# G-SIRS Model with Logistic Growth and Nonlinear Incidence

Authors: Ping He; Defei Zhang
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

We present the stochastic SIRS model in theG-expectation space as follows: dXt=rX1−X/K−βXY/1+αY+δZdBt+σ¯1XdBt, dYt=βXY/1+αY−ρ+ϑ+γYdBt+σ¯2YdBt, dZt=γY−μ+δZdBt+σ¯3ZdBt, where σ¯1,σ¯2,σ¯3 are three intensities of the G-Brownian motion and disturb the three variables, and B1 follows G-Normal distribution, namely, B1∼N0,π¯2,π¯2. For any initial condition X0,Y0,Z0∈D∗, we prove the new model admits a unique solution Xt,Yt,Zt for t≥0 and the solution Xt,Yt,Zt satisfies vω:Xt,Yt,Zt∈R+3,t≥0=1.


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## Body

## 1. Introduction

Liu [1] presented the SIRS model without random perturbation as follows:(1)dXt=rX1−XK−βXY1+αY+δZdt,dYt=βXY1+αY−ρ+ϑ+γYdt,dZt=γY−μ+δZdt,where Xt reflects the susceptible number, Yt is the infected number, and Zt denotes the recovered number at time t. Model (1) took into account logistic growth and nonlinear incidence. The parameters (α,β,ρ,ϑ,K,γ,δ,μ) in model (1) have practical significance, please refer to reference [1].The possible region of (1) is X,Y,Z∈R+3:X+Y+Z≤K=D∗. The basic reproduction number for system (1) is ℜ0=βK/ρ+ϑ+γ.In the actual environment, various diseases are disturbed by random factors, and there are many models that reflect this stochastic phenomenon, for example, [2–6]. Rajasekar and Pitchaimani [7] assumed this random interference is described by three independent Wiener processes. Specifically, they proposed the following SIRS model:(2)dXt=rX1−XK−βXY1+αY+δZdt+σ1XdW1t,dYt=βXY1+αY−ρ+ϑ+γYdt+σ2YdW2t,dZt=γY−μ+δZdt+σ3ZdW3t.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].


## 2.G-SIRS Model

We consider the stochastic SIRS model in theG-expectation space and propose the G-SIRS model (GSIRSM for short) as follows:(3)dXt=rX1−XK−βXY1+αY+δZdBt+σ¯1XdBt,dYt=βXY1+αY−ρ+ϑ+γYdBt+σ¯2YdBt,dZt=γY−μ+δZdBt+σ¯3ZdBt,where σ¯1,σ¯2,σ¯3 are three intensities of the G-Brownian motion, which disturb the three variables, and Bt satisfies B1∼N0,π¯2,π¯2,^B12=π¯2,^−B12=−π¯2. Note that model (3) has nonlinear incidence. We denote vc=infP∈QEP1c and Vc=supP∈QEP1c, where ^⋅=supP∈QEP⋅.It is very important to prove that the solutionX,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.Theorem 1.
∀X0,Y0,Z0∈D∗ and t≥0, Xt,Yt,Zt in (3) are unique and satisfy(4)vω:Xt,Yt,Zt∈R+3,t∈0,+∞=1.Proof.
Since the coefficients of (3) are locally Lipschitz continuous, then ∀X0,Y0,Z0∈D∗, there exists a local solution Xt,Yt,Zt on t∈0,λe quasi surely (q.s.), where λe represents the explosion time. To show λe=+∞ q.s., we prove Xt,Yt,Zt does not explode to infinity in a finite time. Suppose k0>1 is large enough such that (s.t) X0,Y0,Z0 lies in the interval 1/k0,k03. For k≥k0, define(5)λk=inft∈0,λe:Xt∉1k,korYt∉1k,korZt∉1k,k,where λk is increasing as k⟶∞. We have λ∞=limk⟶∞λk, therefore λ∞≤λe quasi surely. Suppose we guarantee that λ∞=∞ q.s., then λe=∞ and vω:Xt,Yt,Zt∈R+3=1 q.s. If we assume 0<Vλ∞<+∞, then there exists a pair of constants χ>0 and ε∈0,1 s.t.(6)Vλ∞≤χ≥vλ∞≤χ≥ε.
Then,∃k1≥k0 s.t.(7)VΠk:=Vλk≤χ≥ε,for allk≥k1.
Set a functionU1:R+3⟶R+ by(8)U1X,Y,Z=X−lnX+Y−lnY+Z−lnZ−3.
We note the functiongx:=x−lnx−1≥0 for any x>0. Using the G-Ito lemma for the function U1, we get(9)dU1=∂U1∂xdX+∂U1∂ydY+∂U1∂zdZ+12∂2U1∂x2dX2+∂2U1∂y2dY2+∂2U1∂z2dZ2=1−1XdX+1−1YdY+1−1ZdZ+12σ¯12+σ¯22+σ¯32dBt=:LU1dBt+ΘX,Y,ZdBt,where(10)LU1=X−1rK−rXK−βY1+αY+δX−1ZX+Y−1βX1+αY−ρ+ϑ+γ+Z−1γYZ−Z−1μ+δ+12∑i=13σ¯i2=r+rKX−rKX2−βX1+αY+βY1+αY−ρY−ϑY−γZY−δXZ−μZ−r+ρ+ϑ+γ+μ+δ+12∑i=13σ¯i2,ΘX,Y,Z=σ¯12X−1+σ¯22Y−1+σ¯32Z−1.
We note that the regionD∗=X+Y+Z≤K and all the parameters are positive, then we have(11)LU1≤r+rKX+βY1+αY+ρ+ϑ+γ+μ+δ+12∑i=13σ¯i2≤r+rKX+βY+ρ+ϑ+γ+μ+δ+12∑i=13σ¯i2≤ΛK+ρ+ϑ+γ+μ+δ+12∑i=13σ¯i2,where Λ=:maxr+r/K,β. We denote(12)C=ΛK+ρ+ϑ+γ+μ+δ+12∑i=13σ¯i2.
Therefore,(13)dU1=:LU1dBt+ΘX,Y,ZdBt≤CdBt+ΘX,Y,ZdBt.
Integrate (13) from 0 to λk∧χ,(14)U1Xλk∧χ,Yλk∧χ,Zλk∧χ≤U1X0,Y0,Z0+C⋅Bλk∧χ+∫0λk∧χΘXt,Yt,ZtdBt,and take the G-expectation,(15)^U1Xλk∧χ,Yλk∧χ,Zλk∧χ≤U1X0,Y0,Z0+C⋅^Bλk∧χ=U1X0,Y0,Z0+C⋅π¯2⋅λk∧χ.
Note the setΠkω:=ω:λkω≤χ and (7), then VΠkω≥εfor allk≥k1. We see that the definition of λk, then for everyω∈Πkω, there exist at least Xλkω or Yλkω or Zλkω equals to k or 1/k. For example, if Xλkω=k or Xλk=1/k, then U1Xλk,Yλk,Zλk=k−1−lnk+Yλk−1−lnYλk+Zλk−1+lnZλk≥k−1−lnk, or U1Xλk,Yλk,Zλk=1/k−1+lnk+Yλk−1−lnYλk+Zλk−1+lnZλk≥1/k−1+lnk. Thus,(16)U1Xλk,Yλk,Zλk≥min1k−1+lnk,k−1−lnk.
From (7) and (14)–(16), we have(17)^IΠkωU1Xλk,Yλk,Zλk=^IΠkωU1Xλk∧χ,Yλk∧χ,Zλk∧χ≤U1X0,Y0,Z0+C⋅^IΠkωBλk∧χ≤U1X0,Y0,Z0+C⋅π¯2⋅χ<+∞,(18)^IΠkωU1Xλk,Yλk,Zλk≥^1k+lnk−1∧k−1−lnk⋅IΠkω=1k+lnk−1∧k−1−lnk⋅^IΠkω=1k+lnk−1∧k−1−lnk⋅VΠkω≥1k+lnk−1∧k−1−lnk⋅ε.
Therefore, from inequalities (17) and (18), we have(19)1k+lnk−1∧k−1−lnk⋅ε≤^IΠkωU1Xλk,Yλk,Zλk<+∞.
Lettingk⟶∞, we find out inequality (19) is a contradiction. Thus, Vλ∞<+∞=0, namely, vλ∞=+∞=1 and vω:Xt,Yt,Zt∈R+3,t≥0=1.


## 3. 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 Theorem 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.



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*Source: 1430105-2020-07-28.xml*