Stochastic Dynamics of Cholera Epidemic Model: Formulation, Analysis and Numerical Simulation

In this paper, we describe the two different stochastic differential equations representing cholera dynamics. The first stochastic differential equation is formulated by introducing the stochasticity to deterministic model by parametric perturbation technique which is a standard technique in stochastic modeling and the second stochastic differential equation is formulated using transition probabilities. We analyse a stochastic model using suitable Lyapunov function and Itô formula. We state and prove the conditions for global existence, uniqueness of positive solutions, stochastic boundedness, global stability in probability, moment exponential stability, and almost sure convergence. We also carry out numerical simulation using Euler-Maruyama scheme to simulate the sample paths of stochastic differential equations. Our results show that the sample paths are continuous but not differentiable (a property of Wiener process). Also, we compare the numerical simulation results for deterministic and stochastic models. We find that the sample path of s a SI I R B − stochastic differential equations model fluctuates within the solution of the s a SI I R B − ordinary differential equation model. Furthermore, we use extended Kalman filter to estimate the model compartments (states), we find that the state estimates fit the measurements. Maximum likelihood estimation method for estimating the model parameters is also discussed.


Introduction
It is well known that diseases have impacts on people's health. Therefore, it is necessary to study the mechanism by which the disease spread, conditions for the disease to have minor or major outbreak and get the knowledge on how to control the diseases such as cholera, Ebola, etc. In this study we are mainly concern with cholera epidemic. Cholera is an infectious disease that causes severe watery diarrhea, which leads to dehydration and even death if untreated. According to WHO report [1], about 1.4 to 4.3 million cases of cholera are reported each year worldwide and more than 140,000 deaths per year are reported due to cholera. As a result of this, several mathematical models have been developed to model cholera epidemics, through these models one can predict the behaviour of the disease and control the particular epidemic. Any epidemic of an infectious disease can be modelled by using either deterministic or stochastic models. The deterministic models are formulated as a system of ordinary different equations and are preferred by many researchers since its analysis is simple compared to stochastic models. However, the shortcomings of deterministic models are: they give less information, rely on the law of large numbers, difficulty to do estimation, when the population is very small it becomes difficult to do analysis and also, experimentally the measured trajectories do not behave as predicted due to some random effects that disturb the system [2].
Due to these limitations of ordinary differential equations in modelling infectious diseases, stochastic modelling of infectious diseases in both heterogeneous and homogeneous population emerged as an alternative to deterministic model and alleviated some of the problems of deterministic models in modelling epidemic diseases. In reality many phenomena in nature are usually affected by stochastic noise and the ordinary differential Equation (ODEs) models ignore the stochastic effects [2]. The stochasticity can be added to the ODEs by including the random terms or elements by parametric perturbation technique. This technique introduces other parameters to the model known as noise intensities.
Many of the models that have been employed in water-borne settings have been deterministic, thus ignoring the possible effects of randomness; see, e.g. [3] [4] [5] [6] [7]. These models incorporate an environmental pathogen component that is the concentration of the vibrios into a SIR (susceptible-infected-recovered) and SI s I a R (susceptible-symptomatic infected-asymptomatic infected-recovered) epidemic framework. Some of these models only considered compartment I as a single compartment (individuals with severe symptoms only) (e.g. [3] [4] [5] [6]) instead of splitting it into two heterogeneous groups that are symptomatic and asymptomatic infected individuals to study the transmission dynamics of cholera epidemic. Also, some of these models considered only direct transmission of disease i.e., human to human and other models ignored the feedback loop from infected individuals to the environment reservoir. However, in their studies no stochastic models considered, so as to keep track where the disease is at continuous time and not only at discrete time as shown in their studies.
In [7] they developed deterministic model by extending the work by [6]. The work in [6] considered infected individuals (I) as a homogeneous group that is people with severe symptoms like vomiting and diarrhoea. Therefore, [7] divided infected individuals (I) into symptomatic infected (I s ) and asymptomatic infected (I a ), in order to observe the contribution of concentration of Vibrio cholerae in the environment through excretion from each compartment and hence how it leads to the transmission dynamics of cholera epidemic.
In the stochastic modeling of cholera epidemic few papers emphasized on cholera stochastic models. [8] developed a deterministic model and further, extended it stochastic differential equations, the limitations to their paper are: no numerical analysis considered in their paper and compartment I is considered as single compartment, it could be better to split it into two groups, individuals with severe symptoms (symptomatic infected individuals) and those with mild symptoms (asymptomatic infected individuals) so as to observe the contribution of these two groups to the concentration of Vibrio cholerae through excretion.
[9] [10] developed a simple deterministic and stochastic model to discuss the spread of cholera. In their paper, they described the spread of cholera by modeling the bacteria population in contaminated water and human interaction with the bacteria in the water supply. The limitation to their paper is the absence of enough theoretical and numerical analysis. In [11] proposed a deterministic model that described the interaction among the two types of vibrios and viruses. The deterministic model proposed was further extended to include the random effects and from the stochastic model formulated it indicated that there is always a positive probability of disease extinction within the human host.
In this paper, we extend the deterministic model developed in [7] by formulating an equivalent stochastic differential Equation (SDEs). The formulated stochastic differential Equation (SDEs) models will be analyzed theoretically using suitable Lyapunov functions, Itô formula and some stochastic techniques. Numerical simulation will be carried using Euler-Maruyama scheme and extended Kalman filter. Also, maximum likelihood estimation method will be used to estimate the model parameters.
In the next section, we present a deterministic cholera epidemic model with water treatment, which is a model formulated by [7]. In Section 3, the corresponding two different stochastic models are formulated using parametric perturbation technique and transition probabilities approach. In Section 4, the formulated stochastic differential equations are analyzed by proving the existence and positivity of solutions, stochastic boundedness, global stability in probability, moment exponential stability, and almost sure convergence. In Section 5, the numerical results from the deterministic and stochastic simulations are presented, discussed and compared. In Section 6, we provide a brief discussion to

Cholera Deterministic Model
From the description of the dynamics of cholera as shown in Figure 1, we the following set of ordinary differential equation system as in [7].
with suitable initial conditions. The total human population is given by b is the birth rate or recruitment rate, β is rate of exposure to contaminated water, κ is the concentration of Vibrio cholerae in water, 1 α is the contribution Figure 1. The interaction of cholera epidemic transmission dynamics between compartments is shown. The key parameter in epidemiology is the basic reproduction number, which is defined as the average number of secondary infectious cases transmitted by a single primary infectious cases introduced into a whole susceptible population [12]. This parameter is useful because it helps determine whether an infectious disease will spread within the population or not. To compute 0 R , the next generation matrix approach is used as in [13]. It is obtained by taking the largest  (1). Hence, the basic reproduction number obtained in [7] was as follows:  (2), when 0 1 R < , the highly infectious vibrios will not grow within human host and the environmental vibrios ingested into human body will not cause cholera infection. But when 0 1 R > the human vibrios will grow and persist, and hence leads to human cholera.

Stochastic Differential Equations
In this section we provide two approaches of formulating stochastic differential equations from the deterministic model (1). These approaches are parametric perturbation Itô SDEs and Itô SDEs from the transition probabilities.

Parametric Perturbation Itô SDEs
The idea of parametric perturbation is to choose a parameter of interest from the deterministic model and change it to random variable [14] [15]. In this study we introduce the randomness into the model by replacing parameters β and d by 3) For all times From the deterministic model (1) we get the following system of stochastic differential equations: with suitable initial conditions. The technique of parameter perturbation introduces another parameters 1 σ and 2 σ in a model.

Itô SDEs with Transition Probabilities
This method was proposed by [16] [17], the stochastic differential equations follow from the diffusion process. The nature of stochastic differential equation to be formulated is in this form: x t is an Itô process, On substituting the values of i P , and the co-variance matrix is given by  T  T   1   T  T  T  1  1  1  2  2  2  3  3  3   T  T  T  4  4  4  5  5  5  6  6  6   T  T  T  7  7  7  8  8  8  9  9  9   T  10 10 10 .
Also, by substituting the values of i P , The Itô stochastic differential equation, where global Lipschitz conditions are satisfied to ensure the existence and uniqueness of strong solution, can be written as in Equation (4) as found in [14]. Hence by Equation (5), the Itô stochastic differential equations become:

Analysis of Parametric Perturbation Itô Stochastic Differential Equation Model
In this paper, unless otherwise stated, we let satisfying the usual conditions (i.e., increasing and right continuous also 0  contains all  -null sets). Let : ,

Global Existence and Uniqueness of Positive Solutions
Before proving the existence and uniqueness of positive solutions, let us state the conditions that guarantee the existence and uniqueness of solution of Equation  (3) for all 0 t ≥ , and the solution will remain in 5 +  with the probability 1, namely Proof. Since the coefficients of system (3) satisfy the local Lipschitz condition, then for initial values In this study we define . As a consequence to this, there exist an integer 1 For k t ν ≤ and at each k, , V x t t is an Itô stochastic process with the SDEs given by where Introducing the expectation to Equation (18) and by the Gronwall inequality, leads to again define the indicator function on k Ω denoted by  be defined for all (19) and by Bienayme Chebyshev inequality we have when k → +∞ , this leads to contradiction as , , This completes the proof as required. □

Moment Exponential Stability
The moment exponential stability of the equilibrium solutions of stochastic differential Equation (3) On substituting these inequalities in Equation (19), we get Hence, according to Lemma 2 the proof is complete. □    In .

Almost Sure Convergence
Applying the Itô formula to function V above we get ( ) From Equation (47) Then, dV deduce to ( ) ( ) Applying integration from 0 to t both sides to Equation (52), we get Then, for all time , the quadratic variation of the Itô integral process Then, as 1 2 This leads to contradiction. Hence, This completes the proof of Theorem 6. □

SI I R B
− stochastic differential equations is shown in Figure 6. It is observed that the sample path is continuous but not differentiable (a Wiener process property).
Maximum likelihood estimation method is used to estimate the unknown parameters θ of a SDE (8) by maximizing the likelihood [16]. Journal of Applied Mathematics and Physics . The Euler-Maruyama scheme is used to simulate the sample paths of stochastic differential Equation (3) and the result is graphed in Figure 6. From Figure 6, we observe that the susceptible proportion eventually converges to zero; the entire population becomes infected, and later they recover from the disease. Also, the sample path of s a SI I R B − stochastic differential equations is continuous but not differentiable (a property of Wiener process).
The sample path of s a SI I R B − stochastic differential equations model together with the solutions of ordinary differential equations is graphed in Figure  7. From Figure 7, we find that the sample path of

SI I R B
− ordinary differential equation model.

Discussion
We have proposed a new modeling framework for the dynamics of cholera using both deterministic and stochastic models. Our focus is on the interaction of environmental vibrios to human (which causes the transformation from the environmental vibrios to human) and the infected individuals shedding bacteria into the environment. For deterministic model, we derived the basic reproduction number 0 R . The basic reproduction number is a critical parameter for disease dynamics. In the deterministic model, the value of the basic reproduction number 0 R determines the persistence or extinction of the disease. If 0 1 R < , the disease is eliminated, whereas if 0 1 R > , the disease persists in the population. From the deterministic model we have formulated two stochastic differential equations using parametric perturbation and transition probabilities methods.  We have proved the existence and uniqueness of positive solution, we showed that the solution of stochastic model are stochastically ultimately bounded, we derived that when 0 1 R < , then the infected compartments and bacteria goes to extinction. We carried out numerical simulation using Euler-Maruyama scheme to simulate the sample paths of stochastic differential Equation (3). Our results show that, the sample paths are continuous but not differentiable (a property of Wiener process) Also, we carefully compared the numerical simulation results for deterministic and stochastic models. We find that, the sample path of s a SI I R B − stochastic differential equations model fluctuates within the solution of the s a SI I R B − ordinary differential equation model as seen in Figure 7. However, the model parameters of SDEs are estimated by maximum likelihood estimation method. It is shown that the estimates are close to the true parameter values as seen in Table 2. Also, we used extended Kalman filter to estimate the states (compartments) of stochastic model (8) by recursively computing the transition probability density. It is observed that the state estimates fit the measurements as seen in Figures 2-5. Hence, we find that both models that are deterministic and stochastic models are very useful in understanding the dynamics of cholera epidemic. Nevertheless, Stochastic differential equation models are more important than deterministic models since they incorporate random effects such as environmental stochasticity and this enables us to model different quantities such as probability of extinction, probability of distributions and variances which cannot be captured in deterministic models.

Conclusions
In this paper, two stochastic differential equations models are formulated from the deterministic model using two different approaches: parametric perturbation and Transition probabilities. For deterministic model, the basic reproduction number 0 R determines whether the disease is eliminated or persists in the given population.
For stochastic model, the perturbed stochastic differential equation is first analyzed by proving the existence and positivity of the solutions. Secondly, we looked at the stability aspect of the model; we proved that the number of symptomatic infected, asymptomatic infected and bacteria tends to asymptotically to zero exponentially almost surely. Also, we showed that the equilibrium solution of the SDEs is pth moment exponentially stable and it is usually said to be exponentially stable in mean square. Numerical simulations are carried to simulate the sample paths of stochastic models by Euler-Maruyama scheme and the Kalman filter is used to estimate the states of stochastic model by recursively computing the transition probability density. It is observed that the state estimates fit the measurements. So, we can say that cholera transmission dynamics can be modeled using stochastic differential equations. It is clear that real world problems such as disease are not deterministic in nature so including random effects to the model gives us a more realistic way of modeling cholera epidemics and other epidemic diseases. For example, using stochastic differential equation model we managed to examine the limiting asymptotic distribution of the number of symptomatic infected, asymptomatic infected and bacteria.