ABSTRACT

In this paper, we consider a SVIR-B cholera model with imperfect vaccination. By analyzing the corresponding characteristic equations, the local asymptotically stability of a disease-free equilibrium and an endemic equilibrium is established. We calculate the certain threshold known as the basic reproduction number R_v. If R_v < 1, we obtain sufficient conditions for the global asymptotically stability of the diseasefree equilibrium, the diseases will be eliminated from the community. By comparison arguments, it is pro_ved that if Rv > 1, the unique endemic equilibrium is local asymptotically stable. We perform sensitivity analysis of Rv on the parameters in order to determine their relative importance to disease control and show that an imperfect vaccine is always beneficial in reducing disease spread within the community.

1. INTRODUCTION

Cholera is an acute intestinal infection caused by ingestion of food or water contaminated with the bacterium vibrio cholera. Since Koch found vibrio cholera in 1883, the research for cholera vaccine has more than one hundred years. People have developed a variety of vaccines. However, these vaccines were parenteral, which have short effective protection and big side effects. In 1973, the World Health Organization canceled the vaccine inoculation which attracted a major concern to oral vaccines. At present, there are three kinds of oral vaccine (i.e., WC/BS vaccine, WC/rBS vaccine and CVD₁₀₃-HgR vaccine) have been proved to be safe, effective and immunogenic, which were approved to apply in some countries [1].

In this paper, according to the natural history of cholera, we improve the model of [2] in the following two aspects. Firstly, if the cholera persists for a long time, it will cause the death [3], especially in the area where water and sanitation resources are not adequate [4], a parameter d is added to describe the rate of disease-related death. Secondly, we propose a proportion of the vaccination in susceptible individuals. As is shown in the following differential equations:

(1.1)

The flow diagram of the model is depicted in Figure 1. Since the first three and last equations in (1.1) are independent of the variable R, it suffices to consider the following reduced model:

(1.2)

Here, S, V, I and R refer to the susceptible individuals, vaccinated individuals, infected individuals, and recovered individuals, respectively.

The pathogen population at time t, is given by B(t). The parameter μ₁ denotes the natural human birth and

death rate, α denotes the rate of recovery from the disease, η represents the rate of human contribution to the growth of the pathogen, and μ₂ represents the death rate of the pathogen in the environment. The coefficients β₁ and β₂ represent the contact rates for the human-environment and human-human interactions, respectively. The rate at which the susceptible population is vaccinated is f, and the rate at which the vaccine wears off is θ.

All parameters are assumed non-negative, and the initial conditions of the system (1.2) are assumed as following

(1.3)

The organization of this paper is as follows: the positivity and boundedness of solutions are obtained in Section 2. In Section 3, we firstly calculate the basic reproduction number and obtain the existence of the endemic equilibrium. We get the local and global asymptotically stability of the disease-free equilibrium in Section 4. In Section 5, we show that the local asymptotically stability of the endemic equilibrium. We analyze the sensitivity of R_v on the parameters, and we present the numerical simulation in Section 6. The paper ends with a conclusion in Section 7.

2. POSITIVITY AND BOUNDEDNESS OF SOLUTIONS

In the following, we show that the solutions of the system (1.2) are positive with the non-negative initial conditions (1.3).

Theorem 2.1. The solutions (S(t), V(t), I(t), B(t)) of the model (1.2) are non-negative for all t > 0 with initial conditions (1.3).

Proof. The system (1.2) can be put into the matrix form

where and is given by

We have

Therefore,

Due to Lemma 2 in [5], any solution of the system (1.2) is such that for all. This completes the proof of Theorem 2.1.

Theorem 2.2. All solutions (S(t), V(t), I(t), B(t)) of the model (1.2) are bounded.

Proof. The system (1.2) is split into two parts, the human population (i.e., S(t), V(t), and I(t)) and pathogen population (i.e., B(t)). It follows from the first three equations of the system (1.2) that

then it follows that. From the first equation, we can get

Thus, as. It is easy obtain

, as. From the last equationwe can obtain

.

Hence, , when. Therefore, all solutions (S(t), V(t), I(t), B(t)) of the model (1.2) are bounded.

From above discussion, we can see that the feasible region of human population for system (1.2) is

and the feasible region of pathogen population for system (1.2) is

.

Define. Let denote the interior of Ω. It is easy to verify that the region Ω is a positively invariant region (i.e., the solutions with initial conditions in Ω remain in Ω) with respect to the system (1.2). Hence, we will consider the global asymptotically stability of (1.2) in region Ω.

3. THE EXISTENCE OF EQUILIBRIA

In this section, we investigate the existence of equilibria of system (1.2). Solving the right hand side of the model system (1.2) by equating it to zero, we obtain the following biologically relevant equilibria.

It is easy to see that model (1.2) always has a disease-free equilibrium (the absence of infection, that is,

, , where and

. Let. Then model (1.2)

can be written as

where

We can get

giving

.

is the next generation matrix for model (1.2). It then follows that the spectral radius of matrix is

. According to Theorem 2 in

[6], the basic reproduction number of model (1.2) is

.

In the following, we will discuss the case with R_v > 1. The existence and uniqueness of the endemic equilibrium is established as follows.

The endemic equilibrium can be deduced by the following system:

which gives,

Obviously, when R_v > 1, ,.

Theorem 3.1. The system (1.2) has a unique endemic equilibrium when R_v > 1 and no positive endemic equilibrium when R_v < 1.

4. STABILITY OF DISEASE-FREE EQUILIBRIUM

Now, we will discuss the local and global asymptotically stability of the disease-free equilibrium. From above and [6], we can obtain the following theorem.

Theorem 4.1. The disease-free equilibrium E₀ is locally asymptotically stable for R_v < 1 and unstable for R_v

> 1.

Proof. The Jacobian matrix of the system (1.2) at X = E₀ is

The characteristic polynomial of the matrix is given by

where

If R_v < 1, then

further

After some calculations, if R_v < 1 we have a₁ > 0, a₂ > 0, a₃ > 0, a₄ > 0, a₁a₂ − a₃ > 0, (see Appendix A). Thus, using the Routh-Hurwitz criterion, all eigenvalues of have negative real part, E₀ is local asymptotically stable for the system (1.2). If, then and we show that has at least one eigenvalues with non-negative real part. Consequently, E₀ is not stable.

Theorem 4.2. When R_v < 1 the disease-free equilibrium is globally asymptotically stable.

We will prove the global asymptotically stability of the disease-free equilibrium using Lemma 4.1.

Lemma 4.1. [7] If a model system can be written in the form

where denotes(its components) the number of uninfected individuals and denotes (its components) the number of infected individuals including latent, etc. denotes the disease-free equilibrium of the system.

And assume that

(H1) is globally asymptotically stable;

(H2) for

, where the Jacobian matrix

is an Metzler matrix(the off-diagonal elements of are non-negative) and Ω is the region where the model makes biological sense. Then the fixed point is a globally asymptotically stable equilibrium of cholera model system (1.2) provided that R_v < 1.

We begin by showing condition (H1) as

.

For the equilibrium, the system reduces to

The characteristic polynomial of the system is given by

There are two negative characteristic foots are,. Hence, is always globally asymptotically stable.

Next, applying Lemma 4.1 to the cholera model system (1.2) gives

which is clearly an Metzler matrix. Meanwhile, we find Hence, the disease-free equilibrium is globally asymptotically stable.

5. STABILITY OF THE ENDEMIC EQUILIBRIUM

Now we consider the case with R_v > 1. The stability of the endemic equilibrium is established as follows:

Theorem 5.1. If is locally asymptotically stable.

Proof. Let

The Jacobian matrix at is

The characteristic polynomial of the matrix is given by

where

Based on Eq.3.3 and Eq.3.4, we have . It is then easy to observe that

further,

.

After some calculations, we have

(see Appendix B). Using the well-known Routh-Hurwitz criterion, the proof is thus complete.

6. SENSITIVITY ANALYSIS OF R_v

To facilitate the interpretation of the sensitivity of R_v, we now present some numerical simulations by using the set of parameters values in Table 1.

Now, we regard the vaccinated rate f and the wanning rate θ as the control parameter, while the other parameters are fixed. From Figures 2 and 3, the effects of various parameters, i.e., f and θ on the basic reproduction number R_v have been shown. It is noted that as the parameter f increases, R_v decreases; as θ decreases, R_v decreases. In fact, we can obtain the critical values of f and θ that reduce R_v to 1,

and

Table 1. Estimation of parameters.

In Figure 2, we select θ = 0.07, 0.03, 0.007, 0.0001, corresponding f_v = 2.07, 0.89, 0.21, 0.01, respectively. We can see that when the wanning rate θ has a greater value, f_v has not reasonable value so that when f > f_v, R_v < 1. Similarly, in Figure 3, we select f = 0.01, 0.1, 0.3, 0.6, 0.99, corresponding θ_v = 0.0002, 0.003, 0.01, 0.02, 0.03, respectively. We can see that when f is smaller, θ_v has not reasonable value so that when θ < θ_v, R_v < 1. Thus, the basic reproduction number can not reduces below unity only by increasing θ or decreasing f. The critical values f_v and θ_v play a key role in regulation the infection magnitude. In order to reduce R_v to 1, a greater vaccinated rate than f_v and a smaller wanning rate than θ_v have to be achieved simultaneously. We will deduce R_v below 1 by using both f and θ at the same time, which can control cholera (see Figure 4).

7. CONCLUSION

In this paper, we have conducted stability analysis of a SVIR-B cholera model. The mathematical analysis results show that the basic reproduction number R_v satisfies a threshold property with threshold value 1. R_v in our model include the parameters f and θ which reflect the effect of vaccination. Numerical simulation show also that the vaccination is always beneficial to the eradication of cholera.

However, there are inherent disadvantages towards the vaccination modeling. For cholera with incubation period, it is hard to rapidly identify those with ambiguous symptoms [4]. Moreover, the vaccination does not always work well due to the limitations of medical development level and financial budget (some vaccine is very expensive and some portion of people cannot be covered) [10].

Hence, incorporating some other control strategies, for example, public health improvement, isolation etc, we may consider the more realistic ordinary differential equation model. The theoretical study of cholera models has been in progress, and is an exciting area of future research.

8. ACKNOWLEDGEMENTS

This work is supported by the National Natural Science Foundation of China (no.11071011), Funding Project for Academic Human Resources Development in Institutions of Higher Learning Under the Jurisdiction of Beijing Municipality (no.PHR201107123).

REFERENCES

APPENDIX A

From Section 4, we know that R_v < 1,

and

.

After some algebraic manipulations, we have, Thus, , when R_v < 1.

APPENDIX B

From Section 5, we know that

and

.

After some algebraic manipulations, we have, Thus, , when.