Periodic solution and ergodic stationary distribution of two stochastic SIQS epidemic systems

Stationary Distribution and Extinction in a Stochastic SIQR Epidemic Model Incorporating Media Coverage and Markovian Switching

“…By the similar arguments to those in Theorem 2.1 in [19,26], one can obtain the following theorem Theorem 1. For any initial value (S(0), I(0), Q(0), R(0), r(0)) ∈ R 4 + × M, there exists a unique solution (S(t), I(t), Q(t), R(t), r(t)) of system (2) on t > 0 and the solution remain in R 4…”

“…As is well known, epidemic models are affected inevitably by external environmental noise, which is a very important component in the ecosystem. In order to describe the influence of external environmental noise on infectious diseases, many authors incorporated stochastic perturbation into the epidemic models [16][17][18][19][20]. For example, Qi et al [19] studied a stochastic SIQS epidemic system.…”

“…In order to describe the influence of external environmental noise on infectious diseases, many authors incorporated stochastic perturbation into the epidemic models [16][17][18][19][20]. For example, Qi et al [19] studied a stochastic SIQS epidemic system. Some sufficient conditions for periodic solution and ergodic stationary distribution of the disease are obtained by using the Khasminskii's theory.…”

Asymptotic behavior and threshold of a stochastic SIQS epidemic model with vertical transmission and Beddington–DeAngelis incidence

“…In view of the fact that stochastic systems can better describe practical problems, a large number of stochastic differential equation models have been applied to study various ecosystems in recent years. For instance, the influence of white noise on the contact coefficient is considered in [25][26][27][28][29][30][31][32], and the Markov chain is used in [33][34][35][36][37] to represent the interference of colored noise on the system. In order to describe the dramatic changes in natural environment such as earthquakes and floods, scholars explain these phenomena by adding Lévy jump coupling to deterministic models [38][39][40][41].…”

Stochastic Periodic Solution and Persistence of a Nonautonomous Impulsive System with Nonlinear Self-Interaction

“…In natural world, due to individual life cycle and seasonal variation, the carrying capacity of species, birth rate, and other parameters always present periodic changes for population systems [16][17][18]. For the determinate biological system, the existence of periodic solution is a very important dynamical behavior [10,[19][20][21][22]. Similarly, for stochastic system, it is very interesting to study the existence of stochastic periodic solution (periodic Markovian process).…”