=Paper=
{{Paper
|id=Vol-3038/paper8
|storemode=property
|title=Mathematical Modelling of an Epidemic Based on Covid-19 Spread Functions
|pdfUrl=https://ceur-ws.org/Vol-3038/short1.pdf
|volume=Vol-3038
|authors=Kateryna Molodetska,Yuriy Tymonin
|dblpUrl=https://dblp.org/rec/conf/iddm/MolodetskaT21
}}
==Mathematical Modelling of an Epidemic Based on Covid-19 Spread Functions==
https://ceur-ws.org/Vol-3038/short1.pdf
Mathematical Modelling of an Epidemic Based on Covid-19
Spread Functions
Kateryna Molodetskaa and Yuriy Tymonina
a
Polissia National University, Blvd Stary, 7, Zhytomyr, 10008, Ukraine
Abstract
The study of the mechanisms of epidemic spread is an important way of controlling the disease.
Reducing damage from a coronavirus epidemic is linked to the use of methods and tools for
mathematical modelling of Covid-19 spread. Epidemic wave representations are used to
characterize the spread of Covid-19, which is highly visual and informative. However, this
"wave" representation places increased demands on Covid-19 spread models.
For mathematical modelling of the spread of the Covid-19 epidemic, is considered the
application of specific Covid-19 propagation functions, based on constrained growth functions.
The Covid-19 spread functions show high accuracy in approximating statistical data, which
demonstrates the good adequacy of these functions in principle. Application of the Covid-19
propagation functions makes it possible to quantitatively describe the basic concepts of the
epidemic and conduct a comparative parametric analysis of the epidemic's spread and predict
the development of the epidemic. Comparison of parameter values makes it possible to identify
differences in indicators and growth rates, based on which the results of epidemic control can
be assessed.
Keywords 1
Covid-19 spread functions, approximation of Covid-19 statistical evidence, parametric
analysis
1. Introduction
Mathematical modelling of epidemic spread makes an important contribution to disease control.
Modelling the mechanisms of epidemic spread and predicting its evolution can significantly reduce the
damage caused by a pandemic [1-5]. Quantitative model simulations can provide comparative analysis
and predictive modelling of temporal descriptions of key epidemic categories such as the number of
people who became ill, recovered and died [6-9]. Covid-19 propagation models are therefore subject to
increasing demands, not only for consistency with statistical data but also for the adequacy and accuracy
of the underlying concept descriptions.
As we know from [10-11], the SIR model developed by Kermack and McKendrick in 1927-1933 is
widely used to describe epidemics, which is based on a scheme of epidemic transition of basic variables
from one category to another. The variables used as basic variables are those that denote the number of
individuals: those susceptible (S) become infected (I), then recover (R). The SIR model is represented
by a system of 1st order coupled differential equations that describe the time dependence of the
underlying concepts, where the coupling is given by conditions that stipulate the sum of the variables
and their derivatives.
Models which implement the concept of epidemic transition have gained wide popularity and
development, so the SIR class of models today also contains varieties: SIRS, SEIR, SIS, MSEIR, etc.
However, the experience of applying SIR class models for mathematical modelling of Covid-19 spread
[10, 12-15] has shown insufficient consistency of the calculations of basic variables with statistical data.
IDDM-2021: 4th International Conference on Informatics & Data-Driven Medicine, November 19–21, 2021 Valencia, Spain
EMAIL: kateryna.molodetska@polissiauniver.edu.ua (K. Molodetska); YTimonin45@gmail.com (Y. Tymonin)
ORCID: 0000-0001-9864-2463 (K. Molodetska); 0000-0002-0179-5226 (Y. Tymonin)
©️ 2021 Copyright for this paper by its authors.
Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
CEUR Workshop Proceedings (CEUR-WS.org)
The desk review [15] noted that “An attempt to apply these models (SIR class) to the case of a
coronavirus pandemic in Ukraine showed that they fail when heterogeneous populations, different
routes of transmission and the presence of randomization factors are present”. Therefore, the project
team concluded that any projections derived for Ukraine, with its characteristic heterogeneity, using
SIR models and their derivatives cannot be considered correct and certain coincidences of projected
data may have a random nature. Therefore, the team of the “FORSAIT COVID-19” project applied a
group of methods of different nature and class to conduct a series of studies of the coronavirus
propagation process in Ukraine, based on the consideration that if the results obtained using different
methods are close, the plausibility of the studies is increased.
Thus, the problem with mathematical modelling of the spread of Covid-19 is the lack of adequacy
of SIR models, preventing the accuracy of description, analysis and prediction. The fundamental
shortcoming of SIR models, in our opinion, is that in the epidemic transition concept, the dynamics of
the main variables (ill 𝐼, recovered 𝑆) are defined through the concept of “contact”, which is defined by
the product of the variables. This representation of the interaction of variables severely limits the
modelling capability of the epidemic. The lack of accuracy of SIR models necessitates new approaches
for mathematical modelling of Covid-19 spread. New approaches that can improve the adequacy consist
of having models of the underlying concepts formed as independent constructs.
The nature of the statistics of the Covid-19 coronavirus epidemic shows that they are highly like
logistic functions. Therefore, we note the application of logistic functions to approximate a piece of
given statistical information. A mathematical model of the spread of the Covid-19 coronavirus epidemic
is considered in [16], which uses a simplified logistic model of the form describing the growth in the
number of cases. However, the application of this logistic equation has shown that this model is of low
accuracy. To improve the accuracy, it is suggested that the study range should be divided into regions
with partial logistic functions, which cause significant computational difficulties.
In the article [17], the authors note: “Having realised the complexity of the forecasting task, the
authors decided to restrict themselves to the simplest logistic model”. The low accuracy of the
calculation results obtained in [16, 17] can be attributed to the fact that simplified representations of
logistics models were used for modelling.
The article [18] considers the wave structure of an epidemic, which is represented by a set of
elementary epidemic flows (waves) shifted along the time axis and differing in parameter values. A
constrained growth function based on a generalized logistic model with extended description
capabilities due to additional conditions were used for mathematical modelling of Covid-19
propagation. Based on this model, an analytical description of the epidemic in the form of a complex
flow of epidemic events was obtained, which can be seen as a solution to an approximation problem for
a piece of given statistical information. However, the content of the article is limited to the
approximation task for statistical information describing the flow of events. Since the generalized
logistic model in mathematical modelling of epidemic event fluxes has shown increased adequacy and
a high degree of compliance of the calculations with the original statistical data, it seems appropriate to
apply this model to modelling epidemic propagation functions.
This article aims to develop mathematical models of epidemics in the form of basic event
propagation functions for key epidemic concepts based on a generalised logistic function and to use
these models for analysis and forecasting.
2. Mathematical models of epidemic spread
2.1. Non-linear differential equations of epidemic spread
Review the following basic concepts of the Covid-19 epidemic. Statistics use the following basic
categories to refer to the spread of an epidemic:
1. The number of individuals who became ill (infected).
2. The number of individuals who died (deaths).
The statistics by category are set on an accumulative basis, where intermediate totals are used to
show the total amount of data as it grows over time. By modelling Covid-19 prevalence statistics using
regression relationships, we obtain a representation of the epidemic prevalence functions, which have
the following properties
• the functions are monotonically increasing;
• the growth of the functions is limited to a certain value (threshold, plateau) to which the
function tends asymptotically.
Thus, the epidemic spread functions are S-shaped logistic curves. Therefore, to describe the
epidemic spread functions, we will use the constrained growth functions that have proved themselves
in conflict interaction models [18-21]. In general, constrained growth functions are defined in
algorithmic form as solutions to a 2-nd order nonlinear differential equation [18, 19]:
• for the number of infected individuals 𝑥(𝑡)
𝑑 2 𝑥(𝑡) 𝑑𝑥(𝑡) (1a)
𝑎2 𝑥(𝑡) 2 + (1 + 𝑎1 𝑥(𝑡)) + (𝑎0 𝑥(𝑡) − 𝜑)𝑥(𝑡) = 0;
𝑑𝑡 𝑑𝑡
• for the number of fatal cases 𝑦(𝑡)
𝑑 2 𝑦(𝑡) 𝑑𝑦(𝑡) (1b)
𝑏2 𝑦(𝑡) + (1 + 𝑏1 𝑦(𝑡)) + (𝑏0 𝑦(𝑡) − 𝜙)𝑦(𝑡) = 0,
𝑑𝑡 2 𝑑𝑡
where 𝑥(𝑡), 𝑦(𝑡) are epidemic variables; 𝜑, 𝜙 – growth rates;{𝑎2 ; 𝑎1 ; 𝑎0 }, {𝑏2 ; 𝑏1 ; 𝑏0 } –
phenomenological coefficients, which are treated as epidemic parameters.
Since the epidemic spread functions are monotonically increasing, to represent them we restrict
ourselves to a 1st order nonlinear differential equation at 𝑎2 ≈ 0, 𝑏2 ≈ 0:
• for infected individuals
𝑑𝑥(𝑡) 𝑎0 𝑥(𝑡) − 𝜑 (2a)
+ 𝑥(𝑡) = 0;
𝑑𝑡 1 + 𝑎1 𝑥(𝑡)
• for fatal cases
𝑑𝑦(𝑡) 𝑏0 𝑦(𝑡) − 𝜙 (2b)
+ 𝑦(𝑡) = 0,
𝑑𝑡 1 + 𝑏1 𝑦(𝑡)
Equations (2a) and (2b) can be considered as a generalised representation of the Verhulst logistic
equation [22-25], to which equations (2a) and (2b) can be reduced with the parameters 𝑎1 = 1 and
𝑏2 = 1.
2.2. Functions of the spread of the Covid-19 epidemic
Solutions to equations (2a) and (2b) specify the epidemic spread functions in the form of constrained
growth functions, which are used to describe the spread of Covid-19. The epidemic propagation
functions 𝑥(𝑡) = 𝑓(𝑡, 𝜑, 𝑎0 , 𝑎1 ), 𝑦(𝑡) = 𝑓(𝑡, 𝜙, 𝑏0 , 𝑏1 ) have two equilibrium states:
1. Initial equilibrium – unstable, 𝑥(0) when 𝑡 = 0;
2. Final equilibrium – stable, 𝑋(𝑡) → 𝑋 when 𝑡 → ∞.
A characteristic element of the epidemic propagation functions are expressions for the equivalent
growth rate coefficients:
• for infected individuals
𝜑 − 𝑎0 𝑥(𝑡) (3a)
𝜑̃(𝑡) = ;
1 + 𝑎1 𝑥(𝑡)
• for fatal cases
𝜙 − 𝑏0 𝑦(𝑡) (3b)
𝜙̃(𝑡) = ;
1 + 𝑏1 𝑦(𝑡)
Expressions (3a) and (3b) for equivalent growth rates describe an important property of epidemic
spread functions, namely that equivalent growth rates are not a constant but a function of primary
variables.
The epidemic spread functions vary over a range bounded by equilibrium states. A final steady-state
𝑑𝑥(𝑡) 𝑑𝑦(𝑡)
equilibrium corresponds to the conditions that the derivatives 𝑑𝑡 ≈ 0, 𝑑𝑡 ≈ 0 and expressions for
the coefficients of the equivalent growth rates 𝜑̃ ≈ 0 and 𝜙̃ ≈ 0, are zero.
An important characteristic of epidemic spread functions is the plateau (the upper limit of the
constrained growth function) – the threshold value towards which the constrained growth function tends
to move in a finite steady-state equilibrium. To determine the plateau of the epidemic's spread function,
we formulate equations corresponding to the zero values of the equivalent growth rate
• for infected individuals
𝑎0 𝑋 − 𝜑 = 0; (4a)
• for fatal cases
𝑏0 𝑌 − 𝜙 = 0. (4b)
where 𝑋, 𝑌 – plateau values.
Plateau values are defined as solutions to equations (4a) and (4b):
• for infected individuals
𝜑 (5a)
𝑋= ;
𝑎0
• for fatal cases
𝜙 (5b)
𝑌= .
𝑏0
The plateau of the epidemic spread functions is characterized by the ratio of the growth rate to the
phenomenological coefficient.
Note that the equivalent growth rate of infected individuals (3a) varies in the range from the
exponential growth rate of 𝜑̃(0) = 𝜑, when 𝑡 = 0, to zero 𝜑̃(0)𝑡→∞ ≈ 0 when 𝑡 → ∞.
Correspondingly, the equivalent growth rate of lethal cases (3b) varies in the range from the exponential
growth rate value of 𝜙̃(0) = 𝜙, when 𝑡 = 0, to zero 𝜙̃(0)𝑡→∞ ≈ 0 when 𝑡 → ∞.
2.3. Discrete Covid-19 spreading function
Let us use the Covid-19 spreading function representations as to the solutions to equations (2) for a
discrete-time as our computational expressions:
• for infected individuals
𝑥𝑘+1 = (1 + 𝜑̃(𝑥𝑘 ))𝑥𝑘 ; (6a)
𝜑−𝑎 𝑥
where 𝜑̃(𝑥𝑘 ) = 1+𝑎 0𝑥 𝑘 – equivalent growth rate coefficient.
1 𝑘
• for fatal cases
𝑦𝑘+1 = (1 + 𝜙̃(𝑦𝑘 ))𝑦𝑘 (6b)
𝜙−𝑏 𝑦
where 𝜙̃(𝑦𝑘 ) = 1+𝑏 0𝑦 𝑘 – the equivalent rate of increase in fatalities.
1 𝑘
The equivalent growth rate of infected cases 𝜑̃𝑘 (𝑥𝑘 ) varies in the range from 𝜑̃𝑘 (0) = 𝜑, 𝑘 = 0 to
zero 𝜑̃𝑘 (𝑥𝑘 ) = 0, 𝑘 → ∞. Correspondingly, the equivalent growth rate of lethal cases 𝜙̃𝑘 (𝑦𝑘 ) varies
in the range from 𝜙̃𝑘 (0) = 𝜙, 𝑘 = 0, to zero 𝜙̃𝑘 (𝑦𝑘 ) = 0, 𝑘 → ∞.
3. Calculations of Covid-19 spreading functions for different countries
3.1. Approximation of Covid-19 distribution statistics in different
countries
Equations (6a) and (6b) were used to approximate the statistical data for the spread of Covid-19 in
different countries. Actual data from the first wave of Covid-19 spread in the first half of 2020, which
has no epidemic prehistory, are used as input data.
The countries selected for the calculation of the Covid-19 spreading functions are Ukraine [26], Italy
[27], Spain [28] and France [29]. The definition of Covid-19 propagation functions consists in selecting
parameters for expressions (6a)–(6b):
• 𝜑, 𝑎1 , 𝑎0 – for the propagation function of infected individuals;
• 𝜙, 𝑏1 , 𝑏0 – for the distribution function of lethal cases.
Using the values of the parameters were calculated the integral indices of 𝑋 and 𝑌 – the plateau of
Covid-19 spread functions, which is estimated in persons. For an overall assessment of the spread of
𝑌
the epidemic, the relative indicator was used 𝑊 = 𝑋.
To estimate the approximation error were used the relative mean absolute error – MAPE, which
calculated according to the formula
1
𝛿𝑥 = ∑|𝑥̅ 𝑘 − 𝑥𝑘 | /𝑥̅𝑘 ,
𝑁
𝑁
where 𝑥̅ 𝑘 – statistical data values.
For the selected countries, the MAPE values show a reasonably high approximation accuracy (Table 1).
Table 1
Approximation accuracy
Country
Function type Parameter
Ukraine Italy Spain France
ill 𝛿𝑥 3,0% 2,4%, 1,6 % 1,4 %
deceased 𝛿𝑦 7,0 % 4,6 % 2,2% 2,0%
Ukraine [26] (Figures 1 and 2) and Italy [27] (Figure 3) are chosen as examples to show how the
calculated Covid-19 spread functions correspond to statistical data.
Figure 1: Compliance of the estimated values of Covid-19 infection spread functions with statistical
data for Ukraine for April and May 2020
Figure 2: Compliance of the estimated Covid-19 fatality distribution functions to the statistical data
for Ukraine for April and May 2020
Figures 1 and 2 clearly show a reasonably good correlation between the calculated and actual data,
where the MAPE does not exceed 3% and 7%.
Figure 3: Compliance of the estimated Covid-19 spread functions with the statistics for Italy for March,
April, and May 2020
Figure 3 clearly shows a reasonably good correlation between the calculated and actual data, where
the MAPE does not exceed 3% and 5%.
A comparison of the calculated Covid-19 spread functions for different countries is shown in
Figure 4.
Figure 4: Comparison of estimated Covid-19 distribution functions for different countries for April and
May 2020
Analysis of the results shows that the number of Covid-19 cases for Italy, France and Spain has
almost reached a plateau, with values around the same level. Ukraine has passed the inflexion point of
the Covid-19 prevalence curve and is halfway to the plateau. The number of Covid-19 cases in Ukraine
is markedly lower than in the other countries examined
The curves of the Covid-19 spread functions are smooth and monotonic, which, assuming unchanged
parameter values, allows them to be used for forecasts with a high degree of confidence.
3.2. Parametric analysis of Covid-19 spreading functions
The Covid-19 spreading functions in the different case studies share a common, universal
mathematical design and differ in parameter values, which allows for a comparative parametric analysis
of the spread of Covid-19. The parameter values for comparative analysis of the spread of Covid-19 in
different countries are shown in Table 2.
Table 2
Parameter values for analysis
Function Country
Parameter
type Ukraine Italy Spain France
𝑋 42 000 240 000 250 000 200 000
𝜑 0,2 0,235 0,25 0,2
ill 6
𝑎1 ∙ 10 230 16 15 17
6 4,76 1,00 1,00 1,00
𝑎0 ∙ 10
𝑌 1400 36 500 36 000 50 000
𝜙 0,15 0,34 0,34 0,47
deceased
𝑏1 ∙ 106 4600 220 220 220
6 105 9,32 9,32 9,32
𝑏0 ∙ 10
𝑌
𝑊= 0,03 0,15 0,14 0,25
𝑋
For Covid-19 cases:
• The values of the incidence rates 𝜑 are in the range 𝜑 ∈ {0,2; 0,25} and differ
insignificantly. The growth rate of the epidemic in these countries is about the same.
• For Italy, France and Spain, the phenomenological coefficients 𝑎1 are in the range of
𝑎1 ∙ 10−6 ∈ {15; 17}, 𝑎0 ≈ 10−6 and differ slightly. In Ukraine, the coefficient
𝑎1 ≈ 230 ∙ 10−6 is about 15 times larger than in the other countries. Given that the
coefficient 𝑎1 describes the inhibition effect of the Covid-19 spread, it can be concluded
that Ukraine has a high resistance to the epidemic.
For Ukraine, the coefficient values 𝑎0 ≈ 4,76 ∙ 10−6 are almost 5 times higher than in the
𝜑
other countries. Given that 𝑋 = 𝑎 , the consequence is that the threshold (plateau) 𝑋 of the
0
incidence curve decreases by a factor of almost 5 compared to other countries.
For Covid-19 lethal cases:
• The values of the indicators for the growth of the deceased 𝜙, are in the range of
𝜙 ∈ {0,15; 0,47} and vary considerably. The value for Ukraine is three times lower than
that for France.
• the values of the phenomenological coefficients for Italy, Spain and France are
approximately the same 𝑏1 ≈ 220 ∙ 10−6 , 𝑏0 ≈ 9,32 ∙ 10−6 . For Ukraine, the value of the
phenomenological coefficient 𝑏1 ≈ 4600 ∙ 10−6 is 20 times higher than in the other
countries, which indicates a high resistance to lethal cases;
• for Ukraine the coefficient value 𝑏0 ≈ 105 ∙ 10−6 is 10 times those of other countries. Given
𝜙
that 𝑌 = , the consequence is that the threshold (plateau) 𝑌 of the morbidity curve has
𝑏0
been reduced by a factor of almost 20 compared to other countries.
𝑌
The integral characteristic of an epidemic, defined as the ratio of deaths to cases 𝑊 = 𝑋 , is several
times smaller than in other countries.
As the values of the indicators can be linked primarily to prevention, sanitation, and treatment
interventions, they can be used to assess the results of controlling the epidemic in different countries.
The strategy for controlling the epidemic in terms of Covid-19 spread models is generic and consists of
lowering the threshold (plateau) of the disease as much as possible (model parameters 𝑋, 𝑌). This
requires:
1. Decreasing the epidemic's growth rate (model parameters 𝜑, 𝜙).
2. Increasing resistance to the virus (model parameters 𝑎1 , 𝑏1 ).
3. Reducing the range of the community of people accessible to infection (increase model
parameters 𝑎0 , 𝑏0 ).
Interpretation of these formal requirements is followed by known protective actions.
4. Conclusions
The epidemic spread models considered are fundamentally different from SIR models in the
following respects:
• SIR class models investigate the behaviour of epidemic categories depending on the
interaction between them, which is consistent with the principles of system dynamics.
Representing SIR models as a coupled set of differential category equations limits the
accuracy of the calculations.
• The epidemic spreading model examines the behaviour of epidemic categories as
decentralised agents and how this behaviour determines the behaviour of the system, so the
model is based on an unrelated set of differential category equations. This approach is
consistent with agent-based modelling methodology, which has greater descriptive power
than SIR models but requires a more detailed description of the epidemic categories. Agent-
based models use a dynamic system representation in which the details of the descriptions
are provided by feedbacks
The application of Covid-19 propagation functions based on the generalised logistic function shows
a high approximation accuracy of the statistical data, which demonstrates the good adequacy of these
functions. This correspondence with the original evidence suggests that the main problem of
mathematical modelling of Covid-19 propagation, which boils down to the adequacy of epidemic
models, can be solved by applying Covid-19 spreading models and functions.
The application of Covid-19 propagation functions makes it possible not only to quantitatively
describe the basic concepts of the epidemic but also to construct a reliable forecast, provided that the
parameters are constant. An even more important, in our opinion, the consequence of the application of
Covid-19 propagation functions is a comparative parametric analysis of specific epidemic spread
functions. Comparison of parameter values can reveal differences in growth rates and
phenomenological coefficients, from which conclusions can be drawn about different processes of
epidemic behaviour in different regions and countries. By linking these processes to prevention,
sanitation and treatment interventions, differences in the results of the epidemic can be identified,
analysed and good practices can be disseminated. In general, the application of Covid-19 spread
functions can help to reduce the harm caused by a pandemic.
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