G -SIRS Model with Logistic Growth and Nonlinear Incidence

We present the stochastic SIRS model in the G -expectation space as follows: t ) + σ 3 Z d B ( t ) , where σ 1 , σ 2 , σ 3 are three intensities of the G -Brownian motion and disturb the three variables, and B ( 1 ) follows G -Normal distribution, namely, B ( 1 ) ∼ N ( 0 , [ π 2 , π 2 )) . For any initial condition ( X ( 0 ) , Y ( 0 ) ,Z ( 0 )) ∈ D ∗ , we prove the new model admits a unique solution ( X ( t ) , Y ( t ) , Z ( t )) for t ≥ 0 and the solution ( X ( t ) , Y ( t ) , Z ( t )) satisﬁes v ( ω : ( X ( t ) , Y ( t ) , Z ( t )) ∈ R 3 + , t ≥

In the actual environment, various diseases are disturbed by random factors, and there are many models that reflect this stochastic phenomenon, for example, [2][3][4][5][6]. Rajasekar and Pitchaimani [7] assumed this random interference is described by three independent Wiener processes. Specifically, they proposed the following SIRS model: Peng in [8,9] constructed the interesting G-Brownian motion in nonlinear expectation space, see [10]. Many important properties on G-Brownian motion were investigated, for example, [11]. As far as we know, there is no research on model (1) in the nonlinear expectation space. Some notations and concepts used in this paper are similar to those in references [11,12].

G-SIRS Model
We consider the stochastic SIRS model in the G-expectation space and propose the G-SIRS model (GSIRSM for short) as follows: where σ 1 , σ 2 , σ 3 are three intensities of the G-Brownian motion, which disturb the three variables, and B(t) satisfies It is very important to prove that the solution (X, Y, Z) of model (3) is of global existence and is nonnegative. We first show system (3) is global and positive. Many asymptotic properties of this system (3) deserve further investigation in the future.
Proof. Since the coefficients of (3) are locally Lipschitz continuous, then ∀( where λ e represents the explosion time. To show λ e � +∞ q.s., we prove (X(t), Y(t), Z(t)) does not explode to infinity in a finite time.

Discussion
Although the endemic equilibrium for (1) exists, the endemic equilibrium of the stochastic versions (2) and (3) do not exist. From stochastic stability of Has'minskii [13], Rajasekar and Pitchaimani [7] exemplified that system (2) admits an ergodic stationary distribution. However, in the G-expectation space, we first need to obtain the new ergodic stationary distribution theorem similar to the theory of Has'minskii and use it to show the ergodic property for G-system (3). We also hope to discuss the disease is extinct for a long time in model (3). We need to find sufficient conditions for extinction of the disease for (3). However, because of the lack of a theorem which is similar to eorem 1.16 in [14], we cannot get the corresponding results immediately for G-system (3). We will investigate the existence of ergodic stationary distribution and the sufficient conditions of extinction for G-stochastic system (3) in the future research. By the way, some more realistic and impulsive perturbations models, as well as a nonautonomous case for system (3) are also worth continuing to probe. In addition, numerical simulations for the system will be further investigated.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.