Nonlinear Dyn (2020) 101:1527–1543https://doi.org/10.1007/s11071-020-05863-5

ORIGINAL PAPER

COVID-19: data-driven dynamics, statistical and distributeddelay models, and observations

Xianbo Liu · Xie Zheng · Balakumar Balachandran

Received: 25 May 2020 / Accepted: 29 July 2020 / Published online: 6 August 2020© Springer Nature B.V. 2020

Abstract COVID-19 was declared as a pandemic bythe World Health Organization on March 11, 2020.Here, the dynamics of this epidemic is studied by usinga generalized logistic function model and extendedcompartmental models with and without delays. Fora chosen population, it is shown as to how forecast-ing may be done on the spreading of the infectionby using a generalized logistic function model, whichcan be interpreted as a basic compartmental model. Inan extended compartmental model, which is a mod-ified form of the SEIQR model, the population isdivided into susceptible, exposed, infectious, quaran-tined, and removed (recovered or dead) compartments,and a set of delay integral equations is used to describethe system dynamics. Time-varying infection rates areallowed in themodel to capture the responses to controlmeasures taken, and distributed delay distributions areused to capture variability in individual responses toan infection. The constructed extended compartmentalmodel is a nonlinear dynamical systemwith distributed

X. LiuState Key Laboratory of Mechanical System and Vibration,School of Mechanical Engineering, Shanghai Jiao TongUniversity, Shanghai 200240, Chinae-mail: liuxianbo@sjtu.edu.cn

X. Liu · X. Zheng · B. Balachandran (B)Department of Mechanical Engineering, University ofMaryland, College Park, MD 20742, USAe-mail: balab@umd.edu

X. Zhenge-mail: xiezheng@umd.edu

delays and time-varying parameters. The critical roleof data is elucidated, and it is discussed as to how thecompartmental model can be used to capture responsesto various measures including quarantining. Data fordifferent parts of the world are considered, and com-parisons are also made in terms of the reproductivenumber. The obtained results can be useful for further-ing the understanding of disease dynamics as well asfor planning purposes.

Keywords Dynamics and control of epidemics ·Generalized logistic function · System identification ·SEIQR model · Delay integral equations

1 Introduction

Since the first reported case of novel coronavirus(SARS-CoV-2) in Wuhan, China, toward the end of2019, this highly infectious disease first spread rapidlywithin China and to its neighboring countries [1,2].After China, the next confirmed cases occurred inJapan, South Korea [3], and Thailand in late Jan-uary, and soon thereafter in 2020, the USA reportedits first case in Washington State. In February, Europefaced its first outbreak in Italy [4], with churches andschools closing immediately thereafter and towns beinglocked down [5]. By early March, following the WorldHealth Organization’s declaration of the coronavirusas a pandemic, the USA imposed a ban on travelersfrom Europe and declared a national emergency on

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March 13. As of May 4, more than 253,000 peoplehad died from COVID-19, with 3.6 million infectionsconfirmed in more than 180 countries and territoriesglobally [6]. The USA has taken the hardest hit fromthe pandemic, with 1.22 million confirmed cases andmore than 70,000 deaths, at the time of writing of thispaper. The goal of this work has been to look at thedata available on the number of infections and studythe evolution of the infection dynamics.

In order to facilitate the authors’ quest to pursuedata-driven dynamics based on infection data, a statisti-cal approach based on the generalized logistic functionhas been taken alongwith studies of a delay differentialsystem. The logistic function, which was introduced byPierre Verhulst for population growth modeling [7],is now widely used in various areas of science andengineering. Applications include friction modeling inmechanical systems [8], activation function in neuralnetworks [9], and infectious disease spreading in bio-logical systems [10]. The generalized logistic function,which has an asymmetric form in between the lowerand upper horizontal asymptotes, was used byRichards[11] for modeling plant growth. This function has beenrecently used for studying disease dynamics [12]. Asfor the studies in spreading dynamics, logistic functionsand generalized logistic functions are essentially com-partmental models [13] with a susceptible state and aninfected state (SI model). In reality, there exists a latentperiod, when people are infected but not yet infectious.Consideration of this aspect leads to the SEIR model[14], in which one has susceptible, exposed, infectious,and removed (recovered or dead) states. Moreover, ifmitigation measures are applied, a new state of quaran-tine needs to be considered, which results in the SEIQRmodel. However, in all of the aforementioned models,the derivatives of the different states are only dependenton their current values and assumed to be uniformlydistributed. Here, these models are extended throughintroduction of distributed time delays.

Delay differential equations (DDEs) arise oftenin various science and engineering applications, forinstance, in mechanical engineering [15–17]. Differentfrom ordinary differential equations, to solve DDEs,one needs information on current states and past statesover time intervals in the past [18]. DDEs are criticalfor modeling the spreading of COVID-19, since timedelays can be used to capture the durations of the latent,quarantine, and recovery periods. A SEIR model withtwo constant delays can be found in the work of [19],

a study of the global behavior of SIER with delay canbe found in the studies [20,21], and an SIR model withdelays has been studied in [22]. Furthermore, stabilitystudies on the solutions of an SIRmodel can be found in[23,24], and solutions of an SEIR model can be foundin the work of [25].

A primary objective of this paper is to use data-driven dynamics to further the current understanding ofkey aspects and features associated with the COVID-19 outbreak, as well as to help assess the viability ofcontrol strategies that are applicable for mitigation, forexample “flattening the curve.” The remainder of thepaper is organized as follows. In Sect. 2, based onthe susceptible-infected (SI) populations model, the S-shaped logistic and generalized logistic functions areintroduced to describe the outbreak of the COVID-19among various countries—China, Italy, Germany, andthe USA. With the goal of finding the inflection pointsfor these countries, parameters of the S-curve functionsfor different countries are identified using an optimiza-tion method. In Sect. 3, by considering the incubationperiod and the quarantine state, an improved SEIQRmodel with distributed time delays is developed. Thetime delay is employed here to capture the time gapbetween different (compartmental) states in the newmodel. With data of confirmed cases, parameters areidentified and simulations together with prediction ofdifferent geographic regions are presented. With thedeveloped models, potential mitigation strategies fordifferent scenarios are proposed. Finally, concludingremarks are provided.

2 Generalized logistic function based data-drivendynamics of COVID-19

In this section, the infection data of COVID-19 for theUS, as well as different countries from all over theworld, are gathered by using a web-read function ofMATLAB from the website of The COVID TrackingProject [26] and Worldometers [27], respectively. Thedetermination of unknown parameters in the general-ized logistic function is driven by the data of the totalconfirmed number of positive cases of the COVID-19,and a nonlinear regression algorithm is used. Then, theinflection (peak) point of COVID-19 is predicted andanalyzed for different countries and regions. Followedby that, considering each of the regions as an elementof the overall global system, a composite global model

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Fig. 1 Development of the composite global model and data-driven dynamics

which is comprised of 148 sub-models is developed, forcapturing and tracking the global COVID-19 dynam-ics, as shown in Fig. 1.

2.1 The generalized logistic function

The susceptible-infectious (SI) model, which is alsocalled the simple epidemic model (SEM), is the sim-plest form of all epidemic models, as shown in Fig. 2a.The evolution of the infection number in an SI modelis also known as logistic growth and results in a logis-tic function (sigmoid function). The logistic functionis centrosymmetric about the inflection point, whichmeans the decreasing (saturation) period mirrors thegrowing period. However, for the COVID-19, from thecurrent data, it can be gleaned that the infection growthoccurs as a short outburst, followed by a long satura-tion period. In this scenario, the logistic function isno longer an appropriate descriptive function, sincethe saturation period needs to be slowed down. Thegeneralized logistic function, also known as Richards’curve, originally developed for plant growth modeling,is an extension of the logistic or sigmoid functions,allowing for more flexible S-shaped curves. By varyingonly one parameter in the generalized logistic function[11], the saturation period of the pandemic can be con-trolled, which makes the model and prediction easierand more realistic. Hence, based on current data, theauthors choose the generalized logistic function hereto capture the infection growth dynamics of COVID-19. The generalized logistic function is the solution of

the Richards’ differential equation (RDE), which is anonlinear dynamical system, can be written as follows

dI

dt= β I

S

N= β I

(1 −

(I

N

)ν)(1)

Then, the generalized logistic function is obtained viadirect integral as

I (t) = N

(1 + eβν(t−t0))1ν

(2)

In Eqs. (1), (2), S = S(t) is the number of suscepti-ble individuals in a population while I = I (t) is thenumber of infectious individuals. N = S(t) + I (t) isa constant, which typically represents the total popu-lation. The infection rate β > 0 represents the rate ofspread of a pandemic and is the probability of transmit-ting disease from an infected individual to a susceptibleindividual. ν > 0 is the parameter affecting the positionof inflection point as well as the saturation period. Forν ≡ 1, one has a symmetrical S-curve about the inflec-tion point, that is, the logistic function. For ν > 1, thesaturation period is shorter and the result is a faster satu-ration compared to that notedwith the logistic function.For 0 < ν < 1, the saturation period is longer than thatobtained with the logistic function. Therefore, for theCOVID-19 outbreak, it is expected that with the gen-eralized logistic function and an optimized 0 < ν < 1,one will have a better match with the data. As shown inFig. 2b, the circles represent the number of confirmedpositive cases in the USA fromMarch 1, 2020, to May

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Fig. 2 a SI model and b comparison between statistical fits withthe logistic function and the generalized logistic function for theUS COVID-19 data

4, 2020. The dotted line in blue is the least-squares fit-ting done by using the logistic function (ν = 1), whilethe dashed line in orange represents the least-squaresfitting carriedout byusing thegeneralized logistic func-tion with an optimized ν = 0.011. The goodness of fitof the models for the COVID-19 data shows that theroot mean squared error (standard error) of the gener-alized logistic function is 1.15 × 104, which is muchsmaller than that for the logistic function (3.7 × 104).The generalized logistic function provides a better fitbecause it is both unbiased at the growth-period andsaturation-period and produces smaller residuals. Fourparameters L , k, t , and ν in the generalized logisticfunction need to be identified from the COVID-19 data,by using nonlinear regression algorithms.

2.2 Infection evolution: identified dynamics andinflection point of COVID-19

The generalized logistic function Eq. (1) has fourunknown parameters L , k, t , and ν, which should beidentified and optimized based on the COVID-19 data.The nonlinear regression algorithm adopted here is aform of regression analysis by which the data can bemodeled as any nonlinear function, such as the gener-alized logistic function. The objective of the nonlinearregression algorithm is to find an optimized point in

Fig. 3 Diagram for parameter identification, based on the non-linear regression algorithm in MATLAB

the four-dimensional parameter L − k − t − ν space,through minimization of the root mean squared error(standard error) between the COVID-19 data and themodel predictions.

Although both the linear and nonlinear regressionalgorithms are integrated into the Curve Fitting Tool-box of MATLAB R2018a, finding the global optimalpoint in the four-dimensional parameter space is stillnot an easy problem due to the nonlinearity of the gen-eralized logistic function. Therefore, to avoid a localoptimum, the parameter identification processes withEq. (1) are carried out as shown in Fig. 3. For eachloop as illustrated in the figure, the nonlinear regres-sion algorithm is used to find a local minimum start-ing from widely varying initial values (random) of theparameters. Andfinally, themost extreme one is chosenfrom these local minima as the global minimum whenthe termination criteria are satisfied. Here, the authorshave used maximum loops of 50 and a minimum coef-ficient of determination, called the R-squared value, of0.99 for the termination criteria. If any of the criteriaare satisfied, the identification processes are terminatedand the optimized parameters have peaked, accordingto the minimum of standard error between the data andthe model prediction. It should be noted that the pro-posed criterion of termination here cannot guaranteethat a global optimum will be found, but the process isexpected to greatly increase the probability of realizinga global optimum, instead of a local optimum.

Based on the parameter identification approachdescribed in this section, the COVID-19 infectiondynamics for several countries from North America,South America, Europe, and Asia is found to be cap-tured well by using the generalized logistic function

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Fig. 4 Phase portraits of COVID-19 dynamics in different coun-tries from North America, South America, Europe, and Asia

Fig. 5 Generalized logistic function based COVID-19 dynam-ics: a China, b South Korea, c Italy, and d the USA

as shown in Fig. 4. In this figure, the abscissa is thetotal number of test-positive cases, while the ordinateis the daily increase in the number of positive cases;that is, the derivative of the total cases. Hence, Fig. 4can be considered as being representative of the phaseportraits of the dynamic system. From this plot withlogarithmic coordinates, one can discern two periodsof the COVID-19 pandemic: a) the exponential growthperiod and b) the saturation period.

Besides, the time-domain COVID-19 dynamics ofChina, South Korea, Italy, and the USA are illustratedin Fig. 5a–d, respectively. In these figures, the left axiscorresponds to the total number of test-positive cases,while the right axis corresponds to the daily increasein the number of positive cases. The results show thegeneralized logistic function predictions are consistentwith the COVID-19 data of China, Italy, and the USAwith a R-squared value greater than 0.995; neverthe-less, the R-squared value for South Korea data is 0.981;this is indicative of a rather discernible fitting error forthe generalized logistic function in this case, as shownin Fig. 5b. This is mainly because the generalized logis-tic function cannot be used to capture the plateau fromMarch 10 to April 5 in the daily increments data forSouth Korea. The inflection points of the total cases,that is, the peaks of the daily increments, are identifiedwith good consistency from the data. The identifiedpeak of COVID-19 in China is on February 7, while thepeaks of South Korea, Italy, and the USA are 25 days,51 days, and 67 days later than China, respectively.

Additional model predicted curves for the countriesand regions all over theworld are shown in Fig. 6. Fromthis figure, one can choose the countries and regionswith total confirmed cases more than 20,000 by May4. It should be noted that the authors have treated eachstate of the USA as an individual region due to the largeinfected population in theUSA.Based on the parameteridentification approach, through the curves in this fig-ure, the authors show the model identified (predicted)daily increment cases from late January 2020 when thepandemic outbreak started in China, to the middle ofAugust 2020. From these plots, it can be seen clearlythat there are remarkable time lags among differentregions for the outbreaks of the COVID-19 pandemic.These remarkable time lags indicate that simplemodelswith low degrees of freedom (DOF) are not sufficientto capture the global dynamics of the COVID-19 infec-tions from all over the world. To show the distributionof the outbreak dates of the pandemic, histograms for

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Fig. 6 Model prediction ofdaily case increments forthe countries and regions allover the world withconfirmed COVID-19 case>20K by May 4, 2020

Fig. 7 Histograms for the date distribution of the peaks ofCOVID-19 for different countries and regions all over the world

the date of peaks identified from the current COVID-19data for 148 countries and regions all over the world areshown in Fig. 7. This histogramplot is illustrative of theoutbreak of the pandemic from one region to anotherin quick succession after China and South Korea. Atthe time of writing of the paper, the prediction was thatall the countries and regions worldwide would haveexperienced the peak (at least, the first peak) of theCOVID-19 pandemic, by July 1, 2020.

2.3 Global model of COVID-19 dynamics

Given the significant time lags among different coun-tries and regions for the outbreak of the COVID-19

Fig. 8 Dynamics of COVID-19 across the world using singlegeneralized logistic model

Fig. 9 Dynamics of COVID-19 across the world using the pro-posed composite global model with 148 identified sub-models

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pandemic (Figs. 6, 7), it is natural to expect the globaldynamics of the pandemic to be governed by a high-dimensional system; that is, it is not conceivable tocapture the global infection dynamics by using a simplemodel with low degrees of freedom. Hence, inspired bythe finite element method (FEM) which is widely usedinmechanics, considering each of the regions as an ele-ment of the overall global dynamic system, a compos-ite global model is established to capture the combineddynamics of countries and regions all over theworld.Asshown in Fig. 1, the parameter identification approach(Fig. 3) is utilized for each elemental sub-model, fol-lowing which, the composite global model is con-structed by assembling all of the 148 sub-models fromdifferent regions all over the world. Thus, the globalmodel for the COVID-19 dynamics at hand is a high-dimensional differential system with 148 sub-models.

The outcome of the COVID-19 dynamics model-ing carried out by using a single generalized logisticfunction is as shown in Fig. 8. The notable deviationbetween the COVID-19 data and the model predic-tion with the generalized logistic function confirms theearlier statement that a low-dimensional model cannotbe used to capture the global dynamics. By contrast,the outcome of composite global model shown in Fig.9, which is comprised of 148 identified sub-models,matches theworldwide COVID-19 data with good con-sistency for both the total number of infection casesand daily increments. The prediction from the modelwith single generalized logistic function is that the dailyincrements cases of COVID-19 will drop to 10 thou-sand by late July; on the other hand, the prediction withthe composite global model is that one will have dailyincrements of more than 100 thousand until August,which is worrisome. With the composite global model,one can see from Fig. 9 that there have been two wavesof COVID-19 pandemic spreading across the globe.The first wave started in late January and ended in lateFebruary in Wuhan, China. Subsequently, the secondwave started mainly in Europe and pushed to a surgeinto theUSA.This surge,which has remained over for along period in the USA, is being followed by onewherethe spreading is occurring in Russia, Brazil, and India.

It is important to note that all of the predictionsmadein this section are based on both the generalized logis-tic function and the COVID-19 data (positive testedcases). The elemental sub-model of the system is quitesimple to avoid any over-fitting of the systemdynamics.However, with this data-driven prediction, it is impor-

tant to keep in mind that consideration has not beengiven toother factors, such as the rising temperature dueto the seasonal changes, the differences and couplingbetween the southern hemisphere and northern hemi-sphere,measures and policies taken to counter the virusin different regions. To study the effects of these fac-tors on the pandemic, amore sophisticatedmodel needsto be established taking into a range of aspects, includ-ing but not limited to different perspectives on infectiondynamics, controlmeasures, viral transmission dynam-ics, stability of the dynamics, long-term predictions,and so forth.

3 Prediction and control of COVID-19 using animproved SEIQR model with distributed timedelays

In this section, an improved epidemic model with time-varying parameters and distributed time delays is pro-posed based on the SEIQR model. With this model,quantitative analysis of control measures and quaran-tining is conducted. Based on the global COVID-19data, some key aspects and parameters that are reflec-tive of the effects of the measures and policies taken bydifferent countries are identified and discussed.

3.1 Improved SEIQR model with distributed timedelay

TheSEIQRmodel is a compartmentmodelwidely usedfor epidemiological modeling in disease propagation[13,28] as well as computer virus in the internet [29].Let it be supposed that the total population N is dividedinto five compartments so that S(t) + E(t) + I (t) +Q(t)+ R(t) ≡ N , where the state variables S(t), E(t),I (t),Q(t), and R(t)denote the populationof ‘Suscepti-ble’ (S), ‘Exposed’ (E), ‘Infectious’ (I), ‘Quarantined’(Q), ‘Removed’ (R) classes at any time t , respectively,they are defined by follows

– S(t): Susceptible cases. Initially, S(t) equal to thepopulation N , which is assumed to be a constant

– E(t): Exposed cases, infectedbut not yet infectious,in a latent period

– I (t): Infectious cases, with infectious capacity butneither quarantined nor recovered. One is consid-ered to be infectious if and only if in this stage

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Fig. 10 Construction of improved SEIQR model with delays: flow diagram

– Q(t): Quarantined cases, the infected individualsare confirmedby tests and then isolated fromothers.Quarantining will lead to loss of infectivity

– R(t): Removed cases (recovered or dead)

As depicted in the flow diagram of Fig. 10, trans-missions occur among these four compartments duringthe COVID-19 pandemic. The dynamics of the trans-mission can be described as follows. Individuals in theinfectious (I) stage can transmit infection to their neigh-bors through contacts. Individuals in the susceptible(S) stage become infected, and they are assigned tothe exposed (E) stage immediately once contacts haveoccurred. All the exposed (E) individuals are assumedto become infectious (I) after a (random) latency periodranging from 2 to 14 days. The infectious (I) individ-ual might be quarantined (Q) and isolated once theyshow symptoms and be tested. Finally, quarantined (Q)individuals will be assigned to remove (R) stage aftersome time. However, according to the recent reports[30,31], asymptomatically infected individuals widelyexist, and they can transmit the virus. Consequently, asecondary path of viral transmission needs to be con-sidered; that is, infectious (I) individuals assigned tothe removed (R) stagewithout being quarantined or iso-lated. It is notable that transmissions among these com-partments take some random period for any exposedindividual and these time lags during the transmissionsresult in a multiple distributed delay in the system.Hence, based on the proposed flow diagram of Fig. 10,the dynamical transfer flows among S, E, I, Q, and R

compartments can be written in the form of differentialequations as follows:⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

S = −Δse

E = Δse − Δei

I = Δei − Δiq − Δir

Q = Δiq − Δqr

R = Δir + Δqr

(3)

here ˙(·) is the derivative operation with respect to timet . Δ jk is the flow rate from compartment J to com-partment K ; that is, the subscripts se, ei, iq, ir , and qrdenote the flow rate from S to E, E to I, I to Q, I to R,and Q to R, respectively. Here, the flow rates Δ jk arenonlinear functions, which not only depend on the cur-rent state variables but they are also related to the paststates of the system. Thus, the time-delay effects areintroduced into the system. Intuitively, the time delayhere rises from the period of incubation, the period ofinfection, and the period of quarantine. According tothe flow diagram of Fig. 10, the flow rates among thestatesΔ jk (for jk = se, ei, iq, ir, qr ) can bewritten as⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

Δse(t) = βS

NI

Δei (t) =∫ 2τei

0pei (τ )Δse(t − τ)dτ

Δiq(t) = ζ

∫ 2τiq

0piq(τ )Δei (t − τ)dτ

Δir (t) = (1 − ζ )

∫ 2τir

0pir (τ )Δei (t − τ)dτ

Δqr (t) =∫ 2τqr

0pqr (τ )Δiq(t − τ)dτ

(4)

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Here, β is the infection rate, which is the average num-ber of effective contacts for each infectious individualwith others per unit time. β can be a time-dependentvariable, that is, β(t), and controlled by taking mea-sures against the epidemic. ζ is the quarantine rate,which is assumed to be a constant in thismodel. Besidesthese two parameters, there are four distributed delaysin this integral equation system Eqs. (4): the delayperiod from E to I (τei ), from I to Q (τiq ), I to R (τir ),and Q to R (τqr ). For different individuals, the periodfrom one state to another can be very different. How-ever, in statistics for a large population, the period ofincubation, infection, quarantine, and recovery shouldfollow some form of distribution. This has promptedthe authors to consider a novel model with distributeddelay in this research. In the literature [32,33], Weibulldistribution, gamma distribution, normal distribution,and other positive skewed distribution are used widely.Without the loss of generality, the authors assume all ofthe distributed timedelays, τ jk (for jk = ei, iq, ir, qr ),to follow the normal distribution, as shown in Fig.11. Hence, the probability density function p jk(τ jk)

in Eqs. (4) follows the normal distribution with a meanvalue τ jk and a standard deviation σ jk , and this distri-bution is written as

p jk(τ jk | τ jk, σ2jk) = 1

σ jk√2π

e− 1

2

(τ jk−τ jk

σ jk

)2

for jk = ei, iq, ir , and qr

(5)

Thedomainof theprobability density function p jk(τ jk)

above is all real numbers spanning −∞ to +∞. How-ever, the integral interval of the distributed delay inEqs. (4) is chosen to be [0, 2τ jk] to not include the non-physical interval less than zero andmake the numericalintegration feasible. It should be noted that this trunca-tion causes the integration of the density function overthe entire integral interval, that is, the area of the shadedregion in Fig. 11, to be less than 1. To normalize theprobability density function p jk in Eq. (5), a compen-sation approach can be used by dividing the densityfunction by the shaded area, for instance, dividing p jk

by a compensation factor 0.68 for σ jk = τ jk , by 0.95for σ jk = τ jk/2, and by 0.997 for σ jk = τ jk/3. Here,the authors have chosen σ jk = τ jk/4 with a compensa-tion factor 0.9999 and the resulting probability densityfunction is shown as Fig. 11.

The proposed, improved SEIQR model at hand,given by Eqs. (3), (4) is a time-varying system with

Fig. 11 Construction of improved SEIQRmodel: the distributedtime delay

multiple distributed time delays. Equations (3) form aset of five differential equations, which represents theframework of the SEIQR model. In the first equationin Eqs. (4), Δse is the transition rate from the compart-ment of susceptible individuals to the compartment ofinfectious individuals, and it is also called the force ofinfection [34]. For a susceptible individual, the transi-tion from S to E is simultaneous once the individualmakes an effective contact with an infectious individ-ual. The rest of the four equations in Eqs. (4), which,respectively, represent the flow rate of transition fromS to I, I to Q, I to R, and Q to R, are delay integralequations. These four delay integral equations intro-duce four distributed delays τ jk into the system. Thedistributed delays τ jk obey normal distribution withmean values τ jk and standard deviations σ jk . Due to thetime delay, the system dimension is infinite while thecontinuity and distribution of the delays introduce theinfinite number of time delays into the system. Hence,analytical solutions for the proposed system are impos-sible and numerical simulation is be conducted basedon discretization methods [35]. Comparing with themost recent epidemic model with discrete delays givenin [36], in the proposed model with distributed delays,one takes the individual differences in symptoms intoconsideration. This is expected to result in a more real-istic dynamic response to the virus and help improvethe epidemic model.

3.2 Numerical studies and model comparisons

The proposed dynamical system, which is given byEqs. (3), (4), is a set of time-varying nonlinear differ-ential equations with multiple distributed delays. Dueto the lack of a universal solver for this specific prob-

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Fig. 12 Comparison of predictions from the conventionalSEIQR model and the proposed SEIQR model with distributeddelays: a time-varying infection rate β, b predictions fromSEIQR model, and c predictions from improved SEIQR modelwith distributed delays

lem, numerical methods are developed for the simula-tions and system identification in this research. For thedelay integral system described by Eqs. (4), the com-putations can be done via the convolution algorithm ordirect integral. The time step size of 1 day is chosenfor the numerical simulations since the historical dataof COVID-19 from all the worldwide health organiza-tions are provided in one day intervals. The differentialequations given by Eqs. (3) are then integrated via lin-ear four-step Adams–Bashforth methods [37] with thesame step size of 1 day. All of the numerical algorithmshave been developed based on the MATLAB R2018aplatform.

Numerical studies have been conducted with boththe conventional SEIQR model and the improvedSEIQRmodel with distributed delays, as shown in Fig.12. For this comparison, the quarantine rate ζ is set tozero; this degenerates to the SEIR model without quar-antine. The population N is set to 10 thousand. Themean values of the distributed delays are set as fol-lows τei = 5 days, τiq = 4 days, τqr = 11 days, andτir = 15 days. The standard deviation is set to be one-fourth of the mean values of the distributed delays; thatis, σ jk = τ jk/4.As shown in Fig. 12a, the infection rate

β(t) is set to 0.8 initially and drops to 0.1 at the 30thday (the vertical dot-dashed line in the figure), whichrepresents the day on which measures taken to counterthe epidemic are put in place. In Figs. 12b, c, the y-axis on the left is the accumulated total infected cases,while the y-axis on the right is the daily increments ininfection cases. They can be written as follows:

Total infected cases =∫ t

0Δei (t)dt

Daily increments of infection =∫ t

t−1Δei (t)dt

(6)

The dynamic responses obtained with the conven-tional SEIQR model Fig. 12b and the improved modelFig. 12c are found to be consistent with each otherbefore the measures are taken. However, there are sig-nificant differences between the dynamics of these twomodels after the measures are put in place on the 30thday. In the conventional SEIQR model, an immediatechange in the predicted response can be observed withthe drop of β(t), resulting in a nonsmooth peak. Withthismodel, one notes that themeasures taken to counterthe virus have an immediate effect on the daily increasein infection cases. Intuitively, this immediate responseis not realistic and doesn’t agree with the COVID-19 data either. In contrast, with the improved model,the response continues to increase until it reaches asmooth peak for the daily increase in cases 3.8 dayslater than the date on which measures were initiated.This delayed response is more realistic and conformswith the COVID-19 dynamics worldwide: for instance,in the COVID-19 data in Wuhan, China, one noted apeak of daily increments on February 4, 2020 while thecity locked down on January 23, 2020. Besides, withthe improvedmodel, one notes a small bump around thefortieth day. This feature matches well the COVID-19dynamics in many counties and regions, such as SouthKorea. In summary, when the system is time-invariant,both the conventional SEIQR model and the improvedmodelwith distributed delays can capture the dynamicsof the epidemic well. When the system is time-varying,which is believed to be the scenariowith theCOVID-19pandemic, the improved SEIQRmodel with distributeddelays is more realistic and has significant advantagesin capturing the system responses compared to the con-ventional SEIQR model.

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3.3 Quantitative analysis of control measures

Infection control measures to reduce transmission ofCOVID-19 include universal source control, for exam-ple, covering the nose and mouth to contain respiratorysecretions, early quarantine, identification, and isola-tion of patients with suspected disease, use of appro-priate personal protective equipment, and environmen-tal disinfection [38,39]. In the proposed model in Eqs.(3), (4), all the aforementioned control measures arecollectively captured through two means: (i) control ofinfection rate β and (ii) control of quarantine rate ζ .

Quarantine is used to separate someone who hasbeen exposed to COVID-19 and confirmed to be pos-itive by testing, away from others. Quarantine helpsprevent the spread of disease by isolating infectiousindividuals from others. Depending on the policies indifferent regions, quarantined people may be in isola-tion or just stay at home. Here, the quarantined indi-viduals are those individuals who have tested positiveand are isolated from others. An infectious individ-ual would lose infectivity, once that person is quaran-tined. The quarantine rate ζ is the ratio of quarantinedcases to the infectious cases. ExtensiveCOVID-19 test-ing and screening can increase the quarantine rate ζ ,which requires more coronavirus testing according tothe World Health Organization. The quarantine rate ζ

and the infection rate β are the only two parametersthat the authors can use to control against the spread-ing of the virus in the improved SEIQR model withdistributed time delays, given by Eqs. (3), (4). In thissubsection, control of the infection spread is studied bytuning the infection rate β and the quarantine rate ζ .

Two variables, namely “total confirmed cases” and“daily increments of confirmed,” have been introducedinto the improved SEIQR model since the states S, E ,I , Q, and R in Eq. (3) are almost not realistic to beobserved directly in the real world. The total numberof confirmed cases and its derivative, that is, the dailyincrements in confirmed cases, are the most widelyused data in epidemiology for COVID-19. Accordingto the proposedmodel with distributed delays (Fig. 10),they can be written as

Total confirmed cases =∫ t

0Δiq(t)dt

Daily increments of confirmed =∫ t

t−1Δiq(t)dt

(7)

Fig. 13 Control of infection spread through quarantine, for pop-ulation N = 1 million, infection rate β = 0.5, and differentquarantine rates: a ζ = 0.1, b ζ = 0.5, and c ζ = 1

The total cases and daily increments as defined aboveinclude only the infectious individuals confirmed bytests and they are observable variables, which distin-guishes them from the total infected cases and dailyincrement of infection, as given in Eq. (6).

To show the effectiveness of quarantining in pre-venting an epidemic from spreading, quantitative anal-ysis of the control of an epidemic is carried out. Asshown in Fig. 13, as the quarantine rate ζ is increasedfrom 10 to 50%, and 100%, the number of total infectedcases (dashed blue lines) shows some slight drops fromalmost 1 million to 0.8 million, while the daily incre-ments of infections (dashed orange lines) show signif-icant drops from 56K per day to 22K per day. How-ever, both the total confirmed cases and daily incre-ments of confirmed are increased due to the increaseof COVID-19 testing in quarantine. The results revealthat the total number of infected cases would not havea significant drop by having more quarantining, but thepeak of daily infection of an epidemic can be sloweddown by increasing the quarantine rate.

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Fig. 14 Flattening of the curve of the COVID-19 epidemic byquarantining, for population N = 1 million, infection rate β =0.4, and different quarantine rates ζ range from 0 to 1

More comparisons of daily increments of infectionsfor different quarantine rates ζ (0% to 100%) have beencarried out, and the corresponding results are shown inFig. 14. The curves show that an increase in the quar-antine rate ζ can “flatten the curve” [40], which canbe helpful from the standpoint of a healthcare system.With the effects of “flattening the curve” as illustratedin this figure, the peak value of daily increments ofinfection reduced from 50K per day to 13K per day,and the peak position is also delayed by 17 days, allow-ing more time for healthcare capacity to increase andbetter cope with the patient load. Besides the increasein quarantine rate, decreasing the infection rate β isthe other approach to control against the spreading ofan epidemic. Effective decrease in the infection rate β

is generally due to the control interventions, includinglockdown of a city or region, stay-at-home order, socialdistancemeasures, wearingmasks in public places, andso on. A lot of measures, policies, and orders havebeen taken by local municipalities and governmentsand state governments worldwide since the outbreakof the COVID-19. To capture the effects of these mea-sures, the authors have introduced a time-varying infec-tion rate β(t) into the COVID-19 model.

As shown in Figs. 15a, c, e, a piecewise linear func-tion is used to describe the time-varying infection rateβ(t). The drop of β(t) on the 26th day is reflective ofthe measures and orders that are taken to counter theepidemic. Here, it is assumed that it takes seven daysfor themeasures and directives to completely activated;this results in a slope during the linear drop of the infec-tion rate β(t) as seen in the figures. In Fig. 15a, b, theinfection rate drops from β0 = 1 to β1 = 0.013 aftertaking measures against the epidemic. From the pre-

Fig. 15 Control of an epidemic by taking measures, with pop-ulation N = 1 million and quarantine rate ζ = 0.5, a, c, e threedifferent time-varying infection rates β(t), and b, d, f corre-sponding dynamic responses

dicted system response, it can be observed that the dailyincrements in infection continuously increases until alocal peak is reached 8 days after the measures are ini-tiated. Then, after a short period (five days) of drop, thedaily increments of infection start increasing again andthe system diverges. A rising tail can overwhelm healthsystems, leading to high fatalities, and herd immunityeventually. In Fig. 15c, d with the infection rate dropto β1 = 0.09, the system response can be said to beconvergent and the daily increments of infection dropslowly but with a very long tail. In this scenario, thedaily increments of infection are flattened by takingmeasures, but the total number of infected cases con-tinuously increases almost linearly and this can still

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lead to a tremendous number of infected cases after avery long saturation period. In Fig. 15e, fwith the infec-tion rate drop to β1 = 0.025, the daily increments ofinfection drop rapidly with a very short tail and there isa quick saturation in the total number of infected cases.For the control of COVID-19, this is the best scenariosince the epidemic can be ended within 1 month aftertaking measures. However, to drop the infection ratedrop to 0.025 is a great challenge for both the govern-ment and the people.

With the results of Fig. 15b, d, f, the authors haveillustrated that slightly different infection rates lead toquite different consequences. An undesirable scenariois one with herd immunity. In this case, one can havebreak down of healthcare systems and high fatalities. Aslightly better scenario is the slow drop casewith a verylong tail. In this case, the pandemic is under control andone will eventually converge. However, the sufferingcan be prolonged for a long time. Relatively speaking,the best scenario is the rapid drop situation with a shorttail. In this case, the pandemic is ended quickly withinabout 1 month after taking measures and the sufferingis not as prolonged as compared to the previous situa-tion. By examining the current COVID-19 data from allover the world, one can notice that all the COVID-19curves in different countries and regions can be cat-egorized into these three types mentioned above andthe associated dynamics can also be explained by theSEIQRmodel with distributed delays and time-varyinginfection rates.

3.4 Data driven dynamics of COVID-19 based on thedistributed-delay model

The nonlinear distributed delay systems given by Eqs.(3), (4) provide a feasible mathematical framework todescribe the evolution of COVID-19 infection dynam-ics as well as for modeling the control performancesassociatedwith quarantine and othermeasures. To havea better understanding of the COVID-19 dynamics andto forecast an epidemic’s evolution with high confi-dence, identification of the nonlinear dynamical sys-tem from the time series of COVID-19 data is highlyimportant. Based on the nonlinear regression algorithmand the parameter identification approach used anddescribed in Fig. 3, a similar approach is used to fit thedynamics of the proposed epidemic model Eqs. (3), (4)with the COVID-19 data.

Based on the data from the Web site of The COVIDTracking Project [26] andWorldometers [27], the iden-tified parameters from the data are as follows:

– β0: original infection rate before taking any mea-sures

– β1: infection rate after taking measures– tm : start date for implementation of measures– ζ : quarantine rate– R0: original reproduction number before taking anymeasures

– R1: reproduction number after taking measures

Here, the reproduction numbers R0 and R1 are deducedfrom the stability of the delay system Eqs. (3), (4),after linearization and discretization of the continuousdistributed delay. The reproduction number is a Floquetmultiplier for the simplified system of Eqs. (3), (4), andthis can be approximated as

R0,1 = β0,1(ζ τiq + (1 − ζ )τir ) (8)

where the R0,1 is used to denote R0 or R1 and β0,1 isused to denote β0 or β1. The Floquet multiplier R0,1

determines the stability of the solution obtained for theepidemic dynamics; that is, if the magnitude of R0,1

is greater than 1, the system is unstable and the con-sequence is herd immunity, which is an undesired sce-nario as mentioned in the last paragraph of Sect. 3.3.If the magnitude of R0,1 is 1, the system is criticallystable. If the magnitude of R0,1 is less than 1, the sys-tem is stable. For the COVID-19 pandemic, the initialreproduction numbers before measures are taken R0

are always great than 1 (generally between 3 and 4),as identified in Table 1. The reproduction number aftertaking of measures, R1 is the key to determine the evo-lution of epidemic dynamics. The smaller the magni-tude of R1 is, the faster COVID-19 spreading ends.

With the identified parameters from the COVID-19 data as listed in Table 1, the predicted data-drivendynamics of COVID-19 in these countries is shownin Fig. 16. The results illustrate excellent consistencybetween the COVID-19 data and the predictions fromthe constructed SEIQR model with distributed delays.The proposed model can capture most of the featuresof the COVID-19 dynamics observed from data, suchas the bump on the daily increments of South Koreafrom March 9, 2020, to March 25, 2020, and the con-tinuous linear increase in the total infection cases in theUSAandUK. The identified date tm when themeasures

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Fig. 16 Identified COVID-19 dynamics from data of different countries (upto May 7 data), based on the model with distributed delays

Table 1 Key parameters identified from the COVID-19 data of different countries worldwide, based on the proposed epidemic modelwith distributed delays Eqs. (3), (4)

Countries β0 β1 tm ζ R0 R1

China 0.50 0.022 Jan. 31 0.65 3.90 0.17

South Korea 1.38 0.048 Feb. 19 0.54 12.5 0.43

Italy 0.57 0.140 Mar. 12 0.85 3.22 0.79

Spain 0.83 0.160 Mar. 17 0.91 4.16 0.80

France 0.49 0.078 Mar. 23 0.73 3.41 0.54

Germany 0.63 0.107 Mar. 19 0.84 3.63 0.62

US 0.78 0.200 Mar. 23 0.92 3.81 0.98

UK 0.55 0.160 Mar. 27 0.78 3.50 0.99

Brazil 0.56 0.230 Mar. 23 0.58 4.82 1.98

were initiated also matches the reality, for example, theidentified date tm of China is January 31, 2020, whilethe actual period of lock down of cities and stay-at-home order taken by the Chinese government is in thewindow of January 23, 2020,–January 28, 2020.

From the results shown in Fig. 16 and Table 1, onecan also assess the effectiveness of measures taken tocounter COVID-19 in different countries. As listed inTable 1, the R1 numbers of China and South Koreaindicate that they these countries undertook measures,which helped best control the COVID-19 spread, witha reproduction number being R1 < 0.5. The effec-

tiveness of COVID-19 control in France, Germany,Italy, and Spain were effective but with a little higherreproduction number R1. Comparatively speaking, theCOVID-19 control measures in the USA and UK havenot been as effective at the time of writing this paper,as those undertaken in the previously mentioned coun-tries, since the reproduction number R1 is close to 1;that is, the system has critical stability. In this scenario,the daily infected cases will continue to drop graduallya very long tail and one can reach a tremendously highnumber of total infected cases, as shown in Fig. 16g, h.These results suggest that USA andUKwill experience

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COVID-19 spreading for an extended period of time.Switching to the southern hemisphere, the reproduc-tion number R1 of Brazil is still around 2. This meansan exponential outbreak of the COVID-19 pandemic.If this reproduction number is not controlled to comeunder 1 in the following weeks, the scenario can poten-tially lead to herd immunity, as shown in Fig. 16i. Thisis not a welcome situation for healthcare systems.

4 Concluding remarks

In this work, the spreading of COVID-19 among dif-ferent geographical regions worldwide has been mod-eled and studied based on the concepts of data-drivendynamical systems. First, the authors have used gen-eralized logistic functions to study the local infectionspreading in different regions. Based on a nonlinearregression algorithm, the statistical fit of the gener-alized logistic model has been optimized in a four-dimensional parameter space and the system dynamicsprediction and forecasting is driven by the COVID-19 data. Subsequently, inspired by the notion of thefinite element method from mechanics, a compositeglobal model with 148 elements (sub-models for dif-ferent regions) is established. In this composite globalmodel, each of the regions worldwide is regarded asan element of the overall global system and the globalmodel construction is based on COVID-19 data fromall over the world. This construction of a global modelbased on generalized function based statistical modelsfor local regions, and the use of this model to make pre-dictions and forecasting is one of the original contribu-tions of this work. This methodologymay be employedfor studies on global spreading of other pandemics aswell.

As an extension of the generalized logistic functionmodel, which in essence is a two-compartment model,an extended compartment model is constructed basedon the SEIQR model. In this model, both time- vary-ing parameters such as time-varying infection rates anddistributed time delays to reflect the differences in indi-vidual responses to an infection are introduced. Thisis another important and original aspect of this work.With this model, one is able to quantitatively assess theeffectiveness of different control measures taken suchas lock down of regions and quarantining. Again, themethodology employed here may be adopted for stud-ies of other epidemics. Based on the COVID-19 data,

some key parameters, which can reflect the effects ofthemeasures and policies taken in different regions anddifferent countries, have been identified and discussed.Based on the observed data-driven dynamics, the fol-lowing remarks are made.

(i) There are significant time lags among differentregions, for the outbreak of the COVID-19 pan-demic, as shown in Figs. 6 and 7. Based on thecurrent data and current situation, from the gener-alized logisticmodel prediction, one can glean thatmost of the countries and regionsworldwidewouldhave passed the infection peak of the COVID-19by July. For understanding the global dynamicsof the COVID-19, the proposed composite globalmodel is attractive compared to a model with lowdegrees of freedom. The composite global modelhas helped capture two or more waves of COVID-19 sweeping the globe. The first phase came tonotice in late January in Wuhan, China. Subse-quently, the second phase started in Europe andmoved to the USA. The next spreading phase isexpected to be dominated by the dynamics in theUSA, Russia, Brazil, and India, as shown in Fig.9.

(ii) The improved SEIQR model with time-varyinginfection rate and distributed delays is found to bebetter suited for understanding COVID-19 infec-tion dynamics with and without control measures.With this model predictions, one is not only able tocapture the effectiveness of the control measuresbut also anomalies such as the bump seen in theSouthKorea data. Based on the prediction compar-ison with available COVID-19 infection data, it isclear that with just a conventional SEIQR model,one is not able to capture the COVID-19 dynamicswell. With the new distributed delay model pre-sented here, one can also understand how mea-sures such as quarantining can help observationssuch as “flattening of the curve” (Fig. 14). It is alsoshown that slight differences in infection rates canlead to quite different consequences. To controlthe COVID-19 infection dynamics, the measurestaken against the epidemic need to be strict enoughto guarantee that the infection rate β1 is less thana critical value, as shown in Fig. 15.

(iii) Based on the data-driven COVID-19 dynamicsstudied with the distributed delay model, it is evi-dent the measures taken in countries such as China

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and South Korea were effective in dropping thereproduction number R1 to be below 0.5. Withregard to Europe, in France, Germany, Italy, andSpain, after the measures that were taken, the R1

value dropped to be in the range from 0.6 to 0.8.In the USA and UK, this reproduction number isclose to 1. A reproduction number less than 0.5 isdesirable, as it means a swift end to the spreadingof the epidemic. In general, a number below 1 indi-cates system stability. On the other hand, when thereproduction number is close to 1, where one hascritical stability, the COVID-19 infection dynam-ics can persist for an extended period of time andeven lead to herd immunity. In the southern hemi-sphere, for the current data from Brazil, one hasa reproduction number around 2, which is highlyundesirable.

One needs to temper the observations made in thiswork, by noting that in the modeling undertaken here,many aspects are not captured such as for example,the seasonal variations in temperature. These additionalaspects need to be considered as well for appropriateuse of findings from the current work.

Acknowledgements The author fromShanghai JiaoTongUni-versity gratefully acknowledge the support received through theNature Science Foundation of China, Grant No. 11902194. Thesenior author from the University of Maryland is thankful for thesupport provided through the Minta Martin Professorship, andthe Tools and Technologies for Pandemics Initiative supportedby the Center for Engineering Concepts Development and theNeilom Foundation.

Compliance with ethical standards

Conflict of interest The authors declare that they have no con-flict of interest.

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