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 N2. 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 Nn. For instance, J. Gany introduced a carrier borne epidemic process with two stages of infection on N3 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


[1] Gikhman I.I., Skorokhod A.V. Vvedenie v teoriyu sluchaynykh protsessov [Introduction to the random processes theory]. Moscow, Nauka Publ., 1977. 568 p.

[2] Kalinkin A.V. Markov branching processes with interaction. Russian Mathematical Surveys, 2002, vol. 57, no. 2, pp. 241–304. DOI: 10.1070/RM2002v057n02ABEH000496

[3] Sevastyanov B.A. Vetvyashchiesya protsessy [Branching processes]. Moscow, Nauka Publ., 1971. 436 p.

[4] Epidemii protsess. Matematicheskaya entsiklopediya. T. 5 [Epidemic process. In: Mathematical encyclopedia. Vol. 5]. Moscow, Sovetskaya entsiklopediya Publ., 1985. 623 p.

[5] Weiss G. On the spread of epidemics by carries. Biometrics, 1965, vol. 21, no. 2, pp. 481–490. DOI: 10.2307/2528105 Available at: http://www.jstor.org/stable/2528105

[6] Gani J. Approaches to the modelling of AIDS. Stochastic processes in epidemic theory. Springer, 1990. Pp. 145–154.

[7] Bartlett M.S. Some evolutionary stochastic processes. J. of Royal Statistical Society. Ser. B (Methodological), 1949, vol. 11, no. 2, pp. 211–229.

[8] Kalinkin A.V. Final probabilities for a branching process with interaction of particles and an epidemic process. Theory of Probability and its Applications, 1999, vol. 43, no. 4, pp. 633–640. DOI: 10.1137/S0040585X97977203

[9] Mastikhin A.V. Final distribution for Gani epidemic Markov processes. Mathematical Notes, 2007, vol. 82, iss. 5–6, pp. 787–797. DOI: 10.1134/S0001434607110223

[10] Mastikhin A.V. Riemann functions for some Kolmogorov equations. Inzhenernyy vestnik [Engineering Bulletin], 2014, no. 12 (in Russ.). Available at: http://engsi.ru/doc/745888.html

[11] Vinogradov A.M., Krasilshchik I.S., ed. Simmetrii i zakony sokhraneniya uravneniy matematicheskoy fiziki [Symmetry and energy conservation laws of equations of mathematical physics]. Moscow, Faktorial Press Publ., 2005. 379 p.

[12] Ovsyannikov L.V. Gruppovoy analiz differentsialnykh uravneniy [Group analysis of differential equations]. Moscow, Nauka Publ., 1978. 339 p.

[13] Bitsadze A.V., Kalinichenko D.F. Sbornik zadach po uravneniyam matematicheskoy fiziki [Problem book of mathematical physics equations]. Moscow, Nauka Publ., 1985. 312 p.

[14] Ibragimov N.Kh. Opyt gruppovogo analiza obyknovennykh differentsialnykh uravneniy [Group analysis experience of ordinary differential equations]. Moscow, Znanie Publ., 1991. 48 p.

[15] Kalinkin A.V., Mastikhin A.V. A limit theorem for a Weiss epidemic. J. Appl. Probab., 2015, vol. 52, no. 1, pp. 247–257.