Asymptotic behavior of global positive solution to a stochastic SIR model

Periodic solution of a stochastic SIRS epidemic model with seasonal variation

“…According to the similar arguments in [5], we know that system (1) has a unique global positive solution for any (S(0), I(0), R(0)) ∈ R 3 + . In the following result we determine the threshold for the disease to occur.…”

Stochastic Asymptotic Stability of SIR Model with Variable Diffusion Rates

“…Remark 1 One of the most important quantities in epidemiology is the basic reproduction number R 0 , expected number of secondary infections produced when one infected individual entered a fully susceptible population [14]. It determines whether there is an epidemic or not.…”

“…Since the local Lipschitz-continuous functions F i are arbitrary we have a family of stochastic SIR model. Similar models with specific diffusion terms are discussed in [1,2,4,[6][7][8][9][10][12][13][14]16,17,21].…”

The threshold behavior and periodic solution of stochastic SIR epidemic model with saturated incidence

“…Note that the existence of positive solution of system (1.3) can be obtain by [20], because the independence of B 1 and B 2 plays an unimportant role in the proof. However, the idea in [9], [10], [29] to acquire the asymptotic behavior of system (1.2) is unavailable for system (1.4) because the Fokker−Planck equation corresponding to system (1.4) is of degenerate type. In this paper, one of our aims is to study the stationary distribution of system (1.4) by applying Markov semigroup theory ( [13], [14], [16], [17], [23], [24]) which is different from the idea in [8] and [30].…”