Модифікована SEIRD-модель опису епідемії COVID-19
This article is devoted to mathematical models in epidemiology, in particular SIR, SEIR, and SEIRD models. It explores the importance of these models in predicting the spread of infectious diseases and evaluating the effectiveness of control measures. These models allow for assessing important epide...
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2023
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System research and information technologies| _version_ | 1866302897596137472 |
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| author | Klymenko, Anastasiia Podkolzin, Gleb |
| author_facet | Klymenko, Anastasiia Podkolzin, Gleb |
| author_sort | Klymenko, Anastasiia |
| baseUrl_str | http://journal.iasa.kpi.ua/oai |
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| datestamp_date | 2023-05-24T21:28:17Z |
| description | This article is devoted to mathematical models in epidemiology, in particular SIR, SEIR, and SEIRD models. It explores the importance of these models in predicting the spread of infectious diseases and evaluating the effectiveness of control measures. These models allow for assessing important epidemic parameters such as the speed of infection transmission, the number of people infected, and the number of deaths. This data can help in making decisions regarding the imposition and lifting of quarantine restrictions, opening and closing of schools and other institutions, as well as in developing vaccination strategies and other control measures. In summary, mathematical models such as SIR, SEIR, and SEIRD are important tools in the fight against epidemics. They enable epidemiologists and medical professionals to predict and control the spread of diseases, thus preserving the health and lives of people. |
| doi_str_mv | 10.20535/SRIT.2308-8893.2023.1.04 |
| first_indexed | 2025-07-17T10:28:08Z |
| format | Article |
| fulltext |
A.I. Klymenko, G.B. Podkolzin, 2023
Системні дослідження та інформаційні технології, 2023, № 1 51
UDC 004.942 + 616-036.22
DOI: 10.20535/SRIT.2308-8893.2023.1.04
MODIFIED SEIRD MODEL FOR DESCRIBING
THE COVID-19 EPIDEMIC
A.I. KLYMENKO, G.B. PODKOLZIN
Abstract. This article is devoted to mathematical models in epidemiology, in par-
ticular SIR, SEIR, and SEIRD models. It explores the importance of these models in
predicting the spread of infectious diseases and evaluating the effectiveness of con-
trol measures. These models allow for assessing important epidemic parameters
such as the speed of infection transmission, the number of people infected, and the
number of deaths. This data can help in making decisions regarding the imposition
and lifting of quarantine restrictions, opening and closing of schools and other insti-
tutions, as well as in developing vaccination strategies and other control measures.
In summary, mathematical models such as SIR, SEIR, and SEIRD are important
tools in the fight against epidemics. They enable epidemiologists and medical pro-
fessionals to predict and control the spread of diseases, thus preserving the health
and lives of people.
Keywords: epidemiology, epidemiological models, modified mathematical models,
COVID-19 modeling SEIR, SEIRD model, unvaccinated people, virus, division of
the population, new strains.
INTRODUCTION
Modeling is a widely used tool to support the evaluation of various disease inter-
ventions. The value of epidemiological models lies in their ability to explore
“what if” scenarios and provide decision makers with a priori knowledge of the
consequences of disease emergence and the impact of control strategies.
To be useful, models must be fit for purpose and properly validated and veri-
fied. The complexity and variability inherent in biological systems should limit
the use of models as predictive tools during actual outbreaks. Models will be most
useful when used prior to an outbreak, particularly in the areas of retrospective
analysis of past outbreaks, contingency planning, resource planning, risk assess-
ment, and training. Models are only one tool for providing scientific advice, and
results should be evaluated in conjunction with experimental data, field experi-
ence, and scientific knowledge.
SIR, SEIR, SEIRD MODELS
The severity and global reach of the COVID-19 pandemic has spurred research in
many areas, including disease dynamics modeling, with the goal of using such
models to better understand the impact of intervention strategies on disease con-
trol [1]. Several factors are known to influence disease dynamics, including inci-
dence rates, recovery rates, quarantine strategies, and the impact of awareness [2].
In the literature, classical epidemic models of susceptible infectious diseases
A.I. Klymenko, G.B. Podkolzin
ISSN 1681–6048 System Research & Information Technologies, 2023, № 1 52
(SIR, SEIR, SEIRD models) have been widely used to model infectious dis-
eases [3].
The SIR model is based on the number of susceptible (S), infectious (I), and
recovered (R) individuals. The classical epidemic model is shown below:
, )(
)(
; )(
)()()(
;
)()()(
tI
dt
tdR
tI
N
tItS
dt
tdI
N
tItS
dt
tdS
where )(tS — the number of people who can be infected; )(tI — the number
of infected people; )(tR — the number of people who have been isolated from
transmission (died or recovered); — the transmission rate; — the recovery
rate.
Each group contains a certain number of people each day. However, this
number changes from day to day as people move from one group to another. For
example, people in Group S will move to Group I when they become infected.
Similarly, infected individuals will move to group R after they recover. It is as-
sumed that the total population in the three groups )( RIS always remains
the same.
The SIR model assumes that recovered individuals cannot be reinfected [2].
The SEIR model has an additional group for individuals who become infected and
contagious after the incubation period (E — Exposed). In other words, the SEIR
model includes a latency period.
The SEIR model is described by the following system of equations:
, )(
)(
;)( )(
)(
;)(
)()()(
;
)()()(
tI
dt
tdR
tItE
dt
tdI
tE
N
tItS
dt
tdE
N
tItS
dt
tdS
where — the rate of transition of the disease from the latent period to the overt
stage; — the rate of infection transmission; γ is the rate of recovery; )(tS —
the number of people susceptible to the virus; )(tE — the number of people with
the virus in the incubation period; )(tI — the number of people who get sick;
)(tR — the number of recovered people who have come into contact with the
pathogen and have gained stable immunity.
To describe the COVID-19 epidemic, the most appropriate of the above
models is the SEIRD model, in which group D — Death appears, i.e. this model
takes into account the dead.
The classical SEIRD model is shown below:
Modified SEIRD model for describing the COVID-19 epidemic
Системні дослідження та інформаційні технології, 2023, № 1 53
),(
)(
;)(
)(
;)()()(
)(
;)(
)()()(
;
)()()(
tI
dt
tdD
tI
dt
tdR
tItE
dt
tdI
tE
N
tItS
dt
tdE
N
tItS
dt
tdS
(1)
where — the coefficient that can be interpreted as the probability of contracting
the disease in case of contact between a susceptible individual and an infected
person; — the mortality rate; — the recovery rate; — the rate of transi-
tion from the latent period to the overt stage; )(tS — the number of people sus-
ceptible to the virus; )(tE — the number of people with the virus in the incuba-
tion period; )(tI — the number of people who fall ill; )(tR — the number of
recovered people who have come into contact with the pathogen and have gained
stable immunity.
MODIFIED SEIRD-MODEL
Modified models can take into account more realistic factors such as population
mobility, different variants of the disease, carrier effects, vaccination, and other
factors. For example, the SIR model typically assumes that people in each group
interact with each other, but in some cases there may be groups that interact less
frequently or not at all. In this case, modified models can be used to describe
more complex scenarios.
In epidemiology, it is very important to have a model that covers all stages
of the disease, including incubation, clinical course, recovery, and death. The SIR
model does not include a parameter responsible for the incubation period of the
disease, and SIR and SEIR cannot be used when the epidemic includes mortality
and fertility. For this reason, the SEIRD model has become the most relevant in
epidemiology, since it includes all the parameters necessary to study the different
stages of the disease and allows more accurate predictions of the spread of the
disease and the effectiveness of control measures.
The advantages of the proposed model are related to the fact that the popula-
tion can be divided into vaccinated and unvaccinated. In addition, the modified
SEIRD model takes into account fertility and natural mortality unrelated to dis-
ease mortality, which allows for the most accurate reproduction of the situation,
bringing it as close as possible to real conditions.
The basic SEIRD model is shown in (1). In the modified model, each com-
ponent is divided into two parts: vaccinated and unvaccinated members of the
population (susceptible, exposed, infectious, recovered). The fertility parameter
applies only to unvaccinated susceptibles, because by default, people are born
unvaccinated.
A.I. Klymenko, G.B. Podkolzin
ISSN 1681–6048 System Research & Information Technologies, 2023, № 1 54
So, the model is:
,
;
;
;)(
;)(
;)(
)(
;)(
)(
;
)(
;
)(
unvacvacunvacvac
unvacvacunvacvac
unvacunvacvacvac
vacvacvac
vac
unvacunvacunvac
unvac
vacvacvacvacvac
vac
unvacunvacunvacunvacunvac
unvac
vacvac
vvvuvuvacvac
unvacunvac
vuvuuuunvacunvac
vvvuvuvac
vac
vac
vuvuuuunvac
unvac
unvac
RRII
EESS
II
dt
dD
RI
dt
dR
RI
dt
dR
IE
dt
dI
IE
dt
dI
E
N
IIS
dt
dE
E
N
IIS
dt
dE
N
IIS
S
dt
dS
N
IIS
Sl
dt
dS
(2)
where unvacS — susceptible unvaccinated persons who are not infected but can
become infected through contact with an infected person (unvaccinated or vacci-
nated); vacS — susceptible to the virus are vaccinated persons in the population
who are not infected but can become infected through contact with an infected
person (unvaccinated or vaccinated); unvacE — the number of unvaccinated peo-
ple with the disease in latent mode (they contacted with an infected person);
vacE — the number of vaccinated people with the disease in latent mode (they
contacted with an infected person); unvacI — the number of unvaccinated sick
people who transmit the virus to unvaccinated and vaccinated susceptible persons;
vacI — the number of vaccinated patients who transmit the virus to unvaccinated
and vaccinated susceptible persons; unvacR — the number of unvaccinated survi-
vors who are susceptible to reinfection, although the probability is lower; vacR —
the number of vaccinated survivors who are susceptible to re-infection, although
the probability is lower; D — people who died from the virus and other causes;
unvac — deaths from the virus in infected unvaccinated people; vac — deaths
from the virus in infected vaccinees; uu — the probability of transmission of the
virus from infected unvaccinated persons to unvaccinated persons; uv — the
probability of transmission of the virus from infected unvaccinated persons to
vaccinated persons; vu — the probability of transmission of the virus from in-
fected vaccinated to unvaccinated persons; vu — the probability of virus trans-
mission from infected vaccinated to vaccinated persons; unvac — the probability
of the disease transition from the latent phase to the overt phase in unvaccinated
persons; vac — the probability of the disease transition from the latent phase to
the overt phase in vaccinated persons; unvac — recovery of infected unvaccinated
Modified SEIRD model for describing the COVID-19 epidemic
Системні дослідження та інформаційні технології, 2023, № 1 55
people from the virus; vac — recovery of infected vaccinated persons from the
virus; — mortality not due to infection; l — birth rate.
The main difference between this model and the classical SEIRD model is
the division of the population into vaccinated and unvaccinated individuals. Prob-
ability of infection of vaccinated persons ( vacS ) is much lower than that of un-
vaccinated persons ( unvacS ). Sick vaccinated persons ( vacI ) are less contagious
and less likely to die than unvaccinated infectious people ( unvacI ).
IMPLEMENTING THE MODIFIED SEIRD MODEL FOR COVID-19 IN
UKRAINE IN 2021
Let's model the situation with COVID-19 in Ukraine. To do this, we need to cal-
culate the appropriate coefficients to substitute them. To do this, we will use in-
formation for 2021.
The number of Ukrainians in 2021 (excluding the occupied territories) is 41
million 167 thousand people [4]. The mortality rate for 2021 is 714 263 people, of
which COVID-19 accounts for 86 015 cases [5]. The birth rate for 2021 will be
271 984 children [6].
The number of all deaths per day averages 714 263/365 = 1 956.88 people.
That is, 1 956.88/41 167 300 = 0.0000475349 of the total population dies per day.
This number includes deaths from COVID-19. The death rate from COVID-19 is
(86 015/365)/41 167 300 = 0.0000057244.
Unfortunately, no mortality statistics for vaccinated and unvaccinated people
could be found for Ukraine. However, you can calculate this coefficient yourself
if the data mentioned in an interview with Professor Leanne Wen of the School of
Public Health [14] are true: that vaccinated people are six times less likely to be
infected than unvaccinated people and 11 times less likely to die from coronavi-
rus. In this case, the mortality rate of vaccinated people per day is 0.0000004770,
and that of unvaccinated people is 0.0000052474. Accordingly, we have
0.0000052474unvac 0.0000004770vac .
Accordingly, the mortality rate per day not due to COVID-19 is
0.0000475349 - 0.0000057244 = 0.0000418105, i.e. μ = 0.0000418105.
The birth rate per day is (271 984 /365)/41 167 300 = 0.0000181008, so
l = 0.0000181008.
Since the modeling requires both vaccinated and unvaccinated populations,
we need to provide data on these. In 2021, 15 201,112 people have been vacci-
nated with two vaccines in Ukraine [7]. That is, 15 201 112/41 167,300 =
0.3692521006. Accordingly, if you want to model a population of only 100 peo-
ple, and one of each of the vaccinated and unvaccinated gets sick, 36.1867058563
will be the vaccinated who have not yet gotten sick, and 61.8132941437 will be
the unvaccinated who have not yet gotten sick.
The incubation period is the number of days between the moment of infec-
tion and the moment of symptoms. To calculate the coefficient responsible for the
rate of transition of the virus from the latent period to the fully infected state, we
use formula:
A.I. Klymenko, G.B. Podkolzin
ISSN 1681–6048 System Research & Information Technologies, 2023, № 1 56
incT
1
,
where — the rate of transition of the disease from the latent to the overt phase;
incT — the average incubation time of the virus.
Viruses are constantly changing, sometimes resulting in the emergence of
new strains. Different strains of COVID-19 may have different incubation peri-
ods. On average, symptoms appear in a newly infected person about 5.6 days after
exposure [8]. So, 6.5incT , accordingly, 17858.06.5/1 . According to
studies, the incubation period for vaccinated and unvaccinated individuals is the
same number of days [9], i.e. 17858.0 unvacunvac .
Some studies have shown that it may take the body 2 weeks to recover from
a mild illness, or up to 6 weeks in severe or critical cases [10]. Other sources say
[11] that recovery usually takes one to two weeks. So let's use the average
recovery time of two weeks.
Let's calculate according to the formula for calculating the recovery rate
coefficient:
recT
1
,
where — the recovery rate of infected people from the virus; recT — average
recovery time.
So, recT = 14 days. Accordingly, unvac = 1/14 = 0.0714.
When the population is divided into vaccinated and unvaccinated, the recovery
time will be different. Different sources report different recovery times: some
studies show that the overall recovery time was six to seven days shorter than for
unvaccinated people [12]. Another study from the Centers for Disease Control
and Prevention found that vaccinated participants spent an average of two to six
days less in bed than unvaccinated participants [13]. Let's assume that, on aver-
age, the vaccinated are sick six days less, that is vac = 1/8 = 0.125.
Now let's calculate the probability of disease transmission. To calculate the
data for vaccinated people, we need statistics on the effectiveness of the vaccine
(let's take the Pfizer vaccine). According to [15], Pfizer has 95% protection
against mild COVID-19. This means that you are less likely to get sick if you are
vaccinated:
05.0 vaccinatedareyouifinfectedgetp .
It should also be noted that vaccinated people are less likely to transmit the
disease, even if they become infected. At a press conference in November, WHO
Director-General Tedros Ghebreyesus said that vaccines protect against the
spread of the virus by 60 percent before the delta variant emerges [16, 19]. This
means:
4.0 vaccinatedareyouifdiseasethetransmitp .
According to the same study [16, 17], vaccinated people are ten times less
likely to be infected [18], and the likelihood of me being infected is half, judging
by the above figures.
Modified SEIRD model for describing the COVID-19 epidemic
Системні дослідження та інформаційні технології, 2023, № 1 57
So, 5.010 vaccinatedareyouifsickgetvaccinatednotareyouifinfectedget pp .
А 8.024.0 vaccinatednotareyouifdiseasethetransmitp .
Now we can calculate the disease transmission rates.
vaccinatednotareyouifinfectedgetvaccinatednotareyouifdiseasethetransmituu pp
4.05.08.0 .
vaccinatedareyouifsickgetvaccinatednotareyouifdiseasethetransmituv pp
04.005.08.0 .
vaccinatednotareyouifinfectedgetvaccinatedareyouifinfectedgetvu pp
2.05.04.0 .
vaccinatedareyouifsickgetvaccinatedareyouifinfectedgetvv pp
02.005.04.0 .
Let's simulate the model with these parameters. In Fig. 1 you can see the re-
sults for one hundred days:
Fig. 1 shows how the number of people who have not had the disease is de-
creasing, while the number of people who have had the disease is increasing.
Fig. 1. Modified SEIRD model for 100 days
A.I. Klymenko, G.B. Podkolzin
ISSN 1681–6048 System Research & Information Technologies, 2023, № 1 58
The number of unvaccinated patients is growing faster, which seems logical given
the vaccination rate [13]. The number of deaths is growing relatively slowly. The
number of uninfected, unvaccinated people is decreasing faster because they are
more likely to contract the disease, while vaccinated people are slower to get sick.
If we look at a much longer time period (e.g., 8,000 days), we can see in
Fig. 2 how the number of deaths increases and the number of deaths in the rest of
the population decreases. The number of people who became ill also decreases
because model (2) also takes into account natural mortality, which in this case
does not include deaths from COVID-19.
This situation with fertility and mortality is due to the fact that in Ukraine
the birth rate is lower than the death rate [20]. If we increase the fertility rate
so that it exceeds the mortality rate, the situation looks different (fertility
l = 0.0500181008) (Fig. 3):
Only the number of unvaccinated people who have not yet become ill in-
creases, because this model does not include vaccination and the corresponding
transition from unvaccinated to vaccinated people. The entire birth population is
unvaccinated by default. From Fig. 3, you can also see how the disease process
progresses over time: on the thousandth and two thousandth day, you can see
waves of sick unvaccinated people, and accordingly, the number of unvaccinated
people who have not yet become sick decreases over time.
Fig. 2. Modified SEIRD model for 8000 days
Modified SEIRD model for describing the COVID-19 epidemic
Системні дослідження та інформаційні технології, 2023, № 1 59
CONCLUSIONS
The modified mathematical epidemiological SEIRD model is an important tool
for assessing epidemic outbreaks and implementing disease control strategies. It is
especially important that this model takes into account vaccination and non-
vaccination, as this reflects the real situation with COVID-19 and other diseases
for which vaccines exist. This model also allows for the impact of different vac-
cine variants on the effectiveness of its use.
To effectively combat epidemic outbreaks, it is necessary to know what fac-
tors influence the spread of the disease and which control strategies are most ef-
fective in specific conditions. The developed modified SEIRD model not only
takes into account vaccination and non-vaccination, but also has coefficients re-
sponsible for birth and death, and that is why it can reflect a realistic picture. This
model can help solve these problems by providing scientists and policymakers
with a tool for forecasting, planning, and decision-making to control the epi-
demic.
Also, for the modified SEIRD model, all parameters were calculated from
real statistical data from Ukraine to be able to simulate the situation as close to
reality as possible. These data are important for making decisions on the devel-
opment of medical infrastructure, the provision of medical equipment and medi-
cines, as well as for determining the need for large-scale vaccinations and other
epidemic control measures.
1
1
1
1
1
7
6
9
2
3
5
4
8
Fig. 3. Modified SEIRD model for 5000 days with a birth rate of l = 0.0500181008
A.I. Klymenko, G.B. Podkolzin
ISSN 1681–6048 System Research & Information Technologies, 2023, № 1 60
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Modified SEIRD model for describing the COVID-19 epidemic
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Received 02.02.2023
INFORMATION ON THE ARTICLE
Anastasiia I. Klymenko, ORCID: 0000-0001-9595-8155, National Commission for State
Regulation of Energy and Public Utilities, Ukraine, e-mail: asja653@gmail.com
Gleb B. Podkolzin, ORCID: 0000-0002-7120-2772, Educational and Scientific Institute
for Applied System Analysis of the National Technical University of Ukraine “Igor Sikor-
sky Kyiv Polytechnic Institute”, Ukraine, e-mail: podkolzin.gleb@lll.kpi.ua
МОДИФІКОВАНА SEIRD-МОДЕЛЬ ОПИСУ ЕПІДЕМІЇ COVID-19 /
А.І. Клименко, Г.Б. Подколзін
Анотація. Присвячено математичним моделям в епідеміології, зокрема SIR,
SEIR і SEIRD. Досліджено важливість цих моделей у прогнозуванні поширен-
ня інфекційних захворювань та оцінювання ефективності контрольних заходів.
Ці моделі дають змогу оцінити важливі параметри епідемії, такі як швидкість
поширення інфекції, кількість людей, які зазнають захворювання, та померлих
від цього захворювання. Ці дані можуть допомогти у прийнятті рішень про
введення та зняття карантинних обмежень, відкриття і закриття шкіл та інших
установ, а також у розробленні стратегій вакцинації та інших контрольних за-
ходів. Загалом математичні моделі SIR, SEIR і SEIRD є важливим інструмен-
том з боротьби з епідеміями. Вони дозволяють епідеміологам і медичним пра-
цівникам прогнозувати та контролювати поширення захворювань, що зберігає
здоров’я та життя людей.
Ключові слова: епідеміологія, епідеміологічні моделі, модифіковані матема-
тичні моделі, SEIR-моделювання COVID-19, SEIRD-модель, невакциновані
люди, вірус, поділ популяції, нові штами.
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| id | journaliasakpiua-article-279749 |
| institution | System research and information technologies |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2025-07-17T10:28:08Z |
| publishDate | 2023 |
| publisher | The National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute" |
| record_format | ojs |
| resource_txt_mv | journaliasakpiua/eb/77100b496258c380045ee633dbbcf7eb.pdf |
| spelling | journaliasakpiua-article-2797492023-05-24T21:28:17Z Modified SEIRD model for describing the COVID-19 epidemic Модифікована SEIRD-модель опису епідемії COVID-19 Klymenko, Anastasiia Podkolzin, Gleb epidemiology epidemiological models modified mathematical models COVID-19 modeling SEIR SEIRD model unvaccinated people virus division of the population new strains епідеміологія епідеміологічні моделі модифіковані математичні моделі SEIR-моделювання COVID-19 SEIRD-модель невакциновані люди вірус поділ популяції нові штами This article is devoted to mathematical models in epidemiology, in particular SIR, SEIR, and SEIRD models. It explores the importance of these models in predicting the spread of infectious diseases and evaluating the effectiveness of control measures. These models allow for assessing important epidemic parameters such as the speed of infection transmission, the number of people infected, and the number of deaths. This data can help in making decisions regarding the imposition and lifting of quarantine restrictions, opening and closing of schools and other institutions, as well as in developing vaccination strategies and other control measures. In summary, mathematical models such as SIR, SEIR, and SEIRD are important tools in the fight against epidemics. They enable epidemiologists and medical professionals to predict and control the spread of diseases, thus preserving the health and lives of people. Присвячено математичним моделям в епідеміології, зокрема SIR, SEIR і SEIRD. Досліджено важливість цих моделей у прогнозуванні поширення інфекційних захворювань та оцінювання ефективності контрольних заходів. Ці моделі дають змогу оцінити важливі параметри епідемії, такі як швидкість поширення інфекції, кількість людей, які зазнають захворювання, та померлих від цього захворювання. Ці дані можуть допомогти у прийнятті рішень про введення та зняття карантинних обмежень, відкриття і закриття шкіл та інших установ, а також у розробленні стратегій вакцинації та інших контрольних заходів. Загалом математичні моделі SIR, SEIR і SEIRD є важливим інструментом з боротьби з епідеміями. Вони дозволяють епідеміологам і медичним працівникам прогнозувати та контролювати поширення захворювань, що зберігає здоров’я та життя людей. The National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute" 2023-03-30 Article Article application/pdf https://journal.iasa.kpi.ua/article/view/279749 10.20535/SRIT.2308-8893.2023.1.04 System research and information technologies; No. 1 (2023); 51-61 Системные исследования и информационные технологии; № 1 (2023); 51-61 Системні дослідження та інформаційні технології; № 1 (2023); 51-61 2308-8893 1681-6048 en https://journal.iasa.kpi.ua/article/view/279749/274469 |
| spellingShingle | епідеміологія епідеміологічні моделі модифіковані математичні моделі SEIR-моделювання COVID-19 SEIRD-модель невакциновані люди вірус поділ популяції нові штами Klymenko, Anastasiia Podkolzin, Gleb Модифікована SEIRD-модель опису епідемії COVID-19 |
| title | Модифікована SEIRD-модель опису епідемії COVID-19 |
| title_alt | Modified SEIRD model for describing the COVID-19 epidemic |
| title_full | Модифікована SEIRD-модель опису епідемії COVID-19 |
| title_fullStr | Модифікована SEIRD-модель опису епідемії COVID-19 |
| title_full_unstemmed | Модифікована SEIRD-модель опису епідемії COVID-19 |
| title_short | Модифікована SEIRD-модель опису епідемії COVID-19 |
| title_sort | модифікована seird-модель опису епідемії covid-19 |
| topic | епідеміологія епідеміологічні моделі модифіковані математичні моделі SEIR-моделювання COVID-19 SEIRD-модель невакциновані люди вірус поділ популяції нові штами |
| topic_facet | epidemiology epidemiological models modified mathematical models COVID-19 modeling SEIR SEIRD model unvaccinated people virus division of the population new strains епідеміологія епідеміологічні моделі модифіковані математичні моделі SEIR-моделювання COVID-19 SEIRD-модель невакциновані люди вірус поділ популяції нові штами |
| url | https://journal.iasa.kpi.ua/article/view/279749 |
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