In this book, we will focus on the simplest models that describe the development
of infectious diseases over time. In particular, the initial stages of epidemics in
different regions, which are characterized by an exponential increase in the number
of cases, will be studied. Simple comparisons of the pandemic dynamics in different
countries and its trends will be presented. In these parts of the book, I shall use
some results of articles and preprints, written together with Gerhard Demelmair,
Ihor Kudybyn, Anatolii Nikitin, and Bohdan Shepetyuk, to whom I am also very
grateful for collecting and systematizing statistical information on the number of
registered cases of the disease.
In this book, we will also use the classical SIR model, with three differential
equations for the evolution of the number of susceptible persons—S; infected,
spreading the infection—I and removed persons—R, which is the sum of isolated,
immunized, and deceased persons. This model contains only four parameters, the
values of which can be estimated using a statistical approach developed and successfully applied for investigations of the mysterious children's disease that
occurred in the Ukrainian city of Chernivtsi in 1988.
The SIR model is unable to determine the duration of the incubation period and to
predict separately the number of deaths caused by coronavirus, but allows us to make
adequate predictions of the duration of epidemics in different countries, estimates
of their actual beginning (they may precede the time of registration of the first patient),
to calculate the time dependences of the total number of patients V = I + R and I, to
estimate the probability of meeting an infected person and the effective reproduction
numbers. Corresponding results for the first COVID-19 epidemic waves in mainland
China, USA, Germany, the UK, the Republic of Korea, Austria, Italy, Spain, France,
the Republic of Moldova, Ukraine, the city of Kyiv and for the whole world are
already published in my articles and preprints. In the book, these results are systematized and conclusions are drawn based on current information about the course
of the pandemic. In Chap. 7 we will compare and discuss the characteristics of the first
waves of the COVID-19 pandemic in different countries and WHO regions.
Constant changes in the pandemic conditions (i.e., in the peculiarities of quarantine and its violation, in situations with testing and isolation of patients, in
coronavirus activity due to its mutations, etc.) cause changes in the values of
parameters of the mathematical models and lead to new pandemic waves. In particular, in October 2020 we observed a sharp increase in the daily number of new
cases in many European countries. We will develop simple methods of detecting
these changes and propose a simple method of identifying new pandemic waves.
The numerical differentiation of smoothed dependences of the accumulated number
of cases allows selecting periods with different values of SIR parameters.
To simulate different pandemic waves (periods with more or less constant values
of its dynamics parameters), a general SIR model and its exact solution will be
proposed. The identification procedures for the parameters of the general SIR model
will be described. The characteristics of several pandemic waves in Ukraine and the
world will be calculated and corresponding predictions will be presented.
To have good accuracy of predictions, the pandemic dynamics must be updated
with the use of new data sets. Because of this, a simple method to assess the final
viii
Introduction
of infectious diseases over time. In particular, the initial stages of epidemics in
different regions, which are characterized by an exponential increase in the number
of cases, will be studied. Simple comparisons of the pandemic dynamics in different
countries and its trends will be presented. In these parts of the book, I shall use
some results of articles and preprints, written together with Gerhard Demelmair,
Ihor Kudybyn, Anatolii Nikitin, and Bohdan Shepetyuk, to whom I am also very
grateful for collecting and systematizing statistical information on the number of
registered cases of the disease.
In this book, we will also use the classical SIR model, with three differential
equations for the evolution of the number of susceptible persons—S; infected,
spreading the infection—I and removed persons—R, which is the sum of isolated,
immunized, and deceased persons. This model contains only four parameters, the
values of which can be estimated using a statistical approach developed and successfully applied for investigations of the mysterious children's disease that
occurred in the Ukrainian city of Chernivtsi in 1988.
The SIR model is unable to determine the duration of the incubation period and to
predict separately the number of deaths caused by coronavirus, but allows us to make
adequate predictions of the duration of epidemics in different countries, estimates
of their actual beginning (they may precede the time of registration of the first patient),
to calculate the time dependences of the total number of patients V = I + R and I, to
estimate the probability of meeting an infected person and the effective reproduction
numbers. Corresponding results for the first COVID-19 epidemic waves in mainland
China, USA, Germany, the UK, the Republic of Korea, Austria, Italy, Spain, France,
the Republic of Moldova, Ukraine, the city of Kyiv and for the whole world are
already published in my articles and preprints. In the book, these results are systematized and conclusions are drawn based on current information about the course
of the pandemic. In Chap. 7 we will compare and discuss the characteristics of the first
waves of the COVID-19 pandemic in different countries and WHO regions.
Constant changes in the pandemic conditions (i.e., in the peculiarities of quarantine and its violation, in situations with testing and isolation of patients, in
coronavirus activity due to its mutations, etc.) cause changes in the values of
parameters of the mathematical models and lead to new pandemic waves. In particular, in October 2020 we observed a sharp increase in the daily number of new
cases in many European countries. We will develop simple methods of detecting
these changes and propose a simple method of identifying new pandemic waves.
The numerical differentiation of smoothed dependences of the accumulated number
of cases allows selecting periods with different values of SIR parameters.
To simulate different pandemic waves (periods with more or less constant values
of its dynamics parameters), a general SIR model and its exact solution will be
proposed. The identification procedures for the parameters of the general SIR model
will be described. The characteristics of several pandemic waves in Ukraine and the
world will be calculated and corresponding predictions will be presented.
To have good accuracy of predictions, the pandemic dynamics must be updated
with the use of new data sets. Because of this, a simple method to assess the final
viii
Introduction