17УДК 519.24
Хорольская Ирина Витальевна Horol'skaya Irina Vital'evna
Магистр менеджмента Master of Management старший преподаватель senior lecturer ФГБОУ ВО ТГМУ Pacific State Medical University
ОПТИМАЛЬНЫЕ МАТЕМАТИЧЕСКИЕ МОДЕЛИ В ИЗУЧЕНИИ ЭПИДЕМИЧЕСКИХ ПРОЦЕССОВ ЗАБОЛЕВАНИЙ ГРИППОМ
OPTIMAL MATHEMATICAL MODELS IN THE STUDY OF EPIDEMIC
PROCESSES OF INFLUENZA DISEASES
Аннотация, математическое моделирование дает возможность изучать механизмы, которые лежат в основе эпидемий, проводить анализ распространения заболеваний, разрабатывать программы, дающие возможность бороться с вспышками заболеваемости, следовательно, актуальность его применения в системе здравоохранения очевидна. В работе рассмотрены виды моделей, применяемых для изучения эпидемических процессов заболеваний гриппом, сформулирована проблема использования математических моделей в данных процессах, проанализирован выбор оптимальной модели.
Abstract: mathematical modeling makes it possible to study the mechanisms that underlie epidemics, to analyze the spread of diseases, to develop programs that make it possible to fight outbreaks of morbidity, therefore, the relevance of its application in the health system is obvious. The paper considers the types of models used to study the epidemic processes of influenza diseases, formulates the problem of using mathematical models in these processes, and analyzes the choice of the optimal model.
Ключевые слова, математическое моделирование, математическая модель, прогнозирование, эпидемический процесс.
Key words: mathematical modeling, mathematical model, forecasting, epidemic process.
The problem with the application of mathematical modeling in the health system is that most mathematical models describing the development of epidemic processes use an estimate only of the average values at each time of the expected number of
VМеждународная научно-практическая конференция infected and excluded from the number of susceptible. The difficulty lies in the fact that the development of the epidemic is a random process, and describing it only with the help of average indicators does not allow us to give an objective assessment for the necessary analysis and evaluation of the resources required for the development of appropriate programs and the implementation of therapeutic and preventive measures [1, p.34]. The availability of objective models will allow the development of programs that assess the range of damage and resources needed to eliminate the consequences of the epidemic, as well as to combat outbreaks of disease.
The analysis of the scientific literature allowed us to identify a number of mathematical models used in modeling influenza epidemics, which allow us to study the mechanisms of development and spread of epidemics [2, p.28].
The SIR model. The model is easy to build and use. The SIR model can be represented by a system of the following equations:
dS _ pis m
dt " N ( )
1 = — - yI (2)
dt N
f == Y1, (3)
R(t) - the number of individuals who have been ill at a given time t;
S(t) - the number of susceptible individuals at a given time t;
I(t) - number of infected individuals at a given time t;
в - the coefficient of intensity of contacts of individuals with subsequent infection
Y - the rate of recovery intensity of individuals.
The equation (1), (2), (3) describe this model, demonstrating the transition of the susceptible to the category of infected, describing the dependence of the rate of increase in the number of infected people on the number of recovered and on the number of contacts of infected and healthy people. The problem with using this model is that the
SIR model does not account for mortality (all patients recover, the population size remains the same). If it is necessary to take into account different transmission routes or heterogeneity of the population (for example, different population densities in different areas), then this model does not allow for an objective assessment.
The SEIR model. The model allows us to describe epidemics in which a sufficiently long incubation period prevents the timely detection of the disease. In this case, the model can be represented by the following equations:
Tt=HU-VS-P^S (4)
IT e-NS-(U + o)E (5)
%t = aE-( y + д)/ (6)
f = Yl - HR, (7)
E(t) - number of carriers of the disease at a time t; a — the inverse of the average incubation period; ^ - death rate.
Equation (4) in this model shows that the number of susceptible individuals decreases over time in proportion to the number of contacts with infected individuals. Equation (5) describes the time delay during the transition from the contact state to the sick state. Equation (6) shows the transition from the contact state to the sick state. Equation (7) shows the proportionality of the number of infected and the number of recovered per unit of time.
The SIS model. The model allows you to analyze the spread of diseases to which immunity is not developed (including influenza). The model can be represented by the following equations:
dS BIS T
-=-T-+YI (8)
dI=BIS-yl (9)
dt N 1 v '
Equations (8) and (9) together show that the number of infections is proportional to the number of contacts between sick and healthy individuals.
The results of the study. For the study, official sources were used: Rosstat, as well as statistical data provided by the A. A. Smorodintsev Influenza Research Institute (Table 1).
Table 1. Number of influenza virus infections of type A (subtype H1N1, H3N2)
and type B in Russia in 2020 year
A week Number of infections A week Number of infections
1 143 10 821
2 235 11 750
3 428 12 749
4 543 13 508
5 709 14 334
6 972 15 119
7 1145 16 68
8 949 17 35
9 915 18 4
The study used a simple mathematical model that made it possible to make a forecast of the number of people who fell ill with influenza (Fig. 1).
1 3 Б 7 9 11 13 15 17 19
Figure 1 Comparison of the data obtained in the simulation with the actual number of influenza in Russia in 2020
VМеждународная научно-практическая конференция
It can be concluded that the use of too simple models in epidemiological studies of infectious diseases leads to differences between the actual data on the number of cases and the calculated results, as well as to errors in the assessment of the transmission activity and the average incidence rate. It is difficult to assess such a parameter as the probability of infection, because it depends on such factors as the individual's immunity, population density, frequency of contacts, quarantines, the presence of an incubation period, differences in mentality, belonging to a risk group, etc. Therefore, to accurately assess the incidence, it is necessary to use models that take into account all these factors.
Conclusions. The problem of applying mathematical models in describing the epidemic process of influenza diseases is the difficulty of constructing the most accurate model. The choice of a simple model leads to a difference between the calculated results and the actual data on the number of cases. For an objective assessment of morbidity, it is necessary to use models that will take into account all of the above factors.
List of literature:
1. Desyatkov B.M., Lapteva N.A., Shabanov A.N. Determination of the accuracy of mathematical modeling of the characteristics of the influenza epidemic // Epidemiology and vaccine prevention. 2013. №2. р. 33-39.
2. Lopatin A. A., Safronov V. A., Razdorsky A. S., Kuklev E. V. Current state of the problem of mathematical modeling and forecasting of the epidemic process // Problems of particularly dangerous infections. Saratov, 2010. edition 105. p. 28-30
3. Romaniukha А. А. Mathematical models in immunology and epidemiology of infectious diseases: monograph. М.: Knowledge Lab, 2020. 296 p.
© I.V. Horol'skaya, 2021
