- Research
- Open Access
Construction of positivity-preserving numerical method for stochastic SIVS epidemic model
- Wenrui Li1 and
- Qimin Zhang1Email author
https://doi.org/10.1186/s13662-019-1966-y
© The Author(s) 2019
- Received: 8 October 2018
- Accepted: 15 January 2019
- Published: 24 January 2019
Abstract
In this paper we propose the balanced implicit numerical techniques for maintaining the nonnegative path of the solution in stochastic susceptible–infected–vaccinated–susceptible (SIVS) epidemic model. We can hardly acquire the explicit solution for the SIVS model, so we often use the numerical scheme to produce approximate solutions. The Euler–Maruyama (EM) method is a useful and effective means in producing numerical solutions of SIVS model. The EM method to simulate the stochastic SIVS model often results in the problem that the numerical solution is not positive. In order to eliminate the negative path of the solution in a stochastic SIVS epidemic model, we construct a numerical method preserving positivity for the SIVS model. It is proved that the balanced implicit method (BIM) can preserve positivity and we show the convergence of the BIM numerical approximate solution to the exact solution. Finally, a numerical example is offered to support the theoretical results and verify the availability of the approach.
Keywords
- Euler–Maruyama scheme
- SIVS model
- Balanced implicit method
- Convergence
1 Introduction
Recent global infectious diseases (such as the outbreak of H7N9 influenza in 2013 and Ebola disease in 2014) resulted in a lot of biological deaths and substantial financial ruins. Infectious diseases are a major concern of public. The modeling of infection diseases is extremely important to research the mechanisms of diseases. A mathematical model is considered as an effective way to forecast the outbreak of disease. In particular, stochastic epidemic models have come to play an important role in the control of diseases, which is an extremely significant tool to account for the real world.
For model (1.1), by [7] and [8] existence and persistence were discussed, respectively. In [9], Yang et al. provided the global threshold dynamics of an SIVS model with waning vaccine-induced immunity and nonlinear incidence. Zhao [10] gave the threshold of a stochastic SIVS epidemic model with nonlinear saturated incidence. Wen et al. in [11] remarked that the threshold of a periodic stochastic SIVS epidemic model with nonlinear incidence. But the stochastic SIVS epidemic model (1.1) rarely has an explicit solution. Therefore, numerical schemes or approximation techniques become the most focus problems in the analysis of stochastic SIVS model is their numerical solution. Up to now, the Euler–Maruyama (EM) scheme is prevalent for stochastic differential equations, which is due to the the simple structure and moderate computational cost. There are many articles investigating this method. In particular, Mao [12] made use of the truncated Euler–Maruyama method to discretize the stochastic differential equations. Hu et al. [13] showed a remarkable result of the semi-implicit Euler–Maruyama scheme for stiff stochastic equations. It also should be mentioned that in [14, 15] we find the Euler–Maruyama approximation of solutions to stochastic differential equations.
When we use EM scheme for the initial model (1.1), it is crucial that whether the numerical approximate solution is able to converge to the exact solutions. On the other hand, positivity is the most basic trait in many real world systems. For instance, featuring susceptible individuals in the infectious disease modeling is inherently nonnegative. Therefore, preserving the nonnegative path of the exact solution of stochastic SIVS model is also important.
In fact, many numerical methods have been developed to preserve the positivity of the approximate solution [16–18]. Nevertheless, for the preserving positivity numerical solution issue of stochastic epidemic model, to the best of our knowledge, there is not any result. Hence, the main purpose of this present paper is to structure a new method to maintain the nonnegative path of the solution for a stochastic epidemic SIVS model, which is the balanced implicit method. The main technique we developed is based on Tan’s principle [19].
- (1)
Structuring a balanced implicit method to maintain the nonnegative path of the true solution for the stochastic epidemic SIVS model.
- (2)
The BIM approximate solution will converge to the true solution with order \(\frac{1}{2}(1-\frac{1}{l})\) for the stochastic SIVS epidemic model.
The arrangement of the paper is as follows. In Sect. 2, we give some necessary notations and preliminaries. Then we define the balanced implicit method. To ensure the positivity of the balanced implicit method, we build feedback controls. In Sect. 3, we show that the balanced implicit method solutions can converge to the true solution. Finally, a numerical example is presented to support the theoretical results and verify the availability of the approach.
2 Preliminaries and BIM scheme
2.1 Necessary notations and preliminaries
To begin with, we use some notation. Throughout this paper, let \((\varOmega , \mathcal{F}, \mathbb{P})\) be a complete probability space with filtration \(\{\mathcal{F}_{t}\}_{t\geq 0}\) satisfying the usual conditions (i.e. it is increasing and right continuous while \(\mathcal{F}_{0}\)) contains all \(\mathbb{P}\)-null sets, and let \(\mathbb{E}\) denote the expectation corresponding to \(\mathbb{P}\). Let \(w(t)\) be a scalar Brownian motions defined on the space and T be an arbitrary positive number. Moreover, for any \(a,b \in R\), \(a \vee b := \max \{a,b\}\), and \(a \wedge b := \min \{a,b\}\). If G is a set, its indicator function by \(1_{G}\), namely \(1_{G}(x) = 1\) if \(x \in G\) and 0 otherwise.
Theorem 2.1
System (1.1) has a unique positive solution on \([0,T]\).
Proof
The proof of this theorem is similar to that in [20]. □
2.2 Life time and BIM scheme
The idea of the life time of of numerical scheme was presented by Schurz in [21]. Schurz employed the notion of an algorithm having eternal lifetime, where we utilized this life time for (1.1) as follows.
Definition 2.1
Lemma 2.1
The EM approximation (2.3) started in \({S}_{0}>0\), \({I}_{0}> 0\), \({V}_{0} > 0 \) has a finite life time.
Proof
Therefore, how it is possible to prevent an approximation integration method from becoming negative. In this paper, we construct the BIM to preserve positivity of rigid SDEs. □
Now we give the following theorem for the positivity of the BIM.
Theorem 2.2
The balanced numerical method (2.6) has an eternal life time.
Proof
3 Convergence of the balanced method
In this section, we show the main results for the strong convergence of the balanced method.
To prove the convergence theorem, we first need to consider the following lemma, which reveals that the continuous EM solution \(S_{t}^{E}\), \({I}_{t}^{E}\), \({V}_{t}^{E}\) will converge to the step process \(\overline{S}_{t}\), \(\overline{I}_{t}\), \(\overline{V}_{t}\). Since the drift and diffusion coefficients of model (1.1) do not satisfy the linear growth condition, the traditional theory of convergence is not applicable for model (1.1). To tackle this problem, we introduce a stopping time.
Lemma 3.1
Proof
For \(t\in [0,T]\), let \([\frac{t}{\Delta }]\) be the integer part of \(\frac{t}{\Delta }\). For simplicity, we show that the approximate solution \(S_{t}^{E}\), \({I}_{t}^{E}\), \({V}_{t}^{E}\) are close to \(\overline{S}_{t}\), \(\overline{I}_{t}\), \(\overline{V}_{t}\), respectively.
Now we estimate the main result, that is to say, the convergence of the EM approximate solution \(S_{t}^{E}\), \({I}_{t}^{E}\), \({V}_{t}^{E}\) to the solution \(S(t)\), \(I(t)\), \(V(t)\).
Theorem 3.1
Proof
An application of the Gronwall inequality will lead to the proof. □
In order to estimate if the BIM approximation solution will converge to the true solution, we plan to prove a strong convergence result.
Theorem 3.2
Proof
Remark 3.1
Theorem 3.2 shows that \(S(t)\), \(I(t)\), \(V(t)\) be the true solution and \(S_{t}^{B}\), \({I}_{t}^{B}\), \({V}_{t}^{B}\) be the BIM approximate solution are close to each other in the sense, and the BIM converges to the true solution of stochastic SIVS epidemic model with order \(\frac{1}{2}(1-\frac{1}{l})\).
4 Numerical experiments
The EM and BIM method numerical simulations of solution \(S(t)\), \(I(t)\), \(V(t)\) for model (1.1)
Figure 1 draws the numerical solution obtained from BIM and EM method. We present results for three different variables for the SIVS model. Figure 1(a) delineates the path of the EM solution and BIM numerical solution for \(S(t)\). Figure 1(b) depicts the sample paths of the EM solution and BIM numerical solution for \(I(t)\). Figure 1(c) describes the sample paths of the EM solution and BIM numerical solution for \(V(t)\). Through the comparison between EM and BIM method, we clearly that the numerical solution of BIM can preserve positivity, and the EM path becomes negative. That is to say, the EM method cannot maintain the positivity of Fig. 1.
Percentage of negative paths for the EM scheme and BIM
Number | S | I | V | ||||||
---|---|---|---|---|---|---|---|---|---|
Δ = 0.1 | Δ = 0.05 | Δ = 0.01 | Δ = 0.1 | Δ = 0.05 | Δ = 0.01 | Δ = 0.1 | Δ = 0.05 | Δ = 0.01 | |
EM | 33.15% | 28.67% | 13.59% | 32.55% | 20.77% | 15.35% | 20.68% | 15.56 % | 12.56% |
BIM | 0% | 0% | 0% | 0% | 0% | 0% | 0% | 0% | 0% |
5 Conclusions
The positivity-preserving numerical method for the stochastic SIVS epidemic model has been systematically discussed in this paper. First, we propose a SIVS epidemic model with vaccination and investigate the EM numerical approximate solution to the model (1.1). Meanwhile, we prove the convergence property of EM approximate solution to the true solution. Then we establish the balanced implicit method for the stochastic SIVS model. Preserving positivity of the proposed method is proved. Numerical results reveal that the BIM is verifying the availability of the approach for maintaining positivity.
Another interesting topic should be further conducted to reveal how to construct a numerical method preserving positivity for stochastic age-structured SIVS epidemic model. We regard that as our future work.
Declarations
Acknowledgements
The authors would like to express their sincere thanks to referees and the editor for their enthusiastic guidance and help.
Funding
The research was funded by the ‘Major Innovation Projects for Building First-class Universities in China’s Western Region’ (ZKZD2017009).
Authors’ contributions
All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
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
References
- Shi, R., Jiang, X., Chen, L.: The effect of impulsive vaccination on an SIR epidemic model. Appl. Math. Comput. 212, 305–311 (2009) MathSciNetMATHGoogle Scholar
- Nie, L., Shen, J., Yang, C.: Dynamic behavior analysis of SIVS epidemic models with state-dependent pulse vaccination. Nonlinear Anal. Hybrid Syst. 27, 258–270 (2018) MathSciNetMATHView ArticleGoogle Scholar
- Liu, L., Wang, J., Liu, X.: Global stability of an SEIR epidemic model with age-dependent latency and relapse. Nonlinear Anal. 24, 18–35 (2015) MathSciNetMATHView ArticleGoogle Scholar
- Omondi, O., Wang, C., Xue, X., Lawi, O.: Modeling the effects of vaccination on rotavirus infection. Adv. Differ. Equ. 2015, 381 (2015) MathSciNetMATHView ArticleGoogle Scholar
- Lin, Y., Jiang, D., Wang, S.: Stationary distribution of a stochastic SIS epidemic model with vaccination. Physica A 394, 187–197 (2014) MathSciNetMATHView ArticleGoogle Scholar
- Tornatore, E., Vetro, P., Buccellato, S.M.: SVIR epidemic model with stochastic perturbation. Neural Comput. Appl. 24, 309–315 (2014) View ArticleGoogle Scholar
- Zhao, Y., Jiang, D., O’Regan, D.: The extinction and persistence of the stochastic SIS epidemic model with vaccination. Physica A 392, 4916–4927 (2013) MathSciNetMATHView ArticleGoogle Scholar
- Lu, R., Wei, F.: Persistence and extinction for an age-structured stochastic SVIR epidemic model with generalized nonlinear incidence rate. Physica A 513, 572–587 (2019) MathSciNetView ArticleGoogle Scholar
- Yang, J., Martchev, M., Wang, L.: Global threshold dynamics of an SIVS model with waning vaccine-induced immunity and nonlinear incidence. Math. Biosci. 268, 1–8 (2015) MathSciNetMATHView ArticleGoogle Scholar
- Zhao, Y., Jiang, D.: The threshold of a stochastic SIRS epidemic model with saturated incidence. Appl. Math. Lett. 34, 90–93 (2014) MathSciNetMATHView ArticleGoogle Scholar
- Wen, B., Teng, Z., Li, Z.: The threshold of a periodic stochastic SIVS epidemic model with nonlinear incidence. Physica A 508, 532–549 (2018) MathSciNetView ArticleGoogle Scholar
- Mao, X.: The truncated Euler–Maruyama method for stochastic differential equations. J. Comput. Appl. Math. 290, 370–384 (2015) MathSciNetMATHView ArticleGoogle Scholar
- Hu, Y.: Semi-implicit Euler–Maruyama scheme for stiff stochastic equations. In: Stochastic Analysis and Related Topics V, vol. 38, pp. 183–202 (1996) View ArticleGoogle Scholar
- Milošević, M.: The Euler–Maruyama approximation of solutions to stochastic differential equations with piecewise constant arguments. J. Comput. Appl. Math. 298, 1–12 (2016) MathSciNetMATHView ArticleGoogle Scholar
- Bayram, M., Partal, T., Buyukoz, G.: Numerical methods for simulation of stochastic differential equations. Adv. Differ. Equ. 2018, 17 (2018) MathSciNetMATHView ArticleGoogle Scholar
- Higham, D., Mao, X., Szpruch, L.: Convergence, non-negativity and stability of a new Milstein scheme with applications to finance. Discrete Contin. Dyn. Syst. 8, 2083–2100 (2017) MATHGoogle Scholar
- Kahl, C., Gunther, M., Rossberg, T.: Structure preserving stochastic integration schemes in interest rate derivative modeling. Appl. Numer. Math. 58, 284–295 (2008) MathSciNetMATHView ArticleGoogle Scholar
- Milstein, G., Platen, E., Schurz, H.: Balanced implicit methods for stiff stochastic systems. SIAM J. Numer. Anal. 35, 1010–1019 (1998) MathSciNetMATHView ArticleGoogle Scholar
- Tan, J., Men, W., Pei, Y., Guo, Y.: Construction of positivity preserving numerical method for stochastic age-dependent population equations. Appl. Math. Comput. 293, 57–64 (2017) MathSciNetGoogle Scholar
- Liu, Q., Jiang, D., Shi, N., et al.: The threshold of a stochastic SIS epidemic model with imperfect vaccination. Math. Comput. Simul. 144, 78–90 (2018) MathSciNetView ArticleGoogle Scholar
- Schurz, H.: Numerical regularization for SDEs: construction of nonnegative solutions. Dyn. Syst. Appl. 5, 323–352 (1996) MathSciNetMATHGoogle Scholar
- Mao, X.: Stochasti Differential Equations and Their Application. Horwood Publishing Series in Mathematics and Applications. Horwood, Chichester (1997) Google Scholar