Vaccination Strategies based on a Mathematical Model of Epidemics
Considering the Age Structure of the Population
MACIEJ URBAN1, JULIA JODŁOWSKA2, JOANNA BALBUS3, KRYSTIAN KUBICA2
1Student’s Scientific Group BioModel Department of Biomedical Engineering,
Wroclaw University of Science and Technology,
POLAND
2Department of Biomedical Engineering,
Wroclaw University of Science and Technology,
POLAND
3Department of Pure and Applied Mathematics,
Wroclaw University of Science and Technology,
POLAND
Abstract: - During the COVID-19 pandemic, it is important to promote the skills needed for analyzing the
disease course, including determining the relevance of vaccinations, especially among people who are
unfamiliar with computer programming. This paper describes the basic epidemiological model (SIR), its
extensions that allow vaccinations, and the emergence of renewed waves of disease growth. It also discusses a
literature model, extended SEIRD, which includes a more detailed division of the population into susceptible,
latent, symptomatic, and asymptomatic infected, recovered, and dead in eight age groups. Modifying the
SEIRD model as shown on the basic SIR model, we analyzed five vaccination strategies, considering the
limited vaccine supply, the number of vaccinations performed per day, and their effectiveness. The analysis
was performed for a group of one million people, using the parameters of the model characteristic of the
COVID-19 pandemic and Sweden's generational structure. We analyzed in terms of reducing both the number
of deaths and the incidence of symptomatic infections, which represent the main burden of healthcare.
Key-Words: - SIR, SEIRD, epidemiology, vaccinations, vaccination effectiveness, mathematical model, social
contact matrix, COVID-19.
Received: March 18, 2023. Revised: November 17, 2023. Accepted: December 19, 2023. Published: February 20, 2024.
1 Introduction
The goal of this study is to popularize scientific
research on the course of the pandemic. There are
various methods for studying the course of the
pandemic, including deterministic models such as
those based on ordinary differential equations, [1],
stochastic models, [2] and statistical models, [3].
We chose to employ the system of ordinary
differential equations (ODE). However, it is
challenging to solve the majority of mathematical
models analytically without making additional
assumptions, [4]. Nevertheless, they can be solved
numerically. Since this work is dedicated to
everyone interested in the subject of pandemic
analysis, including those who cannot program on
their own, we created and made available a ready-
to-use program for this purpose. Instructions for
using the program are available at:
https://github.com/BlankTiger/SEIRD_model/releas
es/download/v1.0.3-rust/Instructions.docx
The basic model of the epidemic (SIR)
The basic model used to study the course of the
epidemic is the Kermack–McKendrick model, which
is often referred to as the SIR model, [5]. The SIR
model contains simplifications that do not allow
effective prognosis of disease development in the
case of many diseases, but it enables to the creation
of more complex models. This model describes the
disease development in a closed population of N
people divided into groups as susceptible to infection
(S), infected (I), and recovered (R). If we want to
include deaths due to the studied disease, the number
of fatal cases should be added to the group of
recovered because they are no longer transmitting
the disease. The SIR model generates a system of
three ordinary differential equations that describe the
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kinetics of changes in the S, I, and R groups. This
model is based on the following assumptions:
The probability of direct contact is the same for
each pair of individuals.
The decrease in the number of susceptible cases
(S) depends on their number, the number of
infected (I), and the coefficient describing the
infection rate (here β).
The change in the number of infected people (I)
depends on two terms: the one describing
increasing I, which is proportional to S, I, and β;
and the one describing decreasing I, which is
proportional to I and coefficient recovery rate
(here γ).
The increase in recovered (R) and dead due to
infection is proportional to the number of
infected I and coefficient γ.
The incubation period is negligibly short, so
immediately after the infection the susceptible
person becomes infected and can infect others,
which makes it difficult to realistically assess the
disease variability over time.
In line with these assumptions, the SIR model
can be written as Equations 1–3:
S/dt=-βSI (1)
dI/dt=βSI-γI (2)
dR/dt=γI (3)
To solve these equations, we have to assume the
initial conditions. Most often, they meet the
condition S + I = N, which means that there are no
people in the R group at the beginning. Since this
model should be able to predict the development of
the disease, it is worth assuming that at the initial
moment t0 the given population already had a small
group of infected people, I(t0) > 0. Since the SIR
model cannot be solved analytically (although the
modified SIR model is analytically solvable), [4], it
can be easily solved numerically, yielding time
dependencies in the S, I, and R groups.
To solve the SIR equations, it is also necessary
to assign values to parameters β and γ. If we consider
one day as the unit of time, then parameter γ, which
is the rate of healing, can be defined as the reciprocal
of the recovery time for an individual patient’s
infection. According to Equation 3, the values of γ
are expressed in day–1 units. On the other hand, the
value of parameter β will be expressed in units day–1
* number of people–1 (based on Equation 1). The
appropriate value of β can be found by considering
the changes in the number of infected, as described
by Equation 2. If the number of infected people does
not change, i.e. dI/dt = 0, then, based on Equation 2,
the product βS = γ. If we take the initial value of S =
S (t0), then a comparison of the product of βS(t0)
with γ will allow assessing whether the infection will
develop or be inhibited. If βS(t0) > γ, the epidemic
will continue to develop; otherwise, it will be
inhibited. From the equality βS(t0) = γ, the limit
value of β can be determined for a given S(t0) and γ.
The basic reproduction number R0 is also used to
assess the development or inhibition of the infection,
which for the SIR model is defined by β, γ, and S
values: R0 == βS(t0)/γ. R0 specifies the number of
people who are secondarily infected in the
susceptible group S by one person who was initially
infected at time t0. If R0 > 1, the epidemic will
continue to develop; otherwise, it will be inhibited.
2 Problem Formulation
2.1 Extension of the SIR Model to Include
Vaccinations
The mathematical models allowing the exploration
of the significance of vaccinations in preventive
measures that reduce the risk of epidemic
development have been described in [1]. In this
work, they demonstrate how to determine the
vaccination threshold for the population to curb
infections. The analysis also examines the
importance of the loss of immunity acquired through
vaccinations, the variability of infectious factors, and
the limited capacity for daily vaccinations. A simple
method for modeling the significance of vaccinations
is described in [6], in which the group of recoverers
(R) resulting from natural immunity is separated
from the group acquiring immunity through
vaccination. However, this model specifically
pertains to preventive vaccination conducted in
newborn groups.
The basic SIR model does not explicitly consider
the presence of a subset of the population (S)
exhibiting natural immunity. However, in cases
where the epidemic under consideration does not
result in fatalities, the parameter γ, representing the
recovery rate, can also be interpreted as indicative of
natural immunity. This is because it is through this
parameter that we observe the decline and eventual
extinction of the epidemic. However, epidemics can
be effectively controlled through vaccination
campaigns, which contribute to an increase in the
number of individuals resistant to infection and/or
experiencing mild symptoms. In this work, we have
demonstrated how to easily modify the SIR model to
analyze the importance of vaccinations, considering
their effectiveness and the limited number of
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vaccinations performed each day. Assuming
permanent immunity obtained through vaccination,
vaccinated people should be transferred from group
S to group R, taking into account the effectiveness of
vaccination and the speed of vaccination of the
population. For this purpose, we have modified
Equations 1 and 3. Since in the case of COVID-19
vaccines, the effectiveness ranges from 70.4% to
95%, [7], in the example shown in Figure 1, we
assumed a vaccination effectiveness (ef) of 90%.
Moreover, we had to assume the number of
vaccinations performed each day (vac), and select
the day of the epidemic when vaccination started. To
study the importance of the timing of vaccination
initiation and completion, Heaviside function (Hev)
(Appendix) can be beneficial. If t1 is the day of
vaccination initiation and t2 is the day of vaccination
completion, then Equation 1 should be supplemented
with the expression: –ef * vac * Hev(t, t1) * (1 –
Hev(t, t2)). The same expression should be added to
Equation 3. So the SIR model including vaccination
would take the form of Equations 4–6:
dS/dt=-βSI-ef vac Hev(t,t1)(1-Hev(t,t2)) (4)
dI/dt=βSI-γI (5)
dR/dt=γI+ef vac Hev(t,t1)(1-Hev(t,t2)) (6)
Since the number of people in each group must
be positive, we must select the number of
vaccinations performed per day and the duration of
vaccination accordingly.
2.2 Modeling an Epidemic with Multiple
Incidence Waves
As a result of temporally limited immunity, whether
acquired naturally or through vaccination, there are
recurring waves of increasing numbers of individuals
falling ill. This can be observed firsthand and is also
the subject of modeling studies, [8], [9]. Building
upon the basic SIR model, it is also possible to
illustrate recurring waves of increased infections.
The consequence of temporally limited
immunity, whether acquired naturally or through
vaccination, is the recurrence of waves with
increased numbers of infected individuals. Currently,
every person can observe this phenomenon, which is
also the subject of modeling studies, [8], [9].
Additionally, based on the basic SIR model, it is
possible to illustrate these recurring waves of disease
increases.
To analyze the epidemic development with
multiple waves of increase in the number of infected
based on the SIR model, the following assumptions
should be made:
1. No one dies due to the infection, i.e. there are only
recovered in group R.
2. Recovered individuals lose immunity after tx days
with a certain probability a.
This problem can be solved based on the SIR
model by gluing the solutions after each time unit,
e.g. after each day. After the SIR equations are
solved, the number of recovered R(ti) should be
noted after each day ti. Then, the initial conditions
for the next solution (for the next day) should be
changed. Thus, for ti tx, the equations are solved
with the initial conditions: S(t0), I(t0), and R(t0),
while for ti > tx the initial conditions for the
following step ti are S(t0) = S(ti) + a * R(ti – tx), I(t0)
= I(ti), and R(t0) = R(ti) – a * R(ti – tx).
2.3 An Epidemiological Model with the Age
Structure of the Population and
Intergenerational Contacts (SEIRD)
The SIR model includes many simplifications,
including grouping recovered and deceased
individuals in the same group. In addition, it is not
always true that people who become ill acquire
permanent immunity after their recovery and that
the incubation time is short enough to be skipped.
To make the basic SIR model more appropriate for
analyzing the current state of the COVID-19
pandemic, [10], it is necessary to extend it by
including additional groups, i.e. those in the latent
phase E, and divide the infected group into
symptomatic (Is) and asymptomatic (Ia).
Furthermore, it is important to clearly distinguish
convalescents (R) and individuals who died due to
infection (D). For the inclusion of an additional
latent (E) group, we must take into account an
additional coefficient of transition from this group
to the infected group I. Since the course of the
disease and mortality significantly differ by age
groups, the age structure of the population under
study should also be introduced, due to the need for
differentiation of susceptibility to infection and
mortality. When dividing the population under
consideration into age groups, we also consider the
differences in contacts within a given age group and
between groups. The application of such a model for
the analysis of the course of the COVID-19
pandemic has already been demonstrated in [10].
The analyzed model is a system of six differential
equations (Equations 7–12), for age groups 0–9, 10–
19, 20–29, 30–39, 40–49, 50–59, 60–69, and 70+
years, for n = 1, 2, 3, 4, 5, 6, 7, and 8, respectively.
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, (7)
, (8)
, (9)
, (10)
, (11)
, (12)
where:
n—index denotes age groups 1–8,
Sn—susceptible,
En—latent,
Is,n—symptomatic cases among age group n,
Ia,n—asymptomatic cases among age group n,
Rn—recovered,
Dn—dead,
σn—susceptibility of age group n,
β—transmission coefficient,
kn,m—an element of the contact matrix between age
group n and m,
ε—progression rate from latent to infectious,
fs —symptomatic cases,
γ—recovery rate, and
δn—mortality rate in age groups.
To determine the values of the kn,m parameter
(matrix of contacts of various age groups), data
collected in studies conducted in 152 countries were
used, [11].
As mentioned above, the basic reproduction
number R0 is an important factor in the emergence of
the epidemic. For the discussed SEIRD model, the
value of this parameter for each age group can be
calculated based on Equation 13 (Appendix):
(13)
If R0n is >1 within a given age group n, then an
epidemic will develop. Thus, for the established
values of parameters fs, γa, γs, and δn, size of the Sn(t0)
age of susceptible groups, and social contact matrix,
epidemic development will depend on the product
βϬn.
3 Problem Solution
3.1 SIR model with Vaccination
For the proposed extension of the SIR model, it was
assumed that the vaccinated subjects achieve
sustained immunity. The modified SIR model can be
used to analyze the impact of vaccination rate and
the time of vaccination commencement on epidemic
extinction.
Figure 1 shows two examples of vaccination
with 90% effectiveness, started on day 50 of the
pandemic for parameters S(t0) = 1000,000; I(t0) = 1;
R(t0) = 0; β = 4 x 10–7; days of recovery = 5; R0 = 2:
a) 20,000 vaccinations/day; duration of vaccination
= 21 days; number of vaccinated = 420,000
(solid lines); and
b) 10,000 vaccinations/day; duration of vaccination
= 42 days; number of vaccinated = 420,000
(dotted lines).
Fig. 1: Vaccination effect as measured by the SIR
model. S(t0) = 1000,000; I(t0) = 1; R(t0) = 0; β = 4
10–7; days of recovery = 5; R0 = 2; start of
vaccination = 50 days of the epidemic:
a) 20,000 vaccinations/day; duration of vaccination
= 21 days; number of vaccinated = 420,000 (solid
lines); Imax = 77545 on the64th day; and
b) 10,000 vaccinations/day; duration of vaccination
= 42 days; number of vaccinated = 420,000 (dashed
lines); Imax = 106980 on the 66th day.
Doubling the vaccination period combined with
twice fewer vaccinations per day results in an
increase in the number of I and R and a decrease in
the number of S, while the maximum number of
infected people appears two days later.
3.2 The SIR Model with Multiple Incidence
Waves
Figure 2 shows an example for the initial conditions
S(t0) = 1000,000, I(t0) = 1, R(t0) = 0, for two cases of
loss of immunity with the same probability a = 0.01,
after 30 and 60 days.
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Fig. 2: Impact of the loss of immunity to the
solutions of the SIR model.
S(t0) = 1000,000; I(t0) = 1; R(t0) = 0; β = 4 10–7;
days of recovery = 5; γ = 0.2; R0 = 2;
a) duration of immunity = 30 days with
probability 0.99 (solid line); and
b) duration of immunity = 60 days with
probability 0.99 (dashed line).
For a longer period of natural immunity, there
are greater variations in the number of S, R, and I in
the next wave of the epidemic. If immunity resulting
from vaccination is of similar duration, then similar
changes in numbers in groups S, I, and R should be
expected.
3.3 Analysis of Selected Vaccination
Strategies based on the SEIRD Model
Based on epidemiological data related to the
COVID-19 pandemic, we simulated selected
vaccination strategies in a closed population of 106
people. To start the simulation, we had to give
values to the parameters of the SEIRD model.
According to the World Health Organization
(WHO), [12], the average incubation time of a virus
in an infected organism is 5–6 days, but it may even
extend up to 14 days. We assumed that the time of
transition from latent group E to infected group I is
5.5 days; therefore, coefficient ε = 1/5.5 [day–1]. As
asymptomatic cases may even account for 40–45%
of all infection cases, [13], we assumed that
coefficient fs, which determines symptomatic cases,
can range from 0.55 to 0.6. The original guidelines
of the WHO indicated that an infected person should
be quarantined for 14 days. Thus, it can be assumed
that 14 days is the duration of infection in a patient.
Consequently, we estimated that the value of the
recovery rate coefficient in symptomatically
affected individuals (γs) is 1/14. However,
asymptomatic people who are infected can infect
others for longer than 14 days, [13]. Therefore, we
assumed that the recovery time for this group is 16
days. Accordingly, the estimated value of the
recovery rate coefficient of asymptomatic patients
(γa) is 1/16. The COVID-19 mortality rate δ was
determined for each age group based on the data
from Sweden, [14]. According to Equation 7,
epidemic development is mainly determined by the
product of β and Ϭn; therefore, to differentiate age
groups based on susceptibility to disease
development, different values of Ϭn can be adopted
while maintaining β value constant for all n.
During the first wave of the Covid-19
pandemic, Sweden, compared to other countries,
adopted the least drastic measures to prevent the
spread of the disease, [15]. However, the number of
deaths per 1 million inhabitants in 2020 in Sweden
was lower than in Germany, France, or Spain. It is,
therefore, worth conducting a simulation of selected
vaccination strategies based on detailed data for the
Swedish population, [16]. To evaluate various
vaccination strategies and analyze potential
observed effects, it's necessary to adjust the model
parameters to reflect the characteristics of the
Swedish population accurately. This adjustment
aims to achieve a good fit with the known pandemic
trajectory in Sweden in 2020, a period when
vaccines were not yet available. The parameters ε,
fs, γs, γa take the values 0.18, 0.6, 0.07, and 0.06,
respectively, [11], [12], [13], [14], [15], [16]. The
parameter β = 6 10-6 was selected to align with the
literature data, [15], aiming to observe the highest
number of daily fatal cases approximately 60 days
after the appearance of the first infected person (12
cases per 1 million). The parameter responsible for
mortality in individual age groups was estimated
based on the relationship between crude case-
fatality rates (CFR), mortality rate δn, and recovery
rate γs: CFR = δn/(δn+γs), [10], and the values of
CFR were extracted from [14]. However, the values
of this parameter calculated in this manner resulted
in ten times the overall total number of fatal cases
during the first pandemic wave in 2020. A tenfold
decrease in its value for all age groups leads to a
mortality rate similar to reality. Finally, the
parameter δn takes values of [2.8 10-7, 2 10-6, 5
10-6, 8.58 10-6, 2.22 10-5, 7.58 10-5, 3.4 10-4,
18.7 10-4] for individual age groups n. The
parameter ϭn was also modified. Setting the same
value for all age groups leads to an overestimation
of mortality cases in the oldest age groups 7 and 8.
The best reproduction of the pandemic course in
Sweden in 2020 was achieved by adopting the
following values for this parameter for subsequent
age groups: σn = [1, 1, 1, 1, 1, 1, 0.1, 0.01]. To
perform simulations, the initial values for the
Swedish population, [16], should also be adopted:
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Sn(0) = [119000, 110000, 130000, 128000, 127000,
128000, 109000,148000]
En(0) = [0, 0, 0, 0, 0, 0, 0, 0]
Isn(0) = [0, 0, 0, 1, 0, 0, 0, 0]
Ian(0) = [0, 0, 0, 0, 0, 0, 0, 0]
Rn(0) = [0, 0, 0, 0, 0, 0, 0, 0]
Dn(0) = [0, 0, 0, 0, 0, 0, 0, 0].
The Swedish social contact matrix is provided
in the Appendix.
We assumed that in a group of a million people,
the pandemic begins with a single person in the age
group 4 (Is4) who is symptomatically infected. For
such assumptions, we solved the system of
equations of the SEIR model using our application
https://github.com/BlankTiger/SEIRD_model/releas
es/download/v1.0.3-rust/SEIRD_model.exe.These
solutions are shown in Figure 3.
After 12 days, there is a significant decline in
the number of susceptible cases (Sn) in age groups
1-6 for about 8 days. This decrease is accompanied
by a significant increase in group E for about 6 days
in age groups 1-6, followed by a decrease in the
next 14 days. With a slight delay—i.e., starting from
12 days after the first appearance of Is4—Is begins to
increase until day 22 in age groups 1-6, and then
decreases within 50 days. Similar changes are
observed in the Ia group (1-6).
In the oldest age groups 7 and 8, the growth of Is
begins later, specifically after 17 and 20 days, and
lasts longer, extending to 35 and 45 days,
respectively. Similarly, Ia changes in these age
groups.
From days 15 and 20 of the pandemic for age
groups 7 and 8 respectively, the number of
recoveries also begins to increase, reaching
asymptotic values after about 80 days. Fatalities
begin to occur between days 15 and 20 of the
pandemic.
The eighth age group has the highest mortality
rate of about 0.4%, whereas in groups 1-7, the
mortality rate is significantly lower. The value of
the R0 parameter for age groups 1–7 is >1 and for
group8 is <1.The results of the simulation are
presented in Table 1, where the R0 parameter was
given for each age group, the maximum number of
Isn, asymptomatic Ian, and the number of fatal cases,
Da. The analysis of the results collected in Table 1
indicates the highest mortality in the 8, 7, and 6 age
groups.
Based on the presented analysis of pandemic
development (Figure 3 and Table 1), we compared
five vaccination strategies, assuming the following:
we have 150000 doses of vaccines and can
vaccinate 10000 people per day, and the
effectiveness of the vaccine is 90%. Vaccinations
will be performed from days 1 to 15 (from the
moment the first case of a symptomatically infected
person is noted). Since the previous analysis
revealed the highest mortality in age groups 8, 7,
and 6, it is worth examining the potential effects of
vaccinating these age groups first.
Given the constraints imposed (limited
availability of vaccinations per day), it is valuable to
compare the outcomes with other vaccination
strategies. These strategies may include limiting
vaccinations to only two groups of the oldest people
(8 and 7), vaccinating only group 8, administering
vaccines evenly across all age groups (1 to 8), and
comparing results with vaccinations targeted at age
groups highly active in the labor market and those
maintaining frequent intergenerational contacts,
specifically groups 4 and 5. In summary, we will
compare the results of the following vaccination
strategies:
a) Vaccination of 3333 people per day in age
groups 8, 7, and 6
b) Vaccination of 5000 people per day in age
groups 8 and 7
c) Vaccination of 10000 people per day in age
group 8
d) Vaccination of 1250 people per day in all age
groups
e) Vaccination of 5000 people per day in age
groups 4 and 5.
Because mortality and the number of
symptomatically ill people are the most important
social parameters, the numbers of symptomatic
patients and the number of fatal cases in each age
group are presented in Table 2, Table 3, Table 4,
Table 5 and Table 6. The bold values are lower than
the number of cases in the absence of vaccinations.
The percentage change from the predicted number
of unvaccinated cases is shown in parentheses. From
Table 2, Table 3, Table 4, Table 5 and Table 6, it
can be understood that the greatest total decrease in
deaths (i.e. about 53%) can be achieved by
vaccinating only people in age group 8. However,
this results from a 91.8% decrease in mortality in
this group. A similar high decrease in the number of
symptomatically infected cases can be seen in group
8. By evenly vaccinating age groups 8, 7, and 6 and
8, 7, the total number of deaths can be reduced by
approximately 38.8% (Table 2) and 49.6% (Table
3), respectively. However, by extending the
vaccination campaign to all age groups, a more than
19% decrease in mortality and a decrease in the
number of symptomatic patients from 13% to 23%
in various age groups (Table 5).
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Fig. 3: Solutions of SEIRD model for Swedish
Considering the global decline in the number of
symptomatic patients (who constitute the greatest
burden for medical services), simultaneous
vaccination of all age groups can reduce the total
number of symptomatic cases by 14.7% (Table 5),
which is 12.5% (Table 4) higher compared to the
decline achieved with vaccination of only age group
8. For strategies b (Table 3) and a (Table 2), the
number of symptomatic cases decreases by 7.6 and
10.7%, respectively. When limiting vaccinations to
groups 4 and 5, the most significant reduction in the
number of symptomatic patients (17.4%) is
observed, alleviating the burden on the health
service the most. However, this approach leads to
the lowest decrease in mortality (11.5%) for the
entire population.
Table 1. Summary of the results of the non-
vaccination seird model
Number
of age
group
R0n Maximum
of Isn
Maximum
of Ian
Number
of Dn
1 57.9 38385 27022 0
2 78.1 35581 25039 1
3 90.2 42053 29593 5
4 92.3 41405 29137 9
5 87.4 41072 28906 24
6 84.4 41371 29126 83
7 4.4 27778 19981 313
8 0.42 6606 4931 598
Total 274251 193735 1033
Table 2. Summary of the results of the SEIRD
model solutions taking into account vaccinations of
age groups 8, 7, and 6 (3333 vaccinations per day in
each group with 90% efficiency).
Number of
age group
Maximum of
Isn
Number of Dn
1 38366 0
2 35569 1
3 42031 5
4 41382 9
5 41047 24
6 26769 (–35) 53 (–36)
7 15667 (–43.5) 182 (–42)
8 4012 (–39) 358 (–40)
Total 244843 (–10.7) 632 (–38.8)
Table 3. Summary of the results of the SEIRD model
solutions taking into account vaccinations of age
groups 8 and 7 (5000 vaccinations per day in each
group with 90% efficiency).
Number of
age group
Maximum of
Isn
Number of Dn
1 38384 0
2 35581 1
3 42053 5
4 41404 9
5 41071 24
6 41370 83
7 10265 (–63) 118 (–62)
8 3199 (–51.6) 280 (–53)
Total 253327 (–7.6) 520 (–49.6)
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Table 4. Summary of the results of the SEIRD model
solutions taking into account vaccinations of only age
group 8 (10000 vaccinations per day with 90%
efficiency)
Number of
age group
Maximum of
Isn
Number of Dn
1 38385 0
2 35581 1
3 42053 5
4 41405 9
5 41072 24
6 41371 83
7 27746 312
8 564 (–91.5) 49 (–91.8)
Total 268177 (–2.2) 483 (–53.2)
Table 5. Summary of the results of the SEIRD
model solutions taking into account vaccinations of
all age groups (1250 vaccinations per day in each
group with 90% efficiency)
Table 6. Summary of the results of the seird model
solutions taking into account vaccinations of age
groups 4 and 5 (5000 vaccinations per day in each
group with 90% efficiency)
Number of
age group
Maximum of
Isn
Number of Dn
1 38199 0
2 35477 1
3 41890 5
4 19404 (-53.1) 4 (-55.5)
5 19080 (-53.5) 11 (-54)
6 41179 83
7 25772 (–7.2) 309
8 5383 (–18.5) 501 (-16.2)
Total 226384 (–17.4) 914 (–11.5)
4 Conclusion
The expected vaccination results can be assessed
from the point of view of reducing mortality rates
across various age groups, and/or reduction in the
number of infected in individual groups of a given
population. Considering the availability of vaccines,
their effectiveness, and restrictions on the number of
vaccinations per day, articulating an optimal
vaccination strategy becomes challenging, requiring
consideration of ethical, financial, and social
criteria. The COVID-19 pandemic was declared by
the WHO, [17], on 11th March 2020, and the first
vaccines were only available at the end of 2020.
Therefore, during the period without vaccines,
significant emphasis was placed on isolating
infected individuals from the healthy population.
Due to financial and/or organizational constraints,
different quantities of diagnostic tests were
conducted by healthcare services in various
countries. These tests were intended to form the
basis for the mandatory isolation of infected
individuals and the identification of a group of
healthy but susceptible individuals, who would be
vaccinated, [18]. In many countries, individuals
were qualified for vaccination based on a medical
interview, which did not allow for the identification
of individuals from the latent group or those
infected asymptomatically. This led to significant
variations in individual responses to vaccinations,
including post-vaccination symptoms, [19].
Analyzing the solutions of the SEIRD model
excluding vaccinations, it becomes apparent that
individuals in groups E (who had already come into
contact with the infecting factor) and Ia appear
almost simultaneously with group Is. Thus, without
specialized tests, people from groups S, E, and Ia
will be eligible for vaccination. This suggests that
extensive testing is required to detect groups E and
Ia, for whom vaccination is not advisable.
Analyzing the values of the R0 reproduction
number for all age groups, it should be noted that
only for the oldest age group 8 is its value less than
1 – thus, the expected effect should be the absence
of pandemic development in this group. It should
also be noted that intergenerational contact matrices
were published over 17 years ago during a
pandemic-free period. During this time, there have
also been socio-cultural changes in many countries
around the world, which likely result in changes to
the estimated values of contact matrices.
Since the appearance of vaccines on the market,
it was necessary to establish rules for their global
distribution and carry out widespread awareness
campaigns about the importance of vaccination.
Unfortunately, in various countries, anti-vaccination
movements also developed, leading to the
underutilization of purchased vaccine doses. In the
quest for an optimal vaccination strategy,
considering the limited vaccine supply, the
organization of vaccination campaigns (the possible
number of vaccinations per day) should also take
into account the organizational efficiency of the
healthcare system in a given country. This includes
Number of
age group
Maximum of
Isn
Number of Dn
1 32824 (–14.5) 0
2 30033 (–15.6) 1
3 36495 (–13) 4
4 35848 (–13.4) 8
5 35515 (–13.5) 20 (-16.6)
6 35814 (–13.4) 72 (-13.2)
7 22355 (–19.5) 262 (-16,3)
8 5050 (–23.5) 461 (-23)
Total 233934 (14.7) 828 (–19.8)
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the speed of establishing temporary hospitals, the
capacity for medical transport, monitoring the
isolation of infected individuals, and ensuring the
effectiveness of protecting healthcare workers as a
top priority. Prioritizing vaccinations for groups
with the highest mortality rate (given limited
vaccine supply) results in a smaller decline in the
total number of symptomatically infected
individuals compared to vaccinations carried out
initially in groups with the highest professional and
familial engagement (groups 4 and 5). This may
lead to limited access to medical services unrelated
to COVID-19, ultimately resulting in an increase in
mortality compared to the pre-pandemic period.
In the future, comprehensive literature studies in
search of reliable epidemiological data on COVID-
19 in different countries will allow for the
comparison of outcomes from selected vaccination
strategies, taking into account cultural customs,
social relations, and the level of national income.
According to the data presented in the publication,
[20], the number of COVID-19 vaccine doses varies
from 0 to 120 per 100 people in different countries
and is not correlated with the gross domestic
product (GDP) per capita. Understanding the current
values of the SEIRD model parameters, which vary
with successive waves of epidemic growth caused
by other variants of the SARS-CoV-2 virus, will
enable the analysis of the pandemic's progression
with multiple recurrent waves.
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APPENDIX
Heaviside function
The Heaviside function is defined as:
τtfor τ<tfor
=(t)Hτ1
0
Using this function, it is possible to control the
course of vaccinations, starting on t1 and ending on
t2, with vaccination rate and effectiveness ef: ef *
vac * Hev(t, t1) * (1 – Hev(t, t2)).
SEIRD model
Denote Rn0 (n=1,2,…,8) as the basic reproduction
number for each age group.
To calculate Rn0 we use the method described in
[21].
The models SIR and SEIRD are the ODEs of the
following type
(1)
where and .
Let be infected population groups of
compartments in .
Let be the rate of appearance of new
infections in the i-th compartment, and let be
the rate of the transition rates in the i-th
compartment.
Then, Equation (1) takes a form:
Define and
for where is the
disease-free equilibrium.
Then, where is the
spectral radius of the matrix .
(the spectral radius of the matrix can be defined as
the largest eigenvalue of the matrix).
The model SIR can be written as:
Define as disease-free equilibrium.
Then, and . Hence
.
The SEIRD model we rewrite as
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for n=1,2,…,8
Let be a disease-free equilibrium.
Then,
(n=1,….,8)
and
(n=1,…,8)
Therefore
(n=1,…,8)
Matrices of contacts between age groups (social
contact matrix)
Table IA presents social contact matrices for
Sweden—based on the literature data, [11].
n is the number of age groups (each 10-year). The
values of kn,m are average values calculated based on
the contact matrix for the population divided into 5-
year groups (data from publication, [11], —16 16
matrices), e.g. the value k1,1 = 2.583 for contacts of
people from the n = 1 group with people of the same
age group is the mean value calculated based on 16
16 matrix: (k1,1 + k1,2 + k2,1 + k2,2)/4.
Table IA. Swedish Social Contact matrix
n
1
2
3
4
5
6
7
8
1
2.583
0.450
0.399
0.986
0.459
0.217
0.169
0.061
2
0.496
4.615
0.690
0.644
0.912
0.279
0.092
0.046
3
0.292
0.933
2.895
1.443
1.160
0.721
0.104
0.043
4
0.811
0.655
1.267
2.523
1.582
0.780
0.210
0.061
5
0.448
1.091
1.058
1.624
2.175
0.884
0.174
0.076
6
0.479
0.893
1.082
1.289
1.531
1.526
0.322
0.092
7
0.393
0.349
0.448
0.716
0.579
0.587
1.108
0.253
8
0.231
0.353
0.192
0.346
0.474
0.365
0.579
0.606
Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
- Maciej Urban has implemented the SEIRD
algorithm in Python,
- Julia Jodłowska carried out the simulation,
- Joanna Balbus provided mathematical oversight
and derived the formula for the basic reproduction
number for each age group,
- Krystian Kubica conceived and designed research
approved final version of the manuscript.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
No funding was received for conducting this study.
Conflict of Interest
The authors have no conflicts of interest to declare.
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
This article is published under the terms of the
Creative Commons Attribution License 4.0
https://creativecommons.org/licenses/by/4.0/deed.en
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