Complex Dynamics of A Dysentery Diarrhoea Epidemic Model With Treatment and Sanitation Under Environmental Stochasticity: Persistence, Extinction and Ergodicity

To understand the transmission dynamics of diarrhea in random environment, in this paper we propose a stochastically perturbed dysentery diarrhoea epidemic model with treatment and sanitation. Using the theory of stopping time, we first show the existence of global positive solution of the model. Then, we study the stochastic dynamics of the model and present a stochastic threshold R0S which determines the extinction and persistence of the disease. Based on Khasminskii’s theory, we further prove that the model has a unique ergodic stationary distribution under the condition of R0S > 1. Numerical simulations are carried out to verify the analytical results, showing that the white noise, and the constant treatment and sanitation may have certain inhibitory effects on disease transmission. Lastly, the model is further extended to include colored noise and seasonal fluctuation to study the long-term transmission dynamics of disease. It is found that the method proposed in this paper is universal.


I. INTRODUCTION
Diarrhoea is an ancient disease that continues to cause epidemics despite ongoing efforts to limit its spread [1]- [4]. It is well known that dysentery is a typical diarrhoeal disease caused by Shigella (S. flexneri and S. dysenteriae), which is associated with contaminated water and poor sanitation, and is transmitted through direct person-to-person contact and environment-to-person interaction. As a result, the disease often occurs in refugee camps. The elderly, the weak and the malnourished are particularly vulnerable to this disease and cause serious death. According to the World Health Organization (WHO) [5], diarrhoeal disease is the second leading cause of death in children under five years old, killing around 525000 children every year, of which dysentery and cholera contribute most of the cases. Available information indicates that about 15% of diarrhoea under five years old is dysentery, but up to 25% of all diarrhoea deaths [6].
Over the past few decades, WHO has launched a series of measures to prevent and control dysentery [7], such as advocating exclusive breastfeeding, using improved sanitation, rotavirus immunization, good personal and food hygiene, health education about how infections spread, etc., and has made remarkable achievements. However, for developing countries, dysentery remains a significant public health burden due to economic constraints and continues to receive worldwide attention [8]- [14]. Stemming from this motivation, to understand the transmission dynamics of dysentery Berhe et al. [15] proposed the following dysentery diarrhoea (1) where S, I, R stand for the susceptible, infected and recovered individuals respectively, and B is the concentration of pathogen population (or concentration of shigella in the environment). Model (1) is based on the assumptions that disease transmission is multiple pathway and the population is homogeneously mixed. Namely, the incidence is composed of two parts: one is modeled by a logistic response curve β 1 B K+B representing the environment-to-human transmission and the other is modeled by β 2 I representing the human-to-human interaction. All parameters in model (1) are positive and have the following biological significance: β 1 and β 2 denote the rates of ingesting shigella from a contaminated environment and through human-to-human interaction respectively; K is the pathogen concentration that yields 25-50% chance of catching dysentery diarrhoea [16]; Λ denotes the recruitment rate of susceptible humans; µ represents the natural death rate of all human classes; k is the rate of infected individuals contribution to shigella; γ is the natural recovery rate of diarrhea; α is the relapse rate of the recovered humans to the susceptible class; d is the disease-induced death rate; δ is the net death rate of the pathogen population in the environment; u 1 is the rate of treatment; e 2 u 2 is the rate of sanitation; e i (i = 1, 2) are the efficacy of treatment and sanitation respectively; ρ 1 is the proportion of the naturally recovered ones who go to the recovered class; ρ 2 denotes the proportion of the recovered individuals due to treatment who move to a temporary immune state. For model (1), the dynamic behavior is completely determined by the basic reproduction number R 0 = Λβ 1 k µ(µ + d + γ + e 1 u 1 )(δ + e 2 u 2 )K + Λβ 2 µ(µ + d + γ + e 1 u 1 ) , that is, the disease-free equilibrium E 0 = ( Λ µ , 0, 0, 0) is globally asymptotically stable in the feasible region if R 0 < 1; while for R 0 > 1, there has a unique endemic equilibrium E * = (S * , I * , R * , B * ) which is globally asymptotically stable (see Ref. [15]). The model was successful in explaining some basic effects of constant controls treatment and sanitation on disease transmission.
As an effective tool to predict and control disease outbreaks, stochastic epidemic models (including models driven by white noise [17]- [23], Markov switching [24]- [29] and Lévy jumps [30]- [33]) have been increasingly favored by many scholars in recent years. Unlike deterministic models, the random transmission dynamics of disease is usually based on probabilistic analysis, which provides a new perspective on the evolution of disease. However, due to the complexity of noise effect, there are still many problems to be studied, such as persistence and extinction of the disease under environmental noise and ergodicity of stochastic epidemic system. To this end, we introduce randomness into model (1) by assuming that stochastic perturbations are of the white noise type which are directly proportional to S, I, R and B (please refer to Ref. [34] for details), resulting the following Itô type model: where W i (t) are mutually independent standard Brownian motions defined on this complete probability space (Ω, F , {F t } t≥0 , P) with a filtration {F t } t≥0 satisfying the usual conditions, σ i ≥ 0 denote the intensity of the white noise, i = 1, 2, 3, 4.
A natural question is: How does environmental noise affect the dynamic behavior of stochastic model (2), especially the persistence, extinction and ergodicity? To answer this question, we give some theoretical results in Section II, including the existence of a unique global positive solution for model (2), the exponential stability and persistence of the disease, and the existence of a unique ergodic stationary distribution. Strict mathematical proofs of the main results are presented in Section III. Later, some numerical simulations are carried out to confirm our theoretical analysis in Section IV. Finally, further discussions are provided in Section V to conclude our study. Our major contributions of the paper are: (i) This paper is the first attempt to consider a dysentery diarrhoea epidemic model with treatment and sanitation under environmental stochasticity. Although there are some stochastic epidemic models such as [19]- [28], model (2) is essentially different due to the complexity its structure; (ii) A new stochastic tool for studying extinction of the disease is used, which is different from the existing literatures [21]- [23], [25] and is universal; (iii) A stochastic threshold R S 0 is given, which determines the persistence and extinction of the disease. Meanwhile, a meaningful result that environmental noise and constant treatment and sanitation may have certain inhibitory effects on disease transmission is obtained.

II. MAIN RESULTS
Some main results are presented in this section, including the existence and uniqueness of global positive solution, the exponential stability and persistence in mean, and the existence of a unique ergodic stationary distribution. Before This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Here, where (υ 1 , υ 2 ) is any positive solution of the following matrix equation and . Now we are in the position to state our main results. Theorem 1. Given initial valve (S(0), I(0), R(0), B(0)) ∈ R 4 + , model (2) admits a unique positive solution (S(t), I(t), R(t), B(t)) for t ≥ 0; furthermore, the solution will remain in R 4 + with probability one. Theorem 2. Let (S(t), I(t), R(t), B(t)) be the solution of model (2) with any given initial value (S(0), where η is defined as (14).
Remark 1. (i) Theorem 2 gives the sufficient conditions for the infected individual to approach zero with probability one and to persist. It is worthy to note that, when σ i = 0 (i = 1, 2, 3), R 0 = R S 0 , which shows that R S 0 is the stochastic version of R 0 ; when σ i = 0 (i = 1, 2, 3), R S 0 < R 0 , implying the possibility that R S 0 < 1 < R 0 . This means that environmental noise may suppress the spread of disease. Furthermore, it is interesting to find that the stochastic version of R S 0 is independent of the noise intensity σ 4 . , .
Biologically, R S 1 and R S 2 are the basic reproduction numbers corresponding to the ingesting shigella from a contaminated environment and through human to human, respectively. (iii) After direct calculation, we easily obtain Thus, the basic reproduction number R S 0 decreases with implementing the controls u 1 and u 2 . Similarly, the basic reproduction number increases with β 1 , β 2 k and Λ.
Theorem 3. If R S 0 > 1, then model (2) admits a unique stationary distribution π(·) and it has the ergodic property.
Remark 2. Theorem 3 tells us that although there is no positive equilibrium point in model (2), under the condition of R S 0 > 1, there exists a unique ergodic stationary distribution, which to some extent reflects that the solution of the model is weakly stable and persistent in mean.

III. PROOFS OF MAIN RESULTS
In this section, we provide the detailed proofs of the main results illustrated in Section II.
The rest of the proof is standard and can be referred to Theorem 3.1 in [17]. So we omit it here.

B. PROOF OF THEOREM 2
Below two lemmas are very useful in proving Theorem 2, where the proof of Lemma 1 is omitted because it is standard (see Lemma 1 in [35]).
where τ 1 and τ 2 are defined as Theorem 2. Then, and lim sup Proof. It can be seen from model (2) which shows that Then, Here, which follows from strong law of large numbers and Lemma 1 that lim Disease extinction of stochastic model (2) In this subsection, we prove Theorem 2(i) about the extinction of disease for model (2).
Proof. Applying Itô's formula to U(I, B) defined in Lemma 2 leads to where Integrating (6) from 0 to t and then dividing by t on both sides, we have where Following the strong law of large numbers yields Notice that It then follows from (7) and Lemma 2 that lim sup which is the desired assertion. Next we assume diffusion process R(t) defined by , which together with stochastic comparison theorem implies that By (10), for arbitrary ε 1 > 0 and ω ∈ Ω, there exists a T 1 = T 1 (ω) such that I(t) ≤ 1 τ 1 exp((ρ + ε 1 )t), ∀t > T 1 . Hence, when t > T 1 , That means that lim sup a.s. Therefore, lim sup
2) Disease persistence of stochastic model (1.2) In this subsection, we prove Theorem 2(ii) about the persistence of disease for model (1.2).

Proof. Let
where a 1 , a 2 and a 3 are positive constants to be determined later. By Itô's formula, we obtain Choose a 1 , a 2 and a 3 such that where a 4 and a 5 are positive constants to be determined later. Then, which together with (13) yields that Choose a 4 and a 5 such that a 4 µ + 1 2 σ 2 1 = β 2 Λ µ+ 1 2 σ 2 1 and β 1 (a 1 +a 4 ) K − a 5 (δ + e 2 u 2 ) = 0, then . And hence (15) Integrating (15) from 0 to t and then dividing by t on both sides, we have . According to Lemma 1 and the strong law of large numbers for martingale, we get lim t→∞ ϕ(t) t = 0, which further implies that lim inf This completes the proof.

C. PROOF OF THEOREM 3
In this subsection, we provide a complete proof of Theorem 3. Before that, we give an important lemma about the existence of ergodic stationary distribution. Let X(t) be a regular time-homogeneous Markov process in R l described by the stochastic differential equation The diffusion matrix is defined as follows A(x) = (a i j (x)), a i j (x) = ∑ r=1 g i r (x)g j r (x). Then, one has the following lemma about the existence of stationary distribution established by Khasminskii [37]. (a2) There exists a non-negative C 2 -function V and a positive constant L such that L V ≤ −L, for any X ∈ R l \ D. Then the Markov process X(t) has a unique stationary distribution π(·), and for any integrable function f (·) with respect to the measure π we have P lim With the help of Lemma 3, we now complete the proof of Theorem 3 about the existence of ergodic stationary distribution for model (2).
Proof. Let us construct a C 2 function V 2 : R 4 + → R as where V 21 , V 22 are defined in Theorem 2, and .
On the other hand, R S 0 = 0.87 < 1, and ρ = 1 . This together with Theorem 2 yields that under random perturbation, all solution trajectories representing infected humans with dysentery diarrhoea drop to zero and the pathogen population is also eliminated from the community. Figure 1 confirms these theoretical results, see the red lines. Comparing the above two situations, we find that large white noise may suppress the spread of disease, while small white noise can not change the original persistence of disease. VOLUME 4, 2016 Case 3. Let u 1 = u 2 = 0 and noise intensities are same as Case 2. As pointed in Remark 2, treatment and sanitation may reduce the reproduction number, which implies that it is possible to decrease the infected and pathogen population using controls. Figure 3 confirms these analysis.

V. CONCLUDING REMARKS
To better understand the spread of disease in random environment, in this paper, we proposed a stochastic dysentery diarrhoea epidemic model with controls disturbed by white noise. We first showed the existence of global positive solution of the model. Then we gave a stochastic threshold R S 0 , which determines the extinction and persistence of the disease. Lastly, based on Khasminskii's theory, we proved that the model has a unique ergodic stationary distribution under the condition of persistence in mean. Theoretical analysis and numerical simulations show that: (i) Large white noise may suppress the spread of disease while small white noise can not change the original persistence of disease; (ii) Constant treatment and sanitation have important positive role in controlling the spread of disease.
It is especially necessary to point out that the method proposed in this paper is universal and can be used to study other types of stochastic epidemic models, for example, stochastic HIV-1 infection model with logistic growth or Beddington-DeAngelis incidence rate, stochastic tuberculosis model with varying total population size and so on. Here are two examples to illustrate the universality of this method from other perspectives.
For the sake of argument, we assume that the generator Q = (q i j ) N×N of the Markov chain is governed by where ∆t > 0 and q i j is the transition rate from state i to state j and q i j ≥ 0 if i = j while q ii = − ∑ i = j q i j . In addition, we further assume that the Markov chain r(t) is irreducible and hence has a unique ergodic stationary distribution π = (π 1 , π 2 , · · · , π N ) satisfying πQ = 0, ∑ i∈N π i = 1 and π i > 0, ∀i ∈ N. d(i) + γ(i) + e 1 (i)u 1 (i) + 1 2 σ 2 2 (i)) N ∑ i π i (δ (i) + e 2 (i)u 2 (i) + 1 2 σ 2 4 (i)). Using the method proposed in this paper, the persistence in mean and existence of an ergodic stationary distribution for hybrid model (27) can be investigated easily. Therefore, the following conclusions are valid. . Theorem 5. If R C 0 > 1, then model (27) admits a unique stationary distribution and it has the ergodic property.