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Epidemic analysis of COVID-19 in China by dynamical modeling

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Abstract and Figures

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.
ABSTRACT
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
1
. 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
4,49
. 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
10
, undetected international cases
11
, or the
actual infected cases in Wuhan and Hubei province based on statistics models
12
, or the epidemic outside Hubei province and
overseas
6,1315
. 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
16,17
and stochastic simulation
18,19
.
Compared with statistics methods
20,21
, mathematical modeling based on dynamical equations
15,2224
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
25
. Based on SEIR model, one can also assess the effectiveness of various measures
since the outbreak
23,24,2628
, 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)
1
compare the effects of lock-down of Hubei province on the transmission dynamics in Wuhan and Beijing
29
. 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
30
. 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
23,24
. The extension of classical SEIR model with delays
31,32
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
predictions.
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
(Hubei
), 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
progress.
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
2329
by introducing seven different states, i.e.
{S(t),P(t),E(t),I(t),Q(t),R(t),D(t)}
denoting at time
t
the
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
R
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).
Constant
N=S+P+E+I+Q+R+D
is the total population in a certain region. The coefficients
{α,β,γ1
,δ1
,λ(t),κ(t)}
2/10
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
α
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
λ
and the mortality rate
κ
are both time dependent. As confirmed in Fig. 2a-d,
the cure rate
λ(t)
is gradually increasing with the time, while the mortality rate
κ(t)
quickly decreases to less than
1%
and
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
Q(t)
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
λ(t)
and
κ(t)
, there are four unknown coefficients
{α,β,γ1
,δ1}
and two
initial conditions
{E0,I0}
about the hidden variables (other initial conditions are known from the data) have to be extracted from
the time series data
{Q(t)}
. 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
γ1
, which is generally estimated within several days
5,33,34
. And then for each
fixed
γ1
, we explore its influence on other parameters (
β=1
nearly unchanged), initial values, as well as the population
3/10
dynamics of quarantined cases and infected cases during best fitting. From Fig. 3a-b, to produce the same outcome, the
protection rate
α
and the reciprocal of the quarantine time
δ1
are both decreasing with the latent time
γ1
, 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
E0
and
I0
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
days),
E0
and
I0
have to be close to each other, which makes only their sum
E0+I0
matters during the fitting. An additional
important finding is that in all cases
β
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
patients5,33,34.
As a summary, we conclude that once the latent time
γ1
is fixed, the fitting accuracy on the time series data
{Q(t)}
basically depends on the values of
α
,
δ1
and
E0+I0
. 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
coefficients
35,36
. 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
6
, both show a positive correlation with the infection rate
β
and
the quarantined time
δ1
and a negative correlation with the protection rate
α
. These facts agree with the common sense and
highlight the necessity of self-protection (increase
α
and decrease
β
), timely disinfection (increase
α
and decrease
β
), early
quarantine (decrease
δ1
), etc. An exception is found for the initial total infected cases. Although a larger value of
E0+I0
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)
,
i.e.
, 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.
4/10
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
α
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.
E0
and
I0
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
95%
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
5/10
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:
1.
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
6/10
up at the beginning of April.
2.
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.
3.
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.
4.
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.
5,33,34
. And in this stage (from Jan. 1st to Jan. 20th), the number of total
7/10
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
R
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.
Acknowledgment
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
All related data and code are publicly available online at https://github.com/THU-ZCAM/2019-nCoV and https://github.com/THU-
ZCAM/SEIRHD-difference-model.
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Supplementary resource (1)

... Then we introduce a new approach to the logistic growth model (SLE), which is highly related to an anomalous diffusion introduced in complex systems. 20 Note, however, that the present statistical approach (SLE) does not have predictive capability, unlike dynamic approaches such as the susceptible-exposedinfected-recovered (SEIR) model, 2,21,22 which will be discussed in Sec. II. ...
... A dynamic model known as the susceptible (S)-exposed (E)-infected (I)-recovered (R) (SEIR) model is an epidemiological compartmental model, where the total time-dependent population N(t) is divided into S, E, I, and R [N(t) = S(t) + E(t) + I(t) + R(t)]. 2,21,22 We should solve four differential equations simultaneously to obtain N(t), under some assumptions. As will be discussed later, the COVID-19 pandemic in each country shows several waves, and it is not easy to cover all steps. 2 On the other hand, it is known that the logistic model 23 has led us to intuitively understand the growth dynamics of many microbiological populations, plants, and animals. ...
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The COVID-19 pandemic has presented unprecedented challenges globally, necessitating a deep understanding of its complex dynamics for effective mitigation strategies. We find that the intricate pattern of the pandemic is replicated well by the stretched logistic equation (SLE), which is a modification of the traditional logistic equation (TLE). The intrinsic infection rate involved in the logistic equation decreases with time (time-dependent) in the SLE, while it is a constant in the TLE. It is suggested that an anomalous sub-diffusion of the virus related to complex human activities is the main reason for the time-dependent spreading reaction rate. The SLE is compared with the compressed logistic equation, which can be applied to the closed system, such as bacterial growth, and the other approaches applied to COVID-19.
... The simulator is a member of the SEIR family of models (Kermack & McKendrick, 1927), widely used during the COVID pandemic (Currie et al., 2020). The SEIR model we employ follows the works of Peng et al. (2020) in China and Lu and Borgonovo (2023) in Italy and the United States, respectively. The code and data are available from https://github. ...
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Several graphical indicators have been recently introduced to help analysts visualize the marginal effects of inputs in complex models. The insights derived from such tools may help decision‐makers and risk analysts in designing interventions. However, we know little about the adequacy and consistency of different indicators. This work investigates popular marginal effect indicators to understand whether they yield indications consistent with the properties of the quantitative model under inspection. Specifically, we examine the notions of monotonicity, Lipschitz, and concavity consistency. Surprisingly, only PD functions satisfy all these notions of consistency. However, when selecting the indicators, in addition to consistency, analysts need to consider the risk of model extrapolation. For situations where such risk is under control, we utilize individual conditional expectations together with PD plots. Two applications, on a NASA space risk assessment model and a susceptible exposed infected recovered (SEIR) model for the COVID‐19 pandemic illustrate the insights obtained from these indicators.
... A similar approach has been applied to model the spread of measles in Nigeria [62], analyze varicella-zoster virus dynamics [63], and investigate COVID-19 spread [64]. An extended SEIR model has been employed to model COVID-19 spread in China [65] and to study the spread of the Ebola virus [66]. ...
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As digitalization and artificial intelligence advance, cybersecurity threats intensify, making malware—a type of software installed without authorization to harm users—an increasingly urgent concern. Due to malware’s social and economic impacts, accurately modeling its spread has become essential. While diverse models exist for malware propagation, their selection tends to be intuitive, often overlooking the unique aspects of digital environments. Key model choices include deterministic vs. stochastic, planar vs. spatial, analytical vs. simulation-based, and compartment-based vs. individual state-tracking models. In this context, our study assesses fundamental infection spread models to determine those most applicable to malware propagation. It is organized in two parts: the first examines principles of deterministic and stochastic infection models, and the second provides a comparative analysis to evaluate model suitability. Key criteria include scalability, robustness, complexity, workload, transparency, and manageability. Using consistent initial conditions, control examples are analyzed through Python-based numerical methods and agent-based simulations in NetLogo. The findings yield practical insights and recommendations, offering valuable guidance for researchers and cybersecurity professionals in applying epidemiological models to malware spread.
... A similar approach has been applied to model the spread of measles in Nigeria [61], analyse varicella-zoster virus dynamics [62], and investigate COVID-19 spread [63]. An extended SEIR model has been employed to model COVID-19 spread in China [64] and to study the spread of the Ebola virus [65]. ...
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Full-text available
As digitalization and artificial intelligence advance, cybersecurity threats intensify, making malware—a type of software installed without authorization to harm users—an increasingly urgent concern. Due to malware's social and economic impacts, accurately modeling its spread has become essential. While diverse models exist for malware propagation, their selection tends to be intuitive, often overlooking the unique aspects of digital environments. Key model choices include deterministic vs. stochastic, planar vs. spatial, analytical vs. simulation-based, and compartment-based vs. individual state-tracking models. In this context, our study assesses fundamental infection spread models to determine those most applicable to malware propagation. It is organized in two parts: the first examines principles of deterministic and stochastic infection models, and the second provides a comparative analysis to evaluate model suitability. Key criteria include scalability, robustness, complexity, workload, transparency, and manageability. Using consistent initial conditions, control examples are analyzed through Python-based numerical methods and agent-based simulations in NetLogo. The findings yield practical insights and recommendations, offering valuable guidance for researchers and cybersecurity professionals in applying epidemiological models to malware spread.
... On the other hand, the instrumental view does not view public participation as an "end" but rather "a means to an end", which implies an approach to achieving some goals [13]. Advocates of this perspective on participation point to many positive ends (results) of public participation in governance and development processes. ...
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The COVID-19 pandemic presented unmatched challenges for citizen participation worldwide in local government budgeting processes. Zimbabwe in particular was grappling with the pandemic’s effects and pre-existing governance issues. The pandemic ignited the central government to put COVID-19 public restrictions militating on citizen engagement in the budgeting process. The implemented measures were aimed at curbing the spread of the virus. While these measures were necessary for public health, they unintentionally constrained citizens’ ability to be unreservedly involved in local government decision-making. The paper employed a qualitative case study research design, employing interviews, Google Forms, and document analysis, to gather data exploring the experiences and perspectives of citizens, civil society organizations, and local government officials regarding the repercussions of constrained citizen participation in local government budgeting during the pandemic. The argument was hinged on the participatory theory based on two broad views: the normative and the instrumentalist perspectives. The paper notes that COVID-19 restrictions severely constrained citizen participation, limiting public input, deliberation, and accountability opportunities. The inability to convene public meetings, consultations, and workshops weakened the citizen-government engagement process, hindering transparency and the ability of citizens to influence resource allocation and stewardship. Furthermore, the restricted participation by marginalized communities exacerbates existing inequalities due to the technological divide hindering their ability to voice their concerns and interests in the budgeting processes. The paper calls for the exigent need for inventive methodologies to guarantee comprehensive and evocative citizen participation, by leveraging digital technologies and promoting alternative channels for engagement to augment citizen participation in local government budgeting.
... The inherent and complex character of COVID-19, as well as the unpredictability and dynamics of this pandemic, cannot be revealed by a single factor alone. The Table 2 [13], [14], [15], [16], [17], [18], [35] Outbreak characteristics and its transmission Regression, probabilistic compartmental models, time-age dependent compartmental Newly infected cases, recovered cases, rate of transmission, morbidity and mortality rates 2020 [19], [20], [21], [22], [23], [24], [25], [26] [27], [28], [29], [30], [31], [32], [33] ...
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The novel coronavirus caused by the Severe Acute Respiratory Syndrome Coronavirus 2 was originated in the Chinese city of Wuhan in 2019. Firstly, identified as an outbreak, as it was related to the original coronavirus that caused severe acute respiratory syndrome, but SARS-CoV-2 spreads faster and farther than the others. This is likely because of how easily it transmitted from person to person, even from asymptomatic carriers of the virus. But with due course of time, the genetic material of the virus has gone through innumerable changes and mutations and gradually swept into the globe affecting each and every sphere of human life like social, economic and physical as well as the mental wellbeing of any person. Majority of experts have agreed to the fact that unlike the previous pandemics in human history, the corona virus is of zoonotic origin i.e., the virus spreads between animals and the people. Although numerous ideas, including a number of conspiracy theories, have been put forward regarding the virus's origin but the lack of evidence for this explanation makes it considerably less credible. In this study, we compared the many elements that contributed to the corona virus's spread. It was observed that many socioeconomic and geographic elements were present along with these characteristics. By observing the alterations and the onset of symptoms, we were able to determine how COVID-19 differed from the ordinary flu. Studying these contrasts gave us a better understanding of how modeling COVID-19 with various models may be highly beneficial in combating the virus. In order to comprehend this, data and information from research publications that dealt with the issues and obstacles experienced by different people, communities etc. were taken into consideration.
... Compartmental models have emerged as a robust computational framework and have demonstrated remarkable success in the fight against COVID-19 disease. These models have been used to understand the dynamics of COVID-19 [5][6][7], assess intervention strategies algorithm. Our model involves estimating vaccination rates for Pfizer, Moderna, and Janssen vaccines. ...
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Compartmental models have emerged as robust computational frameworks and have yielded remarkable success in the fight against COVID-19. This study proposes a vaccination-based compartmental model for COVID-19 transmission dynamics. The model reflects the specific stages of COVID-19 infection and integrates a vaccination strategy, allowing for a comprehensive analysis of how vaccination rates influence the disease spread. We fit this model to daily confirmed COVID-19 cases in Tennessee, United States of America (USA), from June 4 to November 26, 2021, in a Bayesian inference approach using the Hamiltonian Monte Carlo (HMC) algorithm. First, excluding vaccination dynamics from the model, we estimated key epidemiological parameters like infection, recovery, and disease-induced death rates. This analysis yielded a basic reproduction number (R0) of 1.5. Second, we incorporated vaccination dynamics and estimated the vaccination rate for three vaccines: 0.0051 per day for both Pfizer and Moderna and 0.0059 per day for Janssen. The fitted curves show reductions in the epidemic peak for all three vaccines. Pfizer and Moderna vaccines bring the peak down from 8,029 infected cases to 5,616 infected cases, while the Janssen vaccine reduces it, to 6,493 infected cases. Simulations of the model by varying the vaccination rate and vaccine efficacy were performed. A highly effective vaccine (95% efficacy) with a daily vaccination rate of 0.006 halved COVID-19 infections, reducing cases from 8,029 to around 4,000. The results also show that the model’s prediction accuracy for new observations improves with the number of observed data used to train the model.
... Such a macro-modeling approach is particularly valuable in the early phase of a disease outbreak when health administrations aim to develop nationwide macro-intervention protocols. Even during the COVID-19 epidemic, numerous research works have proposed the generalized susceptible, exposed, infectious, removed model to predict the inflexion point for the growth curve (Peng, Yang, Zhang, Zhuge, & Hong, 2020). Additionally, Yang et al. (Yang et al., 2020) modified the proposed model and considered public health interventions in predicting the trend of COVID-19 in China. ...
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The level of surveillance and preparedness against epidemics varies across countries, resulting in different responses to outbreaks. When conducting an in-depth analysis of microinfection dynamics, one must account for the substantial heterogeneity across countries. However, many commonly used statistical model specifications lack the flexibility needed for sound and accurate analysis and prediction in such contexts. Nonlinear mixed effects models (NLMMs) constitute a specific statistical tool that can overcome these significant challenges. While compartmental models are well-established in infectious disease modeling and have seen significant advancements, Nonlinear Mixed Models (NLMMs) offer a flexible approach for handling heterogeneous and unbalanced repeated measures data, often with less computational effort than some individual-level compartmental modeling techniques. This study provides an overview of their current use and offers a solid foundation for developing guidelines that may help improve their implementation in real-world situations. Relevant scientific databases in the Research4life Access initiative programs were used to search for papers dealing with key aspects of NLMMs in infectious disease modeling (IDM). From an initial list of 3641 papers, 124 were finally included and used for this systematic and critical review spanning the last two decades, following the PRISMA guidelines. NLMMs have evolved rapidly in the last decade, especially in IDM, with most publications dating from 2017 to 2021 (83.33%). The routine use of normality assumption appeared inappropriate for IDM, leading to a wealth of literature on NLMMs with non-normal errors and random effects under various estimation methods. We noticed that NLMMs have attracted much attention for the latest known epidemics worldwide (COVID-19, Ebola, Dengue and Lassa) with the robustness and reliability of relaxed propositions of the normality assumption. A case study of the application of COVID-19 data helped to highlight NLMMs’ performance in modeling infectious diseases. Out of this study, estimation methods, assumptions, and random terms specification in NLMMs are key aspects requiring particular attention for their application in IDM.
... The SIR model is a straightforward dynamic model that depicts how illness spreads among communities [12]. Numerous researchers have utilized this approach to gain insights into disease transmission [5,13,[16][17][18]23]. The SIR model, which may examine disease transmission within a community, is the cornerstone of epidemiological modeling. ...
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In this work, the Backward Euler technique and the Adams-Bashforth 2-step method-two numerical approaches for solving the SIR model of epidemiology are compared for performance. An essential resource for comprehending the transmission of infectious illnesses like COVID-19 in the SIR model. While the explicit Adams-Bash forth 2-step approach is well known for its computing efficiency , the implicit Backward Euler method is noted for its stability. The study evaluates the accuracy, strength, and computing cost of the two approaches to determine which approach is best for simulating the spread of infectious illnesses. The SIR Model was easily solved using the Adams Bashforth 2-step analysis and the Backward Euler method. The approaches' solutions are close to the exact requirements. There are important distinctions between the two-step Adams Bash-forth and backward Euler procedures. The running time of the Adams Bashforth 2-step backward Euler method is shorter than that of the backward Euler method.
Article
The aim of this work is to create the SEIQR model for COVID-19 in Saudi Arabia. The inclusion of a quarantine compartment in the model’s architecture is crucial in halting the transmission of disease to the vulnerable class. Simulation had been run in two phases: Phase 1, which ran from January 4, 2020 to June 13, 2020, and phase 2, which ran from June 14, 2020 to March 6, 2021. The SEIQR model analysis yields local stability at the fundamental reproduction number and the disease-free equilibrium point when the next generation matrix approach is used. The reproduction number was determined to be 6.81 when γ\gamma was 2.0×1092.0 \times 10^{-9}, 7.49 when γ\gamma was 2.2×1092.2 \times 10^{-9} and 8.17 when γ\gamma was 2.4×1092.4 \times 10^{-9}. The outcomes of the simulation unambiguously show that phase 2 is the point at which the optimal condition is reached. The most important thing for any disease is to have control methods. Sensitivity analysis has been done as part of control strategies, and after that, a fuzzy reproduction number control approach has been put into practice.
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Outbreak to pandemic In response to global dispersion of severe acute respiratory syndrome–coronavirus 2 (SARS-CoV-2), quarantine measures have been implemented around the world. To understand how travel and quarantine influence the dynamics of the spread of this novel human virus, Chinazzi et al. applied a global metapopulation disease transmission model to epidemiological data from China. They concluded that the travel quarantine introduced in Wuhan on 23 January 2020 only delayed epidemic progression by 3 to 5 days within China, but international travel restrictions did help to slow spread elsewhere in the world until mid-February. Their results suggest that early detection, hand washing, self-isolation, and household quarantine will likely be more effective than travel restrictions at mitigating this pandemic. Science , this issue p. 395
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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.
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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.
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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.
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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
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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.
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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
In this paper, we propose a novel dynamical system with time delay to describe the outbreak of 2019-nCoV in China. One typical feature of this epidemic is that it can spread in the latent period, which can therefore be described by time delay process in the differential equations. The accumulated numbers of classified populations are employed as variables, which is consistent with the official data and facilitates the parameter identification. The numerical methods for the prediction of the outbreak of 2019-nCoV and parameter identification are provided, and the numerical results show that the novel dynamic system can well predict the outbreak trend so far. Based on the numerical simulations, we suggest that the transmission of individuals should be greatly controlled with high isolation rate by the government.