Group Method in Searching Riemann Function for some Epidemic Equations
Authors: Mastikhin A.V., Shevchenko M.N. | Published: 12.04.2018 |
Published in issue: #2(77)/2018 | |
DOI: 10.18698/1812-3368-2018-2-12-22 | |
Category: Mathematics | Chapter: Substantial Analysis, Complex and Functional Analysis | |
Keywords: infinitesimal symmetry generator, Markov process, first Kolmogorov equation, Riemann function, exponential (double) generating function |
In the stochastic theory of epidemics, the general stochastic epidemic (i. e. Bartlett ---Mac-Kendric epidemic process) and the simple stochastic epidemic (i. e. Weiss epidemic process) are the basic models; both of them are considered on the set of states N^{2}. In Bartlett --- Mac-Kendric epidemic process an ineracting pair of infected and susceptible particles transform into a pair of two infected particles (or two carriers). This process is rather complicated; some results were obtained by asymptotic methods. In Weiss epidemic process an interacting pair of infected and susceptible particles transform into one carrier, i. e. one infected individual is removed from the population. This Markov process is more accessible to study; there are many generalizations of Weiss model on the case of N^{n}. For instance, J. Gany introduced a carrier borne epidemic process with two stages of infection on N^{3} which may be of some relevance to the spread of AIDS. We consider the first stationary Kolmogorov equation for exponential (double) generating function of transitions probabilities of Markov epidemic process. For all three processes considered above we have the same stationary Kolmogorov equation. It is possible to solve it by using Riemann function as the solution of a characteristic value problem. In present paper we study this hyperbolic equation by the symmetry method. The 4-dimentional Li algebra was obtained in the theorem 1. It contains a subalgebra, which is invariant due to the characteristic problem. So, we can reduce the hyperbolic equation to Bessel differential equation. Then, it is easy to construct the solution in an integral form and to obtain limit theorems. We also discuss the possibility of using Riemann method in the case of general epidemic process
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