Differential equations in prevention of diseases

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National technical university of Ukraine «Igor Sikorsky Kyiv polytechnic institute»

Differential equations in prevention of diseases

Lystopadova Valentyna Viktorivna,

candidate of physical and mathematical sciences, associate professor

Khalaim Diana Sergiivna,

student of the faculty of engineering and chemistry

Kyiv

Abstract

Today, logical and critical thinking, which can be achieved by studying the exact sciences, is extremely important in today's world. Mathematics is an unsurpassed fundamental science that structures all acquired knowledge. All other exact sciences are based on it. Studying it by scientists has been able to prevent chaos and terrible catastrophes around the planet. Differential equations are an important part of mathematics - one of the most powerful and harmonious ways of learning about the universe and solving various life problems. They have the ability to predict the world around them. Used in a variety of disciplines such as biology, economics, physics, chemistry, applied mechanics and many others and in particular is one of the most important tools of epidemiology.

One of the important issues that can be predicted and solved with the help of differential equations is the spread of infectious diseases. This problem is very relevant now, so understanding this topic will be useful and practical.

The aim of the article is to study the application of differential equations to the problems of epidemiology.

It has been established that epidemiological diseases are one of the most dangerous and exciting problems of all generations, which requires further study and research.

The algorithm for compiling compartment (SIR) models based on differential equations is studied.

It has been proven that it is best to use a compartment model to study the spread of infectious diseases, which allows to estimate the time of spread and the number of lesions.

The basic assumptions are presented and the equation by which the mathematical model of the actual disease - COVID-19 is constructed is solved.

Keywords: differential equations; compartment model; spread of the disease; COVID-19.

Анотація

Листопадова Валентина Вікторівна кандидат фізико-математичних наук, доцент, Національний технічний університет України «Київський політехнічний інститут імені Ігоря Сікорського», проспект Перемоги, 37, м. Київ

Халаїм Діана Сергіївна студентка інженерно-хімічного факультету, «Київський політехнічний інститут імені Ігоря Сікорського», проспект Перемоги, 37, м. Київ

Диференціальні рівняння у запобіганні розповсюдженні захворювань

Сьогодні вкрай важливим у сучасному світі є логічне та критичне мислення, яке можна досягти за допомогою вивчення точних наук. Математика - неперевершена фундаментальна наука, яка структурує всі отримані знання. Саме на ній базуються всі інші точні науки. Вивчення її науковцями, змогло запобігти хаосу та жахливим катастрофам на всій планеті. Важливим розділом математики є диференціальні рівняння - одні з найпотужніших та найгармонічніших шляхів пізнання всесвіту та розв'язання різних життєвих задач. Вони мають здатність передбачати навколишній світ. Використовуються в найрізноманітніших дисциплінах таких як: біологія, економіка, фізика, хімія, прикладна механіка та багатьох інших і зокрема є одним з найважливіших інструментів епідеміології.

Одним із важливих питань, яке можна передбачити та вирішити за допомогою диференціальних рівнянь, є розповсюдження інфекційних захворювань. Ця проблема є неабияк актуальною зараз, тому розуміння цієї теми буде корисним та практичним.

Метою статті є вивчення застосування диференціальних рівнянь до проблем епідеміології.

Встановлено, що епідеміологічні захворювання є однією з найбільш небезпечних та хвилюючих проблем усіх поколінь, яке потребує подальшого вивчення та дослідження.

Вивчено алгоритм складання компартментних (SIR) моделей, на основі диференціальних рівнянь.

Доведено, що для дослідження розповсюдження інфекційних захворювань найкраще застосовувати компартментну модель, яка дає змогу оцінити час поширення та кількість ураження.

Викладено основні припущення та розв'язано рівняння, за допомогою якого побудовано математичну модель актуальної хвороби - COVID-19.

Ключові слова: диференціальні рівняння; компартментна модель; поширення захворювання; COVID-19.

Main part

Formulation of the problem. Mankind has experienced many different catastrophes during its existence, which have destroyed and maimed their lives, ranging from epidemics of incurable diseases to large-scale wars. In order to survive, it was necessary to research and develop one's own interest in how a problem arises and how to build the right algorithm of actions to prevent its occurrence. With the help of the exact sciences, humanity has been exploring its own planet more and more every year and making many unsurpassed discoveries that we, the people of the twenty-first century, still use today. In order to make a certain assumption, it was necessary to delve into the specifics of the phenomenon under study, make a thorough analysis and only then share your own judgments with others and prove them.

Gradually, various unsurpassed sciences emerged that could explain the peculiarities of certain natural events, medicine developed, and the standard and quality of life improved significantly.

The problem of the spread of infectious diseases has always worried mankind. Mass epidemics, one of the most devastating events, took the lives of entire settlements and forced the population to fear and suffer. For example, the so-called Black Death pandemic, which engulfed Asia, Europe and Africa, destroyed a third of the population, and the Spanish flu in 1918 killed even more people than in World War I! In 2009, humanity panicked over the Swine Flu pandemic, and coronavirus infection remains a serious problem around the world, forcing people to wear masks and keep their distance.

Analysis of recent research. The main formulations and ideas regarding differential equations were covered by such scientists as I. Newton, L. Euler, J. Liouville, A. Poincare. A large number of well-known scientists from all over the world are studying the modern disease COVID-19, in particular, Academician of the National Academy of Sciences of Ukraine S. Komisarenko describes his research «known from [1]. At the beginning of the disease, namely in 2020, the scientist in his own article described the origin of the virus, substantiated the current situation at the time and was able to systematize the research using a mathematical model. Solving the problem of the spread of this disease requires further detailed study. Some of my own opinions and research are presented in the article.

The purpose of the article is to learn about the preconditions for the emergence of differential equations. Analyze the spread of diseases through their use. Based on these studies to form a compartmental model of COVID-19 distribution.

Presenting main material. In the study of various processes and phenomena containing elements of motion, mathematical models are often used in the form of equations, which, in addition to independent quantities and dependent on the desired functions, also include derivatives of the desired functions. Such equations are called differential. The term was introduced in 1576 by Leibniz «considered in [2, С. 421]».

Differential equation - an equation that contains derivatives of the desired function and may contain the desired function and an independent variable. The independent variable, the derivative of which is included in the differential equations, is denoted by the letter x or t, because in many cases its role is played by time. The unknown function is denoted by y (x) or y (t) «known from [3, p. 352]». The order of the highest derivative included in the differential equation is called the order of this equation. In the general case, the first-order differential equation can be written as:

F (x, y, y ') = 0

The preconditions for the emergence of the theory of differential equations developed in the second half of the seventeenth century, when mathematicians came close to realizing the mutually inverse nature of the two basic operations of infinitesimal analysis - differentiation and integration.

The so-called «inverse tangent problems» relevant at that time were one of the first to be reduced to solving differential equations «considered in [4, p. 6]».

Epidemics were one of the most devastating events in human history. Since the population began to suffer from various life-threatening diseases, man has developed his curiosity about how diseases spread. Is it possible that such small pathogens can spread with such a big speed? Many people and families died before anyone could answer this question.

The subject of epidemiology models the behavior of diseases and the steps we can take to control diseases. One of the most important tools of epidemiology is differential equations. With them, we can model how the disease behaves over time, taking into account factors such as the number of susceptible and non-susceptible people in the population. These indicators focus on the study of how differential equations accurately model different diseases and help us learn a lesson about the importance of vaccination and collective immunity.

In order to better predict the spread of disease, use the so-called compartment (SIR) model. It allows to understand the complex dynamics and main features of this problem. The SIR model is an epidemiological model that calculates the theoretical number of people who become ill with an infectious disease over time. The name of this class of models comes from the fact that they include equations that relate the number of susceptible people S (t), the number of people infected with I (t), and the number of people who have recovered from R (t). It often makes sense to study the proportions of infected and susceptible people than the original figures. Therefore, we define the new variables as follows: «known from [5]»

where N - the total number of persons.

Since this model uses a closed set, at any time t, we have s (t) + i (t) + r (t) = 1. The rate at which the terms S(t) change will depend on the total infected persons.

Now suppose that Я-contacts are detected per unit time. This is enough to spread the disease. This means that there will be ЯS (t) of new infected individuals per unit time. The number of recovering people increases in proportion to the number of infected people. This ratio is called the recovery rate a.

The only way an infected person can improve is to recover and develop immunity to the disease. We assume that there will be a fixed number of people I (t) recovering on the parameter a, which is responsible for the rate of recovery and vaccinated or people with immunity R (t). This model can be shown in Figure 1.

Fig. 1. Implementation of a mathematical model.

Based on these assumptions we can to write down the system in this way:

These three nonlinear ordinary differential equations lead to the following:

The solution of this equation is:

Based on the above, we will compile a compartment model for the current disease - COVID-19. Let's mark:

S = S (t) - the number of susceptible persons at time t, I = I (t) - the number of infected persons at time t, R = R (t) - the number of cured persons at time t.

Let's first make a few assumptions to solve this problem. The first assumption is that the epidemic will be quite short and the general population will remain constant. The second assumption in the model concerns the mode of disease transmission. Let the rate of increase of infected persons be proportional to the contact between susceptible and infected and this happens at a constant rate. The third assumption concerns the recovery factor a. Suppose there is a constant rate of mortality or recovery. Then we write down the equations that will be used for the model.

So we start with the susceptibles. The rate of change over time of the number of susceptibles is Я, then based on our second assumption we expect that it's going to decrease as people become infective. So the rate of change of the number of susceptibles is going to be equal to minus because it's decreasing the rate of contact R which was in our second assumption and we said that it was proportional to the number of infectives I and the number of susceptibles S. S and I being here together is symbolizing a contact between the number of infectives and number of susceptibles and the R here is this rate of contact or transmission between them. So, the number of susceptible individuals at time t:

For the infectives we have a similar equation. And we want to know the rate of change of I over time. So now this equation will change because of the people moving from susceptible into infective. But now it is going to be plus, because susceptibles are moving to become infected. And now we also have by assumption three that infectives are cover or die at a constant rate a. So if you are an infective then you can move into the third category or the removed category. So here we have minus this constant rate a times the number of infectives. So, the number of infected people at time t:

The final equation which is the rate of change of those removed in the population must be equal to the gain from a and I. So as people are lost from the infective category, they can move into the removed category at the same rate. So, the number of cured individuals at time t:

So we have a system of differential equations.

To solve the system of equations we need to get some initial data. The way we do this, we define the initial number of susceptible people in the population. So we say that is going to equal S=So. Than we say the initial number of infectives will also be specified. Let's call that Io, and it is going to equal I= Io. In the very start of the outbreak we don't expect there to be anybody in removed section because nobody has yet recovered or died from the disease. That is why we can say that the initial value of R will be zero. Now we are going to talk about our first assumption in the context of our equations. The population must remain constant during the epidemic. It means that the rate of change of susceptibles plus infectives plus removed or added together must be zero, because the total population is given by S plus I plus R.

We can solve this equation because we know the initial conditions for the population. So if the total population doesn't change with respect to time, we just take the initial value to be a starting point which is the value of the population at the beginning. As time progresses, it can not change because it has a constant value. So it is always equal to that initial value.

We have an initial number of infected people given by I0 at the beginning of the epidemic. What we want to know is will that grow, because if the number of infective starts to grow, then you have a spread of a disease through a population. So what we are interested in is going to be the rate of change of the number of infectives. But before we do that, we actually want to start on the (1.1) equation. This tells us that the rate of change of the number of susceptibles is equal to a negative value, because Я is some positive constant it is a transmission rate. I is a number in a population as is S. So all of these three things are positive and the change of S is always negative. This tells us that S must always be smaller than it's initial value. At the very beginning of the outbreak everyone in the population in is susceptible to the disease. If S is always going to decrease because it's rate of change is negative, then this tells us that S must be less than or equal to its initial value So.

Now we can take So and plug it in to our (1.2) equation. So, we have an inequality now in our rate of change for the number of infected. And we said an epidemic will occur if the size of I increases from the initial value of Io. To answer our question will the disease spread, let's look on this equation:

mathematics differential equation algorithm

Solving this equation we can say that:

,

which means that the disease will spread, because this equation is

Also known as contact ratio. Contact ratio is the fraction of the population that comes into contact with an infected individual during the period when they are infectious. We can also rearrange this inequality to get a slightly different version of the same condition for whether or not an epidemic will occur. Now we can create a new parameter which is called the basic reproductive number or basic reproductive ratio (R₀). This condition tells us that we will have an epidemic if R₀ > 1. It means:

Based on all of the above, we can conclude that the rate of spread of coronavims infection can be described by this solution: .

The basic reproductive number or basic reproductive ratio (Ro) represents the number of secondary infections in the population caused by one initial primary infection.

Summarizing the studied material, we can conclude that differential equations as one of the ways to solve various problems of life today are actively used in the fields of epidemiology, mechanics, engineering and other sciences. They are closely connected with human life and deserve our attention. They can be used to create models of many processes and explore their dependencies. The result was the implementation of a compartment model that shows the rate of spread of infectious diseases.

References

1. Komisarenko, S.V. (2020). Poliuvannia vchenykh na koronavirus SARS-COV-2, shcho vyklykaie COVID-19: naukovi stratehii podolannia [Hunting scientists for coronavirus SARS - COV-2, which causes COVID-19: scientific strategies to overcome]. VisnykNAN Ukrainy - Bulletin of the NAS of Ukraine, 8, 29-33. Retrieved from http://www.visnyk-nanu.org.ua/ sites/ default/files/ files/Visn.2020/8/Visn_8-2020% 2B6_Komisarenko.pdf [in Ukrainian].

2. Dubovik, V.P., & Yurik, I.I. (2006). Vyshcha matematyka [Higher mathematics]. Kyiv: A.S.K. [in Ukrainian].

3. Garashchenko, F.G., Matvienko, V.T., & Kharchenko, 1.1. (2008). Dyferentsialni rivniannia dlia informatykiv [Differential equations for computer scientists]. Kyiv: Vydavnycho - polihrafichnyi tsentr «Kyivskyi universytet» [in Ukrainian].

4. Samoilenko, A.M., Perestyuk, М.О., & Parasiuk I.O. (2003). Dyferentsialni rivniannia [Differentialequations]. Kyiv: «Lybid» [in Ukrainian].

5. Sait vilnoi entsyklopedii «Vikipediia» [Free encyclopedia site «Wikipedia»]. uk.wikipedia.org Retrieved from https://uk.wikipedia.org/wiki/Полігамm_моделі_в_еmдеміолоriї#Модель_SIR [in Ukrainian].

Література

1. Комісаренко С.В. Полювання вчених на коронавірус SARS-COV-2, що викликає COVID-19: наукові стратегії подолання [Електронний ресурс] / С.В. Комісаренко // Вісник НАН України. - 2020. - №8. - С. 29-33. - Режим доступу: http://www.visnyk - nanu.org.ua/sites/default/files/files/Visn.2020/8/Visn_8-2020% 2B6_Komisarenko.pdf.

2. Дубовик В.П. Вища математика: підручник. / В.П. Дубовик, І.І. Юрик. - Київ: С.К., 2006. - 648 с.

3. Диференціальні рівняння для інформатиків: підручник / Ф.Г. Гаращенко, В.Т. Матвієнко, І.І. Харченко. - Київ: Видавничо-поліграфічний центр «Київський університет», 2008. - С. 6.

4. Диференціальні рівняння: підручник / А.М. Самойленко, М.О. Перестюк., І.О. Парасюк. - Київ: «Либідь», 2003. - С. 8.

5. Сайт вільної енциклопедії «Вікіпедія» [Електронний ресурс]. - Режим доступу: https://uk.wikipedia.org/wiki/Полігамні_моделі_в_епідеміології#Модель_SIR

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