The asymptotic behavior of a stochastic SIS epidemic model with vaccination
- Yanan Zhao^{1, 2},
- Qiumei Zhang^{1, 2} and
- Daqing Jiang^{1, 3}Email author
https://doi.org/10.1186/s13662-015-0592-6
© Zhao et al. 2015
Received: 5 January 2015
Accepted: 3 August 2015
Published: 22 October 2015
Abstract
In this paper, we discuss a stochastic SIS epidemic model with vaccination. We investigate the asymptotic behavior according to the perturbation and the reproduction number \(R_{0}\). When the perturbation is large, the number of infected decays exponentially to zero and the solution converges to the disease-free equilibrium regardless of the magnitude of \(R_{0}\). Moreover, we get the same exponential stability and the convergence if \(R_{0}<1\). When the perturbation and the disease-related death rate are small, we derive that the disease will persist, which is measured through the difference between the solution and the endemic equilibrium of the deterministic model on average in time if \(R_{0}>1\). Furthermore, we prove that the system is persistent in the mean. Finally, the results are illustrated by computer simulations.
Keywords
MSC
1 Introduction
Epidemiology is the study of the spread of diseases with the objective to trace factors that are responsible for or contribute to their occurrence. Significant progress has been made in the theory and application of epidemiology modeling by mathematical research. In a simple epidemic model, there is generally a threshold, \(R_{0}\). If \(R_{0} \leq1\), the disease-free equilibrium is a unique equilibrium in this type of epidemic model and it is globally asymptotically stable; if \(R_{0}>1\), this type of model has also a unique endemic equilibrium, which is globally asymptotically stable. Therefore, the threshold \(R_{0}\) determines the extinction and persistence of the epidemic.
Controlling infectious diseases has been an increasingly complex issue in recent years. Vaccination is an important strategy for the elimination of infectious diseases [1–3]. The vaccination enables the vaccinated to acquire a permanent or temporary immunity. When the immunity is temporary, the immunity can be lost after a period of time. It is used in many references [4–7] where one assumes the process of losing immunity is in the exponential form.
A: the constant input of new members into the population per unit time;
q: the fraction of vaccinated new-borns;
β: transmission coefficient between compartments S and I;
μ: the natural death rate of the S, I, V compartments;
p: the proportionality coefficient of vaccinated cases for the susceptible;
γ: the recovery rate of infectious individuals;
ε: the rate of losing their immunity for vaccinated individuals;
α: the disease-caused death rate of infectious individuals.
All parameter values are assumed to be nonnegative and \(\mu,A> 0\).
However, the deterministic approach has some limitations in the mathematical modeling of the transmission of an infectious disease and it is quite difficult to predict the future dynamics of a system accurately. This happens due to the fact that deterministic models do not incorporate the effect of a fluctuating environment. Stochastic differential equation models play a significant role in various branches of applied sciences including infectious dynamics, as they provide some additional degree of realism compared to their deterministic counterpart. In reality, parameters involved in the modeling approach of ecological systems are not absolute constants, and they always fluctuate around some average value due to continuous fluctuations in the environment. As a result, parameters in the model never attain a fixed value with the advancement of time and rather exhibit a continuous oscillation around some average values. Many authors have introduced parameters of random perturbation into epidemic models and have studied their dynamics.
Jing Fu et al. introduced stochasticity into a multigroup SIS model in [25]. They presented the sufficient condition for the exponential extinction of the disease and proved that the noises significantly raise the threshold of a deterministic system. In the case of persistence, they proved that there exists an invariant distribution which is ergodic.
In [26], Golmankhaneh et al. applied the homotopy analysis method (HAM) successfully for solving second-order random differential equations, homogeneous or inhomogeneous. Expectation and variance of the approximate solutions were computed. Several numerical examples were presented to show the ability and efficiency of this method. Jafarian et al. in [27] solved linear second kind Fredholm and Volterra integral equations systems by applying the Bernstein polynomials expansion method. Illustrative examples were provided to demonstrate the preciseness and effectiveness of the proposed technique.
This paper is organized as follows. In Section 2, we show there is a unique positive solution of system (1.3) for any positive initial value. In Section 3, we show that the positive solution of system (1.3) converges to \(P_{0}\) exponentially as the perturbation is large. In Section 4, we get the same exponential stability and the convergence when \(R_{0}<1\). On the other hand we investigate the asymptotic behavior of the solution of the system (1.3) according to \(R_{0} > 1\) although the solution of system (1.3) does not converge to \(P^{*}\). When the perturbation is not large, we consider the disease to persist. Moreover, we show system (1.3) is persistent in the mean. The key to our analysis is choosing appropriate Lyapunov functions. In Section 5, we make simulations to conform our analytical results. In Section 6, we give a short conclusion. Finally, in order to be self-contained, we have an Appendix containing a lemma used in the previous sections.
2 Existence and uniqueness of positive solution
To investigate the dynamical behavior, the first concern is whether the solution has a global existence. Moreover, for a population dynamics model, whether the value is positive is also considered. Hence in this section we first show that the solution of system (1.3) is global and positive. As we know, in order to get a stochastic differential equation which has a unique global (i.e. no explosion in finite time) solution for any given initial value, the coefficients of the equation are generally required to satisfy the linear growth condition and local Lipschitz condition (cf. [20]). However, the coefficients of system (1.3) do not satisfy the linear growth condition, though they are locally Lipschitz continuous, so the solution of system (1.3) may explode at a finite time. In this section, using the Lyapunov analysis method (mentioned in [11–13]), we show the solution of system (1.3) is positive and global.
Theorem 2.1
There is a unique solution \((S(t),I(t),V(t))\) of system (1.3) on \(t\geqslant0\) for any initial value \((S(0),I(0),V(0))\in\mathbb{R}_{+}^{3}\), and the solution will remain in \(\mathbb{R}_{+}^{3}\) with probability 1, namely, \((S(t),I(t),V(t))\in\mathbb{R}_{+}^{3}\) for all \(t\geqslant0\) almost surely.
Proof
Remark 2.1
From now on, we always assume that \((S(0),I(0),V(0))\in\Gamma^{*}\).
3 Exponential stability in large perturbation
In this section, we show that the large perturbation forces the number of infected cases to zero exponentially regardless of the magnitude of \(R_{0}\).
Theorem 3.1
Proof
Remark 3.1
When the perturbation is very large, the number of infected decays exponentially to zero. It makes sense in the point that the extinction of epidemics can be caused by the occurrence of a large perturbation.
4 Asymptotic behavior around \(P_{0}\) and \(P^{*}\)
When studying epidemic dynamical systems, we are interested in two problems. One is when the disease will die out, and the other is when the disease will persist. In the section, we shall investigate the two problems according to the threshold \(R_{0}\).
4.1 Asymptotic behavior around \(P_{0}\)
When \(R_{0}\leqslant1\), \(P_{0}\) of system (1.1) is globally stable, which means the disease will die out. We shall show in Theorem 4.1 below, the same exponential stability of system (1.3), obtained in Theorem 3.1, continues valid when \(R_{0} < 1\). Moreover, in this case, we prove the solution to (1.3) converges to \(P_{0}\), a.s.
Theorem 4.1
Proof
4.2 Asymptotic behavior around \(P^{*}\)
In the deterministic models, the second problem is usually solved by showing that the endemic equilibrium to the corresponding model is a global attractor or is globally asymptotically stable under some conditions. But there is no endemic equilibrium of system (1.3). How can one measure whether the disease will persist? In this section, we show that the difference between the solution of system (1.3) and \(P^{*}\) is small if white noise is weak, reflecting that the disease is prevalent.
Theorem 4.2
Proof
Remark 4.1
From the result of Theorem 4.2, we conclude system (1.3) is persistent, which also reflects that the disease is prevalent. Chen et al. in [21] proposed the definition of persistence in the mean for the deterministic system. Here, we also use this definition for the stochastic system.
Definition 4.1
Theorem 4.3
Proof
5 Numerical simulations
6 Conclusions
This paper studies the extinction and persistence in the mean of a stochastic SIS epidemic model with vaccination. When the perturbation is very large, the number of infected decays exponentially to zero, which means the disease will die out. A similar conclusion to the system (1.2) is obtained in Theorem 4.3 in [12], but there one simplified system (1.2) into a single equation without vaccination and \(\alpha=0\). The present paper is the first attempt, to the best of our knowledge, of such a study of a large perturbation that surpasses the effect of \(R_{0}\) as a threshold value in a high-dimensional system. Theorem 4.1 showed \(P_{0}\) is exponentially stable if \(R_{0}<1\). The result is better than the asymptotic stability of \(P_{0}\) in Theorem 3.1 in [11]. From Theorems 4.2 and 4.3, when the perturbation and the disease-related death rate are small, if \(R_{0}>1\), we see that the disease will persist in the mean. In such a case, \(R_{0}\) plays a role similar to the threshold of the deterministic model. Hence, a large perturbation surpasses the effect of \(R_{0}\) as a threshold value, and a small perturbation retains some role of \(R_{0}\) in a stochastic sense. Also, it is interesting to study the threshold \(\tilde{R}_{0}\) of a stochastic SIR model, and these investigations are in progress.
Declarations
Acknowledgements
The work was supported by Program for NSFC of China (No: 11371085,11426060), and the Fundamental Research Funds for the Central Universities (No: 15CX08011A), the Scientific and Technological Research Project of Jilin Province’s Education Department (No: 2012244) and the Education Science Research Project of Jilin Province (No: GH150104).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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