*Corr. Author’s Address: Jožef Stefan Institute, Jamova cesta 39, 1000 Ljubljana, Slovenia, matjaz.leskovar@ijs.si 213
Strojniški vestnik - Journal of Mechanical Engineering 68(2022)4, 213-224 Received for review: 2022-02-09
© 2022 The Authors. CC BY 4.0 Int. Licensee: SV-JME Received revised form: 2022-03-03
DOI:10.5545/sv-jme.2022.50 Original Scientific Paper, Special Issue: SARS-CoV-2 Accepted for publication: 2022-03-03
Robust and Intuitive Model for COVID-19 Epidemic in Slovenia
Leskovar, M. – Cizelj, L.
Matjaž Leskovar
*
 – Leon Cizelj
Jožef Stefan Institute, Slovenia
The main goal of epidemic modelling is to support the epidemic management through forecasts and analyses of past developments. With 
this in mind a robust and intuitive SEIR (Susceptible, Exposed, Infectious, Recovered) type model has been developed, applied and validated 
during the multiple waves of the COVID-19 epidemics in Slovenia since March 2020. The model parameters were based on the general 
characteristics of the COVID-19 disease reported globally for the entire planet and refined with the aggregate data available mostly on a 
daily basis in Slovenia, as for example the number of confirmed cases, hospitalized patients, hospitalized patients in intensive care units and 
deceased. The Slovenian aggregate data was also used to estimate the degree of immunisation due to past infections and vaccination, which 
reduces the number of susceptible persons for the disease.
Examples of the model application are presented to illustrate its robustness and intuitiveness in both the forecasts and analyses of past 
developments. The analyses of past developments provided specific estimates of modelling parameters for Slovenia and quantified the effects 
of pharmaceutical and non-pharmaceutical interventions and various events on the development of the epidemics as measured through the 
reproduction number R. This empirically obtained information was then applied in the forecasts. Accurate forecasts are a great support for 
decision makers and for hospitals to plan appropriate actions in advance. The inherent uncertainties in the model and data were quantified 
through intuitive sensitivity analyses represented as different scenarios. The observed accuracy of the forecasts was impressively good also 
in demanding conditions, when various complex processes influencing the spread of the disease were going on in parallel. This demonstrates 
the robustness and relevance of the proposed model.
Keywords: epidemic, COVID-19, modelling, SEIR, reproduction number, public health interventions
Highlights
• 	 Robust and intuitive susceptible, exposed, infectious, recovered (SEIR) type model has been developed.
• 	 Model has been applied for analysis of COVID-19 epidemic in Slovenia.
• 	 Some examples of model application are presented.
• 	 Accuracy of forecasts is impressively good.
0  INTRODUCTION
The COVID-19 epidemic might have caught many 
countries and governments by surprise since its 
beginnings in November 2019 in Hubei province, 
China. Till today a huge amount of data and 
information have appeared online [1] to [4] and in the 
literature [5] to [9]. In addition, several models aiming 
at forecasting the spread have been developed [10] to 
[14].
The main goal of epidemic modelling is to 
support the epidemic management in real time and 
with the available data being far from optimal. It is 
crucial to get a reliable insight in the trends as early 
as possible to enable timely and appropriate actions 
by decisionmakers and for the hospitals to prepare in 
time.
Traditional approach for the modelling of 
epidemics including the COVID-19 includes 
compartment models. In particular, the susceptible, 
exposed, infectious, recovered (SEIR) model appears 
to be widely used [15] to [17]. In this approach the 
observed population (e.g., in Slovenia) is divided into 
four compartments, the susceptible, the exposed, the 
infectious and the recovered. The transition between 
compartments is modelled with transition rates, 
which are proportional to the membership of the 
compartments. The SEIR model may be extended 
with additional compartments, like the hospitalized, 
the hospitalized in intensive care units (ICU) and the 
deceased. The time development of the membership 
in the compartments is usually governed by a set of 
ordinary differential equations, which need to be 
solved numerically.
Our interest was mainly in the aggregate results 
such as the number of infected, the number of 
hospitalized, the number of deceased, the reproduction 
number R characterizing the dynamics of the epidemic 
[18], and similar. Therefore, we decided to address 
the epidemic in Slovenia at the aggregate level and 
provide aggregate forecasts. Focus was on the time 
dependence of the aggregate results, but implicitly 
considering also local phenomena, including the age 
structure and the occasional localisation of infections. 
For this purpose, the SEIR type modelling approach is 
in principle well suited.
To perform well in realistic conditions, it is 
important that the model is robust and intuitive. In 

Strojniški vestnik - Journal of Mechanical Engineering 68(2022)4, 213-224
214 Leskovar, M. – Cizelj, L.
this way the uncertainties in data, like changes in the 
testing regimes, impact of limited testing capacity, 
changes in the hospitalization criteria, changes in data 
reporting, missing data etc. can be most appropriately 
considered through quantitative and qualitative 
approaches, as for example the modeler’s expert 
judgment and holistic view.
With this in mind we developed an extended 
SEIR type model, formulated in integral form. In 
this framework the modelling becomes very intuitive 
and simple. The results, which are not distorted by 
numerical diffusion, become intuitively predictable 
and consequently strengthen the accuracy of the 
forecasts.
The developed robust and intuitive epidemic 
model is presented in the following Section 1, and then 
in Section 2 some examples of the model application 
are provided for illustration purpose.
1  MODELING
1.1  Epidemic Dynamics
The spread of a contagious disease in its early stages 
is an exponential process characterized by
 Nt NN R
tt t
 
00
2
2
/ /
.

 (1)
N(t) is the number of exposed persons at time 
t, N
0
 the number of exposed persons at t = 0, t
2
 the 
time in which the number of exposed doubles, R the 
reproduction number and τ the characteristic infection 
time. The reproduction number R is an average 
number of persons infected by a single person during 
his/her infectiousness period [18].
Epidemics will only develop if R > 1. When 
developing, the R will change with time. Gradual 
changes are usually consequence of immunisation, 
while steep changes may be related to the non-
pharmaceutical interventions enforced in the observed 
population. If R > 1 we have an exponential growth, 
if R = 1 we have stagnation and if R < 1 we have an 
exponential decay (Fig. 1).
The exponential growth of the number of exposed 
persons will be, usually with some delay, followed by 
exponential growth of other aggregate parameters, 
e.g., number of confirmed cases, the number of 
hospitalized patients, the number of hospitalized 
patients in ICU and the number of deceased.
Early stages of the exponential growth are not 
easily recognized by our intuition, which appears to 
be better adapted to linear or linearized phenomena. 
Fig. 2 shows an exponential growth with doubling 
time 7 days, where the aggregate parameters, e.g., 
the number of confirmed cases, will double every 
week. During the first month one could barely see 
anything happening, then the values start to rise and 
all of a sudden, the curve turns steeply upwards. 
This illustrates why it is so important to detect the 
exponential growth at the earliest possible stage.
Fig. 1.  Dynamics of epidemic for various reproduction numbers R
Fig. 2.  Exponential growth for reproduction number R > 1
Fig. 3 shows the same curve as in Fig. 2 in 
logarithmic scale. We see that the red exponential 
growth curve is a rising straight line already from the 
beginning, and thus we know already from the very 
beginning where this curve will lead us if we take no 
action. In logarithmic scale an epidemic runs along 
more or less straight lines. If R > 1 the epidemic will 
grow. This growth will not stop by itself as long as 
there are enough susceptible persons for the disease, 
and at that time the capacity of the hospitals may 
have been already exceeded by more than an order 
of magnitude. Therefore, the growth of an epidemic 
must be stopped in time with interventions, reducing 
the spread of the disease, to prevent the collapse of the 
health care system. And here the model forecasts can 
be of great help.

Strojniški vestnik - Journal of Mechanical Engineering 68(2022)4, 213-224
215 Robust and Intuitive Model for COVID-19 Epidemic in Slovenia
Fig. 3.  Dynamics of epidemic for various reproduction numbers R 
in logarithmic scale
1.2  Model Setup
In this subsection the basics of the epidemic model 
are presented. It was strived to develop a knowledge-
driven model as far as possible, with only the model 
parameters data-driven [19] and [20]. Therefore, the 
model was based on the exponential function, which 
is well suited for describing the spreading of an 
infectious disease. Eq. (1) can be written in difference 
form as
 

Nt Nt
tt
  

21
2
/
, (2)
where ΔN(t) is the change of quantity N(t) during 
the time interval Δt. Based on the form of Eq. 
(2) we derived the fundamental equation of our 
epidemiological model as
 


Et It
tt
  

21
2
/
'
,
 (3)
where ΔE(t) is the change of the cumulative number 
of exposed persons E(t) in time interval Δt, I(t) the 
number of infectious persons and t
2
' a modelling 
parameter. I(t) is calculated by
 Et tE tE t        , (4)
 It Et Et
inci nc inf
    

  , (5)
where τ
inc
 is the incubation time and τ
inf
 the infectious 
period, as depicted in Fig. 4. 
Fig. 4.  Incubation time and infectious period after infection
In Fig. 5 the course of the disease is presented. Let 
us assume that ΔE persons are exposed at time t. After 
a time τ
T
 a fraction α
T
 of them are tested positive ΔT. 
After a time τ
H
 a fraction α
H
 of them are hospitalized 
ΔH. After a time τ
ICU
 a fraction α
ICU
 of them are 
hospitalized in intensive care units ΔICU. And after 
a time τ
D
 a fraction α
D
 of them decease ΔD. At the 
current stage of the modelling, the average values 
of all time delays τ and fractions α were considered. 
They may nevertheless change with time due to for 
example the inherent variability of different strains of 
SARS-CoV-2 virus.
Fig. 5.  Course of disease
From the number of exposed persons ΔE in a 
time interval Δt the time shifted number of cases 
ΔT, hospitalized persons ΔH, hospitalized persons in 
intensive care units ΔICU and deceased persons ΔD 
are calculated based on the equations presented in 
Fig. 5. The empirical estimation of parameters α and τ 
is described in section 1.6.
For more accurate modelling the equations in 
Fig. 5 may be written and solved separately for each 
age group i
 
 Tt Et
iT
i
T
   ,
 (6)
  Ht Et
iH
i
H
   , (7)
  ICUt Et
iI CU
i
ICU
   , (8)
  Dt Et
iD
i
D
   . (9)
The number of currently hospitalized persons 
H
cur
 is calculated from the number of cumulative 
hospitalized persons H as
 Ht tH tH t        , (10)
 HtHt Ht
curH dur
      , (11)
where τ
Hdur
 is the average duration of hospitalization. 
Similarly, the number of currently hospitalized 
persons in intensive care units ICU
cur
 is calculated as
 ICUt tI CU tI CU t        , (12)
 ICUtICUt ICUt
curI dur
      , (13)
where τ
Idur
 is the average duration of hospitalization 
in ICU.

Strojniški vestnik - Journal of Mechanical Engineering 68(2022)4, 213-224
216 Leskovar, M. – Cizelj, L.
1.3  Reproduction Number
Eq. (3) can be expressed as a function of the time 
dependent reproduction number R(t) as
   



Et ItRt
t
Rt
Et
It t
inf
inf
     

 

,, (14)
assuming that the infection rate during the infectious 
period τ
inf
 is constant. The incubation time and 
the infectious period in this simplified approach 
were adjusted to result in the correct serial interval, 
i.e. the time between successive cases in a chain of 
transmission [21]. Considering Eq. (5) the reproduction 
number can be calculated from
 Rt
Et
Et Et
t
inci nc inf
inf


   



 

. (15)
As the number of confirmed cases ΔT, 
hospitalized persons ΔH, hospitalized persons in 
intensive care units ΔICU and deceased persons ΔD 
is a linear function of the number of exposed persons 
ΔE (see equations in Fig. 5), the reproduction number 
can be calculated from these quantities by substituting 
them in Eq. (15). Thus, based on the number of daily 
ΔT and cumulative T cases one obtains the following 
expression
     Rt
Tt
Tt Tt
t
T
inci nc inf
inf
  

  


 
 

, (16)
where the reproduction number is being estimated 
for the time τ
T
 in the past. The current estimate of R 
from the number of confirmed cases is therefore valid 
for the time in the past, characterized by an (average) 
time needed from infection to a confirmation by test. 
Similarly, the reproduction numbers can be estimated 
for the past also from the data of hospitalizations, 
hospitalizations in ICU and deceased, with a time 
delay from infection till the data, replacing in Eq. (16) 
T with H, ICU or D, and τ
T
 with τ
H
, τ
ICU
 or τ
D
.
Due to the predominantly weekly rhythm of the 
society, the data shows strong daily variations with 
a weekly period. Weekly averages of the aggregate 
parameters in the equations could therefore make the 
results more stable and reliable.
1.4  Immunity
Immunity may be obtained by infection or vaccination. 
The number of exposed people is reduced by the 
fraction of people which obtained immunity due to 
past infections according to
  Et
eEt
N
Et
inf
E
pop
 









 1, (17)
where e
E
 is the efficiency of immunization due to past 
infections and N
pop
 is the number of all people in the 
considered population (e.g., 2.1 million for Slovenia). 
Similarly, the number of exposed people is reduced by 
the fraction of people which obtained immunity due to 
vaccination as
  Et
eV t
N
Et
vac
V
pop
 









 1, (18)
where e
V
 is the efficiency of immunization due to 
vaccination and V is the number of vaccinated people. 
The efficiency of immunisation due to past infections 
and vaccination is obtained from global data available 
in the literature. The population with past infection 
and the vaccinated population partly overlap. This is 
considered with the following general equation 
  Et
t
N
Et
im
pop
 









 1
IM
, (19)
where IM is the equivalent number of totally 
immunized people, which may be calculated from E 
and V considering their efficiencies of immunisation 
and overlapping.
Vaccination does not reduce only the number 
of susceptible people but may also change the age 
structure of the exposed. The vaccination was namely 
prioritizing the more vulnerable elders. The fractions 
of exposed, which need hospitalization, intensive 
care or die, depend strongly on the age structure of 
the infected, which changes due to vaccination. The 
influence of vaccination therefore depends on the 
fraction of vaccinated v
i
 in each age group i and the 
relative contribution of this age group, which may be 
described with weights w
i
. Thus, the time dependent 
fraction of exposed people who need hospitalization 
considering the vaccination dynamics may be 
calculated as
 
Hvac H
i
n
H
i
i
tw vt   


1
1, (20)
where w
H
i
 are the weights for hospitalisation, which 
may be obtained from Eq. (7) as
 w
H
i
H
i
H
 /. (21)
Similarly, the influence of vaccination on the 
corresponding fractions for hospitalizations in ICU 
and deaths can be calculated by replacing in Eq. (20) 
and Eq. (21) α
H
 with α
ICU
 or α
D
, α
H
i
 with α
ICU
i
 or α
D
i
, 
and w
H
i
 with w
ICU
i
 or w
D
i
.

Strojniški vestnik - Journal of Mechanical Engineering 68(2022)4, 213-224
217 Robust and Intuitive Model for COVID-19 Epidemic in Slovenia
1.5  Virus Variants
The spreading of various virus variants is considered 
by modelling each variant separately and then 
summing the results. The variants share the same 
susceptible population or part of it, depending on the 
cross-immunity of the variants. The characteristics of 
the variants are obtained from global data available in 
the literature. For practical reasons, in the model the 
characteristics of the variants are expressed by the 
characteristics of the chosen basic variant multiplied 
by a constant proportionality factor to enable a 
straightforward fitting of the model parameters to 
data. Consequently, only the characteristics of the 
basic variant are independent variables and so the 
same fitting procedure can be applied as when treating 
only a single variant. Typically, only two variants are 
of interest at the same time. Thus, the equations are 
presented for the parallel treatment of two variants, 
but the approach could be easily extended for the 
general parallel treatment of more variants.
If variant v1 is chosen as the basic variant, 
then the characteristics of the other variant v2 are 
expressed by the characteristics of variant v1 as 
follows. Different contagiousness of both variants is 
considered by linking the reproduction numbers R
v2
 
and R
v1
 of both variants as
 Rr R
vv v 22 1
= , (22)
where r
v2
 is a constant proportionality factor and R
v1
 
is the independent variable. Different severities of the 
disease for both variants are considered by linking 
the fractions of the exposed for hospitalization α
H
, 
hospitalisation in ICU α
ICU
 and death α
D
 for both 
variants as
 
H
v
H
v
H
v
A
22 1
 , (23)
 
ICU
v
ICU
v
ICU
v
A
22 1
 , (24)
 
D
v
D
v
D
v
A
22 1
 , (25)
where A
H
v2
, A
ICU
v2
 and A
D
v2
 are the corresponding 
constant proportionality factors and α
H
v1
, α
ICU
v1
 and  
α
D
v1
 are the independent variables. Similarly, also 
different time delays from infection to hospitalisation, 
hospitalisation in ICU and death may be considered 
for both variants.
The aggregate results are then obtained by 
summing the results over all considered variants vi, 
presented here for the exposed, as
  Et Et
vi
vi
 

. (26)
Similarly, the aggregate results for the other 
considered quantities are obtained by replacing in Eq. 
(26) ΔE with ΔT, ΔH, ΔICU or ΔD.
The time development of fraction f
vi
 of variant vi 
is obtained from
 ft Et Et
vi vi
    /. (27)
1.6  Model Parameters
If the data is presented in logarithmic scale, we 
realise that the data curves for the daily positive tests, 
hospitalized, hospitalized in ICU and the deceased 
are in the simplified case just shifted curves of the 
infected (Fig. 6). The slope of all these curves is the 
same and depends on the reproduction number R. 
From the slope of these curves we can estimate the 
reproduction number, as presented in Section 1.3.
Fig. 6.  Presentation of data in logarithmic scale; example with 
straight lines
The curves are shifted vertically and horizontally. 
They are shifted vertically due to the fractions α in 
the equations presented in Fig. 5, i.e. the fraction of 
the infected α
T
 that are tested positive, the fraction of 
the infected α
H
 that are hospitalized, the fraction of 
the infected α
ICU
 that are hospitalized in ICU, and the 
fraction of the infected α
D
 that die. Further, they are 
shifted horizontally due to the time delays τ, i.e. the 
time τ
T
 from infection to a positive test, the time τ
H
 
from infection to hospitalization, the time τ
ICU
 from 
infection to hospitalization in ICU and the time τ
D
 
from infection to death.
It may be observed in Fig. 6 that the fractions 
α and time delays τ cannot be determined uniquely, 
because e.g. from the green curve of infected we can 
come to the blue curve of positive tests with different 
combinations of vertical and horizontal shifts. These 
shifts could be determined uniquely only if there 

Strojniški vestnik - Journal of Mechanical Engineering 68(2022)4, 213-224
218 Leskovar, M. – Cizelj, L.
would be a change in the slope of the curves defining 
a reference point.
Fig. 7 presents an example, where the curves 
are kinked due to the sudden introduction of non-
pharmaceutical interventions, which reduce the 
reproduction number R to 1 and change the slope of 
the curves from an exponential growth to horizontal 
stagnation. In this case both, the fractions α and time 
delays τ can be determined uniquely, as there is a 
reference point, i.e. the kink in the curves, which 
occurs when the non-pharmaceutical interventions are 
implemented. The fractions α can be determined from 
the vertical shifts of the kinks in the curves and the 
time delays τ can be determined from the horizontal 
shifts.
Fig. 7.  Presentation of data in logarithmic scale; example with 
kinked lines due to non-pharmaceutical intervention
The easiest way to estimate the fractions α is 
when the epidemic stagnates, as we are not limited 
only to the kinks in the curves, but we can determine 
it in the entire stagnation area (Fig. 7). The time 
delays τ are most easily estimated when very stringent 
non-pharmaceutical interventions are implemented 
maximizing the change in the slope of the curves.
With this approach the fractions α can be 
established also for age groups, as addressed in Section 
1.2. This information is also needed to consider 
the influence of age group specific vaccination, as 
presented in Section 1.4.
Once we estimate the fractions α for age groups 
one can use Eq. (6) to Eq. (9) for short-term estimations 
of hospitalizations, hospitalizations in ICU and deaths 
based on the known number of confirmed cases in age 
groups in the last period as
  Ht Tt
i
n
H
i
T
i HT
   


1


, (28)
  ICUt Tt
i
n
ICU
i
T
i ICUT
   


1


 , (29)
  Dt Tt
i
n
D
i
T
i DT
   


1


. (30)
The model parameters were estimated based 
on the following daily and cumulative available 
aggregate data for Slovenia [4], [22] to [26]:
• Number of positive tests, share of positive tests, 
number of PCR and rapid antigen tests, age 
structure of persons with confirmed infection.
• Daily admissions and discharges from hospitals 
and ICU, current and cumulative number of 
hospitalized patients and hospitalized patients in 
ICU, age structure of hospitalized patients and 
hospitalized patients in ICU.
• Number of deceased, age structure of deceased.
• Share of SARS-CoV-2 virus variants.
• Concentration of SARS-CoV-2 virus in sewage.
The ratio of confirmed cases and all cases α
T
 was 
estimated based on seroprevalence data of SARS-
CoV-2 virus for Slovenia [27] and expert judgement. 
The fractions α and time delays τ were estimated 
regularly at changes of the trend of the epidemic, 
considering also the age structure. The duration of 
hospitalizations and hospitalizations in ICU were 
estimated based on the known data of daily admissions 
and the current number of hospitalized patients 
and hospitalized patients in ICU. The reproduction 
number was estimated based on the daily number of 
positive tests, admissions in hospitals and ICU, and 
deceased. When estimating the model parameters, we 
tried to consider all available data in a holistic way, 
considering also soft data, like the social climate 
affecting the consistent implementation of non-
pharmaceutical interventions.
2  RESULTS
The developed model was applied for the analysis of 
the COVID-19 epidemic situation in Slovenia and to 
forecast the epidemic development [28]. In this section 
some examples of the model application are provided 
for illustration purpose.
The forecasts were performed in the following 
way. If no major change in people's behaviour in 
near future influencing the disease transmission was 
expected, it was assumed that the model parameters 
will not change. In this case the disease transmission 
is influenced only by changes in the immunity of the 
population due to vaccination and past infections, as 
described in Section 1.4.
Planned implementation or release of non-
pharmaceutical interventions and important changes 

Strojniški vestnik - Journal of Mechanical Engineering 68(2022)4, 213-224
219 Robust and Intuitive Model for COVID-19 Epidemic in Slovenia
in people's behaviour, such as the beginning or end 
of holidays, are taken into account by changing the 
reproduction number R based on past experience in 
Slovenia or other countries, or expert judgement. 
Similarly, also the influence of weather, where e.g. 
in colder conditions people stay indoors enhancing 
disease transmission, was considered. We tried 
to consider as far as possible also soft data, i.e. 
information that is not given in form of numbers, 
like the social climate affecting the consistent 
implementation of non-pharmaceutical interventions, 
or that the virus entered a retirement home generating 
an outbreak within a specific age structure.
The initial part of the forecast is checked with the 
short-term prediction based on the known number of 
confirmed cases in age groups in the last period, as 
described in Section 1.6.
2.1  Scenarios
Fig. 8 shows a typical example of an epidemic forecast 
with different scenarios, when approaching the peak 
of the second wave in Slovenia end of year 2020. 
On October 26, 2020 stringent non-pharmaceutical 
interventions were implemented [4]. Before that 
some non-pharmaceutical interventions were already 
implemented and the reproduction number was 
estimated to be R = 1.5.
In the forecast three scenarios are presented for the 
number of hospitalized patients, hospitalized patients 
in ICU and deceased. The dotted curves present the 
scenario without additional non-pharmaceutical 
interventions. In this case the epidemic continues 
to growth with the reproduction number R, that 
gradually decreases due to the build up of immunity 
in the population caused by past infections. When the 
reproduction number is reduced to R = 1 the peak is 
reached and after that the curves start to decrease, 
except for the deceased, where the cumulative number 
is shown.
The dashed line shows the scenario with 
successful non-pharmaceutical interventions, reducing 
the reproduction number to R = 0.9. Because R < 1, the 
epidemic starts to decrease and consequently with the 
time delays τ also the number of hospitalized patients 
and hospitalized patients in ICU decreases.
The full line presents the forecast, where it is 
estimated that the non-pharmaceutical interventions 
Fig. 8.  Forecast with different scenarios for the second wave in Slovenia end of year 2020; the following scenarios are presented: without 
any additional non-pharmaceutical interventions (dotted line), with successful non-pharmaceutical interventions (dashed line) and forecast 
(full line); the curves show the number of hospitalized patients (orange curves), hospitalized in ICU (red curves) and deceased (gray curves)

Strojniški vestnik - Journal of Mechanical Engineering 68(2022)4, 213-224
220 Leskovar, M. – Cizelj, L.
the epidemic grew during all that time. As a result, 
the number of confirmed cases became so large, that 
the capacity for epidemiological contact tracing was 
gradually exceeded, which significantly increased R. 
In addition, during this period the share of the more 
contagious variant B.1.258 significantly increased, 
further contributing to the rise of R. As a consequence, 
the epidemic started to grow extremely fast with 
a doubling time of only about one week. With a 
series of increasingly stringent non-pharmaceutical 
interventions Slovenia finally managed to stop the 
growth, but the numbers remained high. The epidemic 
then began to slowly decline due increased immunity 
of the population caused by past infections. During 
the Christmas and New Year holidays, the epidemic 
started again to rise due to more contacts among 
people. After that it started to decline faster and 
faster due to mass testing with rapid antigen tests, 
past infections and vaccination. First the vulnerable 
population was vaccinated, which significantly 
reduced the number of hospitalizations. The decline 
of the epidemic was then slowed down by some 
relaxation of non-pharmaceutical interventions. Then 
the more contagious alpha variant occurred and the 
epidemic started again to rise. The rise was so fast that 
in April 2021 a short lockdown had to be implemented, 
reduce the reproduction number to R = 1.2. In this case 
the epidemic still rises, but the peak is significantly 
lower than without any additional non-pharmaceutical 
interventions.
The purpose of presenting such scenarios is to get 
an impression about the uncertainties of the forecasts, 
the characteristic times, the possible maximal values 
of the presented results, the possible course of events 
etc. The calculated curves serve primarily for the 
qualitative and quantitative orientation of the possible 
development of the epidemic.
2.2  Analysis
As an example of an epidemic analysis the second 
wave in Slovenia end of year 2020 and beginning 
of year 2021 is analyzed in Fig. 9. The orange dots 
and curves show the number of hospitalized patients 
and the blue curve shows the estimated reproduction 
number R.
Already at the end of August 2020 the 
reproduction number R was bigger than one due to 
the end of holidays. It then increased sharply due 
to the start of the school year and then declined due 
awareness rising and some non-pharmaceutical 
interventions, but remained above 1, meaning that 
Fig. 9.  Analysis of second wave in Slovenia end of year 2020 and beginning of year 2021; the orange dots and curves show  
the number of hospitalized patients; the blue curve shows the estimated reproduction number R

Strojniški vestnik - Journal of Mechanical Engineering 68(2022)4, 213-224
221 Robust and Intuitive Model for COVID-19 Epidemic in Slovenia
which successfully contained the epidemic. Without 
the alpha variant the epidemic would just continue 
to decline and no non-pharmaceutical interventions 
would be needed.
On the basis of such analyses it is possible 
to find out how pharmaceutical (e.g, vaccination) 
and non-pharmaceutical interventions (e.g. masks, 
lockdowns) and various events (e.g. holidays) affect 
the reproduction number R. Such empirically obtained 
information was successfully applied in forecasts, as 
presented in the next Section.
2.3  Accuracy of Forecasts
To get an impression how accurate the forecasts are, 
some examples are provided for the third wave in 
Slovenia in the year 2021.
Fig. 10 presents the share of the alpha variant 
in Slovenia, which displaced the B.1.258.17 variant. 
Based on the data from Denmark [1], the model 
parameters of the alpha variant were determined. It 
was assumed that the ratio of the contagiousness of 
the alpha variant and the at that time dominant variant 
in Slovenia is the same as it was in Denmark and that 
consequently the dynamics of the alpha variant share 
will be the same in Slovenia as it was in Denmark. 
Based on that the green curve of the alpha variant 
share was calculated. When the data for Slovenia 
became available [24] and [25], the calculated curve 
was fitted to that data by accordingly time shifting it 
horizontally. Based on our calculations we forecasted 
already end of January 2021 that the third wave in 
Slovenia will probably start mid March, which turned 
out to be true. It may be seen that the agreement 
of the green curve with the data is nearly perfect, 
which means that the alpha variant share increased 
in Slovenia like it did in Denmark. This is also the 
reason why our forecast was so accurate.
Fig. 10.  Share of alpha variant in the year 2021
Figs. 11 and 12 show a demanding forecast for 
the third wave in Slovenia for hospitalizations and 
hospitalizations in ICU, considering the implemented 
non-pharmaceutical interventions, the spreading of the 
alpha variant, and the immunity due to past infections 
and vaccination. The forecast was prepared on  
March 31, 2021 before the 11 days lockdown, which 
was implemented on April 1, 2021 [4]. Based on past 
experience, where a similar set of non-pharmaceutical 
interventions was released as was now implemented 
in the lockdown, we estimated that during the 
lockdown the reproduction number R will be reduced 
by 25 % and then after the lockdown increased by  
10 % according to the Slovenian COVID-19 traffic 
light non-pharmaceutical intervention system.
The epidemic modelling in this period was very 
demanding as various processes were going on in 
parallel: non-pharmaceutical interventions were 
implemented, the share of the alpha variant was rising, 
the people were extensively vaccinated and the rate 
of infections was high, both importantly increasing 
the immunity of the population. Due to all these 
complex parallel processes it was anticipated that the 
uncertainties of the forecasts will be large. To capture 
them, in the presented scenarios the reproduction 
number R was varied by approximately ±10 % in 
regard to the projection, which had a large effect on 
the results, as may be seen on the figures.
In Figs. 11 and 12 with green dots the actual 
data after the forecast was prepared is plotted. We see 
that till the time, when it is anticipated that the non-
pharmaceutical interventions have an influence, i.e. 
the point when the curves for various scenarios start to 
separate, the forecasts are in nearly perfect agreement 
with the data. Also later, the agreement is impressively 
good, especially for the hospitalizations in ICU. Thus, 
it seems that with an appropriate modelling approach 
it is possible to make quite accurate medium-term 
forecasts also for a few months. It may be observed 
that without the alpha variant (and later new variants) 
the epidemic would end already in May 2021.
During our epidemic modelling we realized that 
at an extensive epidemic epistemic uncertainties 
dominate, like the influence of interventions and 
events significantly affecting people’s behaviour, e.g. 
start of holidays or school. The impact of aleatory 
uncertainties, such as random infections, is much 
smaller than one would expect based on the observed 
large daily variations of data. Namely, these daily 
variations are often only a consequence of the weekly 
rhythm of the society, and if we average the data over 
one week the average values become very smooth and 
much more predictable.

Strojniški vestnik - Journal of Mechanical Engineering 68(2022)4, 213-224
222 Leskovar, M. – Cizelj, L.
Fig. 11.  Forecast for third wave in Slovenia in the year 2021 for hospitalizations, considering the implemented non-pharmaceutical 
interventions, spreading of alpha variant, and immunity due to past infections and vaccination; the green dots show data  
after the forecast was prepared, which are in good agreement with the projection
Fig. 12.  Forecast for third wave in Slovenia in the year 2021 for hospitalizations in ICU, considering the implemented non-pharmaceutical 
interventions, spreading of alpha variant, and immunity due to past infections and vaccination; the green dots show data  
after the forecast was prepared, which are in good agreement with the projection

Strojniški vestnik - Journal of Mechanical Engineering 68(2022)4, 213-224
223 Robust and Intuitive Model for COVID-19 Epidemic in Slovenia
3  CONCLUSIONS
An epidemic is a complex nonlinear time dependent 
phenomenon. While it appears to be governed by 
a number of random processes and events, it is also 
reasonably predictable on the aggregate (e.g., country) 
level using purely deterministic models.
The main goal of epidemic modelling is to 
support the epidemic management through forecasts 
and analyses of past developments. Modelling during 
a developing epidemic must rely on the data available 
in real time, which may not be always optimal and 
might also change with time significantly. A robust and 
intuitive SEIR type epidemic model was developed, 
applied and tested in the realistic conditions of the 
COVID-19 epidemic developments in Slovenia since 
March 2020.
The model, formulated in the integral form for 
better robustness and intuitiveness, is described. 
Examples of applications include the analysis of the 
past epidemic situation in Slovenia and a forecast of 
the epidemic development.
Analyses of past epidemic developments help to 
explain the effect of non-pharmaceutical interventions 
and various events on the development of the 
epidemics characterized through the reproduction 
number R. This empirically obtained information was 
then used in the simulation of different scenarios, 
providing valuable qualitative and quantitative insight 
of the possible future development of the epidemic.
The accuracy of the forecasts of the number of 
hospitalized patients and hospitalized patients in ICU 
was impressively good also in demanding conditions, 
when various complex processes influencing the 
spread of the disease were going on in parallel. This 
demonstrates the robustness and relevance of the 
proposed model. 
Accurate forecasts offered valuable support for 
decision makers and for hospitals to plan appropriate 
actions in advance. 
4  ACKNOWLEDGEMENTS
The authors acknowledge the financial support of 
the Slovenian Research Agency within the research 
program Reactor engineering (P2-0026).
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