1. Introduction

The modeling of communicable diseases with stochastic differential equation (SDE) has gained grounds recently due to its wide range of applications and its ability to reflect reality in epidemiology [3]. Diseases outbreak in a population of susceptibles logically follows stochastic processes, but not the idea of the robustic deterministic as perceived [12]. Stochastic process arises naturally in many physical applications where randomness is to be included in the mathematical model [5, 8]. Stochastic models are adopted when a small number of reacting molecules is present in a modeling system. In such instants of small numbers reacting molecules, fluctuation becomes inevitable and deterministic models become inappropriate to use [2]. In recent years, major studies on stochastic differential equations (SDEs) that have been published by researchers have identified the growing importance of investigating the stability of stochastic positive equilibrium, as well as the global stability of the disease-free and the endemic equilibrium [1, 4, 9].

In this paper, we consider the SI vaccination model proposed by Gardon et al. [7] as follows

(1)

Where and represent unvaccinated susceptible, vaccinated susceptible, unvaccinated infectives and vaccinated infectives respectively. The model answers one important underlying research subjects; the determination of the existence of the threshold parameter which hints on the spreading or dying out of an invading epidemic into a population of susceptible, as studied by various authors [15, 16]. Our motivation lies in the works of Maroufy et al. [14], Adnani et al. [6], Kiouach and Omari [13] and Mukherjee et al [10], who extended their deterministic models to stochastic versions, and studied the stability of the stochastic models. In this research article, we first study the positivity and boundedness of the system (1). The basic reproduction ratio is determined. Applying the hypothetical theorem of the Lyapunov functional, we determine the global stability of the two equilibria for system (1). We extend our stability analysis to the stochastic system (5), which is obtained by random perturbation of the deterministic system (1) and find the stability of its positive equilibrium. Finally, numerical examples which shows the dynamics of systems (1) and (5) are given, which gives the explicit difference in the dynamics of the models.

2. Positivity, Boundedness and Basic Reproduction Ratio

2.1. Positivity and Boundedness

The theory of ordinary differential equations requires that, for every set of initial conditions the state variables of the solution must remain non-negative.

Proposition 2.1. Let be the solution of the system (1).

(a) Given the initial condition then there exist a unique positive solution for every such that the solution will remain in with probability of one.

(b) The solution is defined in the interval and where

Proof:

In (a) we let Evidently, the coefficients of system (1) are locally Lipschitz continuous. Hence, for any given initial condition there exist a unique local solution for every where is the final time. Here, it can be deduced that for every Summing the total population of system (1) gives Suppose is the solution of the differential equation where Hence, by comparison theorem; as required.

Again, we can verify in (b) that

(2)

Integrating inequality (2) gives for every which implies It can therefore be verified that the solution is bounded within the interval This implies for every Hence Hence, employing the same intuition used in proving proposition 2.1, we see that system (1) with non-negative initial conditions has a non-negative solution defined in and the set is invariant by system (1).

2.2. The Basic Reproduction Ratio

The basic reproduction ratio is defined as an infections originating from an infected individual that invades a population originally of susceptible individuals. Our model calculation would be based on the approach of Diekmann and Heesterbeek ([17]). Here, the functions (F) and (V) denote the rate of new infection term and the rate of transfer into and out of the unvaccinated infectives and vaccinated infectives respectively.

The disease compartments are

Hence and

The deterministic system (1) has a unique disease-free equilibrium, given by where

The matrices evaluated at are given by

and

The matrix is a rank one matrix, and its next generation matrix also has rank one ([16]). The spectral radius of a rank one matrix is its trace.

Hence

(3)

2.3. Global Stability of the Disease-free Equilibrium

Theorem 2.3: The disease-free equilibrium is globally asymptotically stable in whenever

Proof: We consider the Lyapunov function

where are constants that would be chosen in the course of the proof.

Hence, calculating the rate of change of along the solution of gives,

Choosing gives the following

It follows that is positive definite and is negative definite. It can therefore be ascertained that the function is a Lyapunov function for system (1). Hence by Lyapunov asymptotic stability theorem [9], the equilibrium is globally asymptotically stable.

2.4. The Global Stability of the Endemic Equilibrium

Theorem 2.4: The unique endemic equilibrium is globally asymptotically stable in whenever

Proof: We consider the Lyapunovfunction

are constants to be chosen in the course of the proof. The derivative of along the solution of (1) gives

(4)

Choosing and such that and Then, the relation (4) can be expressed as

a positive definite and is negative definite. Therefore the function is a Lyapunov function for system (1) and consequently, by Lyapunov asymptotic stability theorem [13], the equilibrium state is globally asymptotically stable. Hence this completes the proof.

3. The Stochastic Model

Here, we introduce stochastic perturbations in the main parameters of the deterministic model (1). Thus we permit stochastic perturbations of the variable around their values at positive equilibrium

Hence, we assume that the white noise of the stochastic perturbations of the variable around values of are proportional to the distances of from Hence the stochastic version of model (1) is

(5)

With where are real constants, and are independent wiener processes. We investigate the asymptotic stability behavior of the equilibrium of the stochastic equation (5) and compare results with the deterministic model (1).

3.1. Stochastic Stability of the Positive Equilibrium

It can be shown clearly that, the deterministic model (1) has one disease-free equilibrium which is globally asymptotically stable when However, when the disease-free equilibrium is unstable . Obviously, there is also a unique positive endemic equilibrium

This equilibrium is globally asymptotically stable. The stochastic system (5) has the same equilibria as the deterministic system (1). Assuming that we investigate the stability of the endemic equilibrium of (5). The stochastic differential equation (5) can be centered at its positive equilibrium by the change of variables

(6)

The linearized system of the stochastic model (5) around takes the form

(7)

Where and equals

Clearly, the endemic equilibrium corresponds to the trivial solution in (7). We denote to be the differential operator associated with (7), defined for the family of nonnegative functions such that it is continuously differentiable with respect to and twice with respect to

According to Afanas’ ev et al [11], the differential operator for a function is given by

(8)

Where and

Where and are the transposition and trace respectively. With reference to Afanas’ ev et al [11], the following results hold.

Theorem 3.1: Suppose a function exist, satisfying the following inequalities

(9)

Where and Then the trivial solution of (7) is moment exponentially stable. Again, given that the trivial solution is said to be exponentially stable in mean square and the equilibrium is globally asymptotically stable.

From theorem 3.1, the conditions for stochastic asymptotic stability of trivial solution of (7) are given theorem 3.2.

Theorem 3.2: Suppose and hold, then the zero solution of (7) is asymptotically mean square stable.

Proof: We consider the Lyapunov

With non-negative constants that will be chosen in the course of the proof. It can be easily ascertained that inequality (9) hold true when

Applying the operator on gives

(10)

Further

(11)

Now remark that

and

(12)

Now, from equation (10), if we choose

and then from equation (10), it is easy to verify that,

Hence, according to theorem 3.1, the proof is completed.

4. Numerical Examples of the Models

Here, we illustrate with figures the dynamics of the systems (1) and (5), and gives an explicit difference in the models by carrying out numerical simulation for the hypothetical set of parameter values. To demonstrate the differences, we simulate the systems (1) and (5) by using the following set of parameter values; However, some of the parameters were allowed to change in the course of the simulations in order to bring out the dynamics of the models. The differences in the dynamics of the models are therefore given by the diagrams in the following:

Example 1:

Here, the dynamic behaviors of the four classes of individuals of the deterministic and its stochastic version are plotted against time. Here we assumed the following set of hypothetical parameter values; Calculating based on theses parameter values gives To confirm the deterministic plot of figure1a, we choose white noises and of equal strength and shows the fluctuations in the trajectories of the plot of the stochastic system (5). We can see that the trajectories of the stochastic plot displayed on our graphs are the same as the trajectories of its deterministic model (1) during a finite time frame.

Figure 1a.

Figure 1b.

Example 2:

Here, we choose the same choice of parameter values: except for and the same strength of white noise Again, calculating on these parameter values gives We observe that the path of the trajectories of the systems (1) and (5) are eventually absorbed in the stable point (see figure 1a, 1b, 2a, 2b).

Figure 2a.

Figure 2b.

Examples 3

Choosing and we observe that system (1) and (5) are stable (see figures 3a and 3b). Again, the path of the stochastic processes leaves the trajectories and is absorbed in the equilibrium (see figure 3b).

Figure 3a.

Figure 3b.

5. Conclusions

In this paper, the dynamics of deterministic SI vaccination model and its stochastic variant are presented. The stability analyses of the deterministic model were investigated. Suitable Lyapunov functions were constructed for the global stability of the two equilibria. We constructed the stochastic version of the model by employing the idea of Mukhere et al [4, 9, 10].

Our main purpose of the study was to investigate the asymptotic stability behavior of the endemic equilibrium of the stochastic version of the deterministic SI model proposed by Gardon et al [7]. The numerical simulation for the models shows that the trajectories of the stochastic plots were the same as the trajectories of the deterministic model. Further, from our stochastic plots, the simulation shows an initial random fluctuation of the stochastic trajectories, until they eventually approach asymptotic level. We have demonstrated that our stochastic system is globally asymptotically stable in probability when the densities of white noise are less than certain threshold parameters. However, if these densities of white noise are zero, it means there are no stochastic environmental factors on the population and hence no stochastic perturbation. Hence theorem 3.2 conditions would be reduced to the condition which implies a nonlinear stability condition for the deterministic system (1). In our future research, we would consider how control strategies may be devised for the model.