A stochastic model for the transmission dynamics of hepatitis B virus

ABSTRACT In this paper, we formulate a stochastic model for hepatitis B virus transmission with the effect of fluctuation environment. We divide the total population into four different compartments, namely, the susceptible individuals in which the disease transmission rate is distributed by white noise, the acutely infected individuals in which the same perturbation occur, the chronically infected individuals and the recovered individuals. We use the stochastic Lyapunov function theory to construct a suitable stochastic Lyapunov function for the existence of positive solution. We also then establish the sufficient conditions for the hepatitis B extinction and the hepatitis B persistence. At the end numerical simulation is carried out by using the stochastic Runge–Kutta method to support our analytical findings.


Introduction
The environmental variation has a critical influence on the infectious diseases, which is the second leading cause of death in the world [3,14,16]. Biological phenomenon are always effected due to the environmental noise. West Nile virus, Dengue, Malaria and hepatitis B etc., are some of the best examples, for which the nature of epidemic growth and spread is inherently random due to the unpredictability of person-to-person contacts and population is subject to a continuous spectrum of disturbances [1]. The contagious disease of hepatitis B causes inflammation of liver results from hepatitis B virus infection in which the epidemics growth and spread are random due to the unpredictability of person-to-person contacts [2]. Together with these realistic patterns, theoretical tactics for taking environmental variation in the epidemiological models deliver a useful framework for discovering the role of environment in population ecology.
Mathematical modelling is an influent tool to describe the dynamical behaviour of various infectious diseases in the real world problem. Several mathematicians and ecologists have developed different epidemic models to realize and control the spread of transmissible diseases in the community. In the past two decades the field of mathematical modelling has been used frequently for the study of communication of different types of infectious diseases (see e.g. [13,15,17,20]). There are two types of epidemic models viz deterministic epidemic models and the stochastic epidemic models. But for modelling the biological phenomena stochastic differential equation is more reasonable then deterministic [4]. Stochastic models produce more valuable output then deterministic models as by running a stochastic model several time, we can build up a distribution of the predicted outcomes, e.g. the number of infected classes at time t, whilst a deterministic model will just give a single predicted value [6][7][8]12,21,22]. Many epidemic models have been discovered for the description of viral dynamic of hepatitis B, which are mostly using the deterministic approach (see for detail [5,9,11,18,23]). Recently Khan et al. [10] presented a stochastic model for the transmission dynamic of hepatitis B.
In the resent study, we proposed a stochastic epidemic model with a varying population environment by extending the work of Khan et al. [10]. Keeping in view the characteristic of hepatitis B, we divided the total population in four different compartments, the susceptible population in which the transmission rate is distributed by white noise, the infected individuals are further divided into two stages, namely the acute, the chronic hepatitis B individuals and the recovered population. For the existence and uniqueness of positive solution to the proposed stochastic hepatitis B model, we use the Lyapunov function theory and construct a suitable stochastic Lyapunov function. We then discuss the hepatitis B extinction and the hepatitis B persistence and derive the sufficient condition for the hepatitis B. Moreover, we carry out the numerical simulation by using the stochastic Runge-Kutta method to support our analytical findings.

Stochastic hepatitis B model
In this section, we present the stochastic epidemic model for the transmission dynamic of hepatitis B with varying population environment. We place the following assumption on the model: (X 1 ). The total population N(t) at time t is divided into four compartments: the susceptible individuals, S(t), the acutely infected with hepatitis B individuals, A(t), the chronically infected with hepatitis B individuals, C(t) and the recovered individuals, . All parameters and state variables of the proposed model are non-negative. (X 3 ). The contact of susceptible individuals with acutely and chronically infected hepatitis B individuals primarily leads to acutely infected class. (X 4 ). For the effect of randomly fluctuating environment taking, is the standard Brownian motion with the property B i (0) = 0 and with the intensity of white noise η 2 i > 0.
The assumption (X 1 )-(X 4 ) leads to the stochastic hepatitis B epidemic model, which is represented by the following system of four stochastic differential equations: where represents the per capita constant birth rate. β i for i = 1,2 represent the transmission rate of hepatitis B. The natural and disease mortality rates are denoted by μ 1 and μ 0 respectively. v represents the vaccination rate of hepatitis B. γ represents the moving rate of acutely infected individuals to chronic stage. γ 1 and γ 2 are respectively described the constant recovery rate of acutely and chronically infected hepatitis B individuals.

Preliminaries
Let ( , F, P) be a complete probability space with a filtration {F t } t≥0 satisfying the usual conditions (i.e. it is increasing and right continuous while F 0 contains all P-null sets), B i (t) (i = 1, 2, 3, 4) are defined on this complete probability space, we also let R n + = {x ∈ R n : x i > 0, 1 < i ≤ n}. Consider the n-dimensional stochastic differential equation with initial value x(0) = x 0 ∈ R n and B(t) represents an n-dimensional standard Brownian motion on the complete probability space ( , F, {F t } t≥0 , P). The differential operator L associated with Equation (2) is defined by where Using the Itô formula, if x(t) ∈ R n , then

Existence and uniqueness
In this section, we discuss the solution of the stochastic hepatitis B model (1).

s. (almost surely).
Proof: The co-efficient of the equation is locally Lipschitz continuous for any given initial size of population (S(0), where τ e is the explosion time (for detail see references [4,5]). To show that this solution is global, we prove that τ e = ∞ a.s. Let k 0 ≥ 0 be sufficiently large, so that S(0), I 1 (0), I 2 (0) and R(0) all lie within the interval [1/(k 0 ), k 0 ]. For each integer k ≥ k 0 , define the stopping time In this paper, we set infφ = ∞, φ denotes the empty set. According to the definition, s., for all t ≥ 0. In other words to complete the proof, we need to show that τ e = ∞ a.s. If this statement is false, then there exist a pair of constants T > 0 and ∈ (0, 1), such that Hence there is an integer k 1 ≥ k 0 , such that Let The solution of Equation (10) yields Now, we define a C 2 -function F : Clearly the function F is non-negative, which can be seen from u − 1 − log u ≥ 0, for all u > 0. Let k ≥ k 0 and T ≥ 0 be arbitrary. Applying the Ito formula on Equation (12), we get dF(S, I 1 , where LF : R 4 + → R + is defined by the following equation: Therefore Setting k = τ k ≤ T for k ≥ k 1 and by Equation (9), P( k ) ≥ . Note that for every ω ∈ k , there exist at least one S(τ k , ω), I 1 (τ k , ω), I 2 (τ k , ω) R(τ k , ω) that equal k or 1/k, and hence F(S(τ k ), It is then follows from Equations (9) and (15) that where 1 (ω) is the indicator function of . Letting k → ∞ leads to the contradiction ∞ > F(S(0), I 1 (0), I 2 (0), R(0)) + MT = ∞, which implies that τ ∞ = ∞ a.s.
Next, we investigate the solution of the second equation of system (1), which looks like Similarly it can be shown that I 2 (t) ≥ 0 and R(t) > 0. Hence (S(t), I 1 (t), I 2 (t), R(t)) ∈ R 4 + , for all t ≥ 0.

Extinction and persistence
In this section, we derive the condition for the extinction and persistence of the hepatitis B. For convenience, we introduce the following notation and definition. Let The parameter R 0 is defined to be the basic reproduction number R 0 for the proposed stochastic hepatitis B model (1), which is given by the following equation: where

Extinction
For the extinction of the disease, we have the following result.

Theorem 5.1: Let (S(t), I 1 (t), I 2 (t), R(t)) be the solution of the proposed hepatitis B model (1) with initial value (S(0),
Proof: The integration of the proposed stochastic hepatitis B model (1) leads to the following system of equations: We compute from Equation (28) that where φ 1 (t) is defined by the following equation: Obviously φ 1 (t) → 0 as t → ∞. Applying the I to formula to the second equation of model (1), which yields (30) Integrating Equation (30) from 0 to t and dividing by t, which leads to the following equation: Similarly it can be shown that lim t→∞ I 2 (t) = 0.
Next, we prove the assertion (27), so from the proposed stochastic hepatitis B model (1), we have The solution of Equation (37) yields that Applying the L'Hospital rule together with the application of Equation (37), we obtain Thus, we have lim t→∞ (S(t) + R(t)) = μ 0 a.s.
According to the model (1), the first equation with limiting system becomes Similarly, we obtain a.s.

Persistence
For the persistence of the disease, we have the following result.
Proof: Again, we compute from Equation (29) that where φ 2 (t) is defined by the following equation: Using Equation (41) in Equation (31), we obtain ≤ μ 0 + γ + γ 1 + The inequality (46) with the application of Equation (35) and then taking the limit inferior of both side leads to the following equation: Thus from Equations (44) and (47), we have the assertion (40), i.e.

Numerical simulation
In this section, the stochastic Runge-Kutta method is used to find the numerical simulation of the proposed stochastic hepatitis B epidemic model (1) for the verification of our analytical findings. Clearly, we observe the influence of noise intensity on the hepatitis B transmission. We perform simulation of the model, see and therefore, the acutely infected with hepatitis B and chronically infected with hepatitis B population exponentially tends to zero with probability one as shown in Figure 1(b,c). This indicates the extinction of the hepatitis B. Similarly, in the case of persistence the parameter values chosen are = 0.5, β 1 = 0.6, β 2 = 0.5, μ 0 = 0.1, v = 0.3, μ 1 = 0.02, γ = 0.1, γ 1 = 0.4, γ 2 = 0.3, η 1 = 0.5 and η 1 = 0.6, which ensure the condition for the hepatitis B persistent, i.e. Theorem 5.2 holds. In other words, we have R 1 = 1.277213353 > 1, β 1 (μ 0 + v) = 0.24 > η 2 1 = 0.0625. Thus the model (1) maintains the persistence (see Figure 2).

Conclusion
Real world problems are not deterministic but including stochastic effect. In this paper, we formulated the stochastic epidemic model for the hepatitis B virus transmission and studied the dynamic with varying population environment. We then showed the existence of positive solution and established a suitable stochastic Lyapunov function by using the concept of Lyapunov functions theory. We also discussed the extinction as well as the persistence and derived the sufficient condition for the hepatitis B extinction and the hepatitis B persistence. The conditions of disease extinction and disease persistence are in the form of expressions involving the system parameters and intensities of noise terms. Clearly, we observed the influence of noise intensity on the hepatitis B transmission. The extinction of the hepatitis B infected individuals increasing with the increasing noise strength on the susceptible population. Similarly the disease persisting decreases with the increasing noise strength. Finally, we performed the numerical simulation for supporting our analytical findings. Our work shows that stochastic epidemic models give another option to model the phenomena of viral dynamic, which give us a more realistic way.