In order to solve the problem that the least squares estimation of the matrix autoregressive model (MAR) is easily affected by outliers or thick-tailed distribution errors and deviates from the true value, this paper improves the loss function. Based on the projection method and iterative least squares method, RE-PROJ and RE-ILS methods are proposed to solve the robust estimation of the MAR model. Both methods use the idea of M estimation in the process of constructing estimators, and the simulation data show that the above two methods can better resist the influence of outliers on parameter estimation. This paper further discusses the BIC criterion for order selection of the MAR(p) model and a robust estimation method with the assumption of rank reduction when the matrix observations are correlated. The actual analysis results show that the robust estimation proposed in this paper has more advantages in fitting and prediction than the least squares estimation in the matrix-valued observation data with many outliers or sharp peaks and thick tails.

We build an estimator of calendar effect in market microstructure noise using high-frequency financial data. The associated asymptotic theory and feasible inference are established. This is, to our knowledge, the first (feasible) asymptotic theory ever built for the estimation of calendar effect in market microstructure noise. This research may lead to a better understanding of market microstructure noise and hence the market mechanism.

BAR (broker adaptive ridge) is a new method to replace $L_0$ penalty regression. Based on a reweighted $L_2$ penalty, BAR penalty combines the advantages of two different penalties, $L_0$ and $L_2$, and avoids the shortcomings of separately using these two penalties. In this paper, the BAR method is extended to linear regression model with robust loss function, and the coordinate descent algorithm is used to estimate the parameters. To characterize the robustness of the proposed method, we provide an influence function for robust BAR estimation. Under appropriate conditions, we theoretically establish the variable selection consistency and Oracle property of BAR estimation, and provide proof. We compared the proposed method with other existing methods through numerical simulation and analysis of real data, further verifying that the new method is more effective in terms of robustness and variable selection performance.

In this paper, we explain the solution of the non-autonomous stochastic delay lattice equation driven by white noise generating a continuous cocycle. Uniform estimation and tail estimation are performed, and it is proved that the system has a $ \mathcal{D}$-pullback random attractor. We prove the double upper semicontinuity of random attractors for delay system when the delay and a certain parameter converge simultaneously.

For the aim of exploring the influencing factors of periodic outbreaks of spruce budworm, we proposed a delayed diffusive spruce budworm model with Holling II predation function under the homogeneous Neumann boundary condition. As is known to all, the evolution of many populations is relate with not only the present situation, but also with the past situation. This implies that it is essential to take over the effect of delay on reflecting the phenomena. For this aim, choosing the delay as bifurcating parameter, we investigated the stability of the positive equilibrium and the existence of Hopf bifurcation deduced by single delay or two delays by using the characteristic equation and mathematical analysis skills. Finally, through performing numerical simulations, we obtained the stable periodic solutions of this model, which provided the theoretical basis for periodic outbreaks of spruce budworm. Moreover, the numerical results indicated that the critical value of Hopf bifurcation deduced by two delays is smaller than one of Hopf bifurcation deduced by single delay.

Mathematical analysis of pharmacokinetic models plays an important role in drug researches. In this paper, considering a one-compartment pharmacokinetic model with parallel first-order and Hill (n = 2) elimination under different dosage designs with periodic intravenous bolus administrations, we have mathematically studied the steady-state pharmacokinetics. As a result, we have proved that the pharmacokinetic model, represented by an impulsive differential equation, admits a unique steady-state periodic solution. Then we have derived the analytical formulas for two important pharmacokinetic indexes: steady-state drug exposure and steady-state average plasma concentration. Moreover, different to the existing the pharmacokinetic model with parallel first-order and Michaelis elimination pathways, we have been able to discover, both numerically and theoretically, the diversity in the steady-state average plasma concentration for different dose regimens. That is, by increasing the dosing frequency, three circumstances can occur for the steady-state average drug concentration: (i) monotonically decreases and converges to a limit value; (ii) monotonically increases and converges to a limit value; and (iii) decreases first and then increases and eventually converges to a limit value. Finally, we have applied the results to a real drug model of recombinant granulocyte colony-stimulating factor (Filgrastim), for which we have provided the analytical formulas of the steady-state average plasma concentrations using different dose regimens and calculated the steady-state average plasma concentration and minimum plasma concentration.

The SEIAV infectious disease model containing factors such as incubation period, saturation cure, secondary infection, transmission of free virus was developed by combining the effect of spatial and temporal diffusion and the incidence of Holling-IV in the process of infectious disease transmission. The stability of the disease-free equilibrium for R₀<1 and the persistence of the endemic disease equilibrium point for R₀>1 are obtained, and the global attractiveness of the disease-free equilibrium point of the model for R₀=1 is further discussed. The fitness of the model is proved using operator semigroup theory, and the eigenvalue problem is also constructed to give a general calculation of the basic regeneration number using the property of the existence of principal eigenvalues. Finally, the effect of spatial diffusion on the spread of the infectious disease model is verified by numerical simulations.

In this paper, we consider the dynamics of a second-order rational difference equation. Through the linear stability analysis, the parameter conditions are gained to guarantee the existence and stability of the equilibrium. By applying the center manifold theorem, the Normal Form theory and the bifurcation theory, we derive the critical parameter values of the fold bifurcation, transcritical bifurcation, flip bifurcation and Neimark-Sacker bifurcation respectively. In order to identify chaos among regular behaviors, we calculate the maximum Lyapunov exponents and fractal dimensions. These results obtained in this paper are illustrated by numerical simulations. From the simulations, we can see some complex dynamic behaviors, such as period doubling cascade, periodic windows, limit cycles, chaotic behaviors and so on. Interestingly, with the selection of parameters, the dynamic behaviors of the system are completely symmetric.

The Oscillation of a class of second order nonlinear neutral delay differential equation with damping term is studied in this paper. By using the generalized Riccati transformation technique and some special techniques, some new oscillation criteria for the equation are obtained in the noncanonical. The oscillation results in several recent literatures are improved, generalized and unified. Finally, some examples demonstrate the applicability of the results of this paper.

This article proposes a hypothesis testing method to detect serial correlation and ARCH effect in high-dimensional time series based on $L_{2}$ norm and Spearman's correlation. In this article, We study the asymptotic behavior of our test statistic and provide a bootstrap-based approach to generate critical values, we prove our test can control Type-I errors. Our test is dimensional-free, which means it is independent of the dimension of the data, hence our test can be used for high dimensional time series data. Our test does not require tail properties of data, hence it can be used for heavy-tailed time series. The simulation results indicate that our new test performs well in both empirical sizes and powers and outperforms other tests. The practical usefulness of our test is illustrated via simulation and a real data analysis.

This work is concerned with a stochastic evolutionary equation with a multiplicative noise. Verifying the convergence from the solution of the stochastic evolutionary equation to one of its Wong-Zakai approximation and applying the exponential martingale argument, the Kallianpur-Striebel formula and Itô formula, we prove that the nonlinear filter generated by the stochastic evolutionary equation converges to one generated by its Wong-Zakai approximation in the observation system with colored noise.

In this paper, we extend the network oligopoly with finitely many markets to the model with infinitely many markets, and propose the network oligopoly model with continuous-variable markets. Under the fully noncooperative hypothesis, we first prove the existence of Cournot-Nash equilibria. Furthermore, we assume that there exists a partition of the set of firms. By defining the coalitional cost function and coalitional profit function of every coalition, we establish a noncooperative game basing the coalition structure, and prove the existence of Nash equilibria. New network oligopoly models and Nash equilibrium existence theorems are our main contributions.

Malaria is a vector-borne disease caused by a pathogen. In order to study the multiple effects of spatial heterogeneity, vector-bias effect, and seasonality on disease transmission, this study proposes a temporal periodic reaction-diffusion model for malaria transmission. Firstly, the basic reproduction number ($R_{0}$) of the model is introduced. Then it is proved that if $R_{0}\leq1$, the disease-free periodic solution is globally asymptotically stable; while if $R_{0}>1$, the system possesses a globally asymptotically stable positive periodic solution. These proofs utilize the monotone dynamical system theorem, the theory of periodic semiflows, and the chain transitivity theory. Numerical studies on malaria transmission in Maputo Province, Mozambique are carried out to validate the theoretical analysis results. The impact of key parameters in the model is discussed, and it is concluded that neglecting the diffusion of human and mosquito populations and the vector-bias effect underestimates the risk of disease transmission. In addition, the impact of medical resources on disease transmission is analyzed from two aspects: quantity and distribution. It is found that increasing medical resources would reduce the risk of disease transmission. If medical resources are fixed, reducing the variability in the distribution of medical resources would also decrease the risk of disease transmission.

In this paper, we study the dynamics of a predator-prey system with delay of fear effect. First we discuss the positivity, boundedness and permanence of the solution, then base on center manifold theorem and normal form, we study Hopf bifurcation caused by delay of fear effect. Furthermore, we discuss the global existence of bifurcated periodic solutions. At last, some simulations are given to support our results.

This paper studies the equilibrium strategy of a fluid queue with breakdowns and working vacations in the fully observable case, where the buffer switches alternately in the busy period, working vacation period, and fault maintenance period. Based on the renewal process and the standard theory of linear ordinary differential equations, the Nash equilibrium behavior of the fluid and the steady-state probability distribution are obtained, then the expected buffer content can be derived by using the classical Laplace-Stieltjes Transform (LST) method. Based on the economics and utility theory, the expected social benefit function is constructed reasonably and the global balking thresholds that maximize the social welfare can be obtained by some numerical examples. The admission fee revenue model can be constructed with the globally optimal thresholds and admission fees as joint decision variables, and the effect of the global optimal thresholds on the maximal admission fee revenue can be illustrated. The parameter optimization strategies and social benefit problems are crucial for improving the secure transmission performance of cognitive wireless networks, where the nodes fail and become semi-dormant. Simulation results can provide a theoretical basis for the optimal allocation of limited wireless network resources.

We consider the existence of weak solutions to a class of elliptic equations with nonstandard growth condition and zero order term. The main tools are the weak convergence method for PDEs and Young measure method. The perturbed problem is treated as a starting point. In view of the assumptions on the zero order term and the integrability of the source term, we pick up some appropriate test functions, which are vital to some necessary a priori estimates and the limit process. Thanks to the Young measure method, the weak limit of the nonlinear term is identified. The existence of a weak solution is obtained by passage of the limit.

Based on the spatial heterogeneity, the asymptomatic hosts and the multiplicity of pathogen transmission routes, a model of reactive diffusion host-pathogen with asymptomatic hosts and multiple infection routes is proposed, which is discussed the existence and uniqueness of the global positive solution of the model by using semigroup theory. Further, according to the spectral radius method of the next generation operator, the basic reproduction number $\mathcal{R}_{0}$ of the model is given, and the extinction and persistence of the disease are described. That is, if $\mathcal{R}_0<1$, the disease-free steady state is globally asymptotically stable; while if $\mathcal{R}_0>1$, the disease is uniformly persistent and the model admits at least one endemic steady state. In addition, the global asymptotic stability of the disease-free and endemic equilibrium states of the model in a spatially homogeneous environment is obtained by constructing suitable Lyapunov functions. Finally, some numerical simulations are conducted to explain the main theoretical results and to explore the influence of diffusion rates on the distribution of infected hosts.

In order to handle the coexistence of spatial heterogeneity and spatial correlation in spatial data, This paper considers a class of error spatial autocorrelation ~GWR~ models, This model is considered an effective fusion of classical GWR model and spatial error model.~the existing estimation methods of the GWR model cannot obtain a consistent estimate of the parameters, due to the endogeneity problem caused by the spatial lag of the error term in the model. Therefore this article propose a new estimation method for the model by combines the ideas of local linear estimation and profile least square estimation metod.Discussed the asymptotic properties of estimators, evaluated the performance of the proposed method under limited samples through data simulation, and compared our method with existing methods.The data simulation results indicates that the method proposed in this paper, it can provide more accurate estimates of model parameters and coefficient functions by contrast . Finally, an empirical case is presented to demonstrate the practicality of the proposed method.

The discounted Hamilton-Jacobi equation is a special form of the contact Hamilton-Jacobi equation. Hence,the study of the discounted Hamilton-Jacobi equation is more intuitionistic.In this paper,we mainly study the convergence of the viscosity solutions of time-periodic discounted Hamilton-Jacobi equation by variational method. For the evolutionary Hamilton-Jacobi equation $u_t+\lambda u+H\left(x, D_x u, t\right)=0$, under the assumptions that $H(x, p, t)$ is a Tonelli Hamiltonian,and $H(x, p, t)$ is 1-period in the variable $t$. we can get the 1-period solution $\bar{u}_\lambda(x, t)$ and the 1-periodic solution is unique under the given conditions. Furthermore,we prove $\lim _{n \rightarrow \infty} u_\lambda(x, t+n)=\bar{u}_\lambda(x, t)$, where $u_\lambda(x, t+n)$ is the viscosity solution of $u_t+\lambda u+H\left(x, D_x u, t\right)=0$. At last,we explain the conclusion with a specific example of Hamilton-Jacobi equation.

In this paper, we consider the non-collapsing property on a class of planar convex curves deformed by non-linear curvature flow.Firstly, we give the definition of the inner non-collapsing property and the outer non-collapsing property on planar curves, then by defining a function $Z$, we prove that the inner non-collapsing property on planar curves is equivalent to the non-negativity of $Z$ and the outer non-collapsing property on planar curves is equivalent to the non-positivity of $Z$. Finally, we use maximum principle to prove the non-collapsing property on the planar convex curve shortening flow is maintained under some conditions.

Ecological researches show that there exist dominance ranks of individuals in many species. Moreover, natural populations are actually subject to seasonal fluctuations which makes their habitats often undergoes some periodic changes. Motivated by these considerations, in this paper, we investigate the optimal harvesting problem for a hierarchical age-structured population system in a periodic environment. Here the objective functional represents the net economic benefit yielded from harvesting. Firstly, by means of frozen coefficients and fixed point theory we show that the state system is well posed if the reproducing number is less than one. Meanwhile, it is shown that the population density depends continuously on control parameters. Similarly, we show that the adjoint system is also well posed. Then, the optimality conditions given by the feedback forms of state variable and adjoint variable are obtained by using the adjoint system and tangent-normal cone techniques. The existence of optimal harvesting policy is verified via Ekeland's variational principle and fixed point reasoning. Finally, we use numerical simulations to verify the main results and find other dynamic properties of the system. The results in this paper generalize and improve the previous related results.

This paper is concerned with the time decay rates of strong solutions for the one-dimensional isentropic compressible micropolar fluid model with density-dependent viscosity. The pressure $p(\rho)=\rho^\gamma$ and the viscosity coefficient $\mu(\rho)=\rho^\alpha$ for some parameters $\alpha,\gamma\in\mathbb{R}$ are considered. By using a priori assumption and some refined energy estimates, we show that the global existence and large-time behavior of strong solutions with large initial data for the Cauchy problem under the perturbation of the constant state. Furthermore, by using the anti-derivative and time weighted energy method, the algebraic time decay rates for the specific volume $v(t,x)$ and the velocity $u(t,x)$ are also established.

In this paper, the exact periodic traveling wave solutions of modified short pulse (mSP) equation are systematically discussed. With the aid of new variables, the mSP equation is converted to an ordinary differential equation of first order. By means of elliptic integrals and Jacobi elliptic functions, this equation is solved and the periodic traveling solutions of the mSP equation are constructed. Through exploring the limits of these periodic solutions, the 1-cuspon solution and a new solution are obtained.

In this paper, the global Calderón-Zygmund estimation of a class of elliptic equation boundary value problems with variable exponential logarithmic growth is concerned. By comparing estimates and utilizing methods such as iterative coverage, we obtain the global Calderón-Zygmund estimates of the problem in non-smooth regions, where operator $\mathbf{a}$ satisfies some appropriate conditions and the given vector function satisfies appropriate growth conditions.

In this paper, a new nonmonotone modified L-M algorithm for solving nonlinear equations is proposed by combining the nonmonotone line search technique with the modified Levenberg-Marquardt algorithm (L-M algorithm). In each iteration of the new algorithm, a modified step is introduced, and the gradient norm of the value function is used to update the L-M parameters. If the trial step is not accepted, the nonmonotone line search technique is used to obtain the new iteration point. Under certain assumptions, the global convergence and local convergence of the algorithm are proved. Numerical experiment results show that the algorithm is feasible and effective.

Based on the expected utility maximization criterion, this paper studies the reinsurance and investment problem. There is an insurer and a reinsurer in the market. By using extrapolation deviation method, the insurance model under correlated claims is obtained. The joint interests of the insurer and the reinsurer are reflected by maximizing the weighted of their respective wealth. In the framework of ambiguity aversion, the robust stochastic optimization problem is established. By using stochastic control and stochastic dynamic programming theory to solve the robust stochastic optimization problem, the explicit solution of the robust optimal reinsurance and investment strategy is obtained. Finally, the effects of model parameters on the robust optimal reinsurance and investment strategy are explained by numerical experiments, and the practical guiding significance of the research results is pointed out.

In the realm of financial data analysis, where data often exhibits volatility clustering, heteroscedasticity, and asymmetry, capturing these inherent features authentically requires the adoption of conditional heteroscedasticity models within a skewed distribution framework. This study addresses the Bayesian statistical diagnostic challenges associated with skew-normal GARCH models. The research begins by employing the Griddy-Gibbs algorithm for effective parameter estimation within the skew-normal distribution embedded in the GARCH model. The investigation considers three distinct sources of perturbations: disruptions from prior assumptions, anomalies within the data, and variations in the model itself. To facilitate comprehensive statistical diagnosis, the study leverages three objective functions: Bayes factor, Kullback-Leibler divergence, and posterior mean, enhancing the precision of the diagnostic process. Empirical validation is achieved through rigorous numerical simulations, conclusively establishing the method's efficacy and resilience. This is further supported by an empirical application involving GARCH modeling for Chevron Stock. By utilizing the skew-normal distribution to encapsulate weekly logarithmic returns, the study empirically underscores the distinct advantages of Bayesian local influence analysis, demonstrating superior results.

In this paper, a multi-fidelity Monte Carlo (Multi-fidelity Monte Carlo, MFMC) estimation method based on data-driven low-fidelity models is proposed for the multi-parameter uncertainty quantitative analysis of the numerical solution of the Advection-Diffusion-Reaction equation. In our method, a high-fidelity model of numerical solution is first obtained according to the finite element method. Then, two types of low-fidelity models of DEIM and POD-DEIM are given respectively based on the POD reducing dimension method of the finite element discrete linear equations and the DEIM interpolation method for parameter space. Finally, through numerical experiments, the mean value and sensitivity of the multi-parameter uncertainty are analyzed for the ADR equation. The results show that, compared with the standard Monte Carlo method, the MFMC estimation method based on the data-driven low-fidelity model can effectively reduce the computational cost and the relative mean square error.

The nonlinear matrix equation ${{X}^{m}}-{{A}^{*}}{{X}^{-s}}A+{{B}^{*}}{{X}^{-t}}B=Q$ form computational physics and optimal control, etc., the presence of parameters and positive and negative hybrid terms, it is difficult to solve the equation. In this paper, the iterative methods of the Hermite positive definite solution of the equation are discussed under certain conditions. First, the original problem is transformed into an equivalent matrix equation by matrix transformation. Then, the coefficient matrix and its partial order are used to construct the existence interval of the solution of the equation and three iterative schemes. According to the characteristics of each iterative sequence, the residual norm of the solution and the monotonic boundedness of the iterative sequence is used to prove that the given iteration converges to the Hermite positive definite solution of the original equation, and the error estimation formula of the solution is obtained. Finally, two numerical examples demonstrate the effectiveness and feasibility of the proposed method.

Delay differential equation is a very important problem in the study of infinite-dimensional dynamical system and application. Recently, delay differential equation has attracted the attention and exploration of many scholars. In this paper we study the existence of global attractors and generalized exponential attractors in two-dimensional beam equations with state delay. By applying the Banach fixed-point theorem and the operator semigroup theory, we prove the existence and uniqueness of the mild solutions and the continuous dependence on the initial data. Further combining with the quasi-stability property we show the existence of generalized exponential attractors and global attractors with finite fractal dimensions.

In this paper, we study a new (2+1) dimensional modified Camassa-Holm equation, which is integrable since it admits bi-Hamiltonian structure. The aim of this paper is to study the existence of peakons for such equation. The suitable definitions of weak solutions both on real line and circle are defined. The single-peakon, two-peakon, and periodic peakon of such equation are derived. A new type of two-peakon is obtained, and the reason why such phenomena may not occur in (1+1) dimension case is also illustrated.

The monostable traveling waves for a class of asymmetric system with nonlocal diffusion and delay(the system can be non-quasimonotone) are investigated. Firstly, the existence of traveling waves is transformed into the fixed point problem of a nonlinear operator by constructing a pair of super-auxiliary system and sub-auxiliary system with quasi-monotonicity, and thus the existence of monostable traveling waves under the non-critical and critical wave speed are established by using Schauder’s fixed point theorem and limiting argument, respectively. Secondly, the non-existence and asymptotic behaviors of monostable waves are discussed by analysis technique and Ikehara’s Tauberian theorem. Finally, concrete examples and numerical simulations for the obtained theoretical results are included.

This paper deals with the uniqueness of a finite-order meormophic solution $f(z)$ of some delay differential equation $$ f(z+1)-f(z-1)+a(z)\frac{f'(z)}{f(z)}=b(z) $$ sharing $0, 1, \infty$ CM with a meromorphic function $g(z)$, where $a(z), b(z)$ are nonzero rational functions, then either $f(z)\equiv g(z)$ holds, either $f(z)=Ce^{ik\pi z}$ and $f(z)g(z)\equiv1$ holds, where $C$ is a nonzero constant, $k$ is a nonzero integer.

In statistical parameter estimation problems, how well the parameters are estimated largely depends on the sampling design used. In this article, we consider maximum likelihood estimator(MLE) of the parameter of Topp-Leone distribution and its properties under ranked set sampling(RSS), maximum ranked set sampling with unequal samples(MaxRSSU) and minimum ranked set sampling with unequal samples(MinRSSU). Theoretical results of asymptotic efficiencies of the above MLE show that the MaxRSSU estimator and the simple random sampling estimator have the same efficiency, the RSS estimator and the MinRSSU estimator are more efficient than the simple random sampling estimator.

In this paper, a nonlocal dispersal dynamic model of HIV latent infection with spatial heterogeneity is studied. We overcome the difficulty of non compactness caused by the nonlocal dispersal operator, and obtain the functional expression of the next generation operator $\mathcal{R}$ by using the renewal equation. Then, the basic reproduction number $R_0$ of the model is obtained, which is defined by the spectral radius of the next generation regeneration operator $\mathcal{R}$. Finally, the threshold dynamics of the system is analyzed. Specifically, by constructing appropriate Lyapunov functional, it is proved that the uninfected steady state is globally asymptotically stable when $R_0<1 $; Applying the consistent persistence theory of point dissipative systems, we prove that the system is uniformly persistent and has at least one positive steady state when $R_0>1$.

In this paper, we consider a singular Rayleigh equation with Minkowski-curvature operator $$ \Big(\frac{u'(t)}{\sqrt{1-(u'(t))^{2}}}\Big)'+f(t,u'(t))+g(u(t))=e(t), $$ $g$ is a continuous function and has a singularity at $u=0$. By using Mawhin continuity theorem and some analytical methods, we obtain the existence of a positive periodic solution for this equation.

In this paper, we study the existence of mild solutions for a class of Hilfer fractional stochastic evolution equations with nonlocal conditions on the positive half-axis. Firstly, by applying fractional calculus, semigroup properties of operators, stochastic analysis theory, the generalized Ascoli-Arzela theorem and Krasnoselskii’s fixed point theorem, sufficient conditions for the existence of the mild solutions on the positive half-axis are derived in the case that the associated semigroup is compact. Then relying on the Kuratowski noncompactness measure, we further discuss the existence of the mild solutions on the positive half-axis when associated semigroup is noncompact. At last, an example is provided to illustrate the applicability of the obtained results.

In this paper, we undertake a comprehensive study the orbital stability of standing waves for the inhomogeneous nonlinear Schrödinger equation $$ \begin{cases} i\partial_t\psi+\Delta \psi+|\psi|^{p}\psi+|x|^{-b}|\psi|^q\psi=0, &\quad (t,x)\in\mathbb{R}\times\mathbb{R}^N, \\ \psi(0,x) = \psi_0 (x), &\quad x\in\mathbb{R}^N, \end{cases} $$ where $N\geq3$, $\psi:\mathbb{R}\times\mathbb{R}^N\rightarrow\mathbb{C}$, $\frac{4-2b}{N}<q<\frac{4-2b}{N-2}$, $0<b<2$. In the case of $0<p<\frac{4}{N}$, the energy functional corresponding to this equation has a local minimizing structure. Therefore, we introduce a local minimizing problem. By studying the compactness of the minimizing sequence of this minimizing problem, we prove the existence of the minimizer of this minimizing problem, and finally obtain that the set of minimizer is orbitally stable.

In this paper, we study a water-borne epidemic model with multiple transmission ways. Firstly, we show the existence and uniqueness of global positive solution of the stochastic model by using suitable Lyapunov function. Moreover, by applying the Has'minskii theory, we obtain the existence of a ergodic stationary distribution of the positive solution of the model system under certain sufficient conditions. At last, we carry out numerical simulations to verify the analytical results. The results show that Random noise has a great influence on the spread of infectious diseases, and larger noise is beneficial to control the outbreak and spread of infectious diseases.