Finite-time contraction stability of a stochastic reaction-di ff usion dengue model with impulse and Markov switching

: From the perspective of prevention and treatment of dengue, it is important to minimize the number of infections within a limited time frame. That is, the study of finite time contraction stability (FTCS) of dengue system is a meaningful topic. This article proposes a dengue epidemic model with reaction-di ff usion, impulse and Markov switching. By constructing an equivalent system, the well-posedness of the positive solution is proved. The main result is that su ffi cient conditions to guarantee the finite time contraction stability of the dengue model are acquired based on the average pulse interval method and the bounded pulse interval method. Furthermore, the numerical findings indicate the influences of impulse, control strategies and noise intensity on the FTCS.


Introduction
Dengue fever is one of the most dangerous diseases spread by mosquitoes around the world. According to the statistics from the WHO, almost 50% of people on earth reside in areas at risk of dengue. Each year, up to 400 million people get infected with dengue, of which 100 million exhibit clinical symptoms of varying severity, resulting in about 40,000 deaths [1]. In recent years, the incidence of dengue fever has increased dramatically, from 505,430 cases in 2000 to 4.2 million cases in 2019 [2]. As a result, dengue fever has seriously endangered people's safety and hindered economic growth. Understanding the pathogenesis of dengue is essential for early intervention.
Mathematical modeling has become an extremely effective method to understand the mechanisms of dengue transmission and predict the development trend of dengue. Recently, various dengue models considering different factors have been widely proposed [3][4][5][6]. For example, Zhu and Xu [5] devel-oped a dengue model that takes into account both spatial heterogeneity and temporal periodicity to discuss the asymptotic stability of periodic solutions. In [4], Li et al. studied a study on the optimal control problem of a dengue system that incorporates reaction-diffusion. Unfortunately, the above models ignored the effect of spraying insecticides and sterile mosquitoes for dengue. These human behaviors can be characterized by impulsive differential equations. So far, some results have been obtained for modeling impulsive dengue models [7][8][9][10]. For instance, Pang et al. [7] investigated a stage-structured impulsive model to explore the feasibility of controlling wild mosquito populations by periodically releasing sterile mosquitoes. Yang and Nie [8] conducted a dengue model that involved culling mosquitoes to investigate the effect of the impulsive strategies. However, they did not discuss the influence of noise.
In fact, it is widely recognized that external environmental factors have a significant impact on the spread of dengue fever. The use of random differential equations driven by white noise to construct dengue models have attracted widespread attention [11][12][13][14]. For example, Liu and Din [11] introduced the random disturbance and information intervention factors into the dengue model to analyze the stability of the positive solution. Chang et al. [12] established a stochastic dengue model related to white noise to investigate the stationary distribution and optimal control. However, white noise can only be applied to continuous random disturbances and cannot describe sudden changes in the environment. Dengue may be impacted by unforeseen weather changes, like temperature and rainfall, which will lead to switching from one environmental system to another. Frequently, the switch occurs without memory and the waiting period until the next transition is determined by the exponential distrnibution [15]. Therefore, some scholars have introduced continuous-time Markov chains to describe the regime switch [16][17][18][19]. Nevertheless, Markov switching is hardly considered in the dengue models, only in [16] Liu et al. analyzed the stationary distribution of a stochastic dengue model considering immune responses and Markov switching.
It is worth noting that the existing dengue models only consider the long-term dynamical behavior. When addressing the concern of controlling dengue, it makes more sense to focus on limiting the number of infected individuals within a specific time frame to reduce within a specified threshold and eventually make it lower than the original number. This dynamic property is referred to as finitetime contraction stability (FTCS). As far as we know, there have been intriguing findings concerning FTCS [20][21][22][23]. However, no studies have analyzed the finite-time contraction stability of dengue.
In view of the above discussion, a stochastic dengue epidemic model with reaction-diffusion, impulse and Markov switching is constructed. Compared with the latest research results on the dengue model that only consider the long-term behaviors of the system [24,25], our results not only consider FTCS of dengue model for the first time but also investigate the effects of control strategies, impulse and noise intensity on FTCS. The coexistence of diffusion, pulse and Markov switching makes the model complex, which brings challenges to analyze the stability of the system by Lyapunov stability theory. Moreover, it is difficult to get a suitable range of control variables for the system to achieve FTCS. To solve these problems, we use the average and bounded pulse interval method combined with the comparison method to effectively obtain sufficient conditions for FTCS. Especially, the highlights are listed as follows: • Considering the use of pesticides and the release of sterile mosquitoes, as well as parameters uncertainties, a hybrid impulse dengue model with diffusion and Markov switching is proposed, which is an extension of the model in [12].
• Less conservative sufficient conditions for FTCS of the stochastic dengue system are presented based on two impulsive representations. In addition, the effects of control strategies, impulse and noise intensity on FTCS are also analyzed.
The remainder of this document is structured in the following manner: In Section 2, Some preliminary preparations are introduced. According to an equivalent system, Section 3 yields the wellposedness of positive solution of the system. In Section 4, the average and bounded pulse interval method are used to provide the criteria to ensure FTCS. Section 5 gives various numerical simulations to effectively illustrate the theoretical discoveries, as well as to display the impacts of impulse, control strategies and noise intensity on FTCS. The last section gives the conclusion of this paper and discusses further work.

Model development and preliminary preparation
For the purpose of subsequent analysis, a reaction-diffusion dengue model with impulse and Markov switching is established and give some notations.

Model development
The rapid development of transportation has created convenient conditions for the spread of mosquitoes and people, which increases the risk of contracting the virus. Considering the existence of spatial diffusion, Chang et al. [12] proposed the following system: (2.2) and boundary condition Below is an explanation of the variables used in the equation. At the same time, there is evidence that the vector control techniques (e.g., the development of Wolbachia-infected mosquitoes, the use of insecticides and so on) are effective ways to avoid the spread of dengue fever [26,27]. Therefore, we consider the following control measures on the basis of system (2.1).
(i) We define ∅ 1k and ∅ 2k as the impulse intensities that affect susceptible mosquitoes and infected mosquitoes, respectively. Among them, ∅ 1k > 0 indicates the release of sterile mosquitoes, and ∅ 1k < 0, ∅ 2k > 0 indicates the killing of mosquitoes. Because of the need to maintain ecological stability, mosquitoes cannot be completely killed. Thus, we have reason to assume that −1 < ∅ 1k < ∅ 1m , 0 < ∅ 2k < ∅ 2m , where ∅ 1m and ∅ 2m are the maximum allowable impulse on susceptible mosquitoes and infected mosquitoes, respectively.
(ii) Treating infected individuals and spraying insecticides on mosquitoes can reduce the number of infected. δ 1 is the recovery rate of dengue infection individuals due to treatment, δ 2 is the culling rate of mosquitoes, δ 1 (x)I H (x, t) indicates the proportion of infected individuals recovered and δ 2 (x)S v (x, t), δ 2 (x)I v (x, t) represents the proportion of mosquitoes eliminated.
Based on the above discussions, we tend to unintentionally introduce pulse perturbations into system (2.1), and obtain the following equations with control variables: (2.4) which with initial values (2.2) and boundary condition (2.3). We set the admissible control set Impulsive sequence {t k }, k ∈ N + , represents the spraying of pesticides and the release of sterile mosquitoes at time point t = t k , satisfies 0 = t 0 < t 1 < t 2 < . . . < t k < . . ., as well as lim k→+∞ t k = +∞ and z(x, t + k ) = lim t→t + k z(x, t).
In real life, it is widely understood that there are a variety of environmental factors (for instance rainfall, temperature change and so on) that can affect the spread of dengue fever. Assume that the transmission rate β H and β v are subject to stochastic fluctuations. That is where B i (t)(i = 1, 2) is standard Brownian movements, and ρ i is its intensity. Furthermore, these sudden environmental changes can also cause the parameters in the system to not be fixed constants. These parameters are assumed to be a stochastic process that satisfies Markov switching. Let {ζ t } t≥0 be a right-continuous Markov chain taking values in a finite state S = {1, 2, ..., M}. Its generator o = (q i j ) M×M is given by the following formula.
Then, system (2.4) can be written as: (2.5) Next, in order to study subsequent conclusions, the following preparatory knowledge needs to be given.

Preliminaries
Assign + as a group of functions that are both bounded and continuous. (Ω, F , {F t } 0≤t≤T , P) be a complete probability space equipped with a filtration {F t } 0≤t≤T satisfying the usual conditions (i.e., it is right continuous and F 0 contains all P-null sets). E denotes the probability expectation corresponding to P. Let z( . These functions are equipped with a norm denoted by ∥ · ∥. For any function z(x, t) in M + , the norm is defined Before continuing, let's first determine the following symbols:

Well-posedness
To start with, using a method similar to [28], to prove the well-posedness of global positive solution of system (2.5), it is imperative that we first scrutinize the ensuing system: and boundary condition And, h i (t), i = 1, 2, are left continuous can be expressed as follows: In fact, the relationship between system (2.5) and system (3.1) has the following lemma. Proof. Denote It is easy to get that Moreover, for t k ≥ 0, , t), we can follow the same process as we did for S v (x, t).
It can be shown that the well-posedness of positive solution for the system (3.1) by proving system (2.5) has a unique positive solution. The required lemma is provided below. The proof procedure is omitted, the interested reader can refer to [29]. Based on this Lemma 3.2, the following theorem holds.
The proof is given in the Appendix. This is the basis of the whole paper, which makes the subsequent analysis meaningful.

Finite-time contraction stability
Before giving sufficient conditions to ensure FTCS, the following necessary lemma is introduced.
Lemma 4.1. [30] Assume that there exist positive constants g r , r = 1, 2, . . . , s, such that x ∈ Γ and |x r | < g r . Given a function z(x) ∈ R n which belongs to C 2 (Γ) and vanishes on ∂Γ, has Furthermore, to facilitate subsequent studies, it is necessary to give the following definitions. Moreover, For any T ≥ t ≥ 0, the average impulsive interval of the sequence t k (k ∈ N + ) on (t, T ] is l, when N 0 ∈ N + and l ∈ N, there is where n(T, t) denotes the number of pulse moments.
Remark 4.4. Note that for a given pulse sequence, N 0 is not unique. As N 0 increases, the number of impulsive instant sequences it contains also increases. In particular, for the case N 0 = 1, it is required that there be no less than one pulse control input on each interval of l.
Definition 4.5. [22] For any t ≥ s, t, s ∈ [0, T ], it can be concluded that the pulse number function n(t, s) of the bounded pulse interval satisfies where B(·) and ζ(·) are the d-dimensional Brownian motion and the right continuous Markov chain, respectively, and f (·, ·) : R n × S → R n , g(·, ·) : where g : R + × R + → R is continuous, lim (t,y)→(t + k ,x) g(t, y) = g t + k , x exists and Ψ k : R + → R + is nondecreasing. Let u(t) be the maximal solution of system (4.3) For any t ≥ t 0 , V(t 0 , x 0 ) ≤ u 0 is established, then V(t, x(t)) ≤ u(t).
Remark 4.7. When dengue fever occurs, some parameters are known, such as the use of pesticides and the intensity and frequency of release of sterile mosquitoes. According to the result of this paper (Theorem 4.6), it can be judged whether the spread of dengue fever can be controlled within a limited time, which has a certain guiding role in the prevention of the disease. Moreover, it can be seen from the Theorem 4.6 and Corollary 4.1 that if the system is FTCS then the system must be FTS.

Finite-time contractive stability with bounded pulse interval
In this part, based on the bounded pulse interval method, a criterion for FTCS about (B 1 , B 2 , B 3 , ω, T ) is given. FTCS about (B 1 , B 2 , B 3 , ω, T ), when φ ≥ 1, if one of the following conditions is satisfied where φ, r 1 and r 2 have the same definitions as in Theorem 4.6.
Similarly, the FTS of the bounded pulse interval method can be deduced from (4.1). , r 2 + ln φ l m }.
Remark 4.9. From Definition 4.5, it can be seen that the impulsive sequence is related to the minimum impulsive interval l m . Then, combined with Theorem 4.8, it is known that for an unstable impulse effect, i.e., φ ≥ 1, when other parameters remain the same, the larger minimal impulsive interval l m (fewer pulses) means that finite-time contraction stability is easier to achieve. Table 1. Description of parameters in system (2.5).

Parameters Values Parameters Values Parameters Values Parameters Values
This section aims to demonstrate the validity of the theoretical findings we obtained in the previous sections. We assume that r  Figure 1). The parameters of system (2.5) for the two states are given in Table 1 as in Refs. [29,33,34].

Analysis of FTCS
In this part, the FTCS for system (2.5) are explained through two examples. = 0.526, this implies that C2 in Theorem 4.6 holds. Hence, system (2.5) is FTCS w.r.t (6.5,7,5.5,1,3). Figure 2 displays the initial concentrations of populations are less than B 1 and not more than B 2 in [0, T ] and reach B 3 in [T − ω, T ]. Namely, before reaching the terminal time, the system can reach a certain threshold which is smaller than the initial threshold, which means that the spread of dengue can be controlled effectively within a limited period of time. = 0.462 and r 2 + ln φ l = 0.526, this implies that C ′ 2 in Theorem 4.8 holds. Thus, system is FTCS w.r.t (6.5,7,5.5,1,3) (see Figure 3).  Figure 3. The trajectories of system (2.5) with initial value (8, 5, 0, 4.5, 4) with control and the bounded impulsive interval.

The influence of pulse
To investigate the impact of pulse on (2.5), the same parameter values shown in Figure 2 are used and let I ik = 0. By simple calculation, it can be found that all conditions of Theorem 4.8 are not satisfied, i.e., system (2.5) about (6.5,7,5.5,1,3) is not FTCS, as shown in Figure 4. For this case, compared to the conditions of Figure 2, we just alter the impulsive condition, but discover that the system cannot be FTCS without impulsive effects. It shows that under the same circumstances, impulsive perturbation is important to the system's stability for a finite period.

The effect of the control variables
According to Theorem 4.6 and Theorem 4.8, it can be noticed that the values of control variables are key to the FTS and FTCS of system (2.5). The following discussion is divided into two cases. Case 1: In this part, we can see by Figure 5 that if without controls (i.e., δ 1 = δ 2 = 0 and other parameters of the system are as same as in Figure 2), then it is neither FTS w.r.t (6.5, 7, 3) nor FTCS w.r.t (6.5, 7, 5.5, 1, 3).
Case 2: We choose the same parameters as in Figure 2 except δ 1 (1) = 0.2, δ 2 (1) = 0.3, δ 1 (2) = 0.1, δ 2 (2) = 0.2. Then system is FTS about (6.5, 7, 3) and not FTCS about (6.5, 7, 5.5, 1, 3) (see Figure  6). Case 1 implied that system (2.5) itself is impossible to be stable in a finite period of time without control. Case 2 illustrates that if the control intensity is low, dengue fever cannot be finally controlled at a lower level (below the initial value). It can be observed that the stochastic dengue epidemic model can be FTCS only by choosing appropriate values of control variables.

Conclusions and discussion
This article investigated the FTCS of a reaction-diffusion dengue model with impulse and Markov switching. Sufficient conditions with respect to control variable for finite-time contraction stability are obtained via two representations of pulse sequences (i.e., the average pulse interval and bounded pulse interval) make use of the Lyapunov functional method and inequality techniques. All these conditions show the impacts of environmental noise intensity (ρ 1 , ρ 2 ) and impulsive factor (φ, N 0 , l, l m ) on the FTCS. FTCS can be realized only when appropriate control measures are selected, that is, the spread of dengue fever can be effectively controlled within a limited time. Moreover, delay [35][36][37][38] is also a widespread phenomenon that can lead to dramatic changes in dynamic behavior. It is challenging to consider the impact of time delays on FTCS, which will also be our future research direction.

Use of AI tools declaration
The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.
Appendix A: The proof of Theorem 3.1 Proof. Due to the coefficients of (3.1) are locally Lipschitz continuous, for (3.2), system (3.1) admits a unique solution y(x, t) on [0, τ e ), where τ e is the explosion time.