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![Hypergeometrical distribution of duration of recovery hτ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h_{\tau }$$\end{document} depending on lag τ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau$$\end{document} for W¯R(s)=k/A(s)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{W}_{R} (s) = {k \mathord{\left/ {\vphantom {k {A(s)}}} \right. \kern-0pt} {A(s)}}$$\end{document}, k=1-q11-q21-q3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k = \left( {1 - q_{1} } \right)\left( {1 - q_{2} } \right)\left( {1 - q_{3} } \right)$$\end{document}, A(s)=1-q1s1-q2s1-q3s\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A(s) = \left( {1 - q_{1} s} \right)\left( {1 - q_{2} s} \right)\left( {1 - q_{3} s} \right)$$\end{document}, q1=0,8\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q_{1} = 0,8$$\end{document}, q2=0,61\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q_{2} = 0,61$$\end{document}q3=0,2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q_{3} = 0,2$$\end{document}](profile/Arnold-Korkhin/publication/368986140/figure/fig3/AS:11431281183215497@1692760022726/Hypergeometrical-distribution-of-duration-of-recovery-htdocumentclass12ptminimal.png)
Hypergeometrical distribution of duration of recovery hτ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h_{\tau }$$\end{document} depending on lag τ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau$$\end{document} for W¯R(s)=k/A(s)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{W}_{R} (s) = {k \mathord{\left/ {\vphantom {k {A(s)}}} \right. \kern-0pt} {A(s)}}$$\end{document}, k=1-q11-q21-q3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k = \left( {1 - q_{1} } \right)\left( {1 - q_{2} } \right)\left( {1 - q_{3} } \right)$$\end{document}, A(s)=1-q1s1-q2s1-q3s\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A(s) = \left( {1 - q_{1} s} \right)\left( {1 - q_{2} s} \right)\left( {1 - q_{3} s} \right)$$\end{document}, q1=0,8\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q_{1} = 0,8$$\end{document}, q2=0,61\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q_{2} = 0,61$$\end{document}q3=0,2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q_{3} = 0,2$$\end{document}
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Models of epidemic dynamics in the form of systems of differential equations of the type SIR and its generalizations, for example SEIR and SIRS, have become widespread in epidemiology. Their coefficients are averages of some epidemic indicators, for example the time when a person is contagious. Statistical data about spreading of the epidemic are k...
Citations
In this paper, we explore the complex dynamics of a discrete-time SIS (Susceptible-Infected-Susceptible)-epidemic model. The population is assumed to be divided into two compartments: susceptible and infected populations where the birth rate is constant, the infection rate is saturated, and each recovered population has a chance to become infected again. Two types of mathematical results are provided namely the analytical results which consist of the existence of fixed points and their dynamical behaviors , and the numerical results, which consist of the global sensitivity analysis, bifurcation diagrams, and the phase portraits. Two fixed points are obtained namely the disease-free and the endemic fixed points and their stability properties. Some numerical simulations are provided to present the global sensitivity analysis and the existence of some bifurcations. The occurrence of forward and period-doubling bifurcations has confirmed the complexity of the solutions.
The chapter proposes a method for modeling the epidemic dynamics, which allows to fit the epidemic model to known time series of indicators, and not to unknown continuous functions of time. The methodology for constructing an epidemic dynamics model is based on taking into account the delayed influence of some variables on others in the form of distributed lag models. The construction of such models is based on the dual nature of non-negative integer random variables. Their distribution can also be considered as an impulse transition function of a discrete system. This consideration makes it possible to obtain a general view of such a distribution law and construct a dynamics model in the form of blocks that reproduce the lag, given by transfer functions, and blocks of the interaction of variables in the form of their products. The proposed approach makes it possible to obtain various models and analyze them as linear dynamic systems with time-varying coefficients. Statements about the relationship between the transfer functions of different blocks are proved. It is shown that the discrete analogue of the SIR model follows from the obtained general model with discrete time. A theorem is proved on a sufficient condition for the stability of a population to infection in relation to the general form of a model with discrete time. Based on the theoretical solutions obtained, various models were built as examples, taking into account the latent period as well as quarantine measures. Their stability was proven. The problem of identifying one such model based on the data on one wave of coronavirus in relation to Ukraine was formulated and solved. Two evaluation criteria are analyzed. The resulting model has satisfactory accuracy despite the high noise level. As a result of the evaluation, unobservable values such as the duration of the latent period, wait time for the test results, and the proportion of asymptomatic diseases were determined.
The article describes the results of the approbation of the method of constructing a piecewise linear trend, which can have breaks at the switching points as well as be continuous at these points, i.e., represent a linear spline. An example of applying the method for constructing a linear switching regression, which has two independent variables with a trend, is considered. The problems of spline approximation of the time series of logarithms of the number of people infected with COVID-19 in Ukraine are solved.
The article describes the results of the approbation of the method of constructing a piecewise-linear trend, which can have breaks at the switching points as well as be continuous at these points, i.e., represent a linear flow. An example of applying the method for constructing a linear regression with switches, which has two independent variables with a trend, is considered. The problem of spline approximation of the time series of logarithms of the number of infected people with COVID-19 in Ukraine is stated and solved. Keywords: trend, regression, switch point, spline, real-time calculation.
