=Paper=
{{Paper
|id=Vol-3302/paper17
|storemode=property
|title=Mathematical Models of Epidemic Dynamics to Simulate the Distribution of COVID-19
|pdfUrl=https://ceur-ws.org/Vol-3302/paper10.pdf
|volume=Vol-3302
|authors=Yuriy Tymonin,Kateryna Molodetska
|dblpUrl=https://dblp.org/rec/conf/iddm/TymoninM22
}}
==Mathematical Models of Epidemic Dynamics to Simulate the Distribution of COVID-19==
https://ceur-ws.org/Vol-3302/paper10.pdf
Mathematical Models of Epidemic Dynamics to Simulate
the Distribution of COVID-19
Yuriy Tymonina, and Kateryna Molodetskaa
a
Polissia National University, Blvd Stary, Zhytomyr, 10008, Ukraine
Abstract
Mathematical modelling of the COVID-19 epidemic is based on system dynamics and SIR
models, which are not considered adequate. To overcome the shortcomings of modelling, a
non-classical discipline, epidemic dynamics, is proposed. The epidemic should be viewed as
an open, self-replicating dynamic system in epidemic dynamics. Epidemic dynamics models
are based on a dynamic system model with an extended network of inverse relationship. This
non-classical approach allows the tools of non-linear and non-equilibrium dynamics to be used
and models of epidemic dynamics to be represented in the form of non-linear and non-
stationary differential equations. The solutions of the equations are special COVID-19
distribution functions – functions of the flows and accumulation levels of the infected and the
dead. The COVID-19 distribution functions show high accuracy in approximating the
statistics, demonstrating the excellent adequacy of these functions in principle. The application
of COVID-19 distribution functions makes it possible to quantitatively describe the basic
concepts of an epidemic to carry out comparative parametric analysis of the distribution of
diseases and predict the development of an epidemic.
Keywords 1
Epidemic dynamics models, approximation, parametric analysis of epidemic distribution
functions, epidemic forecasting.
1. Introduction
Mathematical modelling of epidemic processes and the search for new drugs, vaccination and
preventive measures contribute to disease control. In addition, quantitative model simulations can
provide comparative analysis and predictions of temporal descriptions of key epidemic categories, such
as the number of people who fall ill, recover, and die. Therefore, COVID-19 prevalence models are
highly demanding to match statistical data and ensure that epidemic mechanisms and underlying
conceptual descriptions are adequate.
The SIR model, developed by A. Kermack and W. McKendrick in 1927-1933, is based on a scheme
of epidemic transition of essential variables from one category to another: those susceptible (S) become
infected (I), then recover (R). The SIR model is represented by a system of coupled first-order
differential equations describing the basic concepts' time dependence. Models implementing the
concept of epidemic transition have gained wide popularity and development, so the class of SIR models
today also contains varieties: SIRS, SEIR, SIS, and MSEIR [1-5].
Experience with SIR models [1-5] has shown poor fit of baseline variable calculations to statistical
data and poor accuracy in predicting epidemic processes. An analytical review [5] noted that such
models perform poorly in heterogeneous populations, different routes of transmission and the presence
of randomization factors. In our opinion, the shortcomings of SIR class models lie in the concept of
epidemic transition of basic concepts, which implies the search for new concepts.
A retrospective analysis of approaches to epidemic modelling in [6] shows that a high level of
complexity characterized the types of deterministic and stochastic epidemic models developed. In turn,
IDDM 2022: 5th International Conference on Informatics & Data-Driven Medicine, November 18–20, 2022, Lyon, France
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)
©️ 2022 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)
they have been poorly linked to the formulation and solution of practical epidemiological problems.
Therefore, their use was inefficient. The author [6] reviewed a new methodology for mathematical
modelling of epidemics – “EPIDDINAMICS”. This methodology is based on the method of scientific
analogy in mapping the epidemic process with the process of "transfer" of matter in the equations of
mathematical physics. During an epidemic, a complex self-sustaining process of "transferring" a
population of a pathogen to a community of susceptible individuals is formed. The epidemic process in
this concept is described by a system of non-linear partial derivative equations, very "similar" to the
equations of hydrodynamics. However, analogies with hydrodynamics do not sufficiently reveal the
content of the self-sustaining process of disease spread.
The nature of the COVID-19 coronavirus distribution statistics shows that the epidemic's dynamics
are highly like the logistic functions. Therefore, we note the application of logistic functions to
approximate a piece of given statistical information. The articles [7-10] use logistic-type models to
describe the spread of COVID-19. In [7], a mathematical model of the spread of the COVID-19
coronavirus epidemic is considered using a simplified logistic model describing the increase in cases.
The low accuracy of calculation results obtained in [7-8] can be attributed to the simplified
representations of logistic models used for modelling.
Articles [9, 10] discuss the concept of modelling the spread of COVID-19 based on constrained
growth functions. In [9], the epidemic's wave structure, represented by a set of elementary epidemic
flows shifted along the time axis, is considered. However, the content of the articles does not sufficiently
reveal the mechanism of epidemic development, which reduces the accuracy of epidemic forecasts.
This issue can be explained by the classical methods of mathematical investigation of epidemic
processes. However, a fuller description of the epidemic development mechanism requires modern non-
classical methods for studying complex systems involving non-linear and non-equilibrium dynamics.
The article aims to develop mathematical models for the COVID-19 epidemic based on the non-
classical approach using the methods and tools of non-linear and non-equilibrium dynamics. It allows
for the presentation of epidemic dynamics models in the form of non-linear and non-stationary
differential equations and dynamic models of system with an extended network of inverse relationship.
2. Models of Epidemic Dynamics
The reason for the shortcomings of epidemic modelling is the simplification and inadequacy of the
mechanistic representations of system dynamics characteristic of classical mathematical modelling
methodology. In classical modelling methodology, it is common to represent the object of study using
the means of system dynamics [11-13] as epidemic dynamics. At the same time, an epidemic, a
progressive spread of infectious disease among people capable of causing an emergency, should be seen
as a complex systemic entity.
2.1. Main approaches
Epidemic dynamics is a scientific discipline in the study of disease transmission processes that views
the epidemic as a complex open system, as a system dynamic viewed from a non-classical methodology.
The core of epidemic dynamics is its model. Causality in large self-regulating systems reduces to
the action of a self-regulation program as a goal that ensures the reproduction of the system [Stepin]. It
enables the model of epidemic dynamics to be conceived of as a procedural reproduction system. A
model of epidemic dynamics in a non-classical methodology should have these properties:
• Openness means self-regulating systems are always open and exchange energy and substance
with the external environment (metabolism), due to which the processes of local order and self-
organization occur;
• Providing links with the environment through a network of linear and non-linear inverse
relationship;
• Conditions of equilibrium as a state of crisis;
• Non-equilibrium functioning outside of the equilibrium conditions
• Reproduction of the system through positive inverse relationship.
The methods and tools of the theory of non-linear non-equilibrium dynamical systems are used to
describe the epidemic dynamics model. The epidemic dynamics model is based on a dynamical system,
a mathematical model of an object, process or phenomenon that neglects fluctuations and all other
statistical phenomena. A dynamic system is a system with a state. In this approach, a dynamical system
describes (in general) the dynamics of some process, namely, the process of a system moving from one
state to another. An epidemic dynamics model consists of abstract elements representing some
properties of the modelled system. The following elements are distinguished: integrator, adder, and
inverse relationship chains, which link the variables – flows and quantities of accumulations.
2.2. Variable patterns in epidemic dynamics
Epidemics are the transmission of viruses from ill people to those who are healthy and susceptible to
the disease. Statistics keep track of new cases – the number of people infected per day. An epidemic
process is characterized by an increase in the total number of infected people, counted cumulatively at
a specific date. Growth is limited by the number of people susceptible to the disease. The epidemic
dynamics model derives from a dynamic system model [16], which uses a linear dynamic system [19]
with inverse relationship. The main variables of the model:
• 𝑥(𝑡) - an independent variable denoting the rate, flow of infected, those number of infected per
unit time;
• 𝑋(𝑡) - the dependent variable of the level of accumulation of infected (system state), denoting
the total number of infected over some time.
A phenomenological inverse relationship coefficient complements the dynamic system model 𝜑.
Next model is an isolated design, usually used for linear systems
𝑀 = 〈𝑥, 𝑋, 𝜑〉, 𝑋 = ∫ 𝑥𝑑𝑡 , 𝑥 = 𝑋 ′ . (1)
To use this model for open systems, it is necessary to: increase the number of inverse relationships and
relate them to the environment parameters. To extend the modelling capabilities of open systems, we
use a deployed scheme in the form of multi-circuit inverse relationship. Such a detailed dynamic model
of an open system provides a process description in abstract form and allows the construction of non-
linear differential equations.
2.2.1. Deployment of a dynamic model
The principle of dynamic model deployment is to decompose: the total flow of infected 𝑥 into partial
variables and, similarly, the total phenomenological coefficient 𝜑
𝑥 = ∑ 𝑥𝑖 , 𝜑 = ∑ 𝜑𝑖 . (2)
The complete flow of the infected as distributors of infection consists of elements. Following the
decomposition principle (2), we distinguish two groups of elements based on facilitating the distribution
of infection. Then the total flow of the infected equals the difference of the incumbent carriers minus
the flow of loss of carriers
𝑥 = 𝑥 ∓ − 𝑥̃, (3)
∓
where 𝑥 - streams of carriers who facilitate the distribution of the infection; 𝑥̃ - loss streams of carriers
who counteract the distribution of the infection.
2.2.2. The flow of carriers of infection
The carrier flow (3) is also divided into two parts. One part relates to active spreaders and the other to
inactive spreaders, those who have died. This approach makes it possible to distinguish between fatal
cases and to consider the actual spreaders of infection. The flow of carriers is then equal to the difference
of the flow of infected minus the flow of deceased
𝑥 ∓ = 𝑥 + − 𝑥 −, (4)
where 𝑥 + - is the flow of active carriers, reflecting new cases of infected persons in the process of
spreading infection (inflow of infected persons); 𝑥 − - the flow of lethal cases (outflow of infected
persons).
The flow of fatal cases is represented as a function of current infectious carriers 𝑥 − = 𝑓(𝑥 + ).
2.2.3. Flow patterns
Flow models (4) describe the dependencies of the rate of infection on the number of people infected.
𝑥𝑖 = 𝑓𝑖 (𝑋), (5)
These dependencies reflect the main patterns of epidemic processes. We describe the relationship
between flows and the environment using phenomenological coefficients. The relationship with the
environment can be linear and non-linear. For linear relationships, the flow (5) is proportional to the
number of infected with the inverse relationship coefficient
𝑥𝑗 = 𝜑𝑗 𝑋. (6)
For non-linear relationships, the inverse coefficient is a function of the derivatives of the
accumulation level variables
𝑥𝑖 = 𝜑𝑖 (𝑋)𝑋. (7)
2.2.4. Patterns of carrier flows
The pattern of processes here is that the flows of carriers (4) are proportional to the number of spreaders
of infection. Carrier flow patterns reflect a monotonic increase in the number of infected. We restrict
the group of linear (6) relationships to two types of inverse relationships, positive and negative 𝜑∓ =
= 𝜑+ − 𝜑−
1. The growth of an epidemic is mainly determined by the inflow of infected persons, which is
proportional to the number of people spreading the infection
𝑥 + (𝑡) = 𝜑+ 𝑋(𝑡), (8)
+
where 𝜑 is the positive inverse coefficient, a measure of the growth in the number of infected.
It follows from equation (8) that the number of people infected, and consequently the infectious
+
flow, grows exponentially without limit 𝑋(𝑡) = 𝑒 𝜑 𝑡 . This growth can be ensured if there is an
unlimited number of susceptible individuals regarded as the source of the infection. However, in
practice, the number of susceptible persons is limited, which poses the problem of a limited growth
model.
2. The flow of deaths, which reduces the growth of the epidemic, is proportional to the number of
people infected
𝑥 − (𝑡) = 𝜑− 𝑋(𝑡), (9)
−
where 𝜑 is the growth rate of fatal cases.
The growth rates in (8) and (9) are constant coefficients, so these carrier flow models are linear
relationships. The flow of carriers (4) is then equal to the difference between the flow of infected and
dead
𝑥 ∓ = (𝜑 + − 𝜑− )𝑋(𝑡).
2.2.5. Carrier loss flows
Limit it to three types of loss and write an expression for the total flow
𝑥̃ = 𝑥̃0 + 𝑥̃1 + 𝑥̃2 . (10)
where 𝑥̃𝑖 loss flows of certain types of carriers.
The pattern of processes here is that loss flows are proportional to the number of spreaders, but this
relationship is not linear (7), and the coefficients of proportionality depend on the number of spreaders
𝑥̃𝑖 (𝑡) = 𝜑̃𝑖 (𝑋)𝑋(𝑡). (11)
The principle behind this non-linear relationship is that loss rates are proportional to the derivatives
of the number of infectious spreaders (7), so growth rates are functions that depend on the
phenomenological coefficients
𝜑̃𝑖 (𝑡) = 𝑎𝑖 (𝑋 𝑖 )(𝑡). (12)
Then the total loss flow equation (10) with (11) and (12) will appear as
𝑥̃(𝑡) = 𝑎0 𝑋 2 (𝑡) + 𝑎1 𝑋(𝑡)𝑋 ′ + 𝑎2 𝑋(𝑡)𝑋 ′′ . (13)
Consider the features of the loss components in the total loss flow equation (13) that describe the
dispersal of infectious agents in the environment.
2.2.6. Zero-order loss flow models
Zero-order loss stream is a function proportional to the number (zero-order derivative) of infectious
spreaders, where the coefficient of proportionality depends on the number of spreaders 𝜑̃0 (𝑡) =
= 𝑎0 𝑋 (0) (𝑡) (12). The flow expression is then a non-linear relationship of the form
𝑥̃0 (𝑡) = 𝑎0 𝑋 2 (𝑡). (14)
Zero-order flow models (14) solve the problem of limited epidemic growth, which results from the
limited source of the infected population and consists of the fact that the growth of the infected
population is limited to the number of persons 𝑋̂ susceptible to the epidemic. This value can be regarded
as the source of infection in the environment. The introduction of a source limits the number of infected
to 𝑋 ≤ 𝑋̂. When the limit is reached 𝑋 = 𝑋̂ , the flow value falls by leaps and bounds to zero. Then the
expression for the flow of loss takes the form of a discontinuous function
𝜑̃ 𝑋(𝑡), 𝑋(𝑡) ≤ 𝑋̂; (15)
𝑥̃0 (𝑡) = { 0
0, ̂
𝑋(𝑡) ≥ 𝑋.
A gap in the expression for the loss flow is considered a catastrophe, leading to uncertainty in the
spread of infection. Given that flow gaps are not observed in practice, it is necessary to introduce a
source of infection into the model (14) to exclude a catastrophe (15). To do this, consider the infected
1
source capacity 𝑎̅0 = 𝑋̂𝑇 and the source action time 𝑇. Considering that 𝑎̅0 = 𝑎 we rewrite the
0
expression for the flow (14) in the form of
𝑋 2 (𝑡) (16)
𝑥̃0 (𝑡) = .
𝑋̂ 𝑇
The expression for the loss flow (16) considers the source of the infected, is a continuous function
of the number of infected and can be used to describe the limited growth of an epidemic.
2.2.7. First-order loss flow models
According to (12), the growth rate is a function proportional to the first-order derivative of the number
of distributors 𝜑̃1 (𝑡) = 𝑎1 𝑋 (1) (𝑡). It follows that the flow expression is a non-linear relationship of the
form
𝑥̃1 (𝑡) = 𝑎1 𝑋(𝑡)𝑋 (1) (𝑡). (17)
The proportionality factor 𝑎1 reflects the resistive properties of the environment, which inhibit the
spread of the flow of the infected.
2.2.8. Second-order loss flow models
The second-order loss rate is a function proportional to the second-order derivative of the number of
distributors 𝜑̃2 (𝑡) = 𝑎2 𝑋 (2) (𝑡), then the flow expression is a non-linear relationship of the form
𝑥̃2 (𝑡) = 𝑎2 𝑋(𝑡)𝑋 (2) (𝑡). (18)
The proportional coefficient 𝑎2 can be considered as the elasticity of the environment and reflects
the ability to generate oscillations during flow distribution.
2.2.9. Differential equations of epidemic dynamics
Given the expressions for the elements (8), (9) and (16)-(18), we write the differential equation for the
total flow of infectious agents
𝑥 = ((𝜑+ − 𝜑− ) − (𝑎0 𝑋 (0) + 𝑎1 𝑋 (1) + 𝑎2 𝑋 (2) ))𝑋. (19)
Given 𝑥(𝑡) = 𝑋(𝑡), let us rewrite differential equation (19) for the number of infected in the standard
form
𝑋2 (20)
𝑎 𝑋𝑋 ′′ + (1 + 𝑎 𝑋)𝑋 ′ + = (𝜑+ − 𝜑− )𝑋.
2 1 𝑋̂ 𝑇
Model (20) describes epidemic dynamics as a nonlinear, non-stationary differential equation. This
model reflects epidemic dynamics as a process system reproduces stably due to interaction with the
environment. Equation (20) has no analytical solutions, and numerical methods are used to solve it.
Solutions to equation (20) give the epidemic episode functions:
• The total infectiousness flow functions have a bell-shaped form;
• The functions of the number of infected (infection rate) are S-shaped.
The epidemic dynamics functions have two equilibrium states:
• Initial equilibrium – unstable 𝑥(0) = 0, 𝑋(0) at 𝑡 = 0;
• Final equilibrium – stable, 𝑥(𝑡) → 0, 𝑋(𝑡) → 𝑋̅ at 𝑡 → ∞.
The epidemic dynamics functions vary over a range bounded by the equilibrium states, and then the
epidemic processes in the range are non-equilibrium and irreversible.
Equilibrium of the system means that the growth of infected individuals stops, and the derivatives
tend towards zero. We obtain the equilibrium equation from the equation of epidemic dynamics (20)
with zero derivatives 𝑋 ′′ (𝑡) = 0, 𝑋 ′ (𝑡) = 0 and 𝑋(𝑡) ≠ 0
𝑋̅ (21)
= 𝜑+ − 𝜑− .
̂
𝑋𝑇
The stable equilibrium equation is a parametric equation that relates the parameters of the COVID-
19 distribution functions. The epidemic threshold is described by the expression
𝑋̅ = (𝜑 + − 𝜑− )𝑋̂ 𝑇. (22)
In many practical cases, it is possible to restrict oneself to first-order epidemic dynamics models,
which are derived from (20) at 𝑋 ′′ (𝑡) = 0
𝑋2 (23)
(1 + 𝑎 𝑋)𝑋 ′ + = (𝜑+ − 𝜑 − )𝑋.
1 𝑋̂ 𝑇
In particular cases, equation (23) admits analytical solutions, but numerical finite-difference
methods are generally used to solve it. The solution of finite difference equations constructed by (23)
leads to discrete COVID-19 distribution functions.
2.2.10. Discrete distribution functions
In the discrete COVID-19 distribution functions, the relations of the variables are described by the
discrete 𝑋𝑘+1 = 𝑋𝑘 + 𝑥𝑘+1 , 𝑥𝑘+1 = 𝜑𝑘+ 𝑋𝑘 , where 𝜑𝑘+ = (𝜑+ − 𝜑− ) ×
𝑋
1− 𝑘
× 1+𝑎 𝑋̅𝑋 is the equivalent variable for the growth rate of infected persons. Expressions for the discrete
1 𝑘
function of the infector flow and the number of people infected
𝑋 𝑋
1− 𝑘 1− 𝑘
𝑥𝑘+1 = (𝜑 + − 𝜑− ) 1+𝑎 𝑋̅𝑋 𝑋𝑘 ; 𝑋𝑘+1 = (1 + (𝜑+ − 𝜑 − ) 1+𝑎 𝑋̅𝑋 ) 𝑋𝑘 ;
1 𝑘 1 𝑘 (24)
𝑋̅ = (𝜑 + − 𝜑− )𝑋̂ 𝑇.
The discrete COVID-19 distribution functions fit the statistics well, so they are used to model the
epidemic. Similarly to (24), the discrete flow and number of deaths functions are
𝑌 𝑌
1− 𝑘 1− 𝑘
𝑦𝑘+1 = (𝜑+ − 𝜑 − ) 1+𝑎 𝑌̅𝑌 𝑌𝑘 ; 𝑌𝑘+1 = (1 + (𝜑+ − 𝜑− ) 1+𝑎 𝑌̅𝑌 ) 𝑌𝑘 ;
1 𝑘 1 𝑘 (25)
𝑌̅ = (𝜑 + − 𝜑− )𝑌̂ 𝑇.
3. Approximation of Statistical Data Covid-19 Distribution
Discrete COVID-19 distribution functions are used to analyse and predict epidemics. These functions
are obtained by approximating statistical data. Two problems can be solved by approximating the
statistics:
• The total flow of infected people;
• The function of the number of infected.
Ukraine [16] is chosen as an example to show how the calculated values of COVID-19 distribution
functions correspond to statistical data [16].
3.1. Approximations of Data on the Number of Infected
Figure 1and Figure 2 show the correspondence between the calculated functions for the number of
infected (24) and deaths (25) and the first wave statistics for April and May 2020.
Figure 1: Correspondence of estimated numbers of COVID-19 infected with statistics for Ukraine for
April and May 2020
Figure 2: Correspondence between the estimated COVID-19 fatality rates and the statistical data for
Ukraine for April and May 2020.
Figure 1 and Figure 2 clearly show a reasonably good agreement between the calculated and actual
data, where the MAPE does not exceed 3% and 7% respectively.
3.2. Approximations of the Full Flow of Infected
Approximation of the total flow of infected persons was performed using the example of the second
wave of COVID-19 spread in the time interval from 1.04 to 10.09 2020 according to the statistical data
given in [17, 18]. The approximating functions have the character of episodes (27), those of completed
processes, and the total flow is the sum of episodes. Figure 3 shows the results of the decomposition of
the complex flow into 4 episodes, those into 4 elementary f flows.
Figure 3: Correlation of estimated COVID-19 infection flows with statistical data for Ukraine for the
period April - October 2020
The overall flow contains a sequence of four episodes with increasing peaks, resulting in an increase
in the overall epidemic flow.
In general, such methods of epidemic dynamics correspond to the methodology of agent-based
modelling, which, compared with classical models, has greater descriptive power but requires a more
detailed description of epidemic categories. Agent-based models use a dynamical system representation
in which details of descriptions are provided by inverse relationships.
The undoubted advantage of the method is the presentation of epidemiological dynamics models in
an analytical form. This approach allows to study mathematical models and analyze the sensitivity of
the result to changes in the parameters of differential equations. Also, the application of the proposed
mathematical models to the modeling of epidemiological dynamics allows to take into account the
nature of the studied processes of infection spread, in particular their wave nature.
4. Conclusions
Research on the history of epidemic modelling shows that the main reason for the shortcomings of
mathematical modelling of epidemics (SIR, System Dynamics, “EPIDDINAMICS”) can be considered
as the application of the classical methodology characteristic of simple isolated systems. Therefore,
there is a problem associated with the transition of the modelling methodology from classical to non-
classical positions, within which the epidemic should be considered an open self-replicating dynamic
system. This shift is associated with a new scientific approach to the study of disease transmission
processes, where epidemics are viewed as system dynamics from a non-classical methodology. This
scientific direction is called epidemic dynamics, the core of which is a reproducible procedural system.
A model of epidemic dynamics in a non-classical methodology would be open and reflect an exchange
of energy and matter with the external environment to enable reproduction in a nonequilibrium
functioning mode. The methods and tools of the theory of non-linear nonequilibrium dynamical systems
are used to describe the model of epidemic dynamics.
The epidemic dynamics models are based on a dynamic system model with an extended feedback
network. This non-classical approach allows epidemic dynamics models are represented as non-linear
and non-stationary differential equations. Solutions to the equations - COVID-19 distribution functions
– are used for mathematical modelling and investigation of epidemic processes. Importantly, these
functions describe elementary complete epidemic processes - "epidemic episodes". The modelling
process begins by approximating epidemic statistics with discrete functions describing two "epidemic
episodes" types: flows and accumulations. Next, superpositions of unrelated epidemic episodes shifted
in time and described complex processes.
This approach is consistent with agent-based modelling methodology, which has greater descriptive
power than classical models but requires a more detailed description of epidemic categories. The
application of COVID-19 distribution functions shows high accuracy in approximating statistical data,
demonstrating the excellent adequacy of these functions in principle. Applying COVID-19 distribution
functions enables quantitative description and analysis of epidemic processes and reliable forecasting.
Overall, applying COVID-19 distribution functions can help reduce the harm caused by a pandemic.
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