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In this paper has been formulated and solved the problem of identifying unknown parameters of the mathematical model describing the spread of COVID-19 infection in Kazakhstan, based on additional statistical information about infected, recovered and fatal cases. The considered model, which is part of the family of modified models based on the SIR model developed by W. Kermak and A. McKendrick in 1927, is presented as a system of 5 nonlinear ordinary differential equations describing the variational transition of individuals from one group to another. By solving the inverse problem, reduced to solving the optimization problem of minimizing the functional, using the differential evolution algorithm proposed by Rainer Storn and Kenneth Price in 1995 on the basis of simple evolutionary problems in biology, the model parameters were refined and made a forecast and predicted a peak of infected, recovered and deaths among the population of the country. The differential evolution algorithm includes the generation of populations of probable solutions randomly created in a predetermined space, sampling of the algorithm's stopping criterion, mutation, crossing and selection. [ABSTRACT FROM AUTHOR]

The study of a time-periodic solution of the multidimensional wave equation ∂2/∂t2u ~ −Δxu ~ = ƒ ~ (x, t), u ~ ⁢ (x, t) = eiktu(x), over the whole space ℝ3 leads to the condition of the Sommerfeld radiation at infinity. This is a problem that describes the motion of scattering stationary waves from a source that is in a bounded area. The inverse problem of finding this source is equivalent to reducing the Sommerfeld problem to a boundary value problem for the Helmholtz equation in a finite domain. Therefore, the Sommerfeld problem is a special inverse problem. It should be noted that in the work of Bezmenov [I. V. Bezmenov, Transfer of Sommerfeld radiation conditions to an artificial boundary of the region based on the variational principle, Sb. Math. 185 1995, 3, 3–24] approximate forms of such boundary conditions were found. In [T. S. Kalmenov and D. Suragan, Transfer of Sommerfeld radiation conditions to the boundary of a limited area, J. Comput. Math. Math. Phys. 52 2012, 6, 1063–1068], for a complex parameter λ, an explicit form of these boundary conditions was found through the boundary condition of the Helmholtz potential given by the integral in the finite domain Ω: (*)u(x, λ) = ∫Ωε(x − ξ, λ) ⁢ ρ(ξ, λ)𝑑 ξ where ε ⁢ (x − ξ λ) are fundamental solutions of the Helmholtz equation, −Δxε(x) − λε = δ(x), is a density of the potential, λ is a complex number, and δ is the Dirac delta function. These boundary conditions have the property that stationary waves coming from the region Ω to ∂Ω pass ∂Ω without reflection, i.e. are transparent boundary conditions. In the present work, in the general case, in ℝn, n ≥ 3, we have proved the problem of reducing the Sommerfeld problem to a boundary value problem in a finite domain. Under the necessary conditions for the Helmholtz potential (*), its density ρ(ξ, λ) has also been found. [ABSTRACT FROM AUTHOR]

We investigate inverse problems of finding unknown parameters of mathematical models SEIR-HCD and SEIR-D of COVID-19 spread with additional information about the number of detected cases, mortality, self-isolation coefficient, and tests performed for the city of Moscow and Novosibirsk region since 23.03.2020. In SEIR-HCD the population is divided into seven groups, and in SEIR-D into five groups with similar characteristics and transition probabilities depending on the specific region of interest. An identifiability analysis of SEIR-HCD is made to reveal the least sensitive unknown parameters as related to the additional information. The parameters are corrected by minimizing some objective functionals which is made by stochastic methods (simulated annealing, differential evolution, and genetic algorithm). Prognostic scenarios for COVID-19 spread in Moscow and in Novosibirsk region are developed, and the applicability of the models is analyzed. [ABSTRACT FROM AUTHOR]