In this paper, we analyze the quasi-stationary distribution of the stochastic <em>SVIR</em> (Susceptible, Vaccinated, Infected, Recovered) model for the measles. The quasi-stationary distributions, as disc...In this paper, we analyze the quasi-stationary distribution of the stochastic <em>SVIR</em> (Susceptible, Vaccinated, Infected, Recovered) model for the measles. The quasi-stationary distributions, as discussed by Danoch and Seneta, have been used in biology to describe the steady state behaviour of population models which exhibit discernible stationarity before to become extinct. The stochastic <em>SVIR</em> model is a stochastic <em>SIR</em> (Susceptible, Infected, Recovered) model with vaccination and recruitment where the disease-free equilibrium is reached, regardless of the magnitude of the basic reproduction number. But the mean time until the absorption (the disease-free) can be very long. If we assume the effective reproduction number <em>R</em><em><sub>p</sub></em> < 1 or <img src="Edit_67da0b97-83f9-42ef-8a00-a13da2d59963.bmp" alt="" />, the quasi-stationary distribution can be closely approximated by geometric distribution. <em>β</em> and <em>δ</em> stands respectively, for the disease transmission coefficient and the natural rate.展开更多
In the present paper, to build model of two-line queuing system with losses GI/G/2/0, the approach introduced by V.S. Korolyuk and A.F. Turbin, is used. It is based on application of the theory of semi-Markov processe...In the present paper, to build model of two-line queuing system with losses GI/G/2/0, the approach introduced by V.S. Korolyuk and A.F. Turbin, is used. It is based on application of the theory of semi-Markov processes with arbitrary phase space of states. This approach allows us to omit some restrictions. The stationary characteristics of the system have been defined, assuming that the incoming flow of requests and their service times have distributions of general form. The particular cases of the system were considered. The used approach can be useful for modeling systems of various purposes.展开更多
Based on suitable choice of states, this paper studies the stability of the equilibrium state of the EZ model by regarding the evolution of the EZ model as a Markov chain and by showing that the Markov chain is ergodi...Based on suitable choice of states, this paper studies the stability of the equilibrium state of the EZ model by regarding the evolution of the EZ model as a Markov chain and by showing that the Markov chain is ergodic. The Markov analysis is applied to the EZ model with small number of agents, the exact equilibrium state for N = 5 and numerical results for N = 18 are obtained.展开更多
This paper formulates two stochastic nonautonomous SIRI epidemic systems with nonlinear perturbations.The main aim of this study is to investigate stochastic dynamics of the two SIRI epidemic systems and obtain their ...This paper formulates two stochastic nonautonomous SIRI epidemic systems with nonlinear perturbations.The main aim of this study is to investigate stochastic dynamics of the two SIRI epidemic systems and obtain their thresholds.For the nonautonomous stochastic SIRI epidemic system with white noise,the authors provide analytic results regarding the stochastic boundedness,stochastic permanence and persistence in mean.Moreover,the authors prove that the system has at least one nontrivial positive T-periodic solution by using Lyapunov function and Hasminskii’s theory.For the system with Markov conversion,the authors establish sufficient conditions for positive recurrence and existence of ergodic stationary distribution.In addition,sufficient conditions for the extinction of disease are obtained.Finally,numerical simulations are introduced to illustrate the main results.展开更多
文摘In this paper, we analyze the quasi-stationary distribution of the stochastic <em>SVIR</em> (Susceptible, Vaccinated, Infected, Recovered) model for the measles. The quasi-stationary distributions, as discussed by Danoch and Seneta, have been used in biology to describe the steady state behaviour of population models which exhibit discernible stationarity before to become extinct. The stochastic <em>SVIR</em> model is a stochastic <em>SIR</em> (Susceptible, Infected, Recovered) model with vaccination and recruitment where the disease-free equilibrium is reached, regardless of the magnitude of the basic reproduction number. But the mean time until the absorption (the disease-free) can be very long. If we assume the effective reproduction number <em>R</em><em><sub>p</sub></em> < 1 or <img src="Edit_67da0b97-83f9-42ef-8a00-a13da2d59963.bmp" alt="" />, the quasi-stationary distribution can be closely approximated by geometric distribution. <em>β</em> and <em>δ</em> stands respectively, for the disease transmission coefficient and the natural rate.
文摘In the present paper, to build model of two-line queuing system with losses GI/G/2/0, the approach introduced by V.S. Korolyuk and A.F. Turbin, is used. It is based on application of the theory of semi-Markov processes with arbitrary phase space of states. This approach allows us to omit some restrictions. The stationary characteristics of the system have been defined, assuming that the incoming flow of requests and their service times have distributions of general form. The particular cases of the system were considered. The used approach can be useful for modeling systems of various purposes.
基金Project supported by the National Natural Science Foundation of China (Grant Nos 60534080, 60774085, and 70771012)
文摘Based on suitable choice of states, this paper studies the stability of the equilibrium state of the EZ model by regarding the evolution of the EZ model as a Markov chain and by showing that the Markov chain is ergodic. The Markov analysis is applied to the EZ model with small number of agents, the exact equilibrium state for N = 5 and numerical results for N = 18 are obtained.
基金supported by the National Natural Science Foundation of China under Grant No.11371230the Research Fund for the Taishan Scholar Project of Shandong Province of Chinathe SDUST Research Fund under Grant No.2014TDJH102
文摘This paper formulates two stochastic nonautonomous SIRI epidemic systems with nonlinear perturbations.The main aim of this study is to investigate stochastic dynamics of the two SIRI epidemic systems and obtain their thresholds.For the nonautonomous stochastic SIRI epidemic system with white noise,the authors provide analytic results regarding the stochastic boundedness,stochastic permanence and persistence in mean.Moreover,the authors prove that the system has at least one nontrivial positive T-periodic solution by using Lyapunov function and Hasminskii’s theory.For the system with Markov conversion,the authors establish sufficient conditions for positive recurrence and existence of ergodic stationary distribution.In addition,sufficient conditions for the extinction of disease are obtained.Finally,numerical simulations are introduced to illustrate the main results.
基金Acknowledgements This work was supported by the Specialized Research Fund for the Doctoral Program of Higher Education （20120002110045） and the National Natural Science Foundation of China （Grant No. 11271220）. The author was grateful to the referees for the careful reading of the first version of the paper.
