Dynamics of cholera epidemic models in fluctuating environments

Based on our deterministic models for cholera epidemics, we propose a stochastic model for cholera epidemics to incorporate environmental fluctuations which is a nonlinear system of Itô stochastic differential equations. We conduct an asymptotical analysis of dynamical behaviors for the model. The basic stochastic reproduction value Rs is defined in terms of the basic reproduction number R0 for the corresponding deterministic model and noise intensities. The basic stochastic reproduction value determines the dynamical patterns of the stochastic model. When Rs < 1, the cholera infection will extinct within finite periods of time almost surely. When Rs > 1, the cholera infection will persist most of time, and there exists a unique stationary ergodic distribution to which all solutions of the stochastic model will approach almost surely as noise intensities are bounded. When the basic reproduction number R0 for the corresponding deterministic model is greater than 1, and the noise intensities are large enough such that Rs < 1, the cholera infection is suppressed by environmental noises. We carry out numerical simulations to illustrate


Introduction
Cholera is an infection of small intestine caused by the gram-negative bacterium called Vibrio cholerae, characterized by a variety of diarrhea, abdominal cramp and dehydration. It has caused seven global pandemics since 1816 [6,22,24]. The current pandemic started from Indonesia in 1961, spread into Europe, South Pacific and Japan in the late 1970s, reached South America in 1990s, arrived to Africa 2000s, and has continued to the present. The most recent case was the cholera outbreak in Yemen 2018, which caused 1,207,596 infections and 2,510 associated deaths [24]. Cholera has posed a major practical problem in the need to provide models of the epidemic reliably accounting for real-life epidemiological and environmental complexity in a dependable predictive fashion. In fact, there have been many efforts in modeling cholera epidemics. The earliest mathematical model was proposed by Capasso and Paveri-Fontana [5] to study the 1973 cholera epidemic in the Mediterranean region. Codeco extended this model to count for the role of Vibrio cholerae in the aquatic resource [7]. Based on these models, many deterministic models have been proposed, Hartly et al. [10], Mukandavire et al. [18], Jenson et al. [11], Pascual et al. [20], Nishiura et al. [19], to name a few. A good summary was presented in our work [23] which emphasized the global stability of these deterministic cholera epidemic models.
It is known that there are many stochastic elements or uncertainties in modeling of cholera epidemics. Actually, May pointed out in ecosystem modeling that the environmental noise affects population growth rates, carrying capacities, and other parameters [17]. Environments are deeply involved in cholera epidemics. The functional relations among variables such as susceptibles, infected individuals, and concentrations of the pathogen in water resource are subject to environmental fluctuations. Of course, the parameters in the models are not only subject to measurement errors but also to intrinsically environmental noises. Although it is insightful to obtain average dynamics of cholera epidemics from deterministic models, it is needed to have more realistic mathematical models which may provide more complete pictures for cholera epidemics. There are very few stochastic models for cholera epidemics in terms of stochastic differential equations. Gani et al. used a particular Poisson process to model cholera bacteria and infective numbers [8]. Bertuzzo et al. provided an individual-based spatially-explicit stochastic process to model cholera spreading [4]. Azaele et al. randomized infected individuals to obtain an analytically tractable stochastic model for cholera epidemics [2]. In this study, we Cholera epidemics in fluctuating environments will randomize the incidence function and some parameters in a general model we proposed in [23].
In our work [23], we studied the following deterministic cholera epidemic model with a general nonlinear environment-dependent incidence function f (B): where S(t) stands for the susceptible individuals at time t, I(t) represents the infected individuals, R(t) represents the recovered individuals and B(t) stands for the concentration of the vibrios (V. cholerae) in the water resource. The parameter N is the total human population under consideration; b is the natural human birth or death rate; γ is the rate of recovery from cholera; e is the rate of contribution (e.g., V. cholerae shredding) to the aquatic resource; m is the net death rate of vibrios in the environment. To incorporate environmental fluctuations and noises in parameter values (called them errors simply), we may consider as follows. Let P be a parameter or an incidence function (on the average), and P + E be its value with errors. When the system changes from time t to t+dt in a small time subsequent interval dt, the error will be Edt. From the central limit theorem, Edt can be approximated by a normal distribution with mean zero and variance τ 2 dt. This way, Edt can be approximated by τdW (t), where dW (t) = W (t + dt) − W (t) and W (t) is the Brownian motion following the standard normal distribution, and τ is called the intensity of the noise. Loosely speaking, we may use white noise dW (t) dt to model simultaneous errors. In the deterministic model (1.1), the incidence function f (B) encodes the interaction between human and cholera vibrios in the environment such as water resources. The environmental fluctuation will influence the incidence function in several ways. For example, the temperature will impact human drinking behavior and patterns of washing or swimming. The environmental factors also impact the death rate m of cholera vibrios in the water resource. From a long term viewpoint, the human birth or death rate b in a region is also influenced by its environmental situation. Particularly, when a cholera epidemic occurs in the region, the human birth/death rate can be affected. In order to count for the environmental fluctuations and random noises in parameter values, we consider the situation in which the incidence function f (B), the parameter b and m are subject to white noise perturbations. We may consider each infected individual contributes environmental noise with intensity of τ 1 . Then, the incidence function F (B) which is actually a rate of interactions can be replaced by f (B) + τ 1 I N dW1 dt . For the human birth/death rate b, the environmental noise is mostly from infected and susceptible individuals. Similarly, if we consider each individual of infected and susceptible people contributes environmental noise with intensity τ 2 , we may replace b by b + τ 2 For the death rate of cholera vibrios, we consider the environment affects cholera vibrios as a whole in the water. So, we may replace m by m + τ 3 dW3 dt , where τ 3 is the noise intensity in the water. It should be noticed that dW1 dt , dW2 dt , and dW3 dt are independent Gaussian white noises.
Then, we have a stochastic model in terms of Itô stochastic differential equations to describe the epidemics of cholera as follows: where W 1 , W 2 , and W 3 are mutually independent Wiener processes and τ 1 , τ 2 , and τ 3 are their intensities. Here, the total population N becomes a random variable and N = S(t) + I(t) + R(t). Adding the first three equations of the system (1.2) gives This means that dN dt = 0 with probability 1 and hence N is an almost sure constant random variable. For convenience of discussion, we will drop the equation for the recovered individuals R in (1.2) since S(t, ω) + I(t, ω) + R(t, ω) = N (ω) and R(t, ω) can be obtained from this equation almost surely. Thus, in our study, we will focus on the following stochastic system: (1. 3) The analysis of stochastic differential equation systems is difficult in general. It should be noticed that cholera epidemic models are SIR models coupled with infective cholera vibrios from the environment. Analysis of such models can borrow tools from SIR model analysis although there are extra difficulties as we did in [23] and other papers about cholera models. To conduct an analysis of the model (1.3), it is nature to study relevant SIR models in terms of stochastic differential equations in the literature. Lahrouz et al. conducted a study of a stochastic SIRS epidemic model where the white noise is 1-dimensional [15]. In their second paper [14], they studied a similar model with 4-dimensional white noise. Yang et al. studied the ergodicity of two special epidemic models with 3 or 4-dimensional white noise [25]. Recently, Bao et al. conducted a study about the stationary distribution of a stochastic SIRS epidemic model where they provided a stochastic basic production which can characterize the dynamical behaviors of their model [3]. The major analytical tool in these studies is stochastic versions of Lyapunov functions. The advantage is that these models all have specific and explicit incidence functions which allow to construct a specific stochastic Lyapunov function. In our proposed model above, the incidence function is not specific given. This creates an extra difficulty in our study. For example, to circumvent this difficulty in the proof of existence of the stationary ergodic distribution, we construct several stochastic Lyapunov functions and combine them together to form the final stochastic Lyapunov function. If we take noise intensities to be zeros in the model (1.3), our proof gives a new proof of [23, Theorem 2.10] which was proved by using the theory of monotone dynamical systems. We define the basic stochastic reproduction value which is a sum of the basic reproduction number for the deterministic model and the function of noise intensities. From the formula of the basic stochastic reproduction value, we can also easily deduce that the noises can suppress the outbreak of cholera infections.
The rest of the paper is organized as follows. In Sec. 2, we present a basic analysis including existence and uniqueness of positive global solutions, and the condition under which the cholera epidemic will extinct or persist. In Sec. 3, we prove the existence of a unique stationary ergodic distribution. In Sec. 4, we carry out numerical simulations for three cases in both stochastic model and its corresponding deterministic model. The paper closes with Sec. 5.

Basic Dynamics
In this section, we discuss the almost surely positive invariant domain for the system (1.3) in which the solution will exist uniquely and globally for any initial value. We define the basic stochastic reproduction value R s , and point out its epidemic implication. We show the cholera infection will die out if R s < 1 almost surely, while the cholera epidemic will persist almost surely when R s > 1.
Throughout this paper, we specify an appropriate filtered probability space as follows. Let Ω = {ω ∈ C(R, R 3 ), ω(0) = 0}, F be the Borel σ-algebra on Ω and P be the measure induced by {W t } t∈R , a two-sided 3-dimensional Wiener process. The elements of Ω can be identified with paths of the Wiener process ω(t) = W t (ω). We consider the P-completion F P of F , i.e. F P contains all P-null sets of F . The filtration F P t is given by the canonical filtration generated by the Wiener process W t completed by all P-null sets of F P . Denote the probability measure given by the extension of the probability measure P to F P again by P. Therefore, a P-completed filtered probability space (Ω, F P , {F P t } t∈R , P) is obtained. For convenience, we write our stochastic differential equation system (1.3) in the vector form. Let X = (S, I, B) T , Then (1.2) can be written as where W t = (W 1 , W 2 , W 3 ) T is the 3-dimensional Wiener process defined above. We define the diffusion matrix for (2.1) as A(X) = σ(X)σ(X) T = (a ij (X)). We also define the differential operator L associated with (2.1) by The incidence function f (B), depending on the concentration B of cholera vibrios in the environmental water resources, determines the rate of new infection introduced into the population and this function includes multiple transmission pathways. Note that f (0) = 0, f ≥ 0, f ≥ 0, and f ≤ 0 [23]. For example, f (B) = β B K+B with β being the contact rate with the contaminated water, and K the pathogen concentration that yields 50% chance of catching cholera infection.

Existence and uniqueness of global and positive solutions
In the stochastic differential equation setting, our first concern is whether the solution exists globally, and it is important to know where the solution lives in. In order for a general SDE to have a unique global solution (i.e. no explosion within a finite time) for any given initial value, the drift and diffusion coefficients of the equation are generally required to satisfy the linear growth condition and the Lipschitz condition (see [9,12]). Unfortunately, the coefficient F of (2.1) does not satisfy the linear growth condition as the incidence function is nonlinear, so the solution of (1.3) might explode at a finite time (see [1]). In this subsection, using [12, Corollary 3.1], we will show that the solution of (1.3) exists uniquely and globally, and prove that Stoch. Dyn. Downloaded from www.worldscientific.com by 174.28.181.51 on 09/24/20. Re-use and distribution is strictly not permitted, except for Open Access articles.

Cholera epidemics in fluctuating environments
is almost surely positively invariant domain for the system (1.3), where admits a unique solution (S(t), I(t), B(t)) on t ≥ 0, and this solution remains in Δ almost surely.
Hence, the coefficients F and σ satisfy the Lipschitz condition and the linear growth condition on each Δ n . Now, we define a function V : Here, we have used the equality f (B) ≤ f (0)B, which is followed from the fact that f (B) m . Therefore LV ≤cV wherec = κm 2f (0) . Next, we would like to show that, when X = (S, I, B) ∈ Δ\Δ n , V (X) approaches ∞ as n → ∞. Suppose the solution X = (S, I, B) of the system (1.3) is in Δ\Δ n . As n → ∞, the solution X = (S, I, B) will reach the boundary, ∂Δ, of the positive domain Δ. Note that the boundary ∂Δ includes 5 faces S = 0, I = 0, adding the first two equations of the system (1.3) yields Since N is constant a.s., I = 0 and hence V (X) = ∞. Thus the function V (X) will blow up to ∞ when X = (S, I, B) approaches ∂Δ. This completes the proof.

Extinction of cholera infections
Recall that, in [23], the dynamical behavior of the deterministic part of the system (1.3) is determined by the basic reproduction number R 0 = N γ+b f (0) e m . As R 0 < 1, there is only one disease-free equilibrium (DFE) X 0 = (N, 0, 0) in Δ, which is globally asymptotically stable. When R 0 > 1, there are two non-negative equilibria, X 0 and the endemic equilibrium X * = (S * , I * , B * ). X 0 becomes unstable; while X * is globally asymptotically stable, meaning that the deterministic system becomes uniformly persistent and any trajectory of the deterministic model starting at any point in Δ converges to X * . It can be shown that X * uniquely exists without explicit expression since the incidence function f is not explicitly given. However, for the stochastic system (1.3), we derive a sufficient condition for the extinction of cholera infections in terms of the basic reproduction number for the deterministic model, noise intensities, and system parameters, which we call the basic stochastic reproduction value. The basic stochastic reproduction value is given by where δ := δ(τ 1 , τ 2 , τ 3 ) is the function of noise intensities with the property that, as (τ 1 , τ 2 , τ 3 ) approaches (0, 0, 0), δ approaches 0 and, as one of the noise intensities increases, δ will be increasing. Specifically, we define a function of the solutions of the system as follows: Clearly, δ is a random variable which is a function of the noise intensities with the property that δ is increasing as one of the noise intensities increases and, as all the noises approach 0, δ → 0. We claim that δ > 0. It suffices to show that, as the solution (S, I, B) approaches the boundary ∂Δ, h(S, I, B) is still strictly positive. On the face S = 0, the system (1.3) becomes and h(S, . When S is close to 0, the first equation becomes Proof. First, we apply Ito's formula for the function log(I(t) + B(t)) as follows: Here, we have used the equalities Sf (B) ≤ N f (0)B.

T. A. Phan, J. P. Tian & B. Wang
Next, integrating (2.2) from 0 to t and dividing by t, we get By the strong law of large numbers for local martingales, since all the integrands of the last three stochastic integrals in the right-hand side of (2.3) are positive and bounded a.s., these stochastic integrals approach 0 almost surely as t → ∞. Now, we separately consider two cases, R 0 > 1 and R 0 ≤ 1. When R 0 > 1, then .
In order to show that lim t→∞ S(t) = N almost surely when R 0 > 1, we need to utilize the non-negative semi-martingale convergence theorem (see [16, Theorem 3.9 on p. 14]). From the first equation of (1.2), we can write
Remark. It may happen that R 0 > 1 but R s < 1. This means that the cholera infection in the stochastic model (1.3) dies out while the cholera infection in the deterministic part of (1.3) is persistent. Thus, theoretically, white noises can suppress the outbreak of cholera infections. In reality, it may happen that we estimate parameter values and use the deterministic model to predict the cholera infection will spread out, however, the cholera infection may just die out. We will perform numerical simulations to demonstrate this case in Sec. 4. This may remind us we should consider environmental fluctuations.

Persistence of cholera infections
The next theorem will give a condition for the persistence of cholera infections in the stochastic model (1.3).

So, we can conclude that the solution (S(t), I(t), B(t))
does not converge to the disease-free equilibrium (N, 0, 0) with positive probability.
Next, adding the first two equations of (1.2), we get

Cholera epidemics in fluctuating environments
Integrating this from 0 to t and then dividing by t gives

I(s)ds
On the other hand, from the third equation of (1.2), we get
For almost sure, there exists a T > 0 such that t ≥ T implies N − δ < S(t) < N and 0 < B(t) < δ. Now, from the second and third equations of (1.3), Then taking expectation both sides, we have Note that 0 < EB(t) ≤ eN m for all t ≥ T and R 0 > 1 because R s > 1. Then, we can choose > 0 sufficiently small so that d dt E(I(t) + B(t)) > 0 for t ≥ T . It follows that lim inf t→∞ EI(t) > 0. This completes the proof.

Existence of a Stationary Ergodic Distribution
When R s > 1, then R 0 > 1, and the endemic equilibrium X * for the deterministic model (1.1) is globally asymptotically stable [23]. We know that the endemic equilibrium X * does not solve the stochastic model (1.3). However, in the section, we will show there exists a unique asymptotically stationary distribution for the solutions of the stochastic system (1.3) when R s > 1 and noise intensities are small and are bounded by model parameter quantities, which in turn implies the global stability in the stochastic sense.
Precisely, let us denote the probability measure induced by the solution X(t) = (S(t), I(t), B(t)) with initial value x by P (t, x, ·), that is, If there is a probability measure μ(·) on the measurable space (Δ; B(Δ)) such that for any x ∈ Δ we have lim t→∞ P (t, x, B) = μ(B), then we say that the stochastic system (1.3) has a stationary distribution μ.
Based on Khasminskii [12], if there exists a bounded open domain U ⊂ Δ with a regular boundary such that its closureŪ ⊂ Δ, and the following conditions hold:

Cholera epidemics in fluctuating environments
μ, then for all x ∈ Δ, we have P lim Remark. To validate the assumption (i), it suffices to show that there exists a positive number c so that 3 i,j=1 a ij (X)ξ i ξ j ≥ c|ξ| 2 for all X ∈ U and ξ ∈ R 3 (see [9] and Rayleigh's principle in [21]). For the assumption (ii), it is enough to find a non-negative C 2 function V such that, for any X ∈ Δ\U , LV (X) ≤ −C for some constant C > 0 (see [26]).
As we mention that when R s = R 0 − δ m > 1, then R 0 > 1. Hence, there is a unique endemic equilibrium X * = (S * , I * , B * ) for the deterministic part of the model (1.3). The conditions of equilibriums give the following identities which are needed in our proof: Now, we state and prove our main theorem.

Theorem 3.1. Let (S(t), I(t), B(t)) be the solution of the stochastic system (1.3)
with any initial value (S(0), Then there is a unique stationary distribution μ for the stochastic system (1.3) and the solution is ergodic with respect to μ.
Proof. First, we define the positive C 2 function V 1 : R 3 + → R + by

T. A. Phan, J. P. Tian & B. Wang
Applying Ito's formula for this function and using (3.1), we get By Cauchy's inequality, Then, note that S N ≤ 1, I N ≤ 1, and N −S N ≤ 1, Since f (B) and B f (B) are both increasing, This implies that Secondly, we define the positive C 2 function V 2 : By computation and using (3.1), Fourthly, we define the non-negative C 2 functions V 4 : R 2 + → R + and V 5 : Applying Ito's formula for these functions and using (3.1), we get Then Finally, we define the positive definite C 2 function V : R 3 + → R + so that Then, we have

Cholera epidemics in fluctuating environments
From the third equation of (1.3), taking expectation both sides gives which implies that EB ≤ eN m . By using Ito's formula, we have Taking expectation both sides, we get Suppose that τ 2 3 < 2m. Then the above inequality implies that is a Gaussian process with finite expectation and variance. Hence B(t) must be bounded a.s. by some positive constant M . Note that where m 1 , m 2 , m 3 , and ρ are defined as in the statement of the theorem.
Finally, since ρ → 0 as (τ 1 , τ 2 , τ 3 ) → (0, 0, 0), we can choose τ 1 , τ 2 , and τ 3 small enough such that τ 2 3 < 2m and the condition (3.2) is fulfilled. Then the ellipsoid lies entirely in Δ. Take U to be any neighborhood of the ellipsoid with U ⊂ Δ and thus for any x ∈ Δ\U we get LV ≤ −C for some positive constant C > 0. Therefore, we finish the validation of the condition (ii). On the other hand, it is straightforward to see that the diffusion matrix A(X) associated with (1.3) is uniformly elliptic in U , and so fulfills the condition (i). We then complete the proof.
Remark. When the noise intensities τ 1 = 0, τ 2 = 0, and τ 3 = 0, then this proof gives another proof, using Lyapunov functions, for the globally asymptotical stability of the endemic equilibrium X * of the deterministic part of the model (1.3), which is different from the proof using monotone dynamical systems in [23].

Numerical Simulations
To illustrate our analytical results, we conduct some numerical simulations. To be concrete, we take the incidence function f (B) = β B K+B for our cholera model. We compare the solution behaviors of the stochastic model with that of its corresponding deterministic model. The derivative-free Milstein's strong order 1.0 method (see  [13,Chap. 11]) is used to approximate the strong solution of the stochastic system (1.2) with given initial value. The corresponding discretization equations are Δt, where ΔW 1 i , ΔW 2 i , and ΔW 3 i are √ Δt N(0, 1)-distributed independent random variables. The parameters are chosen as in Table 1 (see [7]). The unit for time is day, the unit for populations of human is one person, and for the concentration of V. cholerae is computed in the unit of 10 5 cells per mL.

Case 1.
We demonstrate the situation where the cholera infection in both stochastic model and its corresponding deterministic model will die out when R s < 1 and R 0 < 1. Taking β = 0.05, τ 1 = τ 2 = τ 3 = 0.1, and the values of other parameters in Table 1, we get R 0 = 0.7937 and R s = 0.7853. The initial values for both deterministic and stochastic models are the same (7,000, 1,000, 1.5). Figure 1 shows the solutions for both stochastic and its corresponding deterministic models, where the left column is for the deterministic model and the right column is for the stochastic model. These plots show the disease will die out in both deterministic and stochastic models. We also observe that the in stochastic model the disease dies out faster than that in its corresponding deterministic model.

Case 2.
We demonstrate the situation where the cholera infection in the deterministic model will persist while the infection in the stochastic model will be suppressed by noises when R s < 1 but R 0 > 1. Taking β = 0.15, τ 1 = 1.5, τ 2 = 0.5, τ 3 = 1.3, and using the values of other parameters in Table 1, we get R 0 = 2.381 and R s = 0.9726. The initial values for both deterministic and stochastic models are the same (7,000, 1,000, 1.5). Figure 2 shows the solutions for both deterministic and stochastic models, where the left column is for the deterministic model and the disease persists; the right column is for the stochastic model and the disease dies out.

Case 3.
We demonstrate the situation where the cholera infection in the stochastic model will persist when R s > 1 and R 0 > 1, and there is a stationary distribution for the stochastic model. We take β = 0. 15  in Table 1, we get R 0 = 2.381 as in Case 2. Taking β = 0.15, τ 1 = τ 2 = τ 3 = 0.1, and the values of other parameters in Table 1, we get R s = 2.3726 and the condition (3.2) of Theorem 3.1 is satisfied. The left column in Fig. 3 shows one path of the solutions for the stochastic model. At time t = 300, we simulate our stochastic model 10,000 times, and obtain frequency histograms for each population which show the stationary distributions. The frequency histograms are shown in the right column of Fig. 3.

Conclusions and Discussion
Cholera epidemics have been an important subject in public health, and mathematical modeling has provided some understanding for cholera spreading and controls.
In order to obtain detailed dynamics of cholera infections, the environmental fluctuations should be counted into modeling. Towards this direction, we take white-noise perturbations as a first approximate of environmental fluctuations. Based on a basic general deterministic mathematical model of cholera epidemics proposed in [23], we perturb nonlinear incidence function, the human birth/death rate, and the death rate of cholera vibrios in the aquatic resources with three mutually independent white noises, and then we have a stochastic model for cholera epidemics in terms of stochastic differential equations. We identify almost sure positive invariant domain for our stochastic system, and show the solutions exist globally. We obtain a threshold R s called the basic stochastic reproduction value which serves well as the basic reproduction number in the deterministic epidemic models. If the basic stochastic reproduction value is less than 1, then the solution tends asymptotically to the disease free equilibrium X 0 exponentially almost surely. That is, the cholera infections extinct almost surely. When the basic stochastic reproduction value is greater than 1, then the disease persists, and furthermore we show there is a unique stationary ergodic distribution to which all solutions eventually tend in distribution when the noises are bounded. It is also clear that the noises can suppress cholera infection spreading.
The basic stochastic reproduction value is the sum of the basic reproduction number (deterministic case) and a function of noise intensities and parameters, namely, R s = R 0 − δ m . To have a deep biological understanding, we recall that from the deterministic model to the stochastic model, f (B), b, and m are replaced by f (B) + τ 1 I N dW1 dt , b + τ 2 SI N 2 dW2 dt , and m + τ 3 dW3 dt , respectively. We may think that, each individual contributes environmental noise with intensity of τ 1 or τ 2 . For the incidence function f (B) which encodes interaction between human and cholera vibrios, it is reasonable to consider the contribution from infected individuals. So, we have f (B) + τ 1 I N dW1 dt in stochastic model. Similarly, for the human birth/death rate b, the environmental noise is mostly from infected and susceptible individuals, and for the convenience of analysis, we take the multiplication form b+τ 2 SI N 2 dW2 dt in our stochastic model. For cholera vibrio death rate, the environment noise is encoded by m + τ 3 dW3 dt since we consider the environment affect them as a whole. Somehow, the overall effect δ = δ(τ 1 , τ 2 , τ 3 ) needs to be divided by the death rate of cholera vibios m. Although the explicit form for δ(τ 1 , τ 2 , τ 3 ) is unavailable, we may still deduce some information from its definition. When any one of three noise intensities increases, the basic stochastic reproduction value will decrease. The environmental noises reduce the threshold for the model dynamical patterns. Although we think each different noise should affect the model differently, because of implicit form δ(τ 1 , τ 2 , τ 3 ), we cannot tell how each noise affects the model behaviors differently.
It is clear that there are other ways to include environmental fluctuations into modeling in general. For example, we may simply consider the environmental noise modeled by τ dW dt . However, in order to have a more complete dynamical picture of cholera epidemics, we should perturb other types of models including more realistic 2150011-23 Stoch. Dyn. Downloaded from www.worldscientific.com by 174.28.181.51 on 09/24/20. Re-use and distribution is strictly not permitted, except for Open Access articles.

T. A. Phan, J. P. Tian & B. Wang
incidence functions as proposed in [23], and to verify our thoughts about different noises would affect model dynamics differently.
As we mentioned, this study is our first step toward more realistic models for cholera epidemics which incorporate environmental fluctuations. There are several important aspects we need to study further in the future. First, we will perturb more parameters within any given model. Second, we will consider more general models in cholera epidemics. Third, we will consider how the perturbations affect the dynamical behaviors with different noises in general modeling setting.