Epidemic analysis of COVID-19 in China by dynamical modeling

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DOI: 10.1101/2020.02.16.20023465
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The outbreak of novel coronavirus-caused pneumonia (COVID-19) in Wuhan has attracted worldwide attention. Here, we propose a generalized SEIR model to analyze this epidemic. Based on the public data of National Health Commission of China from Jan. 20th to Feb. 9th, 2020, we reliably estimate key epidemic parameters and make predictions on the inflection point and possible ending time for 5 different regions. According to optimistic estimation, the epidemics in Beijing and Shanghai will end soon within two weeks, while for most part of China, including the majority of cities in Hubei province, the success of anti-epidemic will be no later than the middle of March. The situation in Wuhan is still very severe, at least based on public data until Feb. 15th. We expect it will end up at the beginning of April. Moreover, by inverse inference, we find the outbreak of COVID-19 in Mainland, Hubei province and Wuhan all can be dated back to the end of December 2019, and the doubling time is around two days at the early stage.
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Epidemic analysis of COVID-19 in China by
dynamical modeling
Liangrong Peng1, Wuyue Yang2
, Dongyan Zhang3, Changjing Zhuge3†, Liu Hong2
1College of Mathematics and Data Science, Minjiang University, Fuzhou, 350108, P.R.C.
2Zhou Pei-Yuan Center for Applied Mathematics, Tsinghua University, Beijing, 100084, P.R.C.
3Beijing Institute for Scientific and Engineering Computing, College of Applied Sciences, Beijing University of
Technology, Beijing, 100124, P.R.C.
The outbreak of novel coronavirus-caused pneumonia (COVID-19) in Wuhan has attracted worldwide attention. Here, we
propose a generalized SEIR model to analyze this epidemic. Based on the public data of National Health Commission of China
from Jan. 20th to Feb. 9th, 2020, we reliably estimate key epidemic parameters and make predictions on the inflection point
and possible ending time for 5 different regions. According to optimistic estimation, the epidemics in Beijing and Shanghai
will end soon within two weeks, while for most part of China, including the majority of cities in Hubei province, the success of
anti-epidemic will be no later than the middle of March. The situation in Wuhan is still very severe, at least based on public
data until Feb. 15th. We expect it will end up at the beginning of April. Moreover, by inverse inference, we find the outbreak of
COVID-19 in Mainland, Hubei province and Wuhan all can be dated back to the end of December 2019, and the doubling time
is around two days at the early stage.
1 Introduction
A novel coronavirus, formerly called 2019-nCoV, or SARS-CoV-2 by ICTV (severe acute respiratory syndrome coronavirus
2, by the International Committee on Taxonomy of Viruses) caused an outbreak of atypical pneumonia, now officially called
COVID-19 by WHO (coronavirus disease 2019, by World Health Organization) first in Wuhan, Hubei province in Dec., 2019
and then rapidly spread out in the whole China
. As of 24:00 Feb. 13th, 2020 (Beijing Time), there are over 60, 000 reported
cases (including more than 1, 000 death report) in China, among which, over 80% are from Hubei province and over 50% from
Wuhan city, the capital of Hubei province2,3.
The central government of China as well as all local governments, including Hubei, has tightened preventive measures to
curb the spreading of COVID-19 since Jan. 2020. Many cities in Hubei province have been locked down and many measures,
such as tracing close contacts, quarantining infected cases, promoting social consensus on self-protection like wearing face
mask in public area, etc. However, until the finishing of this manuscript, the epidemic is still ongoing and the daily confirmed
cases maintain at a high level.
During this anti-epidemic battle, besides medical and biological research, theoretical studies based on either statistics or
mathematical modeling may also play a non-negligible role in understanding the epidemic characteristics of the outbreak, in
forecasting the inflection point and ending time, and in deciding the measures to curb the spreading.
For this purpose, in the early stage many efforts have been devoted to estimate key epidemic parameters, such as the basic
reproduction number, doubling time and serial interval, in which the statistics models are mainly used
. Due to the limitation
of detection methods and restricted diagnostic criteria, asymptomatic or mild patients are possibly excluded from the confirmed
cases. To this end, some methods have been proposed to estimate untraced contacts
, undetected international cases
, or the
actual infected cases in Wuhan and Hubei province based on statistics models
, or the epidemic outside Hubei province and
. With the improvement of clinic treatment of patients as well as more strict methods stepped up for containing
the spread, many researchers investigate the effect of such changes by statistical reasoning
and stochastic simulation
Compared with statistics methods
, mathematical modeling based on dynamical equations
receive relatively less
attention, though they can provide more detailed mechanism for the epidemic dynamics. Among them, the classic susceptible
exposed infectious recovered model (SEIR) is the most widely adopted one for characterizing the epidemic of COVID-19
outbreak in both China and other countries
. Based on SEIR model, one can also assess the effectiveness of various measures
since the outbreak
, which seems to be a difficult task for general statistics methods. SEIR model was also utilized to
Those authors contribute equally to this work.
Author to whom correspondence should be addressed. Electronic mail: zcamhl@tsinghua.edu.cn(L.Hong), zhuge@bjut.edu.cn(C.Zhuge)
compare the effects of lock-down of Hubei province on the transmission dynamics in Wuhan and Beijing
. As the dynamical
model can reach interpretable conclusions on the outbreak, a cascade of SEIR models are developed to simulate the processes
of transmission from infection source, hosts, reservoir to human
. There are also notable generalizations of SEIR model
for evaluation of the transmission risk and prediction of patient number, in which model, each group is divided into two
subpopulations, the quarantined and unquarantined
. The extension of classical SEIR model with delays
is another
routine to simulate the incubation period and the period before recovery. However, due to the lack of official data and the
change of diagnostic caliber in the early stage of the outbreak, most early published models were either too complicated to avoid
the overfitting problem, or the parameters were estimated based on limited and less accurate data, resulting in questionable
In this work, we carefully collect the epidemic data from the authoritative sources: the National, provincial and municipal
Health Commissions of China (abbreviated as NHC, see e.g. http://www.nhc.gov.cn/) until the article is completed (Feb. 16th,
2020). Then we follow the routine of dynamical modeling and focus on the epidemic of COVID-19 in five most interested
regions in China, i.e. the Mainland excluding Hubei province (denoted as Mainland
), Hubei province excluding Wuhan city
), Wuhan, Beijing and Shanghai. Such a design aims to minimize the influence of Hubei province and Wuhan city on
the data set due to their extremely large infected populations compared to other regions. Without further specific mention, these
conventions will be adopted thorough the whole paper.
By generalizing the classical SEIR model, e.g. introducing a new quarantined state and considering the effect of preventive
measures, key epidemic parameters for COVID-19, like the latent time, quarantine time and basic reproduction number are
determined in a relatively reliable way. The widely interested inflection point, ending time and total infected cases in hot cities
and regions are predicted and validated through both direct and indirect evidences. Furthermore, by inverse inference, the
starting date of this outbreak are estimated. The analysis of other hot spots in China, as well as overseas countries are still in
2 Model and Methods
2.1 Generalized SEIR model
Figure 1. The epidemic model for COVID-19. The highlighted part shows the classical SEIR model.
To characterize the epidemic of COVID-19 which outbroke in Wuhan at the end of 2019, we generalize the classical
SEIR model
by introducing seven different states, i.e.
denoting at time
respective number of the susceptible cases,insusceptible cases,exposed cases (infected but not yet be infectious, in a latent
period), infectious cases (with infectious capacity and not yet be quarantined), quarantined cases (confirmed and infected),
recovered cases and closed cases (or death). The adding of a new quarantined sate is driven by data, which together with
the recovery state takes replace of the original
state in the classical SEIR model. Their relations are given in Fig. 1and
characterized by a group of ordinary differential equations (or difference equations if we consider discrete time, see SI).
is the total population in a certain region. The coefficients
represent the protection rate, infection rate, average latent time, average quarantine time, cure rate, and mortality rate,
separately. Especially, to take the improvement of public health into account, such as promoting wearing face masks, more
effective contact tracing and more strict locking-down of communities, we assume that the susceptible population is stably
decreasing and thus introduce a positive protection rate
into the model. In this case, the basic reproduction number becomes
BRN =βδ1(1α)T,Tis the number of days.
It is noted that here we assume the cure rate
and the mortality rate
are both time dependent. As confirmed in Fig. 2a-d,
the cure rate
is gradually increasing with the time, while the mortality rate
quickly decreases to less than
becomes stabilized after Jan. 30th. This phenomenon is likely raised by the assistance of other emergency medical teams,
the application of new drugs, etc. Furthermore, the average contact number of an infectious person is calculated in Fig. 2e-f
and could provide some clue on the infection rate. It is clearly seen that the average contact number is basically stable over
time, but shows a remarkable difference among various regions, which could be attributed to different quarantine policies and
implements inside and outside Hubei (or Wuhan), since a less severe region is more likely to inquiry the close contacts of a
confirmed case. A similar regional difference is observed for the severe condition rate too. In Fig. 2g-h, Hubei and Wuhan
overall show a much higher severe condition rate than Shanghai. Although it is generally expected that the patients need a
period of time to become infectious, to be quarantined, or to be recovered from illness, but we do not find a strong evidence for
the necessity of including time delay (see SI for more details). As a result, the time-delayed equations are not considered in the
current work for simplicity.
Figure 2. (Color online) (a)-(b) The cure rate λ, (c)-(d) mortality rate κ, (e)-(f) average close contacts, and (g)-(h) severe
condition rate (see SI for their definitions) are calculated based on the public data from NHC of China from Jan. 20th to Feb.
9th for Mainland, Mainland, Hubei, Hubei, Wuhan, Beijing and Shanghai separately.
2.2 Parameter estimation
According to the daily official reports of NHC of China, the cumulative numbers of quarantined cases, recovered cases and
closed cases are available in public. However, since the latter two are directly related to the first one through the time dependent
recovery rate and mortality rate, the numbers of quarantined cases
plays a key role in our modeling. A similar argument
applies to the number of insusceptible cases too. Furthermore, as the accurate numbers of exposed cases and infectious cases
are very hard to determine, they will be treated as hidden variables during the study.
Leaving alone the time dependent parameters
, there are four unknown coefficients
and two
initial conditions
about the hidden variables (other initial conditions are known from the data) have to be extracted from
the time series data
. Such an optimization problem could be solved automatically by using the simulating annealing
algorithm (see SI for details). A major difficulty is how to overcome the overfitting problem.
To this end, we firstly prefix the latent time
, which is generally estimated within several days
. And then for each
, we explore its influence on other parameters (
nearly unchanged), initial values, as well as the population
dynamics of quarantined cases and infected cases during best fitting. From Fig. 3a-b, to produce the same outcome, the
protection rate
and the reciprocal of the quarantine time
are both decreasing with the latent time
, which is consistent
with the fact that longer latent time requires longer quarantine time. Meanwhile, the initial values of exposed cases and
infectious cases are increasing with the latent time. Since
include asymptomatic patients, they both should be larger
than the number of quarantined cases. Furthermore, as the time period between the starting date of our simulation (Jan. 20th)
and the initial outbreak of COVID-19 (generally believed to be earlier than Jan. 1st) is much longer than the latent time (3-6
have to be close to each other, which makes only their sum
matters during the fitting. An additional
important finding is that in all cases
is always very close to 1, which agrees with the observation that COVID-19 has an
extremely strong infectious ability. Nearly every unprotected person will be infected after a direct contact with the COVID-19
As a summary, we conclude that once the latent time
is fixed, the fitting accuracy on the time series data
basically depends on the values of
. And based on a reasonable estimation on the total number of infected
cases (see Fig. 3c-d), the latent time is finally determined as 2 days.
2.3 Sensitivity analysis
In order to further evaluate the influence of other fitting parameters on the long-term forecast, we perform sensitivity analysis
on the data of Wuhan (results for other regions are similar and not shown) by systematically varying the values of unknown
. As shown in Fig. 3e-f, the predicted total infected cases at the end of epidemic, as well as the the inflection
point, at which the basic reproduction number is less than 1
, both show a positive correlation with the infection rate
the quarantined time
and a negative correlation with the protection rate
. These facts agree with the common sense and
highlight the necessity of self-protection (increase
and decrease
), timely disinfection (increase
and decrease
), early
quarantine (decrease
), etc. An exception is found for the initial total infected cases. Although a larger value of
could substantially increase the final total infected cases, it shows no impact on the inflection point, which could be learnt from
the formula of basic reproduction number.
Figure 3. (Color online) Sensitivity analysis on parameters for the generalized SEIR model. The influence of the latent time
on (a) the protection rate αand quarantine time δ1, (b) the initial values of exposed cases E0and infected cases I0on Jan.
20th, (c) the cumulative quarantined cases, (d) the sum of exposed and infectious cases
E(t) + I(t)
, the currently infected
but not yet quarantined cases. (e) Effects of other parameters on the final total infected cases; (f) and the time period from the
starting point (Jan. 20th) to the inflection point (when the basic reproduction number becomes less than 1). In the top panel, the
value of latent time γ1is varied; while in the bottom panel, γ1is fixed as 2. All calculations are performed with respect to
the data of Wuhan city, with reported data (red circles) obtained from NHC of China from Jan. 20th to Feb. 9th, 2020.
3 Results and Discussion
3.1 Interpretation of the public data
We apply our pre-described generalized SEIR model to interpret the public data on the cumulative numbers of quarantined
cases, recovered cases and closed cases from Jan. 20th to Feb. 9th, which are published daily by NHC of China since Jan. 20th.
Our preliminary study includes five different regions, i.e. the Mainland, Hubei, Wuhan, Beijing and Shanghai.
Through extensive simulations, the optimal values for unknown model parameters and initial conditions, which best explain
the observed cumulative numbers of quarantined cases, recovered cases and closed cases (see Fig. 4), are determined and
summarized in Table 1. There are several remarkable facts could be immediately learnt from Table 1. Firstly, the protection rate
of Wuhan is significantly lower than other regions, showing many infected cases may not yet be well quarantined until Feb. 9th
(the smaller
for Wuhan does not necessarily mean people in Wuhan pay less attention to self-protection, but more likely due
to the higher mixing ratio of susceptible cases with infectious cases). Similarly, although the average protection rate for Hubei
is higher than that of Wuhan, it is still significantly lower than other regions. Secondly, the quarantine time for Beijing and
Shanghai are the shortest, that for Mainland
is in between. Again, the quarantine time for Wuhan and Hubei
are the longest.
Finally, the estimated number of total infected cases on Jan. 20th in five regions are all significantly larger than one, suggesting
the COVID-19 has already spread out nationwide at that moment. We will come back to this point in the next part.
Table 1.
Summary of all constant parameters for the generalized SEIR model.
denote the initial values for exposed
cases and infectious cases separately. The time-dependent cure rate λ(t)and mortality rate κ(t)can be read out from Fig. 2
and are given in SI.
3.2 Forecast for the epidemic of COVID-19
Most importantly, with the model and parameters in hand, we can carry out simulations for a longer time and forecast the
potential tendency of the COVID-19 epidemic. In Fig. 4and Fig. 5a-b, the predicted cumulative number of quarantined cases
and the current number of exposed cases plus infectious cases are plotted for next 30 days as well as for a shorter period of next
13 days. Official published data by NHC of China from Feb. 10th to 15th are marked in red spots and taken as a direct validation.
Overall, except Wuhan, the validation data show a well agreement with our forecast and all fall into the
confidence interval
(shaded area). And we are delighted to see most of them are lower than our predictions, showing the nationwide anti-epidemic
measures in China come into play. While for Wuhan city (and also Hubei province), due to the inclusion of suspected cases
with clinical diagnosis into confirmed cases (12364 cases for Wuhan and 968 cases for Hubei
on Feb. 12th) announced by
NHC of China since Feb. 12th during the preparation
Figure 4.
(Color online) Predictions of the generalized SEIR model on the cumulative quarantined cases (red solid lines), sum
of current exposed and infectious cases (blue solid lines), cumulative recovered cases (purple solid lines), and cumulative
closed cases (green solid lines) in Mainland, Hubei, Beijing, Shanghai, and Wuhan (from top to bottom). The red triangles,
purple asterisks and green circle represent the public data points between Jan. 20th and Feb. 9th, 2020. The shaded area
indicates predictions within 95% confidence interval. With the Euclidean distance k·k2, the average relative error
RE =qkyxk2
kxk2between the prediction yand public data xis evaluated for the cumulative quarantined cases, that is
RE =2.4%,5.6%,1.9%,2.9%,3.8% for Mainland, Hubei, Beijing, Shanghai and Wuhan. Parameters are taken in
accordance with Table 1.
of our manuscript, there is a sudden jump in the quarantined cases. Although it to some extent offsets our original
overestimates, it also reveals the current severe situation in Wuhan city, which requires much closer attention in the future.
Towards the epidemic of COVID-19, our basic predictions are summarized as follows:
Based on optimistic estimation, the epidemic of COVID-19 in Beijing and Shanghai would soon be ended within two
weeks (since Feb. 15th). While for most parts of mainland, the success of anti-epidemic will be no later than the middle
of March. The situation in Wuhan is still very severe, at least based on public data until Feb. 15th. We expect it will end
up at the beginning of April.
The estimated final total infected cases (not only total quarantined cases) for Beijing and Shanghai will be around
four hundred. This number is about 13-16 thousand for mainland (exclude Hubei province), 20-26 thousand for Hubei
province (exclude Wuhan city) and 55 thousand for Wuhan city.
According to the basic reproduction number shown in Fig. 5c, the inflection date for Beijing, Shanghai, mainland
(exclude Hubei) is around Jan. 30th, which is close to the reported Feb. 3rd for the last one based on daily new confirmed
cases. The inflection point for Hubei province (exclude Wuhan city) agrees with the reported Feb. 5th. These facts
indicate that the epidemic is now under control in most cities in China.
The predicted inflection point, ending date and final number of total infected cases are summarized in Fig. 5e. In
particular, the inflection point for Wuhan city is determined as Feb. 12th (data after Feb. 9th are not included into
parameter estimation). By coincidence, on the same day, we witnessed a sudden jump in the number of confirmed
cases due to a relaxed diagnosis caliber, meaning more suspected cases will receive better medical care and have much
lower chances to spread virus. Besides, Wuhan local government announced the completion of community survey on all
confirmed cases, suspected cases and close contacts in the whole city.
Figure 5. (Color online) (a-b) Predicted cumulative quarantined cases in the near future from Feb. 10th to Feb. 22nd, 2020.
The shaded area indicates a 95% confidence interval. The red spots represent the reported data of Wuhan from Feb. 10th to Feb.
15th, 2020 as a validation. Parameters are taken in accordance with Table 1. (c) The basic reproduction number, (d) the
estimated total infected cases at the early stage of COVID-19 epidemic between Dec. 28th, 2019 and Jan. 20th, 2020 by inverse
inference, and (e) a summary on the estimated inflection point, ending date and number of final total infected cases in
Mainland, Hubei, Wuhan, Beijing and Shanghai.
3.3 Inverse inference on the epidemic of COVID-19
Besides the forecast, the early trajectory of the COVID-19 outbreak is also critical for our understanding on its epidemic as well
as future prevention. To this end, by adopting the shooting method, we carry out inverse inference to explore the early epidemic
dynamics of COVID-19 since its onset in Mainland
, Hubei
, and Wuhan (Beijing and Shanghai are not considered due to their
too small numbers of infected cases on Jan. 20th). With respect to the parameters and initial conditions listed in Table 1, we
make an astonishing finding that, for all three cases, the outbreaks of COVID-19 all point to 20-25 days before Jan. 20th (the
starting date for public data and our modeling). It means the epidemic of COVID-19 in these regions is no later than Jan. 1st
(see Fig. 5d), in agreement with reports by Li et al.
. And in this stage (from Jan. 1st to Jan. 20th), the number of total
infected cases follows a nice exponential curve with the doubling time around 2 days. This in some way explains why statistics
studies with either exponential functions or logistic models could work very well on early limited data points. Furthermore, we
notice the number of infected cases based on inverse inference is much larger than the reported confirmed cases in Wuhan city
before Jan. 20th.
4 Conclusion
In this study, we propose a generalized SEIR model to analyze the epidemic of COVID-19, which was firstly reported in Wuhan
last December and then quickly spread out nationwide in China. Our model properly incorporates the intrinsic impact of hidden
exposed and infectious cases on the entire procedure of epidemic, which is difficult for traditional statistics analysis. A new
quarantined state, together with the recovery state, takes replace of the original
state in the classical SEIR model and correctly
accounts for the daily reported confirmed infected cases and recovered cases.
Based on detailed analysis of the public data of NHC of China from Jan. 20th to Feb. 9th, we estimate several key
parameters for COVID-19, like the latent time, the quarantine time and the basic reproduction number in a relatively reliable
way, and predict the inflection point, possible ending time and final total infected cases for Hubei, Wuhan, Beijing, Shanghai,
etc. Overall, the epidemic situations for Beijing and Shanghai are optimistic, which are expected to end up within two weeks
(from Feb. 15th, 2020). Meanwhile, for most parts of mainland including the majority of cities in Hubei province, it will be no
later than the middle of March. We should also point out that the situation in Wuhan city is still very severe. More effective
policies and more efforts on medical care and clinical research are eagerly needed. We expect the final success of anti-epidemic
will be reached at the beginning of this April.
Furthermore, by inverse inference, we find that the outbreak of this epidemic in Mainland, Hubei, and Wuhan can all be
dated back to 20-25 days ago with respect to Jan. 20th, in other words the end of Dec. 2019, which is consistent with public
reports. Although we lack the knowledge on the first infected case, our inverse inference may still be helpful for understanding
the epidemic of COVID-19 and preventing similar virus in the future.
Conflict of interest
The authors declare no conflict of interest.
We acknowledged the financial supports from the National Natural Science Foundation of China (Grants No. 21877070,
11801020), Startup Research Funding of Minjiang University (mjy19033) and the Fundamental Research Funding of Beijing
University of Technology (006000546318505, 006000546319509, 006000546319526). The authors would like to thank Dr.
Yajing Huang for her stimulating discussions.
Author contributions
L.H. designed the project. W.Y. and D.Z. collected the data. All authors analyzed the data. L.P., W.Y., C.Z. and L.H. wrote the
manuscript, and all authors reviewed it.
Additional Information
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Supplementary resource

  • ... where the coefficients α, β, γ −1 , δ −1 , λ, k represent the protection rate, infection rate, average latent time, average quarantine time, cure rate, and mortality rate, respectively. [9] The reproduction number of the model is given by ...
    ... We fix the latent time γ −1 and β directly obtaining them from the available literature. [9] 2) The two remaining parameters α and δ −1 are estimated by fitting our model to the daily data of the quarantined cases Q(t), recovered cases R(t), and deceased cases D(t). ...
    Full-text available
    The novel coronavirus disease, COVID-19, has had an unprecedented impact on life in 2020. To understand and control the disease, it is important to analyze and build an effective mathematical model to aid in the ongoing efforts to contain the virus and the disease. In this paper, we study the SEIRQDP model, in detail. The results of the model and analysis performed on data from Italy give us a direct comparison of the current situation of the virus in Italy. Also, the results predicted by the model demonstrate the effectiveness of measures taken by the Italian government. Through sensitivity analysis, the protection rate was found to be the key parameter for the model. Forecasts for the basic reproduction number presented to show the possible trajectories of the disease and its impact in the immediate future. Finally, the SEIRQDP model was found to be highly adaptable and accurate with the availability of accurate data.
  • ... t −1 latent is the inverse of the latent period of the virus, or the time before an exposed subject becomes infectious. We assume a value of t −1 latent = 0.5 days −1 based on Peng et al. 9 The parameter ρ describes the infectious period for subjects with unconfirmed infections, for which we assume a value of ρ = 0.1 days −1 , based on Rocklöv et al. 28 The rates at which subjects with confirmed infections recover and perish are described by β and µ , respectively. Finally, γ describes the rate at which recovered subjects become susceptible to the disease again. ...
    ... The "seed" of the outbreak is therefore the initial number of exposed subjects e 0 = e(t = 0) , which we include as a parameter in the least-squares problem. We include bounds on the possible parameter values, based on values reported for similar models fitted to data from other regions 9 . Note that, while κ reflects the level of testing, it also affects the predicted asymptomatic ratio, which cannot be controlled. ...
    Full-text available
    The novel coronavirus SARS-CoV-2 and resulting COVID-19 disease have had an unprecedented spread and continue to cause an increasing number of fatalities worldwide. While vaccines are still under development, social distancing, extensive testing, and quarantining of confirmed infected subjects remain the most effective measures to contain the pandemic. These measures carry a significant socioeconomic cost. In this work, we introduce a novel optimization-based decision-making framework for managing the COVID-19 outbreak in the US. This includes modeling the dynamics of affected populations, estimating the model parameters and hidden states from data, and an optimal control strategy for sequencing social distancing and testing events such that the number of infections is minimized. The analysis of our extensive computational efforts reveals that social distancing and quarantining are most effective when implemented early, with quarantining of confirmed infected subjects having a much higher impact. Further, we find that “on-off” policies alternating between strict social distancing and relaxing such restrictions can be effective at “flattening” the curve while likely minimizing social and economic cost.
  • ... At the level of the numerical performance of the present q-SEIR model for the COVID-19 pandemic, it advantageously compares with models including time-dependent coefficients [44][45][46][47]. For instance, the SEIQRDP model [44,45,47] includes seven equations with several coefficients, two of them phenomenologically being time-dependent. ...
    ... At the level of the numerical performance of the present q-SEIR model for the COVID-19 pandemic, it advantageously compares with models including time-dependent coefficients [44][45][46][47]. For instance, the SEIQRDP model [44,45,47] includes seven equations with several coefficients, two of them phenomenologically being time-dependent. It does fit rather well the COVID-19 reported data until a given date. ...
    Full-text available
    We generalize the phenomenological, law of mass action-like, SIR and SEIR epidemiological models to situations with anomalous kinetics. Specifically, the contagion and removal terms, normally linear in the fraction $I$ of infecteds, are taken to depend on $I^{\,q_{up}}$ and $I^{\,q_{down}}$, respectively. These dependencies can be understood as highly reduced effective descriptions of contagion via anomalous diffusion of susceptibles and infecteds in fractal geometries, and removal (i.e., recovery or death) via complex mechanisms leading to slowly decaying removal-time distributions. We obtain rather convincing fits to time series for both active cases and mortality with the same values of $(q_{up},q_{down})$ for a given country, suggesting that such aspects may in fact be present in the evolution of the Covid-19 pandemic. We also obtain approximate values for the effective population $N_{eff}$, which turns out to be a small percentage of the entire population $N$ for each country.
  • ... To date, a large number of mathematical models have been published to understand the transmission dynamics of Covid-19 in a given community, state, or a country [see for example [1][2][3][4]. However, these studies do not consider the influence that the migration effect between neighboring countries can exert, and thus be able to design public policies that best fit the cases registered across the borders. ...
    Full-text available
    We present a mathematical model that would allow one to describe the transmission dynamics of Covid-19 between two neighboring cities or countries. This model is analyzed both analytical and numerically. It is a preliminary model because it assumes that the migration rate and the mortality rate are constant over time. Despite these simplifications, only two of the four equilibrium conditions were deduced from the system of equations proposed in this paper. Finally, we show an example the transmission dynamics between Portugal and Spain according to the cases registered before June 3, 2020.
  • ... The scientific community reacted as never before, and many researchers focused on this urgent topic [3][4][5][6][7][8][9][10]. The mathematical and computer science communities are also studying this challenging problem, and we testimony the recent emergence of new models and algorithmic approaches. ...
    This paper tackles the information of 133 RNA viruses available in public databases under the light of several mathematical and computational tools. First, the formal concepts of distance metrics, Kol-mogorov complexity and Shannon information are recalled. Second, the computational tools available presently for tackling and visualizing patterns embedded in datasets, such as the hierarchical clustering and the multidimensional scaling, are discussed. The synergies of the common application of the mathematical and computational resources are then used for exploring the RNA data, cross-evaluating the normalized compression distance, entropy and Jensen-Shannon divergence, versus representations in two and three dimensions. The results of these different perspectives give extra light in what concerns the relations between the distinct RNA viruses.
  • ... The scientific community reacted as never before, and many researchers focused on this urgent topic [3][4][5][6][7][8][9][10]. The mathematical and computer science communities are also studying this challenging problem, and we testimony the recent emergence of new models and algorithmic approaches. ...
    This paper tackles the information of 133 RNA viruses available in public databases under the light of several mathematical and computational tools. First, the formal concepts of distance metrics, Kolmogorov complexity and Shannon information are recalled. Second, the computational tools available presently for tackling and visualizing patterns embedded in datasets, such as the hierarchical clustering and the multidimensional scaling, are discussed. The synergies of the common application of the mathematical and computational resources are then used for exploring the RNA data, cross-evaluating the normalized compression distance, entropy and Jensen–Shannon divergence, versus representations in two and three dimensions. The results of these different perspectives give extra light in what concerns the relations between the distinct RNA viruses.
  • ... Outcomes gauge the fast spread of SARS. Peng et al. 7 applied and discussed a generalized mathematical model for SEIR. The model includes a new state quarantined with a recovery state, the analysis is based on public data of NHC of china from 20 January to 9 February 2020. ...
    Full-text available
    The global pandemic of COVID-19 has raised several questions and attracted researchers from all of the disciplines of scientific research. Regardless of advances in science and technology, equipped laboratories of virology, high literacy rates, and medical resources in developed countries, several nations and their health care systems completely failed to overcome the disaster. The fast spread is caused by frequent air travel for business, tourism, education, etc. COVID-19 can infect third world countries severely. United States of America has the highest per capita spending of health still 1/3rd of the global burden of COVID-19 has consumed existing resources. The WHO has declared COVID-19 as a pandemic. More than 200 countries and territories have reported infected cases. The quarantine is the most effective way to slow the spread of disease and “Flatting of Curve” is a phenomenon to tackle the surge by health systems. To achieve good results from existing Medical Health Care Systems (MHCS), an accurate prediction for the spread of disease is crucial. This study utilizes the generalized method of SIR to accurately predict the spread of COVID-19 associated infection, recoveries, and deaths in Pakistan. The data from the National Command and Control of Pakistan (NCCP) is utilized. Through multiple cases applied on currently available data, the proposed mathematical models predict that by the end of April about more than 14553 infected and about 310 deaths are in Pakistan. The recovery rate is highest in the region up to 99.87 %.
  • ... A relatively large volume of COVID-19-focused research has been dedicated to predicting when the epidemics will peak. [51][52][53] Noteworthy was the Institute for Health Metrics and Evaluation (IHME) 54 prediction model that provided state-level estimates for the next 4 months using a nonlinear mixed-effects model with an incorporated parametrized Gaussian structure for cumulative error rates. Unlike the IHME, our focus was a short-term county-level analysis on a daily scale. ...
    Purpose There are growing signs that the COVID‐19 virus has started to spread to rural areas and can impact the rural health care system that is already stretched and lacks resources. To aid in the legislative decision process and proper channelizing of resources, we estimated and compared the county‐level change in prevalence rates of COVID‐19 by rural‐urban status over 3 weeks. Additionally, we identified hotspots based on estimated prevalence rates. Methods We used crowdsourced data on COVID‐19 and linked them to county‐level demographics, smoking rates, and chronic diseases. We fitted a Bayesian hierarchical spatiotemporal model using the Markov Chain Monte Carlo algorithm in R‐studio. We mapped the estimated prevalence rates using ArcGIS 10.8, and identified hotspots using Gettis‐Ord local statistics. Findings In the rural counties, the mean prevalence of COVID‐19 increased from 3.6 per 100,000 population to 43.6 per 100,000 within 3 weeks from April 3 to April 22, 2020. In the urban counties, the median prevalence of COVID‐19 increased from 10.1 per 100,000 population to 107.6 per 100,000 within the same period. The COVID‐19 adjusted prevalence rates in rural counties were substantially elevated in counties with higher black populations, smoking rates, and obesity rates. Counties with high rates of people aged 25‐49 years had increased COVID‐19 prevalence rates. Conclusions Our findings show a rapid spread of COVID‐19 across urban and rural areas in 21 days. Studies based on quality data are needed to explain further the role of social determinants of health on COVID‐19 prevalence.
  • Preprint
    Full-text available
    COVID-19 is now in an epidemic phase, with a second outbreak likely to appear at any time. The intensity and timing of a second outbreak is a common concern worldwide. In this study, we made scenario projections of the potential second outbreak of COVID-19 using a statistical-epidemiology model, which considers both the impact of seasonal changes in meteorological elements and human social behaviors such as protests and city unblocking. Recent street protests in the United States and other countries are identified as a hidden trigger and amplifier of the second outbreak. The scale and intensity of subsequent COVID-19 outbreaks in the U.S. cities where the epidemic is under initial control are projected to be much greater than those of the first outbreak. For countries without reported protests, lifting the COVID-19 related restrictions prematurely would accelerate the spread of the disease and place mounting pressure on the local medical system that is already overloaded. We anticipate these projections will support public health planning and policymaking by governments and international organizations.
  • Article
    Full-text available
    In this paper we explore how the COVID-19 pandemic, also known as Coronavirus pandemic, affected the operation of small electric grids, and what can this event teach us on the readiness of such grids in the face of future global health crises. We focus on three major effects: changing patterns of generation and consumption, frequency stability, and the joint impact of low consumption and high share of renewable energy sources. Specifically, we analyze changes in consumption in the Israeli, Estonian, and Finnish grids, and attempt to identify patterns of consumption changes that may be explained by the pandemic. We also analyze changes in voltage and frequency, and show that the low consumption caused significant deviations from the nominal values of both parameters. One main conclusion is that the reduced energy consumption during the pandemic is critical, and has a major effect on the operation of small electric grids. Another conclusion is that since the pandemic pushed the relative share of renewable energy to record highs, this event may help us to better understand the influence of a high share of renewables on small grids, thus offering a glance into a renewable-rich future.
  • Article
    Motivated by the rapid spread of COVID-19 in Mainland China, we use a global metapopulation disease transmission model to project the impact of travel limitations on the national and international spread of the epidemic. The model is calibrated based on internationally reported cases, and shows that at the start of the travel ban from Wuhan on 23 January 2020, most Chinese cities had already received many infected travelers. The travel quarantine of Wuhan delayed the overall epidemic progression by only 3 to 5 days in Mainland China, but has a more marked effect at the international scale, where case importations were reduced by nearly 80% until mid February. Modeling results also indicate that sustained 90% travel restrictions to and from Mainland China only modestly affect the epidemic trajectory unless combined with a 50% or higher reduction of transmission in the community.
  • Article
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    Objective To estimate the serial interval of novel coronavirus (COVID-19) from information on 28 infector-infectee pairs. Methods We collected dates of illness onset for primary cases (infectors) and secondary cases (infectees) from published research articles and case investigation reports. We subjectively ranked the credibility of the data and performed analyses on both the full dataset (n = 28) and a subset of pairs with highest certainty in reporting (n = 18). In addition, we adjust for right truncation of the data as the epidemic is still in its growth phase. Results Accounting for right truncation and analyzing all pairs, we estimated the median serial interval at 4.0 days (95% credible interval [CrI]: 3.1, 4.9). Limiting our data to only the most certain pairs, the median serial interval was estimated at 4.6 days (95% CrI: 3.5, 5.9). Conclusions The serial interval of COVID-19 is close to or shorter than its median incubation period. This suggests that a substantial proportion of secondary transmission may occur prior to illness onset. The COVID-19 serial interval is also shorter than the serial interval of severe acute respiratory syndrome (SARS), indicating that calculations made using the SARS serial interval may introduce bias.
  • Article
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    Background Isolation of cases and contact tracing is used to control outbreaks of infectious diseases, and has been used for coronavirus disease 2019 (COVID-19). Whether this strategy will achieve control depends on characteristics of both the pathogen and the response. Here we use a mathematical model to assess if isolation and contact tracing are able to control onwards transmission from imported cases of COVID-19. Methods We developed a stochastic transmission model, parameterised to the COVID-19 outbreak. We used the model to quantify the potential effectiveness of contact tracing and isolation of cases at controlling a severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2)-like pathogen. We considered scenarios that varied in the number of initial cases, the basic reproduction number (R0), the delay from symptom onset to isolation, the probability that contacts were traced, the proportion of transmission that occurred before symptom onset, and the proportion of subclinical infections. We assumed isolation prevented all further transmission in the model. Outbreaks were deemed controlled if transmission ended within 12 weeks or before 5000 cases in total. We measured the success of controlling outbreaks using isolation and contact tracing, and quantified the weekly maximum number of cases traced to measure feasibility of public health effort. Findings Simulated outbreaks starting with five initial cases, an R0 of 1·5, and 0% transmission before symptom onset could be controlled even with low contact tracing probability; however, the probability of controlling an outbreak decreased with the number of initial cases, when R0 was 2·5 or 3·5 and with more transmission before symptom onset. Across different initial numbers of cases, the majority of scenarios with an R0 of 1·5 were controllable with less than 50% of contacts successfully traced. To control the majority of outbreaks, for R0 of 2·5 more than 70% of contacts had to be traced, and for an R0 of 3·5 more than 90% of contacts had to be traced. The delay between symptom onset and isolation had the largest role in determining whether an outbreak was controllable when R0 was 1·5. For R0 values of 2·5 or 3·5, if there were 40 initial cases, contact tracing and isolation were only potentially feasible when less than 1% of transmission occurred before symptom onset. Interpretation In most scenarios, highly effective contact tracing and case isolation is enough to control a new outbreak of COVID-19 within 3 months. The probability of control decreases with long delays from symptom onset to isolation, fewer cases ascertained by contact tracing, and increasing transmission before symptoms. This model can be modified to reflect updated transmission characteristics and more specific definitions of outbreak control to assess the potential success of local response efforts. Funding Wellcome Trust, Global Challenges Research Fund, and Health Data Research UK.
  • Article
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    Background: As reported by the World Health Organization, a novel coronavirus (2019-nCoV) was identified as the causative virus of Wuhan pneumonia of unknown etiology by Chinese authorities on 7 January, 2020. The virus was named as severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) by International Committee on Taxonomy of Viruses on 11 February, 2020. This study aimed to develop a mathematical model for calculating the transmissibility of the virus. Methods: In this study, we developed a Bats-Hosts-Reservoir-People transmission network model for simulating the potential transmission from the infection source (probably be bats) to the human infection. Since the Bats-Hosts-Reservoir network was hard to explore clearly and public concerns were focusing on the transmission from Huanan Seafood Wholesale Market (reservoir) to people, we simplified the model as Reservoir-People (RP) transmission network model. The next generation matrix approach was adopted to calculate the basic reproduction number (R0) from the RP model to assess the transmissibility of the SARS-CoV-2. Results: The value of R0 was estimated of 2.30 from reservoir to person and 3.58 from person to person which means that the expected number of secondary infections that result from introducing a single infected individual into an otherwise susceptible population was 3.58. Conclusions: Our model showed that the transmissibility of SARS-CoV-2 was higher than the Middle East respiratory syndrome in the Middle East countries, similar to severe acute respiratory syndrome, but lower than MERS in the Republic of Korea.
  • Article
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    The basic reproduction number of an infectious agent is the average number of infections one case can generate over the course of the infectious period, in a naïve, uninfected population. It is well-known that the estimation of this number may vary due to several methodological issues, including different assumptions and choice of parameters, utilized models, used datasets and estimation period. With the spreading of the novel coronavirus (2019-nCoV) infection, the reproduction number has been found to vary, reflecting the dynamics of transmission of the coronavirus outbreak as well as the case reporting rate. Due to significant variations in the control strategies, which have been changing over time, and thanks to the introduction of detection technologies that have been rapidly improved, enabling to shorten the time from infection/symptoms onset to diagnosis, leading to faster confirmation of the new coronavirus cases, our previous estimations on the transmission risk of the 2019-nCoV need to be revised. By using time-dependent contact and diagnose rates, we refit our previously proposed dynamics transmission model to the data available until January 29th, 2020 and re-estimated the effective daily reproduction ratio that better quantifies the evolution of the interventions. We estimated when the effective daily reproduction ratio has fallen below 1 and when the epidemics will peak. Our updated findings suggest that the best measure is persistent and strict self-isolation. The epidemics will continue to grow, and can peak soon with the peak time depending highly on the public health interventions practically implemented. Keywords: Novel coronavirus, Emerging and reemerging pathogens, Mathematical modeling, Basic reproduction number, Effective daily reproduction ratio
  • Article
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    We evaluated effectiveness of thermal passenger screening for 2019-nCoV infection at airport exit and entry to inform public health decision-making. In our baseline scenario, we estimated that 46% (95% confidence interval: 36 to 58) of infected travellers would not be detected, depending on incubation period, sensitivity of exit and entry screening, and proportion of asymptomatic cases. Airport screening is unlikely to detect a sufficient proportion of 2019-nCoV infected travellers to avoid entry of infected travellers.
  • Article
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    Backgrounds: An ongoing outbreak of a novel coronavirus (2019-nCoV) pneumonia hit a major city of China, Wuhan, December 2019 and subsequently reached other provinces/regions of China and countries. We present estimates of the basic reproduction number,R0, of 2019-nCoV in the early phase of the outbreak. Methods: Accounting for the impact of the variations in disease reporting rate, we modelled the epidemic curve of 2019-nCoV cases time series, in mainland China from January 10 to January 24, 2020, through the exponential growth. With the estimated intrinsic growth rate (γ), we estimated R0 by using the serial intervals (SI) of two other well-known coronavirus diseases, MERS and SARS, as approximations for the true unknown SI. Findings: The early outbreak data largely follows the exponential growth. We estimated that the meanR0 ranges from 2.24 (95%CI: 1.96-2.55) to 3.58 (95%CI: 2.89-4.39) associated with 8-fold to 2-fold increase in the reporting rate. We demonstrated that changes in reporting rate substantially affect estimates of R0. CONCLUSION: The mean estimate ofR0 for the 2019-nCoV ranges from 2.24 to 3.58, and significantly larger than 1. Our findings indicate the potential of 2019-nCoV to cause outbreaks.
  • Article
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    A cluster of pneumonia cases linked to a novel coronavirus (2019‐nCoV) was reported by China in late December 2019. Reported case incidence has now reached the hundreds, but this is likely an underestimate. As of 24 January 2020, with reports of thirteen exportation events, we estimate the cumulative incidence in China at 5502 cases (95% confidence interval: 3027, 9057). The most plausible number of infections is in the order of thousands, rather than hundreds, and there is a strong indication that untraced exposures other than the one in the epidemiologically linked seafood market in Wuhan have occurred.