=Paper=
{{Paper
|id=Vol-2753/paper32
|storemode=property
|title=Mathematical Modeling Covid-19 Wave Structure of Distribution
|pdfUrl=https://ceur-ws.org/Vol-2753/paper21.pdf
|volume=Vol-2753
|authors=Kateryna Molodetska,Yuriy Tymonin
|dblpUrl=https://dblp.org/rec/conf/iddm/MolodetskaT20
}}
==Mathematical Modeling Covid-19 Wave Structure of Distribution==
https://ceur-ws.org/Vol-2753/paper21.pdf
Mathematical Modeling Covid-19 Wave Structure of
Distribution
Kateryna Molodetskaa and Yuriy Tymonina
a
Polissia National University, Staryi Blvd., 7, Zhytomyr, 10008, Ukraine
Abstract
For mathematical modeling of the spread of the Covid-19 epidemic, a wave structure, which
represents a complex flow of epidemic events in the form of a set of simple epidemic flows
(epidemic waves) is considered. To represent the wave structure we need to decompose a
complex flow of epidemic events, given by statistics, into elementary flows. Mathematical
models of the wave structure of the epidemic are represented by a set of elementary epidemic
flows (waves) shifted along the time axis and different applications in the values of the
parameters. Application of Covid-19 propagation waves allows not only to describe the basic
concepts of the epidemic quantitatively but also build a reliable forecast of the spread of the
epidemic. An important consequence of the Covid-19 wave pattern is the possibility of
conducting a comparative parametric analysis of specific wave patterns of epidemic spread.
Based on the results of the analysis we can assess the results of the epidemic control. The
calculations of wave structures have been made for two European countries – Ukraine, Italy
as well as the world leaders in the distribution of Covid-19 – the United States, Brazil and
Russia.
Keywords 1
Covid-19 propagation waves; mathematical modeling; data approximation; parametric
analysis; epidemic waves
1. Introduction
Studying of the mechanisms of the spread of epidemics is an important way to control diseases,
along with finding new drugs, vaccination and preventive measures [1-5]. The reduction in damage
from the coronavirus epidemic is connected with the use of methods and tools of mathematical
modeling of the spread of Covid-19. Modeling allows making quantitative calculations, comparative
analysis, and forecasting of temporary descriptions of major categories of the epidemic, such as the
number of people who have become ill, recovered, and died [6-9]. Covid-19 distribution models are
required to be adequate to the descriptions of the basic concepts. They also must comply with
statistical data.
The SIR model developed by A. Kermak and W. McKendrick in 1927-1933 is widely used to
describe the epidemic [10]. The SIR model is based on a scheme of the epidemic transition of the
number of individuals from one category to another: susceptible (S) become infected (I), then they
recover (R). The SIR model is represented by a system of first-order differential equations that
describe the time dependences of variables that reflect basic concepts [11].
The class of SIR models that implements the concept of epidemic transition has gained wide
popularity, development and in addition to the SIR model also contains its variations: SIRS, SEIR,
SIS, MSEIR [11]. The experience of applying SIR class models to mathematical modeling of Covid-
19 spread [11-13] has shown incomplete correspondence of calculations of basic variables to
statistical data.
IDDM’2020: 3rd International Conference on Informatics & Data-Driven Medicine, November 19–21, 2020, Växjö, Sweden
EMAIL: kmolodetska@gmail.com (K. Molodetska); YTimonin45@gmail.com (Y. Tymonin)
ORCID: 0000-0001-9864-2463 (K. Molodetska); 0000-0002-0179-5226 (Y. Tymonin)
©️ 2020 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)
It was proved in the analytical review [14] that any predictions based on the models of the SIR
class and its derivatives cannot be considered correct, and certain coincidences of the predicted data
can be random.
Insufficient accuracy of SIR class models necessitates new approaches to mathematical modeling
of the spread of Covid-19. The nature of the statistical data on the spread of the Covid-19 coronavirus
epidemic shows that they have a high similarity to the wave process.
1. In [15] it is noted that the representative of the WHO European Bureau Oleg Storozhenko
claimed that the world is facing the second wave of coronavirus infection. At the same
time, in some countries, the third wave has begun.
2. To characterize the spread of Covid-19, highly visual and informative representations in
the form of epidemic waves are used. However, this “wave” representation places
increased demands on the Covid-19 propagation models.
The spread of Covid-19 is a complex regular stream of epidemic events that reflect the number of
events happening at each subsequent moment – a day. Cases of infection, recovery, and deaths are
considered as epidemic events. Primary statistical information about regular streams of events is given
in the form of time series.
Epidemic streams have an important distinctive feature, meaning that the stream starts at zero
before the outbreak of the epidemic begins and ends at zero after the outbreak ends. This feature gives
a convex character the descriptions of epidemic flows.
The real stream of epidemic events can be defined as a complex, compositional stream made up of
some elementary streams. The complexity of the epidemic flow prevents both analysis and
projections.
Analytical work requires determining the wave structure of a complex flow, i.e. it is composed of
elementary streams. Therefore, there is an urgent problem, which consists in decomposing a complex
stream of epidemic events into elementary streams. However, the solution of these problems comes
up against the problems of mathematical modeling of simple and complex epidemic flows and their
interaction.
The study aims to develop new mathematical models of simple and complex epidemic flows,
which will enable formalization of the processes of Covid-19 spread, as well as the exploration of
these mathematical models in order to develop and implement effective anti-epidemic measures. To
achieve this aim, such tasks must be completed:
analyse possible ways of mathematical formalisation of epidemiological events;
develop a model of elementary epidemiological flow;
propose an approach to formalising a complex epidemiological flow based on simple
flows;
conduct computational experiments with the proposed models based on statistical data
from different countries.
2. Models and methods
2.1. Approximation problem solving method
Let’s consider a problem that aims at decomposing a complex stream of epidemic events into
elementary ones. Such tasks are solved according to the following stages:
implementation of the approximation of time series, the result of which should be an
analytical model of a complex flow;
defining an analytical model of a simple, elementary stream;
solving the problem of decomposing a complex epidemic stream into elementary streams.
Since the construction of a complex flow model by approximating time series is very problematic,
this task must be solved according to the “bottom-top” scheme, by defining the models of the
elementary flow with their subsequent superposition.
The analytical model of a simple flow is a continuous convex function, bell-shaped in the
foreseeable time interval, i.e. the function starts and ends with a null value. Besides, we introduce an
additional requirement that expands the properties of the model and consists of the asymmetry of the
bell shape. Asymmetry is manifested in the fact that the growth and decay of the function have
different rates. This requirement expands the capabilities of flow modeling, but excludes the use of
well-known symmetric bell-shaped functions, for example, probability distribution functions.
Therefore, there arises the problem of constructing a continuous, asymmetrically bell-shaped function
and using it as a model of an elementary epidemic flow.
2.2. Construction of the epidemic elementary flow model
As a mathematical basis for the model of an elementary epidemic flow, we use the functions of
limited growth, which have proven their effectiveness in the models of conflict interaction [16-18].
Assuming that these models can be used to describe different epidemic events, we will restrict
ourselves to cases of infection.
Then, the number of infected individuals can be set by the function of limited growth, as a
solution to a nonlinear differential equation of the 2nd order [18]:
(1)
where is the growth rate; – phenomenological coefficients, which are
considered as parameters of the epidemic: – an indicator of the number of susceptible to
infection; – coefficient of flow asymmetry; a2 – a level of susceptibility of population to the virus;
, – multiplication coefficients.
To represent the model of an elementary flow, we restrict ourselves to nonlinear
differential equations of the 1st order for and , . Assuming that the
epidemic flow is described by the rate of infection, we present the expression for the flow
using the derivative from (1)
(2)
where phenomenological coefficients have the following meaning: – coefficient of flow
asymmetry; – an indicator of the number of susceptible to infection.
Equation (2) can be considered as a generalized representation of the logistic Verhulst equation, to
which it is reduced at . Discrete functions of the number of infected people are described by
the expression [18]
(3)
Following (2), discrete flow functions can be determined through the increment in the number of
infected .
We can say that the functions of the elementary epidemic flow describe a simple epidemic wave.
In Figure 1 shows asymmetric epidemic waves represented by the functions of elementary flows and
calculated according to expression (3). Calculations were carried out for one value of growth
indicators and different values of the phenomenological coefficient .
In a particular case, when , i.e. for the logistic model, the wave degenerates into a symmetric
one.
Figure 1: Graphs of bell-shaped epidemic waves with pronounced asymmetry
2.3. Construction of a complex epidemic stream model
To build a complex flow model by superposition of elementary flow models, it is required to
determine the permissible operations that can be performed with simple flow models. The algebra of
simple epidemic flows contains the following additive algebraic operations:
addition of a stream with a constant (shift of the function along the value axis)
, ,
multiplying the flow by a constant (scaling)
, ,
time shift of the flow
, .
These operations do not change the parameter values. When adding the streams, we get a complex
stream , where and . When building a
model of a complex flow by superposition, all kinds of operations are used.
3. Experiments
Complex flow wave models were formed by the superposition of sequential selection of epidemic
waves given by elementary flow models. The calculations of the models of the elementary flow of the
wave pattern were carried out using the discrete flow function (3). The approximation error was
estimated by the deviation of the complex flow function from the time series data. To estimate the
approximation error, the relative indicator MAPE was used. The average absolute error calculated by
the formula , where are the values of statistical data.
To calculate the wave models of epidemic flows, two European countries were selected – Italy and
Ukraine, as well as the world leaders in the spread of the epidemic – the United States, Brazil and
Russia. We should consider that, at the same time, another wave is superimposed on the wave picture
of the epidemic, which has its own parameters for different countries. This wave, the last in the
considered time interval, mainly determines the forecast of the epidemic development.
The wave structures of the epidemic in different countries have a different wave pattern of the
spread of Covid-19. The parametric analysis shows the specific nature of the development of the
epidemic in different countries. Parameter values can be associated with preventive and curative
measures.
3.1. Wave pattern of Covid-19 spread in Italy
The spread of Covid-19 in the time interval from 1.03 to 31.08 2020 was considered [19]. Figure 2
shows a waveform representation of the spread of Covid-19 in Italy:
statistical data (curve 1);
the curve of the general, complex flow (curve 2);
four curves of elementary streams (epidemic waves, curves 3).
Figure 2: Epidemic waves of Covid-19 spread in Italy
The error in approximating the statistical data of the complex flow curve does not exceed 20%.
The large value of the error can be explained by the large scatter of the initial data. If we reduce the
spread of data by averaging them, then the error can be reduced to 10%. The figure shows a well-
defined bell-shaped epidemic stream of the first wave. Also, the general flow contains a small second
wave and a noticeable third wave.
Table 1 shows the parameters of the waves of the general epidemic stream, in which the first wave
dominates.
Table 1
Parameters of flows in Italy
Wave parameters 1st wave 2nd wave 3rd wave 4th wave
Growth rate 0,236 0,11 0,17 0,25
20 5000 80 70
0,98 9,2 11 10
Wave peak 5531 99 413
Wave peak date 27.03.2020 10.07.2020 13.08.2020
From Table 1, the following differences between the parameters of the third and the first waves
can be noticed:
decrease in the rate of infection ;
increase in the coefficient of asymmetry ;
increase in the rate of susceptible to infection ;
decrease in peak values of waves.
In general, the wave character of the general flow indicates a decrease in the level of the epidemic
in August-September.
3.2. Wave picture of the spread of Covid-19 in Ukraine
The spread of Covid-19 in the time interval from 1.04 to 10.09 2020 was considered [20, 21].
Figure 3 shows the wave representations of the spread of Covid-19 in Ukraine: the curve of the total,
complex flow (curve 1) and 5 curves of elementary flows (epidemic waves, curves 2).
Figure 3: Epidemic waves of the spread of Covid-19 in Ukraine
The total flow contains the sequence of four waves with growing peaks, as a result of which there
is an increase in the total flow of the epidemic. Table 2 shows the parameters of the four waves of the
total flow.
Table 2
Flow parameters in Ukraine
Wave 1st wave 2nd wave 3rd wave 4th wave 5th wave
parameters
Growth rate 0,2 0,13 0,3 0,55 0,26
230 30 200 500 600
4,7 4,2 2,7 36,7 130
Wave peak 462 706 982 938
Wave peak date 29.04.2020 30.06.2020 13.08.2020 19.08.2020
Table 2 demonstrates the following differences in wave parameters:
increase in the rate of infection ;
close values of the asymmetry coefficient ;
increase in the rate of susceptible to infection ;
increase in peak values of waves.
In general, the wave nature of the general flow indicates an increase in the epidemic.
3.3. Wave pattern of Covid-19 spread in Russia
Russia ranks third in the world ranking of the spread of Covid-19 [22]. The spread of Covid-19 in
the time interval from 1.04 to 31.08 2020 was considered. In Figure 4 shows the wave representations
of the spread of Covid-19 in Russia, including:
total flow curve (curve 1);
three curves of elementary streams (curve 2).
Figure 4: Epidemic waves of the spread of Covid-19 in Russia
The total stream contains a sequence of three waves, where the first and third waves having similar
peaks. The wave picture shows the dominance of the second wave and, in general, there is a decrease
in the overall flow of the epidemic.
Table 3 shows the parameters of the waves of the general flow, in which the second wave
dominates.
Table 3
Wave parameters in Russia
Wave parameters 1st wave 2nd wave 3rd wave
Growth rate 0,36 0,25 0,21
24 77 80
0,65 0,41 0,2
Wave peak 2422 8708 2121
Wave peak date 30.06.2020 10.05.2020 13.08.2020
From Table 3, the following differences in wave parameters follow:
decrease in the rate of infection ;
increase in the coefficient of asymmetry ;
increase in the coefficient ;
decrease in peak values of waves.
In general, the wave character of the general flow indicates a decrease in the epidemic.
3.4. Wave Pattern of Covid-19 spread in Brazil
Brazil ranks third in the global ranking of Covid-19 spread [23]. The spread of Covid-19
in the time interval from 1.05 to 31.08 2020 was considered. Figure 5 shows the wave
representations of the spread of Covid-19 in Brazil:
the curve of the total flow (curve 1)
8 curves of elementary flows (curves 2).
Figure 5: Epidemic waves of the spread of Covid-19 in Brazil
The total stream contains a sequence of three waves, with the first and second waves
having similar peaks. The wave picture is dominated by the second wave and, in general,
there is a decrease in the overall flow of the epidemic.
Table 4 shows the parameters of the waves of the total flow. A large number of epidemic waves
and their similarity should be noted.
Table 4
Parameters of waves in Brazil
Wave 2nd
parameters 1st wave 3rd wave 4th wave 5th wave 6th wave 7th wave 8th wave
wave
Growth rate
0,5 2,1 1,5 1,7 1,9 2,3 1,5 1,0
16 90 100 80 130 110 120 100
2,0 2,3 2,5 2,3 1,3 0,65 15 12,5
Wave peak 11776 18706 11590 16420 12666 938,29 7070 4988
Wave peak
2.05.20 29.05.20 9.06.20 23.06.20 8.07.20 28.07.20 15.08.20 29.08.20
date
Table 4 shows the following differences in wave parameters:
increase in the rate of infection ;
close, in general, values of the coefficient of asymmetry ;;
increase in the rate of susceptible to infection ;
increase in peak values of waves.
In general, the wave nature of the general flow indicates a decrease in the epidemic.
3.5. Wave Pattern of COVID-19 Spread in the United States
The data about the spread of Covid-19 in the time interval from 03.15 to 31.08.2020 was
considered [24]. According to it, the United States is the world leader in the number of Covid-19
cases. Figure 6 shows the wave representations of the spread of Covid-19 in the United States:
a total flow curve (curve 1);
3 elementary flow curves (curves 2).
The total flow contains a sequence of three waves, with the second wave dominating in the wave
picture and, in general, a decrease in the total flow of the epidemic. The first and third waves have
similar peaks.
Figure 6: Epidemic waves of the spread of Covid-19 in the United States
Table 5 shows the parameters of the waves of the total flow, in which the first and second waves
dominate.
Table 5
Wave parameters in the USA
Wave parameters 1st wave 2nd wave 3rd wave
Growth rate 0,54 0,25 0,35
17 3 80
0,098 0,063 0,03
Wave peak 25838 47140 3853
Wave peak date 13.04.2020 15.07.2020 31.08.2020
Table 5 demonstrates the following differences in wave parameters:
- decrease in the rate of infection ;
- increase in the coefficient of asymmetry ;
- decrease in the coefficient ;
- decrease in peak values of waves.
In general, the wave character of the general flow indicates a decrease in the epidemic.
4. Conclusions
For qualitative characteristics, the spread of Covid-19, the concept of “wave” is used, which has
increased visualization. The prevalence of Covid-19 in the world continues to grow, and the next
outbreak of the epidemic is considered as a “second wave”. The quantitative wave representation of
the epidemic requires formalization of the process of the spread of Covid-19. The epidemic process is
a complex stream of epidemic events, such as the cases of infection, recovery, and death.
For the first time in order to analyze the spread of Covid-19, it was proposed to use a wave
structure, which represents a complex stream of epidemic events in the form of a set of simple
epidemic streams (epidemic waves). To represent the wave structure we had to decompose a complex
stream of epidemic events, given by statistical information, into elementary streams.
The derivative of the restricted growth function is used as an analytical model of the elementary
epidemic flow. Elementary epidemic stream (epidemic wave) has an asymmetric bell-shaped
appearance. Asymmetry reflects the fact that the rate of rise and fall of the wave is different. The
asymmetry property fundamentally distinguishes the model of an elementary epidemic flow from the
well-known symmetric bell-shaped functions, in particular, those used for the description of the
probability distribution.
The wave structure of the epidemic is represented by a set of elementary epidemic flows (waves)
shifted along the time axis which differs in the values of the parameters. The wave structure is
determined by the sequential selection of waves and the values of their parameters. Summing up the
values of the elementary epidemic flows, we can obtain an analytical description of the complex flow
of epidemic events. The flow can be considered as a solution to the approximation problem for the
given statistical information.
We have also carried out the approximation calculations for two European countries – Ukraine,
Italy as well as the world leaders in the spread of Covid-19: the United States, Brazil and Russia. The
wave structures of the epidemic in different countries have a different wave pattern of the spread of
Covid-19. The parametric analysis shows the specific nature of the development of the epidemic in
different countries. Parameter values can be closely connected with preventive and curative measures.
In general, wave models of the epidemic have visibility and enhanced capabilities for analyzing
and predicting the spread of Covid-19, which indicates the feasibility of further research in this
sphere.
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