ = 0.14Data

CR(t)CU(t)R(t)U(t)

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40000

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60000

D: Data January 19 - February 21

μ = 0.139Data

CR(t)CU(t)R(t)U(t)

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40000

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E: Data January 19 - February 28

μ = 0.139Data

CR(t)CU(t)R(t)U(t)

20 30 40 50 600

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30000

40000

50000

60000

F: Data January 19 - March 6

μ = 0.139Data

CR(t)CU(t)R(t)U(t)

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Predicting the Cumulative data for China with f = 0.8

Jan 21 Jan 28 Feb 04 Feb 11 Feb 18 Feb 25 Mar 03 Mar 10 Mar 17

2020

0

20000

40000

60000

CR(t)

CU(t)

Cumulative Data

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Predicting the weekly data for China

Jan 21 Jan 28 Feb 04 Feb 11 Feb 18 Feb 25 Mar 03 Mar 10 Mar 17

2020

0

10000

20000

30000I(t)

U(t)

R(t)

DR(t)

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Daily number of cases

The daily number of reported cases from the model can be obtained bycomputing the solution of the following equation:

DR′(t) = ν f I(t)−DR(t), for t ≥ t0 and DR(t0) = DR0. (3.4)

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Predicting the weekly data in China

Jan 21 Jan 28 Feb 04 Feb 11 Feb 18 Feb 25 Mar 03 Mar 10 Mar 17

2020

0

500

1000

1500

2000

2500

3000

3500

4000DR(t)

Daily Data

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Multiple good fit simulations

We vary the time interval [d1, d2] during which we use the data to obtainχ1 and χ2 by using an exponential fit. In the simulations below we vary thefirst day d1, the last day d2, N (date at which public intervention measuresbecame effective) such that all possible sets of parameters (d1, d2, N) willbe considered. For each (d1, d2, N) we evaluate µ to obtain the best fitof the model to the data. We use the mean absolute deviation as thedistance to data to evaluate the best fit to the data. We obtain a largenumber of best fit depending on (d1, d2, N, f) and we plot the smallestmean absolute deviation MADmin. Then we plot all the best fit with meanabsolute deviation between MADmin and MADmin + 5.

Remark 3.3The number 5 chosen in MADmin + 5 is questionable. We use this valuefor all the simulations since it gives sufficiently many runs that are fittingvery well the data and which gives later a sufficiently large deviation.

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Cumulative data for China until February 6 with f = 0.6

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Cumulative data for China until February 20 with f = 0.6

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Cumulative data for China until March 12 with f = 0.6

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Daily data for China until February 6 with f = 0.6

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Daily data for China until February 20 with f = 0.6

Jan 29 Feb 12 Feb 26 Mar 11 Mar 25

2020

0

1000

2000

3000

4000

5000

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Daily data for China until March 12 with f = 0.6

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Estimating 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 5Extinction date (f = 0.6) May 25 May 31 June 12Extinction date (f = 0.4) May 31 June 5 June 17Extinction date (f = 0.2) June 7 June 12 June 24

Table: In this table we record the last day of epidemic.

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Cumulative data for France until Mars 30 with f = 0.4

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Cumulative data for France until April 20 with f = 0.4

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Daily data for France until Mars 30 with f = 0.4

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Daily data for France until April 20 with f = 0.4

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References

Z. Liu, P. Magal, O. Seydi, and G. Webb, Understanding unreported cases inthe COVID-19 epidemic outbreak in Wuhan, China, and the importance of majorpublic health interventions, MPDI Biology 2020, 9(3), 50.Z. Liu, P. Magal, O. Seydi, and G. Webb, Predicting the cumulative number ofcases for the COVID-19 epidemic in China from early data, MathematicalBiosciences and Engineering 17(4) 2020, 3040-3051. .Z. Liu, P. Magal, O. Seydi, and G. Webb, A COVID-19 epidemic model withlatency period, Infectious Disease Modelling (to appear)Z. Liu, P. Magal, O. Seydi, and G. Webb, A model to predict COVID-19epidemics with applications to South Korea, Italy, and Spain, SIAM News (toappear)Z. Liu, P. Magal and G. Webb, Predicting the number of reported andunreported 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 outbreakin mainland China, medRxiv.

IntroductionResultsNumerical Simulations