Impact of environmental noises on the stability of a waterborne pathogen model

Abstract

A stochastic waterborne disease model is analyzed to reveal the effects of the white noise and telegraph noise. For this stochastic model, a critical quantity $R_0^s$ is derived, which depends on not only model parameters but also environmental noises. If $R_0^s>1$, then this model admits an ergodic stationary distribution. One of the most important contributions is the suitable Lyapunov function depending on the state of Markov chains is constructed. Theoretical results and numerical simulations provide a clear insight of the impact of environmental noises on the dynamical behavior of the model.

Appendices

Appendix A

We introduce some results concerning the positive recurrence and stationary distribution. Let (X(t), r(t)) be the diffusion process described by the following equation:

$$\begin{aligned} \mathrm {d}X(t)=b(X(t),r(t))\mathrm {d}t+\sigma (X(t),r(t))\mathrm {d}B(t),\ \ X(0)=x_0,\ r(0)=r, \end{aligned}$$

(17)

where $b(\cdot ,\cdot ):\mathbb {R}^n\times {\mathcal M}\rightarrow \mathbb {R}^n$, $\sigma (\cdot ,\cdot ):\mathbb {R}^n\times {\mathcal M}\rightarrow \mathbb {R}^{n\times n}$, and $D(x,k)=\sigma (x,k)\sigma ^T(x,k)=(d_{ij}(x,k))$. For each $k\in {\mathcal M}$, let $V(\cdot ,k)$ be any twice continuously differentiable function, the operator ${\mathcal L}$ can be defined by

$$\begin{aligned} {\mathcal L}V(x,k)=\sum _{i=1}^{n}b_i(x,k)\frac{\partial V(x,k)}{\partial x_i}+\frac{1}{2}\sum _{i,j=1}^nd_{ij}(x,k)\frac{\partial ^2V(x,k)}{\partial x_i\partial x_j}+\sum _{l=1}^{N}\gamma _{kl}V(x,l). \end{aligned}$$

Definition 1

[26] The process of $X_t^x$ with $X_0=x$ is recurrent with respect to U, if for any $x\not \in U$, $\mathbb {P}(\tau _u<\infty )=1$, where $\tau _u$ is the hitting time of U for the process $X_t^x$. The process $X_t^x$ is said to be positive recurrent with respect to U if $\mathbb {E}(\tau _u))<\infty$ any $x\not \in U$.

From Theorems in [27] it follows the following lemma which gives a criterion for the ergodic stationary distribution of the solution (X(t), r(t)) to system (17):

Lemma 2

If the following conditions are satisfied:

• (A1) $\gamma _{ij}>0$ for any $i\ne j$;

• (A2) for each $k\in {\mathcal M}$, $D(x,k)=(d_{ij}(x,k))$ is symmetric and satisfies $\lambda |\zeta |^2\le \langle D(x,k)\zeta ,\zeta \rangle \le \lambda ^{-1}|\zeta |^2\ \ \ \mathrm {for\ all}\ \zeta \in \mathbb {R}^n,$ with some constant $\lambda \in (0,1]$ for all $x\in \mathbb {R}^n$;

• (A3) there exists a nonempty open set U with compact closure, satisfying that, for each $k\in {\mathcal M}$, there is a nonnegative function $V(\cdot ,k):U^c\rightarrow \mathbb {R}$ such that $V(\cdot ,k)$ is twice continuously differential and that for some $\alpha >0$, ${\mathcal L}V(x,k)\le -\alpha ,\ \ (x,k)\in U^c\times \mathcal {M},$

then the Markov process (X(t), r(t)) is positive recurrent and ergodic. That is to say, there exists a unique stationary distribution.

Lemma 3

[15] Assume that there is a bounded open domain $U\subset \mathbb {R}^n$ with a regular boundary $\Gamma$, which has the following properties:

• (A4) there is a positive number c such that $\sum _{i,j=1}^{n}d_{ij}(x)\xi _i\xi _j\ge c|\xi |^2$ holds, for $x\in U$ and $\xi \in \mathbb {R}^n$;

• (A5) there is a nonnegative $C^2$-function V such that for some $\alpha >0$, ${\mathcal L}V(x)\le -\alpha ,\ \ x\in U^c.$

Then, the Markov process X(t) has a unique ergodic stationary distribution.

Appendix B

We verify ${\mathcal L}V(X_1,X_2,Y,B,k)\le -1, \ \ (X_1,X_2,Y,B,k)\in (\mathbb {R}_+^4\setminus U)\times {\mathcal M}.$

• Case 1. When $(X_1,X_2,Y,B)\in U_1^c$, according to (8), we have

  $$\begin{aligned}{\begin{matrix} {\mathcal L}V\le &{}-M\left( \gamma +\alpha +d+\sum _{k=1}^{N}\pi _k\frac{\sigma _3^2(k)}{2}\right) (R_0^s-1) +M\left( a_2c_2+\frac{K_2\zeta }{\eta }\right) Y\\ &{}-\frac{(1-p)\Lambda }{X_1}-\frac{p\Lambda }{X_2}-\frac{\zeta Y}{B} +\check{\beta }(2+\omega \beta ')B+K_3\\ &{}+K_4-\frac{1}{2}\left( \min \{d,\frac{\alpha +d}{2},\eta \}-\frac{\theta }{2}\max _{i=1,2,3,4}\{\check{\sigma }_i^2\}\right) \\ &{}\times \left( X_1^{\theta +1}+X_2^{\theta +1}+Y^{\theta +1}+\left( \frac{\alpha +d}{2\zeta }B\right) ^{\theta +1}\right) \\ \le &{}-\frac{(1-p)\Lambda }{X_1}+M\left( a_2c_2+\frac{K_2\zeta }{\eta }\right) Y +\check{\beta }(2+\omega \beta ')B+K_3+K_4\\ &{}-\frac{1}{4}\left( \min \{d,\frac{\alpha +d}{2},\eta \}-\frac{\theta }{2}\max _{i=1,2,3,4}\{\check{\sigma }_i^2\}\right) \\ &{}\times \left( X_1^{\theta +1}+X_2^{\theta +1}+Y^{\theta +1}+\left( \frac{\alpha +d}{2\zeta }B\right) ^{\theta +1}\right) \\ \le &{}-\frac{(1-p)\Lambda }{\varepsilon }+K_6\\ \le &{}-1. \end{matrix}} \end{aligned}$$

• Case 2. When $(X_1,X_2,Y,B)\in U_2^c$, we have

  $$\begin{aligned} {\mathcal L}V\le&-M\left( \gamma +\alpha +d+\sum _{k=1}^{N}\pi _k\frac{\sigma _3^2(k)}{2}\right) (R_0^s-1) +M\left( a_2c_2+\frac{K_2\zeta }{\eta }\right) Y\\&-\frac{(1-p)\Lambda }{X_1}-\frac{p\Lambda }{X_2}-\frac{\zeta Y}{B} +\check{\beta }(2+\omega \beta ')B+K_3\\&+K_4-\frac{1}{2}\left( \min \{d,\frac{\alpha +d}{2},\eta \}-\frac{\theta }{2}\max _{i=1,2,3,4}\{\check{\sigma }_i^2\}\right) \\&\times \left( X_1^{\theta +1}+X_2^{\theta +1}+Y^{\theta +1}+\left( \frac{\alpha +d}{2\zeta }B\right) ^{\theta +1}\right) \\ \le&-\frac{p\Lambda }{X_2}+M\left( a_2c_2+\frac{K_2\zeta }{\eta }\right) Y +\check{\beta }(2+\omega \beta ')B+K_3+K_4\\&-\frac{1}{4}\left( \min \{d,\frac{\alpha +d}{2},\eta \}-\frac{\theta }{2}\max _{i=1,2,3,4}\{\check{\sigma }_i^2\}\right) \\&\times \left( X_1^{\theta +1}+X_2^{\theta +1}+Y^{\theta +1}+\left( \frac{\alpha +d}{2\zeta }B\right) ^{\theta +1}\right) \\ \le&-\frac{p\Lambda }{\varepsilon }+K_6. \end{aligned}$$

  From (9) it follows that

  $$\begin{aligned} {\mathcal L}V(X_1,X_2,Y,B,k)\le -1,\ \ (X_1,X_2,Y,B,k)\in U_2^c\times {\mathcal M}. \end{aligned}$$

• Case 3. When $(X_1,X_2,Y,B)\in U_3^c$,

  $$\begin{aligned}{\begin{matrix} {\mathcal L}V\le &{}-M\left( \gamma +\alpha +d+\sum _{k=1}^{N}\pi _k\frac{\sigma _3^2(k)}{2}\right) (R_0^s-1) +M\left( a_2c_2+\frac{K_2\zeta }{\eta }\right) Y\\ &{}-\frac{(1-p)\Lambda }{X_1}-\frac{p\Lambda }{X_2}-\frac{\zeta Y}{B} +\check{\beta }(2+\omega \beta ')B+K_3\\ &{}+K_4-\frac{1}{2}\left( \min \{d,\frac{\alpha +d}{2},\eta \}-\frac{\theta }{2}\max _{i=1,2,3,4}\{\check{\sigma }_i^2\}\right) \\ &{}\times \left( X_1^{\theta +1}+X_2^{\theta +1}+Y^{\theta +1}+\left( \frac{\alpha +d}{2\zeta }B\right) ^{\theta +1}\right) \\ \le &{}-M\left( \gamma +\alpha +d+\sum _{k=1}^{N}\pi _k\frac{\sigma _3^2(k)}{2}\right) (R_0^s-1)+M\left( a_2c_2+\frac{K_2\zeta }{\eta }\right) Y\\ &{} +\check{\beta }(2+\omega \beta ')B+K_3+K_4\\ &{}-\frac{1}{2}\left( \min \{d,\frac{\alpha +d}{2},\eta \}-\frac{\theta }{2}\max _{i=1,2,3,4}\{\check{\sigma }_i^2\}\right) \\ &{}\times \left( X_1^{\theta +1}+X_2^{\theta +1}+Y^{\theta +1}+\left( \frac{\alpha +d}{2\zeta }B\right) ^{\theta +1}\right) \\ \le &{}-M\left( \gamma +\alpha +d+\sum _{k=1}^{N}\pi _k\frac{\sigma _3^2(k)}{2}\right) (R_0^s-1)+M\left( a_2c_2+\frac{K_2\zeta }{\eta }\right) \varepsilon +K_5. \end{matrix}} \end{aligned}$$

  According to (10), we deduce that ${\mathcal L}V\le -1$ for any $(X_1,X_2,Y,B,k)\in U_3^c\times {\mathcal M}$.

• Case 4. When $(X_1,X_2,Y,B)\in U_4^c$, from (11) we can also get that

  $$\begin{aligned} {\mathcal L}V \le&-\frac{\zeta Y}{B}+M\left( a_2c_2+\frac{K_2\zeta }{\eta }\right) Y +\check{\beta }(2+\omega \beta ')B+K_3+K_4\\&-\frac{1}{4}\left( \min \{d,\frac{\alpha +d}{2},\eta \}-\frac{\theta }{2}\max _{i=1,2,3,4}\{\check{\sigma }_i^2\}\right) \\&\times \left( X_1^{\theta +1}+X_2^{\theta +1}+Y^{\theta +1}+\left( \frac{\alpha +d}{2\zeta }B\right) ^{\theta +1}\right) \\ \le&-\frac{\zeta }{\varepsilon }+K_6\\ \le&-1. \end{aligned}$$

• Case 5. When $(X_1,X_2,Y,B)\in U_5^c$,

  $$\begin{aligned}{\begin{matrix} {\mathcal L}V\le &{}-M\left( \gamma +\alpha +d+\sum _{k=1}^{N}\pi _k\frac{\sigma _3^2(k)}{2}\right) (R_0^s-1) +M\left( a_2c_2+\frac{K_2\zeta }{\eta }\right) Y\\ &{}-\frac{(1-p)\Lambda }{X_1}-\frac{p\Lambda }{X_2}-\frac{\zeta Y}{B} +\check{\beta }(2+\omega \beta ')B+K_3\\ &{}+K_4-\frac{1}{2}\left( \min \{d,\frac{\alpha +d}{2},\eta \}-\frac{\theta }{2}\max _{i=1,2,3,4}\{\check{\sigma }_i^2\}\right) \\ &{}\times \left( X_1^{\theta +1}+X_2^{\theta +1}+Y^{\theta +1}+\left( \frac{\alpha +d}{2\zeta }B\right) ^{\theta +1}\right) \\ \le &{}-\frac{1}{4}\left( \min \{d,\frac{\alpha +d}{2},\eta \}-\frac{\theta }{2}\max _{i=1,2,3,4}\{\check{\sigma }_i^2\}\right) X_1^{\theta +1}+M\left( a_2c_2+\frac{K_2\zeta }{\eta }\right) Y\\ &{}+K_3+K_4-\frac{1}{4}\left( \min \{d,\frac{\alpha +d}{2},\eta \}-\frac{\theta }{2}\max _{i=1,2,3,4}\{\check{\sigma }_i^2\}\right) \\ &{}\times \left( X_1^{\theta +1}+X_2^{\theta +1}+Y^{\theta +1}+\left( \frac{\alpha +d}{2\zeta }B\right) ^{\theta +1}\right) \\ \le &{}-\frac{1}{4}\left( \min \{d,\frac{\alpha +d}{2},\eta \}-\frac{\theta }{2}\max _{i=1,2,3,4}\{\check{\sigma }_i^2\}\right) \frac{1}{\varepsilon ^{\theta +1}}+K_6. \end{matrix}} \end{aligned}$$

  From (12) it follows that

  $$\begin{aligned} {\mathcal L}V(X_1,X_2,Y,B,k)\le -1,\ \ (X_1,X_2,Y,B,k)\in U_5^c\times {\mathcal M}. \end{aligned}$$

  If $(X_1,X_2,Y,B)\in U_6^c$ and $(X_1,X_2,Y,B)\in U_7^c$, using the similar calculation as that in Case 5, from (12) we also obtain that

  $$\begin{aligned} {\mathcal L}V(X_1,X_2,Y,B,k)\le -1,\ \ (X_1,X_2,Y,B,k)\in (U_6^c\cup U_7^c)\times {\mathcal M}. \end{aligned}$$

• Case 6. When $(X_1,X_2,Y,B)\in U_8^c$, we have

  $$\begin{aligned} {\mathcal L}V\le&-M\left( \gamma +\alpha +d+\sum _{k=1}^{N}\pi _k\frac{\sigma _3^2(k)}{2}\right) (R_0^s-1) +M\left( a_2c_2+\frac{K_2\zeta }{\eta }\right) Y\\&-\frac{(1-p)\Lambda }{X_1}-\frac{p\Lambda }{X_2}-\frac{\zeta Y}{B} +\check{\beta }(2+\omega \beta ')B+K_3\\&+K_4-\frac{1}{2}\left( \min \{d,\frac{\alpha +d}{2},\eta \}-\frac{\theta }{2}\max _{i=1,2,3,4}\{\check{\sigma }_i^2\}\right) \\&\times \left( X_1^{\theta +1}+X_2^{\theta +1}+Y^{\theta +1}+\left( \frac{\alpha +d}{2\zeta }B\right) ^{\theta +1}\right) \\ \le&-\frac{1}{4}\left( \min \{d,\frac{\alpha +d}{2},\eta \}-\frac{\theta }{2}\max _{i=1,2,3,4}\{\check{\sigma }_i^2\}\right) \left( \frac{\alpha +d}{2\zeta }B\right) ^{\theta +1}\\&+M\left( a_2c_2+\frac{K_2\zeta }{\eta }\right) Y\\&+K_3+K_4-\frac{1}{4}\left( \min \{d,\frac{\alpha +d}{2},\eta \}-\frac{\theta }{2}\max _{i=1,2,3,4}\{\check{\sigma }_i^2\}\right) \\&\times \left( X_1^{\theta +1}+X_2^{\theta +1}+Y^{\theta +1}+\left( \frac{\alpha +d}{2\zeta }B\right) ^{\theta +1}\right) \\ \le&-\frac{1}{4}\left( \min \{d,\frac{\alpha +d}{2},\eta \}-\frac{\theta }{2}\max _{i=1,2,3,4}\{\check{\sigma }_i^2\}\right) \left( \frac{\alpha +d}{2\zeta \varepsilon ^2}\right) ^{\theta +1}+K_6. \end{aligned}$$

  In view of (13), we obtain that ${\mathcal L}V\le -1$ for any $(X_1,X_2,Y,B,k)\in U_8^c\times {\mathcal M}$.

Obviously, from above analysis it follows that ${\mathcal L}V(X_1,X_2,Y,B,k)\le -1, \ \ (X_1, X_2,Y,B,k)\in (\mathbb {R}_+^4\setminus U)\times {\mathcal M}.$