Finite-Time Contraction Stability and Optimal Control for Mosquito Population Suppression Model

Abstract

Releasing Wolbachia-infected mosquitoes into the wild to suppress wild mosquito populations is an effective method for mosquito control. This paper investigates the finite-time contraction stability and optimal control problem of a mosquito population suppression model with different release strategies. By taking into account the average duration of one reproductive cycle and the influences of environmental fluctuations on mosquitoes, we consider two cases: one with a time delay and another perturbed by stochastic noises. By employing Lyapunov’s method and comparison theorem, the finite-time contraction stabilities of these two cases under a constant release strategy are analyzed. Sufficient conditions dependent on delay and noise for these two systems are provided, respectively. These conditions are related to the prespecified bounds in finite-time stability (FTS) and finite-time contraction stability (FTCS) of the system, and FTCS required stronger conditions than FTS. This also suggests that the specified bounds and the delay (or the noise intensity) play a critical role in the FTCS analysis. And finally, the optimal control for the stochastic mosquito population model under proportional releases is researched.

1. Introduction

Releasing Wolbachia-infected mosquitoes to block the transmission of mosquito-borne diseases, such as dengue, Zika, chikungunya, and yellow fever, is a safe and sustainable biological control method. The main vectors of dengue virus transmission, Aedes albopictus and Aedes aegypti, will exhibit greatly impaired ability to transmit dengue once infected with Wolbachia. In recent years, the dynamics of Wolbachia transmission in mosquito populations have attracted the attention of many investigators, and have become a research hotspot in this field.

The earliest studies of the dynamics of Wolbachia transmission in mosquito populations can be traced back to the discrete model established by Caspari and Watson [1] in 1959, which describes the degree of Wolbachia infection in a mosquito population in terms of infection frequency. Afterwards, various mathematical models were developed to evaluate the influences of releasing Wolbachia-infected males to eliminate wild mosquito populations, or to study the dynamics of Wolbachia transmission in mosquito populations, such as discrete models [2,3,4,5], ordinary differential systems [6,7,8], impulsive differential equations [9,10,11], delay differential equations [12,13,14,15], stochastic differential equations [16,17,18,19], and reaction–diffusion systems [20,21].

Previous studies mainly focused on the analysis of long-term stable dynamic properties of mosquito transmission based on Wolbachia-driven mosquito control technology, but few studies consider whether mosquitoes could be controlled within a limited period of time. Recently, Guo and Yu in [19] studied the finite-time stability (FTS) of wild mosquitoes by releasing Wolbachia-infected mosquitoes. FTS pertains to the boundedness of wild mosquito populations within a prespecified time. That is, by releasing Wolbachia-infected mosquitoes, the population of wild mosquitoes is effectively reduced to below a fixed threshold. In fact, the control perspective necessitates not only meeting the requirement of boundedness, but also aiming for reducing the number of wild mosquitoes below the initial value for a limited time. Finite-time contraction stability (FTCS) [22] can be used to describe this characteristic, which means that in addition to staying in a certain threshold in finite time, the state will be within a specified range smaller than the initial state before reaching the terminal time [23].

Compared with FTS, the main difference is that FTCS requires not only “boundedness”, but also the characteristic of “contraction” over a finite time interval. In other words, FTCS can be used to describe the phenomenon of suppressing mosquito density within a safety threshold smaller than the initial value. This study addresses this issue and establishes some sufficient conditions for FTCS of a time delay model and stochastic system in the framework of the Lyapunov method. The highlights of this paper are the following:

•

Novel stochastic mosquito population models with different release strategies are established.

•

The finite-time contraction stabilities of a deterministic time delay model and stochastic mosquito population suppression system under constant release strategy are proved, as well as sufficient conditions to ensure FTCS of these two systems is obtained, respectively.

•

By using the control theory and maximum principle, the optimal control strategy of the stochastic model under proportional releases is proposed and rigorously proved by mathematical theory.

The remaining sections of this paper are structured as follows: Section 2 presents FTCS for the delay differential equation with constant releases, along with the derivation of sufficient conditions dependent on time delay. In Section 3, we derive FTCS for the stochastic differential equation with constant releases, and demonstrate that noise intensity plays a crucial role in FTCS of the mosquito population system. An optimal control strategy for the stochastic mosquito population model is provided by using proportional releases in Section 4. Finally, we conclude and provide further remarks on this study in the concluding Section 5.

2. Finite-Time Contraction Stability for Delay Differential Equation by Constant Releases

In this section, we focus on the derivation and FTCS for the time delay model. Assuming that wild mosquitoes are evenly distributed in sex, let $w\left(t\right)$ and $g\left(t\right)$ be the numbers of wild females/males and Wolbachia-infected males at time t, respectively. Li [24] proposed the following model:

$$\{\begin{array}{l}\frac{\mathrm{d}w\left(t\right)}{\mathrm{d}t}=\frac{aw\left(t\right)}{w\left(t\right)+g\left(t\right)}[1-\xi w\left(t\right)]w\left(t\right)-\mu w\left(t\right),\\ \frac{\mathrm{d}g\left(t\right)}{\mathrm{d}t}=\mathbb{B}(·)-\mu g\left(t\right),\end{array}$$

(1)

where a is the maximum number of surviving offspring produced per mosquito, $\frac{w\left(t\right)}{w\left(t\right)+g\left(t\right)}$ is the mating probability between wild mosquitoes, $\mu$ is the density-independent death rate of wild mosquitoes, $\xi$ is the carrying capacity parameter such that $1-\xi w\left(t\right)$ describes the density-dependent survival probability, and $\mathbb{B}(·)$ is the release rate of the Wolbachia-infected male mosquitoes. Suppose that the Wolbachia-infected male mosquitoes are constantly released such that $\mathbb{B}(·)=b$ (a constant), then model (1) is rewritten as the following mosquito population suppression model:

$$\{\begin{array}{c}\frac{\mathrm{d}w\left(t\right)}{\mathrm{d}t}=\frac{aw\left(t\right)}{w\left(t\right)+g\left(t\right)}[1-\xi w\left(t\right)]w\left(t\right)-\mu w\left(t\right),\hfill \\ \frac{\mathrm{d}g\left(t\right)}{\mathrm{d}t}=b-\mu g\left(t\right).\hfill \end{array}$$

(2)

Since mosquitoes take at least 20 days from mating to the emergence of the next generation [25], in order to provide a more precise and realistic depiction of the situation, it is necessary to incorporate the waiting period from mating to emergence into our model. Let $\tau >0$ denote the average duration of one reproduction cycle. Considering the average duration from the mating of one generation to the production of their adult offspring, we present the following delay model:

$$\{\begin{array}{c}\begin{array}{c}\frac{\mathrm{d}w\left(t\right)}{\mathrm{d}t}=\frac{aw(t-\tau )}{w(t-\tau )+g(t-\tau )}[1-\xi w(t-\tau )]w(t-\tau )-\mu w\left(t\right),\hfill \\ \frac{\mathrm{d}g\left(t\right)}{\mathrm{d}t}=b-\mu g\left(t\right).\hfill \end{array}\hfill \end{array}$$

(3)

According to reference [23], we present the following definitions of FTS and FTCS for delay system (3).

Definition 1.

For any given positive constants $D_1,D_2,T_0$ with $D_2>D_1$, system (3) is said to be the finite-time stability with respect to $(D_1,D_2,T_0)$ if

$$\begin{array}{c}\underset{-\tau \le \theta \le 0}{\sup }[w^2\left(\theta \right)+g^2\left(\theta \right)]\le D_1\Rightarrow w^2\left(t\right)+g^2\left(t\right)\le D_2,\forall t\in [0,T_0].\hfill \end{array}$$

Definition 2.

Given five positive constants $D_1,D_2,D_0,\zeta ,T_0$ with $D_2>D_1>D_0$, $\zeta \in (0,T_0)$, system (3) is said to be the finite-time contraction stability with respect to $(D_1,D_2,D_0,\zeta ,T_0)$ if

$$\begin{array}{c}\underset{-\tau \le \theta \le 0}{\sup }[w^2\left(\theta \right)+g^2\left(\theta \right)]\le D_1\Rightarrow w^2\left(t\right)+g^2\left(t\right)\le D_2,\forall t\in [0,T_0],\hfill \end{array}$$

and moreover,

$$\begin{array}{c}{w}^2\left(t\right)+g^2\left(t\right)\le D_0,\forall t\in [T_0-\zeta ,T_0].\hfill \end{array}$$

Next, we use the Lyapunov method to derive FTCS and present some sufficient conditions for system (3).

Theorem 1.

Assume that there exist positive constants $D_1,D_2,T_0$ with $D_2>D_1$ if one of the following conditions is satisfied:

Case 1. $c_0\le 0,\left(i\right)(a{e}^{-c_0\tau }+c_0)T_0+a\tau e^{-c_0\tau }\le \alpha$, for $\forall t\in [0,T_0]$.

Moreover, (ii) $(a{e}^{-c_0\tau }+c_0)T_0+a\tau e^{-c_0\tau }\le {\alpha }_0,$ for $\forall t\in [T_0-\zeta ,T_0]$.

Case 2. $c_0>0,\left(i\right)(c_0+a)T_0\le \gamma$, for $\forall t\in [0,T_0]$.

Moreover,  $\left(ii\right)(c_0+a)T_0\le {\gamma }_0,$ for $\forall t\in [T_0-\zeta ,T_0]$.

System (3) is finite-time contractively stable with respect to $(D_1,D_2,D_0,\zeta ,T_0)$, where the coefficients are shown as follows:

$$\begin{array}{cc}\hfill & c_0=max\left\{a-2\mu ,1-2\mu \right\},\hfill \\ \hfill & \alpha =\ln D_2-\ln (D_1-\frac{b^2}{c_0}{e}^{-c_0{T}_0}),\hfill \\ \hfill & {\alpha }_0=\ln D_0-\ln (D_1-\frac{b^2}{c_0}{e}^{-c_0{T}_0}),\hfill \\ \hfill & \gamma =\ln D_2-\ln (D_1+\frac{b^2}{c_0}),\hfill \\ \hfill & {\gamma }_0=\ln D_0-\ln (D_1+\frac{b^2}{c_0}).\hfill \end{array}$$

Proof. 

Let $\left(w\left(t\right),g\left(t\right)\right)$ be a solution of system (3), given that

$$\begin{array}{c}\underset{-\tau \le \theta \le 0}{\sup }[w^2\left(\theta \right)+g^2\left(\theta \right)]\le D_1,\theta \in [-\tau ,0].\hfill \end{array}$$

(4)

According to Definition 2, we need to prove that the following formula

$$\begin{array}{c}{w}^2\left(t\right)+g^2\left(t\right)\le D_2,\mathrm{for}\forall t\in [0,T_0]\hfill \end{array}$$

holds. Moreover, we also need to show that for any $t\in [T_0-\zeta ,T_0]$, the following condition

$$\begin{array}{c}{w}^2\left(t\right)+g^2\left(t\right)\le D_0\hfill \end{array}$$

holds. Consequently, we can obtain the sufficient condition of FTCS for system (3). Denote

$$\begin{array}{c}V\left(t\right)=w^2\left(t\right)+g^2\left(t\right).\hfill \end{array}$$

Taking the derivative of $V\left(t\right)$ along system (3) yields

$$\begin{array}{cc}\dot{V}\left(t\right)\hfill & =2w\left(t\right)\{\frac{aw(t-\tau )}{w(t-\tau )+g(t-\tau )}[1-\xi w(t-\tau )]w(t-\tau )-\mu w\left(t\right)\}+2g\left(t\right)[b-\mu g\left(t\right)]\hfill \\ & =2aw\left(t\right)w(t-\tau )\frac{w(t-\tau )}{w(t-\tau )+g(t-\tau )}-2a\xi w\left(t\right)w^2(t-\tau )\frac{w(t-\tau )}{w(t-\tau )+g(t-\tau )}\hfill \\ & - 2\mu w^2\left(t\right)+2bg\left(t\right)-2\mu g^2\left(t\right)\hfill \\ & \le a{w}^2\left(t\right)+a{w}^2(t-\tau )-2\mu w^2\left(t\right)+b^2+g^2\left(t\right)-2\mu g^2\left(t\right)\hfill \\ & =b^2+(a-2\mu )w^2\left(t\right)+(1-2\mu )g^2\left(t\right)+a{w}^2(t-\tau )\hfill \\ & \le b^2+c_{0}V\left(t\right)+aV(t-\tau ).\hfill \end{array}$$

Now, let $Y\left(t\right)$ be the solution of the following equations

$$\{\begin{array}{c}\dot{Y}\left(t\right)=b^2+c_{0}Y\left(t\right)+aY(t-\tau \left(t\right)),\hfill \\ Y\left(\theta \right)=w^2\left(\theta \right)+g^2\left(\theta \right),-\tau \le \theta \le 0.\hfill \end{array}$$

Since $V\left(\theta \right)=Y\left(\theta \right)$ for $-\tau \le \theta \le 0$, according to the comparing theorem, we have

$$\begin{array}{c}V\left(t\right)\le Y\left(t\right).\hfill \end{array}$$

(5)

Utilizing the variational formula of constants, we obtain

$$\begin{array}{cc}Y\left(t\right)=\hfill & e^{c_{0}t}\left[Y\left(0\right)+\frac{b^2}{c_0}\right]-\frac{b^2}{c_0}+a{\int }_0^t{e}^{c_0(t-s)}Y(t-\tau )\mathrm{d}s,0<s\le t\in [0,T_0].\hfill \end{array}$$

(6)

In what follows, we will continue our procedure in two cases.

Case 1: $c_0\le 0$. From Equation (6), we have

$$\begin{array}{cc}Y\left(t\right)e^{-c_{0}t}\hfill & \le \left[Y\left(0\right)+\frac{b^2}{c_0}\right]-\frac{b^2}{c_0}{e}^{-c_{0}t}+a{\int }_0^{t}Y(s-\tau )e^{-c_{0}s}\mathrm{d}s\hfill \\ & \le \left[Y\left(0\right)+\frac{b^2}{c_0}\right]-\frac{b^2}{c_0}{e}^{-c_{0}t}+a{e}^{-c_0\tau }{\int }_0^{t}Y(s-\tau )e^{-c_0(s-\tau )}\mathrm{d}s\hfill \\ & \le \left[Y\left(0\right)-\frac{b^2}{c_0}{e}^{-c_{0}t}\right]+a{e}^{-c_0\tau }{\int }_{-\tau }^{t-\tau }Y\left(s\right)e^{-c_{0}s}\mathrm{d}s\hfill \\ & \le \left[Y\left(0\right)-\frac{b^2}{c_0}{e}^{-c_{0}t}\right]+a{e}^{-c_0\tau }{\int }_{-\tau }^{t}Y\left(s\right)e^{-c_{0}s}\mathrm{d}s.\hfill \end{array}$$

Using the Gronwall inequality yields

$$Y\left(t\right)e^{-c_{0}t}\le \left[Y\left(0\right)-\frac{b^2}{c_0}{e}^{-c_{0}t}\right]exp\left\{a{e}^{-c_0\tau }(t+\tau )\right\}.$$

Then we obtain

$$\begin{array}{cc}V\left(t\right)\le Y\left(t\right)\hfill & =Y\left(t\right)e^{-c_{0}t}{e}^{c_{0}t}\le \left[\underset{-\tau \le \theta \le 0}{\sup }Y\left(\theta \right)-\frac{b^2}{c_0}{e}^{-c_{0}t}\right]exp\left\{a{e}^{-c_0\tau }(t+\tau )+c_{0}t\right\}\hfill \\ & \le \left[\underset{-\tau \le \theta \le 0}{\sup }Y\left(\theta \right)-\frac{b^2}{c_0}{e}^{-c_{0}t}\right]exp\left\{(a{e}^{-c_0\tau }+c_0)t+a\tau e^{-c_0\tau }\right\}.\hfill \end{array}$$

Using condition (i) of Case 1, we get $V\left(t\right)\le D_2$, namely, $w^2\left(t\right)+g^2\left(t\right)\le D_2$. Thus, system (3) is said to be FTS with respect to $(D_1,D_2,T_0)$. In addition, based on the condition (ii) of Case 1, we have $V\left(t\right)\le D_0$ for any $t\in [T_0-\zeta ,T_0]$, which yields $w^2\left(t\right)+g^2\left(t\right)\le D_0$. Hence, system (3) is finite-time contractively stable with respect to $(D_1,D_2,D_0,\zeta ,T_0)$.

Case 2: $c_0>0$. From Equation (6), we have

$$\{\begin{array}{c}Y\left(t\right)\le e^{c_{0}t}\left[\underset{-\tau \le \theta \le 0}{\sup }Y\left(\theta \right)+\frac{b^2}{c_0}\right]+a{\int }_0^t{e}^{c_0(t-s)}Y(s-\tau )\mathrm{d}s.\hfill \end{array}$$

(7)

Let $Z\left(t\right)$ be the solution of the following equations

$$\{\begin{array}{c}Z\left(t\right)=e^{c_{0}t}\left[\underset{-\tau \le \theta \le 0}{\sup }Z\left(\theta \right)+\frac{b^2}{c_0}\right]+a{\int }_0^t{e}^{c_0(t-s)}Z(s-\tau )\mathrm{d}s,t>0,\hfill \\ Z\left(\theta \right)=w^2\left(\theta \right)+g^2\left(\theta \right),-\tau \le \theta \le 0.\hfill \end{array}$$

(8)

Combining Equations (7) and (8), we obtain $0\le Y\left(t\right)\le Z\left(t\right)$ for $t>-\tau$.

When $0<t\le \tau$, we have

$$\begin{array}{cc}Z\left(t\right)-Z(t-\tau )\hfill & \ge Z\left(t\right)-\left[\underset{-\tau \le \theta \le 0}{\sup }Z\left(\theta \right)+\frac{b^2}{c_0}\right]\hfill \\ & =\left[\underset{-\tau \le \theta \le 0}{\sup }Z\left(\theta \right)+\frac{b^2}{c_0}\right](e^{c_{0}t}-1)+a{\int }_0^t{e}^{c_0(t-s)}Z(s-\tau )\mathrm{d}s\ge 0.\hfill \end{array}$$

When $t>\tau$, one can obtain

$$Z\left(t\right)-Z(t-\tau )\ge \left[\underset{-\tau \le \theta \le 0}{\sup }Z\left(\theta \right)+\frac{b^2}{c_0}\right](e^{c_{0}t}-e^{c_0(t-\tau )})+a{e}^{c_0(t-\tau )}{\int }_{t-\tau }^t{e}^{-c_{0}s}Z(s-\tau )\mathrm{d}s\ge 0.$$

Thus, for $t>0$, $Z\left(t\right)\ge Z(t-\tau )$ holds, and from Equation (8), we derive

$$Z\left(t\right)\le e^{c_{0}t}\left[\underset{-\tau \le \theta \le 0}{\sup }Z\left(\theta \right)+\frac{b^2}{c_0}\right]+a{\int }_0^t{e}^{c_0(t-s)}Z\left(s\right)\mathrm{d}s.$$

By using Gronwall inequality, we obtain

$$Z\left(t\right)e^{-c_{0}t}\le \left[\underset{-\tau \le \theta \le 0}{\sup }Z\left(\theta \right)+\frac{b^2}{c_0}\right]e^{at}.$$

Thus,

$$\begin{array}{cc}V\left(t\right)\le \hfill & Y\left(t\right)\le Z\left(t\right)\le \left[D_1+\frac{b^2}{c_0}\right]e^{(c_0+a)t}\le \left[D_1+\frac{b^2}{c_0}\right]e^{(c_0+a)T_0}.\hfill \end{array}$$

By using Case 2 (i), we have $V\left(t\right)\le D_2$, which indicates that $w^2\left(t\right)+g^2\left(t\right)\le D_2$ holds. Moreover, condition (ii) of Case 2 implies that $w^2\left(t\right)+g^2\left(t\right)\le D_0$ for any $t\in [T_0-\zeta ,T_0]$, which means that system (3) is said to be FTCS. This proof is completed. □

3. Finite-Time Contraction Stability for Stochastic Differential Equation by Constant Releases

The development, behavior, and survival of mosquitoes, as well as the spread of diseases, are significantly influenced by environmental factors, such as temperature, rainfall, humidity, etc., [26]. Specifically, temperature impacts the developmental rate of larvae, as well as the biting frequency and mortality rate of adult mosquitoes, thus affecting the transmission of mosquito-borne diseases [27]. Therefore, it is of great practical significance to study mosquito population dynamics by using stochastic differential equations. Assume that the environmental fluctuations are in the form of whites noises that are directly proportional to $w\left(t\right)$ and $g\left(t\right)$. We propose the following stochastic mosquito population suppression model by considering environmental conditions

$$\{\begin{array}{c}\mathrm{d}w\left(t\right)=[\frac{aw\left(t\right)}{w\left(t\right)+g\left(t\right)}\left(1-\xi w\left(t\right)\right)w\left(t\right)-\mu w\left(t\right)]\mathrm{d}t+{\sigma }_{1}w\left(t\right)\mathrm{d}{B}_1\left(t\right),\hfill \\ \mathrm{d}g\left(t\right)=[b-\mu g\left(t\right)]\mathrm{d}t+{\sigma }_{2}g\left(t\right)\mathrm{d}{B}_2\left(t\right),\hfill \end{array}$$

(9)

where ${\sigma }_i\ge 0(i=1,2)$ are the noise intensities and $B_i\left(t\right)$ are independent and standard Brownian motions defined on a complete filtered probability space $(\mathsf{\Omega },\mathcal{F},{\left\{{\mathcal{F}}_t\right\}}_{t\ge 0},\mathbb{P})$ with the filtration ${\left\{{\mathcal{F}}_t\right\}}_{t\ge 0}$ satisfying the usual conditions (i.e., it is right continuous and ${\mathcal{F}}_0$ contains all $\mathbb{P}$-null sets). Let $X\left(t\right)=\left(w\left(t\right),g\left(t\right)\right)\in {\mathbb{R}}_{+}^2$, and its norm is $\left|X\left(t\right)\right|=\sqrt{w^2\left(t\right)+g^2\left(t\right)}$. Denote $\mathbb{E}$ as the probability expectation respect to $\mathbb{P}$.

3.1. Stochastic Boundedness

We first present the existence and uniqueness of the global positive solution of model (9) directly.

Theorem 2.

For any given initial value $\left(w\left(0\right),g\left(0\right)\right)\in {\mathbb{R}}_{+}^2$, system (9) has a unique global positive solution $X\left(t\right)$ on $t>0$, and the solution will remain in ${\mathbb{R}}_{+}^2$ with probability one.

Since the proof is standard and is similar to the statement of Theorem 2.1 in [19], we omit it here.

Theorem 3.

If for all $p>1$, the following inequality

$$\begin{array}{c}p\mu >\frac{p(p-1)}{2}max\left\{{\sigma }_1^2,{\sigma }_2^2\right\}+1\hfill \end{array}$$

(10)

holds, then there exists a positive constant K (which is dependent on p) such that the solution $X\left(t\right)=\left(w\left(t\right),g\left(t\right)\right)$ with initial value $\left(w\left(0\right),g\left(0\right)\right)\in {\mathbb{R}}_{+}^2$ has the following property:

$$\begin{array}{c}\underset{t\to \infty }{lim\; sup}\mathbb{E}{\left|X\left(t\right)\right|}^p<K.\hfill \end{array}$$

That is to say, the solution $X\left(t\right)$ is stochastically ultimately bounded.

Proof. 

For all $p>1$, denote the Lyapunov function as follows:

$$\begin{array}{c}V(w\left(t\right),g\left(t\right))=w^p\left(t\right)+g^p\left(t\right).\hfill \end{array}$$

Using the Itô formula leads to the following equation:

$$\begin{array}{c}\mathrm{d}V(w\left(t\right),g\left(t\right))=LV(w\left(t\right),g\left(t\right))\mathrm{d}t+p{\sigma }_1{w}^p\left(t\right)\mathrm{d}{B}_1\left(t\right)+p{\sigma }_2{g}^p\left(t\right)\mathrm{d}{B}_2\left(t\right),\hfill \end{array}$$

where

$$\begin{array}{cc}LV\left(w\right(t),g(t\left)\right)\hfill & =p{w}^{p-1}\left(t\right)[\frac{aw\left(t\right)}{w\left(t\right)+g\left(t\right)}\left(1-\xi w\left(t\right)\right)w\left(t\right)-\mu w\left(t\right)]+\frac{1}{2}p(p-1){\sigma }_1^2{w}^p\left(t\right)\hfill \\ & + p{g}^{p-1}\left(t\right)[b-\mu g\left(t\right)]+\frac{1}{2}p(p-1){\sigma }_2^2{g}^p\left(t\right)\hfill \\ & \le -\frac{aw\left(t\right)}{w\left(t\right)+g\left(t\right)}p\xi w^{p+1}+pa{w}^p\left(t\right)-p\mu w^p\left(t\right)+\frac{p(p-1)}{2}{\sigma }_1^2{w}^p\left(t\right)\hfill \\ & + pb{g}^{p-1}\left(t\right)-p\mu g^p\left(t\right)+\frac{p(p-1)}{2}{\sigma }_2^2{g}^p\left(t\right).\hfill \end{array}$$

Using the Young inequality to $pa{w}^p$, we have

$$\begin{array}{cc}LV\left(w\right(t),g(t\left)\right)\le \hfill & - [p\mu -\frac{p(p-1)}{2}{\sigma }_1^2]w^p\left(t\right)-[p\mu -\frac{p(p-1)}{2}{\sigma }_2^2]g^p\left(t\right)\hfill \\ & + \frac{p}{p-1}{w}^{p-1}\left(t\right)+pb{g}^{p-1}\left(t\right)+\frac{a^p{p}^p}{p-1}.\hfill \end{array}$$

Let

$$\begin{array}{cc}F\left(w\right(t),g(t\left)\right)=\hfill & - w^p[p\mu -\frac{p(p-1)}{2}{\sigma }_1^2-1]-g^p[p\mu -\frac{p(p-1)}{2}{\sigma }_2^2-1]\hfill \\ & + \frac{p}{p-1}{w}^{p-1}+bp{g}^{p-1}+\frac{a^p{p}^p}{p-1}.\hfill \end{array}$$

It follows from Equation (10) that there exists a positive constant $K_1$ such that $F(w\left(t\right),g\left(t\right))\le K_1$. Hence, we can obtain that

$$\begin{array}{c}LV(w\left(t\right),g\left(t\right))\le F(w\left(t\right),g\left(t\right))-[w^p\left(t\right)+g^p\left(t\right)]\le K_1-V(w\left(t\right),g\left(t\right)),\hfill \end{array}$$

which yields

$$\begin{array}{c}\mathrm{d}V(w\left(t\right),g\left(t\right))\le [K_1-V(w\left(t\right),g\left(t\right))]\mathrm{d}t+p{\sigma }_1{w}^p\left(t\right)\mathrm{d}{B}_1\left(t\right)+p{\sigma }_2{g}^p\left(t\right)\mathrm{d}{B}_2\left(t\right).\hfill \end{array}$$

Integrating both sides of the equation $\mathrm{d}\left[e^{t}V(w\left(t\right),g\left(t\right))\right]$ and taking expectations, we derive

$$\begin{array}{c}\mathbb{E}\left[e^{t}V(w\left(t\right),g\left(t\right))\right]\le V(w\left(0\right),g\left(0\right))+K{e}^t.\hfill \end{array}$$

Then, we have

$$\begin{array}{c}\underset{t\to \infty }{lim\; sup}\mathbb{E}V(w\left(t\right),g\left(t\right))\le K_1.\hfill \end{array}$$

Combining the elementary inequalities, we can compute that

$$\begin{array}{c}\underset{t\to \infty }{lim\; sup}\mathbb{E}{\left|X\left(t\right)\right|}^p\le \frac{K_1}{2^{(1-\frac{p}{2})\wedge 0}}:=K.\hfill \end{array}$$

In particular, when $p=2$, we have

$$\begin{array}{c}\underset{t\to \infty }{lim\; sup}\mathbb{E}{\left|X\left(t\right)\right|}^2\le K_1,\hfill \end{array}$$

which is the second moment of the solution of model (9). Now, for any $ϵ>0$, let $K_0=\sqrt{\frac{K_1}{ϵ}}$. Then, by Chebyshev’s inequality, we obtain

$$\begin{array}{c}\mathbb{P}\left\{\left|X\left(t\right)\right|>K_0\right\}\le \frac{{\mathbb{E}\left|X\left(t\right)\right|}^2}{K_0^2},\hfill \end{array}$$

which implies that

$$\begin{array}{c}\underset{t\to \infty }{lim\; sup}\mathbb{P}\left\{\left|X\left(t\right)\right|>K_0\right\}\le \frac{K_1}{K_0^2}=ϵ.\hfill \end{array}$$

Hence, we have

$$\begin{array}{c}\underset{t\to \infty }{lim\; sup}\mathbb{P}\left\{\left|X\left(t\right)\right|\le K_0\right\}\ge 1-ϵ.\hfill \end{array}$$

Namely, the solution of model (9) is stochastically ultimately bounded. This proof is completed. □

3.2. Finite-Time Contraction Stability

In this section, we focus on the sufficient conditions of FTCS for system (9) by employing the Lyapunov method and stochastic comparison theorem. By replacing the initial state appropriately in [28], we present the following definition of FTCS for stochastic system (9).

Definition 3.

For any given positive constants $N_1,N_2,T$ with $N_2>N_1$, system (9) is said to be finite-time stable with respect to $(N_1,N_2,T)$, if

$$\begin{array}{c}{w}^2\left(0\right)+g^2\left(0\right)\le B_1\Rightarrow \mathbb{E}[w^2\left(t\right)+g^2\left(t\right)]\le B_2,\forall t\in [0,T].\hfill \end{array}$$

Definition 4.

Given five positive constants $N_1,N_2,N_0,\eta ,T$ with $N_2>N_1>N_0$, $\eta \in (0,T)$, system (9) is said to be finite-time contraction stable with respect to $(N_1,N_2,N_0,\eta ,T)$, if

$$\begin{array}{c}{w}^2\left(0\right)+g^2\left(0\right)\le N_1\Rightarrow \mathbb{E}[w^2\left(t\right)+g^2\left(t\right)]\le N_2,\forall t\in [0,T],\hfill \end{array}$$

and,

$$\begin{array}{c}\mathbb{E}[w^2\left(t\right)+g^2\left(t\right)]\le N_0,\forall t\in [T-\eta ,T].\hfill \end{array}$$

Next, we present the following theorem.

Theorem 4.

Assume that there exist positive constants $N_1,N_2,N_0,\eta ,T$ with $N_2>N_1>N_0$ and $\eta \in (0,T)$, if one of the following conditions is satisfied:

Case 1. $c\le 0,\left(i\right)cT\le \kappa$, for $\forall t\in [0,T]$. Moreover, (ii) $cT\le {\kappa }_0,$ for $\forall t\in [T-\eta ,T]$.

Case 2. $c>0,\left(i\right)cT\le \lambda$, for $\forall t\in [0,T]$. Moreover,  $\left(ii\right)cT\le {\lambda }_0,$ for $\forall t\in [T-\eta ,T]$.

System (9) is finite-time contractively stable with respect to $(N_1,N_2,N_0,\eta ,T)$, where the coefficients are shown as follows:

$$\begin{array}{cc}& c=max\left\{2a+{\sigma }_1^2-2\mu ,{\sigma }_2^2+1-2\mu \right\},\hfill \\ & \kappa =\ln (N_2+\frac{b^2}{c})-\ln N_1,\hfill \\ & {\kappa }_0=\ln (N_0+\frac{b^2}{c})-\ln N_1,\hfill \\ & \lambda =\ln N_2-\ln (N_1+\frac{b^2}{c}),\hfill \\ & {\lambda }_0=\ln N_0-\ln (N_1+\frac{b^2}{c}).\hfill \end{array}$$

Proof. 

Let

$$\begin{array}{c}[w^2\left(0\right)+g^2\left(0\right)]\le N_1.\hfill \end{array}$$

(11)

According to Definition 4, if we can show that

$$\begin{array}{c}\mathbb{E}[w^2\left(t\right)+g^2\left(t\right)]\le N_2,\mathrm{for}\forall t\in [0,T]\hfill \end{array}$$

holds, and also prove that for any $t\in [T-\eta ,T]$, the following condition

$$\begin{array}{c}\mathbb{E}[w^2\left(t\right)+g^2\left(t\right)]\le N_0\hfill \end{array}$$

holds. Thus, we can derive the sufficient conditions of FTCS for system (9). Choose

$$\begin{array}{c}\tilde{V}\left(t\right)=w^2\left(t\right)+g^2\left(t\right).\hfill \end{array}$$

Applying the Itô formula, we derive that

$$\begin{array}{cc}\mathrm{d}\tilde{V}\left(t\right)\hfill & =2w[\frac{aw\left(t\right)}{w\left(t\right)+g\left(t\right)}(1-\xi w\left(t\right))w\left(t\right)-\mu w\left(t\right)]\mathrm{d}t+{\sigma }_1^2{w}^2\left(t\right)\mathrm{d}t\hfill \\ & + 2g[b-\mu g\left(t\right)]\mathrm{d}t+{\sigma }_2^2{g}^2\left(t\right)\mathrm{d}t+2{\sigma }_1{w}^2\left(t\right)\mathrm{d}{B}_1\left(t\right)+2{\sigma }_2{g}^2\left(t\right)\mathrm{d}{B}_2\left(t\right)\hfill \\ & = [2w^2\left(t\right)\frac{aw\left(t\right)}{w\left(t\right)+g\left(t\right)}-2\xi w^3\left(t\right)\frac{aw\left(t\right)}{w\left(t\right)+g\left(t\right)}-2\mu w^2\left(t\right)+{\sigma }_1^2{w}^2\left(t\right)]\mathrm{d}t\hfill \\ & + [2bg\left(t\right)-2\mu g^2\left(t\right)+{\sigma }_2^2{g}^2\left(t\right)]\mathrm{d}t+2{\sigma }_1{w}^2\left(t\right)\mathrm{d}{B}_1\left(t\right)+2{\sigma }_2{g}^2\left(t\right)\mathrm{d}{B}_2\left(t\right)\hfill \\ & \le \left[\left(2a+{\sigma }_1^2-2\mu \right)w^2\left(t\right)+\left({\sigma }_2^2+1-2\mu \right)g^2\left(t\right)+b^2\right]\mathrm{d}t\hfill \\ & + 2{\sigma }_1{w}^2\left(t\right)\mathrm{d}{B}_1\left(t\right)+2{\sigma }_2{g}^2\left(t\right)\mathrm{d}{B}_2\left(t\right)\hfill \\ & \le \left[b^2+c\tilde{V}\left(t\right)\right]\mathrm{d}t+2{\sigma }_1{w}^2\left(t\right)\mathrm{d}{B}_1\left(t\right)+2{\sigma }_2{g}^2\left(t\right)\mathrm{d}{B}_2\left(t\right).\hfill \end{array}$$

Now, let $\tilde{Y}\left(t\right)$ be the solution of the following stochastic system

$$\{\begin{array}{c}\begin{array}{c}\mathrm{d}\tilde{Y}\left(t\right)=\left[b^2+c\tilde{Y}\left(t\right)\right]\mathrm{d}t+2{\sigma }_1{w}^2\left(t\right)\mathrm{d}{B}_1\left(t\right)+2{\sigma }_2{g}^2\left(t\right)\mathrm{d}{B}_2\left(t\right),\hfill \end{array}\hfill \\ \tilde{Y}\left(0\right)=w^2\left(0\right)+g^2\left(0\right).\hfill \end{array}$$

Since $\tilde{V}\left(0\right)=\tilde{Y}\left(0\right)$ for $t\le 0$, the stochastic comparing theorem implies that

$$\begin{array}{c}\tilde{V}\left(t\right)\le \tilde{Y}\left(t\right).\hfill \end{array}$$

(12)

Utilizing the variational formula of constants, we obtain

$$\begin{array}{c}\tilde{Y}\left(t\right)=e^{ct}\left[\tilde{Y}\left(0\right)+\frac{b^2}{c}\right]-\frac{b^2}{c}+M\left(s\right),0<s\le t\in [0,T],\hfill \end{array}$$

(13)

where

$$\begin{array}{c}M\left(s\right)=2{\sigma }_1{\int }_0^t{e}^{c(t-s)}{w}^2\left(s\right)\mathrm{d}{B}_1\left(s\right)+2{\sigma }_2{\int }_0^t{e}^{c(t-s)}{g}^2\left(s\right)\mathrm{d}{B}_2\left(s\right).\hfill \end{array}$$

Case 1: $c\le 0$. Since $M\left(s\right)$ is a local martingale, taking the expectation of both sides of Equation (13), we have

$$\begin{array}{c}\mathbb{E}\tilde{Y}\left(t\right)=e^{ct}\left[\tilde{Y}\left(0\right)+\frac{b^2}{c}\right]-\frac{b^2}{c}\le e^{ct}\tilde{Y}\left(0\right)-\frac{b^2}{c}\le e^{ct}{N}_1-\frac{b^2}{c}.\hfill \end{array}$$

By using condition (i) of Case 1, we obtain $\mathbb{E}\tilde{V}\left(t\right)\le N_2$, namely, $\mathbb{E}[w^2\left(t\right)+g^2\left(t\right)]\le N_2$. Thus, system (9) is finite-time stable with respect to $(N_1,N_2,T)$. Furthermore, based on Case 1 (ii), we obtain $\mathbb{E}\tilde{V}\left(t\right)\le N_0$ for any $t\in [T-\eta ,T]$, which yields $\mathbb{E}[w^2\left(t\right)+g^2\left(t\right)]\le N_0$. Hence, system (9) is finite-time contractively stable with respect to $(N_1,N_2,N_0,\eta ,T)$.

Case 2: $c>0$. Taking the expectation of both sides of Equation (13), we obtain

$$\begin{array}{c}\mathbb{E}\tilde{Y}\left(t\right)\le e^{ct}\left[\tilde{Y}\left(0\right)+\frac{b^2}{c}\right]-\frac{b^2}{c}\le e^{ct}\left[N_1+\frac{b^2}{c}\right].\hfill \end{array}$$

On the basis of condition (i) of Case 2, we have $\mathbb{E}[w^2\left(t\right)+g^2\left(t\right)]\le N_2$. Namely, system (9) is finite-time stable with respect to $(T,N_1,N_2)$. Moreover, according to Case 2 (ii), we obtain $\mathbb{E}\tilde{V}\left(t\right)\le N_0$ for any $t\in [T-\eta ,T]$, which means $\mathbb{E}[w^2\left(t\right)+g^2\left(t\right)]\le N_0$. Hence, system (9) is finite-time contractively stable with respect to $(N_1,N_2,N_0,\eta ,T)$. □

Remark 1.

Theorem 4 presents some sufficient conditions associated with the prespecified bounds (i.e., $N_1,N_2$, and $N_0$) for FTS and FTCS of system (9) in the framework of the Lyapunov method. Obviously, FTCS required stronger conditions (the conditions are related to noise intensity) than FTS, which also indicates that the specified bounds and the noise intensity play a critical role in the FTCS analysis.

4. Optimal Control for Stochastic Mosquito Population Model by Proportional Releases

As in [29], in order to investigate the optimal control strategy for releasing sterile mosquitoes in an area where the population size of wild mosquitoes is relatively small, we can use a parallel releasing policy, i.e., the rate of release is proportional to the population size of the wild mosquitoes such that $\mathbb{B}(·)=ug$, where u is a constant. If we want to seek a cost-effective release strategy that minimizes the release of Wolbachia-infected mosquitoes with the intended effect of reducing the wild mosquito population to a minimum and achieving optimal control of wild mosquitoes, we need to set a control variable. Let u be the control variable related to time t, and apply the following control equations:

$$\{\begin{array}{c}\begin{array}{c}\mathrm{d}w\left(t\right)=[\frac{aw\left(t\right)}{w\left(t\right)+g\left(t\right)}(1-\xi w\left(t\right))w\left(t\right)-\mu w\left(t\right)]\mathrm{d}t+{\sigma }_{1}w\left(t\right)\mathrm{d}{B}_1\left(t\right),\hfill \\ \mathrm{d}g\left(t\right)=[u\left(t\right)g\left(t\right)-\mu g\left(t\right)]\mathrm{d}t+{\sigma }_{2}g\left(t\right)\mathrm{d}{B}_2\left(t\right),\hfill \end{array}\hfill \end{array}$$

(14)

where $u\left(t\right)$ is given in the control set $\mathcal{U}[0,T]$ defined by

$$\begin{array}{c}\mathcal{U}[0,T]=\left\{u\left(t\right):u(·)\mathrm{is}\mathrm{measurable}\mathrm{and}{\left\{{\mathcal{F}}_t\right\}}_{t\ge 0}-\mathrm{adapted},t\in [0,T]\right\}.\hfill \end{array}$$

(15)

Define the objective functional as

$$\begin{array}{c}J\left(u(·)\right)=\mathbb{E}\left\{{\int }_0^{T}f\left(w\left(t\right),u\left(t\right)\right)\mathrm{d}t+h\left(w\left(T\right)\right)\right\},\hfill \end{array}$$

(16)

where $f\left(w\left(t\right),u\left(t\right)\right)$ denotes the cost at moment t and $h\left(w\right(T\left)\right)=hw\left(T\right)$ is the expected wild mosquito population that exists at terminal T. We let

$$\begin{array}{cc}f\left(w\left(t\right),u\left(t\right)\right)=\hfill & A_{1}w\left(t\right)+\frac{A_2}{2}{u}^2\left(t\right),\hfill \end{array}$$

where the positive parameters $A_1$ and $A_2$ are weight constants of the wild mosquitoes and the control strategy, respectively. The meaning of the objective functional J is described as follows:

• The term ${\int }_0^T{A}_{1}w\left(t\right)\mathrm{d}t$ denotes the total number of wild mosquito populations over time T.
• The term ${\int }_0^T\frac{A_2}{2}{u}^2\left(t\right)\mathrm{d}t$ shows the total cost of releasing Wolbachia-infected mosquitoes.

Our goal is to seek an optimal control $u^{*}$ such that

$$\begin{array}{c}J\left(u^{*}(·)\right)=min\{J\left(u\left(t\right)\right):u\left(t\right)\in \mathcal{U}[0,T]\}.\hfill \end{array}$$

(17)

Due to the convexity of the objective functional $J\left(u\right(·\left)\right)$ with respect to the control variable and the regularity of system (14), the existence of the optimal control can be derived (see [30,31]), and detailed derivation is not thus given.

Utilizing the maximum principle [31], if $u^{*}$ is optimal for control problem (17) with fixed final time T, then there exist adjoint vectors $p\left(t\right)=(p_1\left(t\right),p_2\left(t\right))$ and $q\left(t\right)=$$(q_1\left(t\right),q_2\left(t\right))$ such that the following adjoint equations hold:

$$\begin{array}{cc}\mathrm{d}{p}_1\left(t\right)=\hfill & \left[p_1\left(t\right)(\mu -\frac{a{w}^2(1-2\xi w)+awg(2-3\xi w)}{{(w+g)}^2})-{\sigma }_1{q}_1\left(t\right)+A_1\right]\mathrm{d}t+q_1\left(t\right)\mathrm{d}{B}_1\left(t\right),\hfill \\ \mathrm{d}{p}_2\left(t\right)=\hfill & \left[p_2\left(t\right)\left(\mu -u\right)-{\sigma }_2{q}_2\left(t\right)\right]\mathrm{d}t+q_2\left(t\right)\mathrm{d}{B}_2\left(t\right),t\in [0,T],\hfill \end{array}$$

(18)

with the transversality condition

$$\begin{array}{c}{p}_1\left(T\right)=h_w\left(w\left(T\right)\right)=-h,p_2\left(T\right)=0.\hfill \end{array}$$

The following theorem presents the necessary optimality condition associated with the control problem (16).

Theorem 5.

Assume the optimal control problem (17) with fixed final time T admits a unique optimal solution $(w^{*},g^{*})$ with respect to an optimal control $u^{*}$ for $t\in [0,T]$. Then, there exist adjoint variables $p_i^{*},q_i^{*}$ (for $i=1,2$) that satisfy adjoint Equation (18) with the following transversality condition:

$$\begin{array}{c}{p}_1^{*}\left(T\right)=-h,p_2^{*}\left(T\right)=0.\hfill \end{array}$$

Furthermore, the corresponding optimal controls are given as follows:

$$\begin{array}{c}{u}^{*}=max\left\{0,\frac{p_2^{*}{g}^{*}}{A_2}\right\}.\hfill \end{array}$$

(19)

Proof. 

The Hamiltonian associated with the control problem (17) is

$$\begin{array}{cc}& H(t,w(t),g(t),u(t),p(t),q(t\left)\right)\hfill \\ \hfill =& p_1\left(t\right)\left[\frac{aw\left(t\right)}{w\left(t\right)+g\left(t\right)}(1-\xi w\left(t\right))w\left(t\right)-\mu w\left(t\right)\right]+p_2\left(t\right)\left[u\left(t\right)g\left(t\right)-\mu g\left(t\right)\right]\hfill \\ & -A_{1}w\left(t\right)-\frac{A_2}{2}{u}^2\left(t\right)+q_1\left(t\right){\sigma }_{1}w\left(t\right)+q_2\left(t\right){\sigma }_{2}g\left(t\right).\hfill \end{array}$$

Based on the stochastic maximum principle [31], there exist adjoint vectors $p\left(t\right)=(p_1\left(t\right),p_2\left(t\right))$ and $q\left(t\right)=(q_1\left(t\right),q_2\left(t\right))$ that satisfy the following equations:

$$\begin{array}{cc}\hfill & \mathrm{d}{p}_1\left(t\right)=-H_w(t,w\left(t\right),g\left(t\right),u\left(t\right),p\left(t\right),q\left(t\right))\mathrm{d}t+q_1\left(t\right)\mathrm{d}{B}_1\left(t\right),\hfill \\ \hfill & \mathrm{d}{q}_2\left(t\right)=-H_g(t,w\left(t\right),g\left(t\right),u\left(t\right),p\left(t\right),q\left(t\right))\mathrm{d}t+q_2\left(t\right)\mathrm{d}{B}_2\left(t\right),\hfill \end{array}$$

with the transversality condition

$$p_1\left(T\right)=-h,p_2\left(T\right)=0.$$

Thus, we obtain

$$\begin{array}{c}{u}^{*}=\frac{p_2^{*}{g}^{*}}{A_2}.\hfill \end{array}$$

From the define of control set (15), we have

$$\begin{array}{c}{u}^{*}=\{\begin{array}{cc}0,\hfill & \mathrm{if}\frac{p_2^{*}{g}^{*}}{A_2}\le 0,\hfill \\ \frac{p_2^{*}{g}^{*}}{A_2},\hfill & \mathrm{if}\frac{p_2^{*}{g}^{*}}{A_2}>0.\hfill \end{array}\hfill \end{array}$$

Hence, they can be equivalently written as (19). This proof is completed. □

5. Concluding Remarks

Compared to long-term stability, the control time of FTCS is limited. However, it has the characteristics of fast implementation speed, strong operability, and more applicability in practical applications. This paper presents sufficient conditions for FTCS of systems under a constant release strategy, including the delay mosquito model and the stochastic model driven by noise. All these conditions demonstrate the influences of time delay and environmental factors on FTCS. Additionally, the optimal control strategies for the corresponding stochastic mosquito model under proportional releases are provided. The waiting period from mating to emergence varies over time due to climate change, making FTS of time-varying systems an intriguing topic. Future studies on FTS for stochastic reaction–diffusion systems will continue to capture our attention.