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- Open Access
Dynamical analysis of a stochastic SIS epidemic model with nonlinear incidence rate and double epidemic hypothesis
- Anqi Miao^{1},
- Xinyang Wang^{1},
- Tongqian Zhang^{1, 2}Email authorView ORCID ID profile,
- Wei Wang^{3} and
- BG Sampath Aruna Pradeep^{4}
https://doi.org/10.1186/s13662-017-1289-9
© The Author(s) 2017
Received: 16 April 2017
Accepted: 19 July 2017
Published: 4 August 2017
Abstract
In this paper, a stochastic SIS epidemic model with nonlinear incidence rate and double epidemic hypothesis is proposed and analysed. We explain the effects of stochastic disturbance on disease transmission. To this end, firstly, we investigated the dynamic properties of the system neglecting stochastic disturbance and obtained the threshold and the conditions for the extinction and the permanence of two kinds of epidemic diseases by considering the stability of the equilibria of the deterministic system. Secondly, we paid prime attention on the threshold dynamics of the stochastic system and established the conditions for the extinction and the permanence of two kinds of epidemic diseases. We found that there exists a significant difference between the threshold of the deterministic system and that of the stochastic system. Moreover, it has been established that the persistent of infectious disease analysed by use of deterministic system becomes extinct under the same conditions due to the stochastic disturbance. This implies that a stochastic disturbance has significant impact on the spread of infectious diseases and the larger stochastic disturbance leads to control the epidemic diseases. In order to illustrate the dynamic difference between the deterministic system and the stochastic system, there have been given a series of numerical simulations by using different noise disturbance coefficients.
Keywords
- stochastic SIS epidemic model
- double epidemic hypothesis
- Beddington-DeAngelis incidence rate
- extinction
- permanence
MSC
- 60H10
- 65C30
- 91B70
1 Introduction
It is well known that random noise factors play an important role in the transmission of infectious diseases. Therefore, many scholars [36–43] have studied the impact of the stochastic epidemic system, various stochastic perturbation approaches have been introduced into epidemic models and have obtained excellent results. For example, the authors of [44–49] have considered a stochastic epidemic model with a Markov transform. A class of epidemic model which shows the effect of the random white noise has been studied by the researchers in the articles [50–64]. Further, in [65–70], the authors studied a class of stochastic epidemic models, in which the stochastic white noise is assumed to be proportional to S, I and R. It can be seen following the literature that the authors of the articles [71, 72] analysed a stochastic epidemic model with two different kinds of perturbation. A stochastic epidemic model with Lévy jumps has been proposed and studied by the researchers [73–75]; the authors investigated stochastic perturbation around the positive equilibria of deterministic models (see, for example, [40, 41, 76–78]). Although there were limited numbers of publications in the recent literature considering time delay and stochastic behavior, the authors in [77] paid attention on the stochastic epidemic model with time delay.
2 Preliminaries and lemmas
In this section, we will give some notations, definitions and some lemmas which will be used for analysing our main results.
Throughout this paper, let \((\Omega,\mathcal{F}, \{\mathcal{F}\}_{t\geq 0}, \mathcal{P})\) be a complete probability space with a filtration \(\{ \mathcal{F}_{t}\}_{t\geq0}\) satisfying the usual conditions (i.e. it is increasing and right continuous while \(\mathcal{F}_{0}\) contains all \(\mathcal{P}\)-null sets \(R_{+}^{3}=\{ x_{i}>0,i=1,2,3\}\)). The function \(B(t)\) denotes a scalar Brownian motion defined on the complete probability space Ω. For an integrable function f on \([0,+\infty)\), define \(\langle f(t)\rangle =\frac{1}{t}\int_{0}^{t}f(\theta)\, \mathrm{d}\theta\).
Definition 2.1
- (i)
The diseases \(I_{1}(t)\) and \(I_{2}(t)\) are said to be extinctive if \(\lim_{t\rightarrow+\infty}I_{1}(t)=0\) and \(\lim_{t\rightarrow+\infty}I_{2}(t)=0\).
- (ii)
The diseases \(I_{1}(t)\) and \(I_{2}(t)\) are said to be permanent in mean if there exist two positive constants \(\lambda_{1}\) and \(\lambda_{2}\) such that \(\liminf_{t\rightarrow+\infty}\langle I_{1}(t)\rangle\geq\lambda_{1}\) and \(\liminf_{t\rightarrow+\infty }\langle I_{2}(t)\rangle\geq\lambda_{2}\).
Lemma 2.1
For any initial value \((S(0), I_{1}(0), I_{2}(0))\in R^{3}_{+}\), there exists a unique solution \((S(t), I_{1}(t), I_{2}(t))\) to system (4) on \(t\geq0\), and the solution will remain in \(R^{3}_{+}\) with probability 1, namely, \((S(t), I_{1}(t), I_{2}(t))\in R^{3}_{+}\) for all \(t\geq0\) almost surely.
Proof
If this statement is false, then there is a pair of constants \(T>0\) and \(\delta\in(0,1)\) such that \(P\{\tau_{0}\leq T\}>\delta\). Hence, there is a positive constant \(\varepsilon_{1}\leq\varepsilon_{0}\) such that \(P\{\tau _{\varepsilon}\leq T\}\) for any positive \(\varepsilon\leq\varepsilon_{1}\).
Lemma 2.2
Denote \(\Gamma=\{(S(t), I_{1}(t), I_{2}(t))\in R_{+}^{3}: S(t), I_{1}(t), I_{2}(t)\leq\frac{A}{d}, t\geq0\}\), then Γ is an invariant set on system (3) or (4).
Proof
By Lemma 2.2 and the strong law of large numbers for martingales [36], we can obtain the following lemma.
Lemma 2.3
3 Dynamics of deterministic system (3)
Theorem 3.1
- (i)
if \(\mathcal{R}_{1}<1\) and \(\mathcal{R}_{2}<1\), then both diseases go extinct and system (3) has a unique stable ‘diseases-extinction’ equilibrium \(E_{0}\);
- (ii)
if \(\mathcal{R}_{1}>1\) and \(\mathcal{R}_{2}<1\), then the disease \(I_{2}\) goes extinct and system (3) has a unique stable equilibrium \(E_{1}\);
- (iii)
if \(\mathcal{R}_{1}<1\) and \(\mathcal{R}_{2}>1\), then the disease \(I_{1}\) goes extinct and system (3) has a unique stable equilibrium \(E_{2}\); and
- (iv)
when condition H holds, and if \(\mathcal{R}_{1}>1\) and \(\mathcal{R}_{2}>1\), then \(E^{*}\) is a unique stable equilibrium, which implies both diseases of system (3) are permanent.
Proof
Similarly, we can show that if \(\mathcal{R}_{1}<1\) and \(\mathcal {R}_{2}>1\), then the equilibrium \(E_{2}\) of system (3) is stable.
Now, let us prove that the positive equilibrium \(E^{*}\) is stable as \(\mathcal{R}_{1}>1\) and \(\mathcal{R}_{2}>1\).
4 Dynamics of stochastic system (4)
4.1 Extinction
In this section, we are going to explore the conditions which lead to the extinction of two infectious diseases mentioned in the system (4) under a white noise stochastic disturbance.
Theorem 4.1
Proof
Remark 4.1
Theorem 4.1 shows that when \(\sigma_{i}>\frac {\beta_{i}}{\sqrt{2(d+\alpha_{i}+r_{i})}}\), \(i=1,2\), two infectious diseases of system (4) die out almost surely, that is to say, large white noise stochastic disturbance can lead to the two epidemics to be extinct. Therefore, we always assume that the white noise stochastic disturbance is not too large in the rest of this paper.
Theorem 4.2
Proof
Remark 4.2
Theorem 4.1 and Theorem 4.2 show that two diseases will die out if the white noise disturbance is sufficiently larger or \(\mathcal{R}_{i}^{*}<1\) and the white noise disturbance is not large. Note that the expressions for \(\mathcal{R}_{i}^{*}\) for \(i=1,2\) which are the threshold values of system (4) are strictly different compared with the thresholds \(\mathcal{R}_{i}\) of system (3), This implies that the conditions which are needed to have \(I_{i}(t)\) for \(i=1,2\) gone in extinction in deterministic system (3) are stronger than in the corresponding stochastic system (4).
4.2 Permanence in mean
Theorem 4.3
- (i)If \(\mathcal{R}_{1}^{*}>1\), \(\mathcal{R}_{2}^{*}<1\) and \(\sigma_{2}\leq \sqrt{\frac{\beta_{2}(A a_{2}+d)}{A}}\), then the disease \(I_{2}\) goes extinct and the disease \(I_{1}\) is permanent in mean, moreover, \(I_{1}\) satisfies$$ \liminf_{t\rightarrow+\infty}\bigl\langle I_{1}(t)\bigr\rangle \geq \frac {(Aa_{1}+d)(d+\alpha_{1}+r_{1})}{\beta_{1}(d+\alpha_{1})+b_{1}d(d+\alpha _{1}+r_{1})}\bigl(\mathcal{R}_{1}^{*}-1\bigr). $$
- (ii)If \(\mathcal{R}_{2}^{*}>1\), \(\mathcal{R}_{1}^{*}<1\) and \(\sigma_{1}\leq \sqrt{\frac{\beta_{1}(A a_{1}+d)}{A}}\), then the disease \(I_{1}\) goes extinct and the disease \(I_{2}\) is permanent in mean, moreover, \(I_{2}\) satisfies$$ \liminf_{t\rightarrow+\infty}\bigl\langle I_{2}(t)\bigr\rangle \geq \frac {(Aa_{2}+d)(d+\alpha_{2}+r_{2})}{\beta_{2}(d+\alpha_{2})+b_{2}d(d+\alpha _{2}+r_{2})}\bigl(\mathcal{R}_{2}^{*}-1\bigr). $$
- (iii)If \(\mathcal{R}_{1}^{*}>1\) and \(\mathcal{R}_{2}^{*}>1\), then two infectious diseases \(I_{1}\) and \(I_{2}\) are permanent in mean, moreover, \(I_{1}\) and \(I_{2}\) satisfywhere$$ \liminf_{t\rightarrow+\infty}\bigl\langle I_{1}(t)+I_{2}(t) \bigr\rangle \geq\frac {1}{\Delta_{\max}}\sum_{i=1}^{2}a_{i}(d+ \alpha_{i}+r_{i}) \bigl(\mathcal {R}_{i}^{*}-1 \bigr), $$$$\Delta_{\max}=\sum_{i=1}^{2} \biggl[\frac{\beta_{1}+\beta_{2}}{d}(d+\alpha _{i})+b_{i}(d+ \alpha_{i}+r_{i}) \biggr]. $$
Proof
By Lemma 2.3, we get \(\lim_{t\rightarrow+\infty}\frac{M(t)}{t}=0\). According to Lemma 2.2, one can see that \(I_{1}(t)\leq\frac {A}{d}\). Thus, one has \(\lim_{t\rightarrow+\infty}\frac{I_{1}(t)}{t}=0\), \(\lim_{t\rightarrow +\infty}\frac{\ln I_{1}(t)}{t}=0\) as \(I_{1}(t)\geq1\) and \(\lim_{t\rightarrow+\infty}\Theta(t)=0\).
Remark 4.3
Theorem 4.3 shows that both diseases will prevail if the white noise disturbances are small enough such that \(\mathcal{R}_{i}^{*}>1\), conversely, if the white noise disturbances are large enough, then both diseases will become extinct. This implies that the stochastic disturbance may cause epidemic diseases to die out.
5 Conclusion and simulations
This paper proposed two \(SIS\) epidemic models with Beddington-DeAngelis incidence rate and double epidemic hypothesis from the point of view of deterministic and stochastic aspect. The threshold dynamics of both two systems were investigated and the conditions for extinction and permanence of both epidemic diseases were obtained. From Theorems 4.1 and 4.2, it can be seen that there is a significant difference between the thresholds of the stochastic system and the deterministic system, from which it can be concluded that the conditions for two epidemic diseases to go to extinction in the stochastic system are weaker than those of the deterministic system.
To illustrate the dynamic difference between the deterministic system and the stochastic system, we next carry out some numerical simulations of these cases with respect to different noise disturbance intensity using the Euler Maruyama (EM) method [36, 79].
Declarations
Acknowledgements
This work is supported by Shandong Provincial Natural Science Foundation (No. ZR2015AQ001), National Natural Science Foundation of China (No. 11371230), Project for Higher Educational Science and Technology Program of Shandong Province (No. J13LI05), Research Funds for Joint Innovative Center for Safe and Effective Mining Technology and Equipment of Coal Resources by Shandong Province and SDUST Research Fund (2014TDJH102).
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
- Hamer, WH: Epidemic disease in England. Lancet 1, 733-739 (1906) Google Scholar
- Brauer, F, Castillo-Chavez, C: Mathematical Models in Population Biology and Epidemiology. Texts in Applied Mathematics. Springer, New York (2001) View ArticleMATHGoogle Scholar
- Bernoulli, D, Blower, S: An attempt at a new analysis of the mortality caused by smallpox and of the advantages of inoculation to prevent it. Rev. Med. Virol. 14(5), 275-288 (2004) View ArticleGoogle Scholar
- Ross, RA: The Prevention of Malaria. Murray, London (1911) Google Scholar
- May, RM, Anderson, RM, McLean, AR: Possible demographic consequences of HIV/AIDS epidemics. I. Assuming HIV infection always leads to AIDS. Math. Biosci. 90(1), 475-505 (1988) MathSciNetView ArticleMATHGoogle Scholar
- Anderson, RM, May, RM: Infectious Diseases of Humans: Dynamics and Control. Oxford University Press, Oxford (1992) Google Scholar
- Zhang, T, Ma, W, Meng, X, Zhang, T: Periodic solution of a prey-predator model with nonlinear state feedback control. Appl. Math. Comput. 266, 95-107 (2015) MathSciNetGoogle Scholar
- Meng, X, Zhang, L: Evolutionary dynamics in a Lotka-Volterra competition model with impulsive periodic disturbance. Math. Methods Appl. Sci. 39(2), 177-188 (2016) MathSciNetView ArticleMATHGoogle Scholar
- Cheng, H, Zhang, T: A new predator-prey model with a profitless delay of digestion and impulsive perturbation on the prey. Appl. Math. Comput. 217(22), 9198-9208 (2011) MathSciNetMATHGoogle Scholar
- Cheng, H, Zhang, T, Wang, F: Existence and attractiveness of order one periodic solution of a Holling I predator-prey model. Abstr. Appl. Anal. 2012, Article ID 126018 (2012) MathSciNetMATHGoogle Scholar
- Zhang, T, Ma, W, Meng, X: Global dynamics of a delayed chemostat model with harvest by impulsive flocculant input. Adv. Differ. Equ. 2017(1), 115 (2017) MathSciNetView ArticleGoogle Scholar
- Xu, X: A deformed reduced semi-discrete Kaup-Newell equation, the related integrable family and Darboux transformation. Appl. Math. Comput. 251, 275-283 (2015) MathSciNetMATHGoogle Scholar
- Dong, H, Zhang, Y, Zhang, X: The new integrable symplectic map and the symmetry of integrable nonlinear lattice equation. Commun. Nonlinear Sci. Numer. Simul. 36, 354-365 (2016) MathSciNetView ArticleGoogle Scholar
- Zhang, Y, Dong, H, Zhang, X, Yang, H: Rational solutions and lump solutions to the generalized-dimensional shallow water-like equation. Comput. Math. Appl. 73(2), 246-252 (2017) MathSciNetView ArticleGoogle Scholar
- Dong, H, Guo, B, Yin, B: Generalized fractional supertrace identity for Hamiltonian structure of NLS-MKdV hierarchy with self-consistent sources. Anal. Math. Phys. 6(2), 199-209 (2016) MathSciNetView ArticleMATHGoogle Scholar
- Fang, Y, Dong, H, Hou, Y, Kong, Y: Frobenius integrable decompositions of nonlinear evolution equations with modified term. Appl. Math. Comput. 226, 435-440 (2014) MathSciNetMATHGoogle Scholar
- Meng, X, Chen, L, Wu, B: A delay sir epidemic model with pulse vaccination and incubation times. Nonlinear Anal., Real World Appl. 11(1), 88-98 (2010) MathSciNetView ArticleMATHGoogle Scholar
- Zhang, T, Meng, X, Zhang, T: Global analysis for a delayed SIV model with direct and environmental transmissions. J. Appl. Anal. Comput. 6(2), 1479-1491 (2016) MathSciNetGoogle Scholar
- Cui, Y: Uniqueness of solution for boundary value problems for fractional differential equations. Appl. Math. Lett. 51, 48-54 (2016) MathSciNetView ArticleMATHGoogle Scholar
- Bai, Z, Dong, X, Yin, C: Existence results for impulsive nonlinear fractional differential equation with mixed boundary conditions. Bound. Value Probl. 2016(1), 63 (2016) MathSciNetView ArticleMATHGoogle Scholar
- Cui, Y, Zou, Y: An existence and uniqueness theorem for a second order nonlinear system with coupled integral boundary value conditions. Appl. Math. Comput. 256, 438-444 (2015) MathSciNetMATHGoogle Scholar
- Bai, Z, Dong, X, Yin, C: Monotone iterative method for fractional differential equations. Electron. J. Differ. Equ. 2016, 6 (2016) MathSciNetView ArticleGoogle Scholar
- Zou, Y, Cui, Y: Existence results for a functional boundary value problem of fractional differential equations. Adv. Differ. Equ. 2013(1), 233 (2013) MathSciNetView ArticleGoogle Scholar
- Zhang, T, Zhang, T, Meng, X: Stability analysis of a chemostat model with maintenance energy. Appl. Math. Lett. 68, 1-7 (2017) MathSciNetView ArticleMATHGoogle Scholar
- Kermack, WO, McKendrick, AG: Contributions to the mathematical theory of epidemics - I. Bull. Math. Biol. 53(1), 33-55 (1991) MATHGoogle Scholar
- Kermack, WO, McKendrick, AG: Contributions to the mathematical theory of epidemics: II. Further studies of the problem of endemicity. Bull. Math. Biol. 53(1), 89-118 (1991) MATHGoogle Scholar
- Xu, R, Ma, Z: Global stability of a SIR epidemic model with nonlinear incidence rate and time delay. Nonlinear Anal., Real World Appl. 10(5), 3175-3189 (2009) MathSciNetView ArticleMATHGoogle Scholar
- Xu, R, Zhang, S, Zhang, F: Global dynamics of a delayed SEIS infectious disease model with logistic growth and saturation incidence. Math. Methods Appl. Sci. 39, 3294-3308 (2016) MathSciNetView ArticleMATHGoogle Scholar
- Zhang, T, Teng, Z: Global asymptotic stability of a delayed SEIRS epidemic model with saturation incidence. Chaos Solitons Fractals 37(5), 1456-1468 (2008) MathSciNetView ArticleMATHGoogle Scholar
- Liu, W, Levin, SA, Iwasa, Y: Influence of nonlinear incidence rates upon the behavior of SIRS epidemiological models. J. Math. Biol. 23(2), 187-204 (1986) MathSciNetView ArticleMATHGoogle Scholar
- Liu, W, Hethcote, HW, Levin, SA: Dynamical behavior of epidemiological models with nonlinear incidence rates. J. Math. Biol. 25(4), 359-380 (1987) MathSciNetView ArticleMATHGoogle Scholar
- Hethcote, HW, Lewis, MA, van den Driessche, P: An epidemiological model with a delay and a nonlinear incidence rate. J. Math. Biol. 27(1), 49-64 (1989) MathSciNetView ArticleMATHGoogle Scholar
- Ruan, S, Wang, W: Dynamical behavior of an epidemic model with a nonlinear incidence rate. J. Differ. Equ. 188(1), 135-163 (2003) MathSciNetView ArticleMATHGoogle Scholar
- Chen, L, Hu, Z, Liao, F: The stability of an SEIR model with nonlinear Beddington-DeAngelis incidence, vertical transmission and time delay. J. Aanhui Norm. Univ. 39(1), 26-32 (2016) MATHGoogle Scholar
- Wang, L, Li, J-Q: Global stability of an epidemic model with nonlinear incidence rate and differential infectivity. Appl. Math. Comput. 161(3), 769-778 (2005) MathSciNetMATHGoogle Scholar
- Mao, X: Stochastic Differential Equations and Applications. Horwood, Chichester (2008) View ArticleGoogle Scholar
- Artalejo, JR, Economou, A, Lopez-Herrero, MJ: On the number of recovered individuals in the SIS and SIR stochastic epidemic models. Math. Biosci. 228(1), 45-55 (2010) MathSciNetView ArticleMATHGoogle Scholar
- Bacaër, N: On the stochastic SIS epidemic model in a periodic environment. J. Math. Biol. 71(2), 491-511 (2014) MathSciNetView ArticleMATHGoogle Scholar
- Meng, X: Stability of a novel stochastic epidemic model with double epidemic hypothesis. Appl. Math. Comput. 217(2), 506-515 (2010) MathSciNetMATHGoogle Scholar
- Beretta, E, Kolmanovskii, V, Shaikhet, L: Stability of epidemic model with time delays influenced by stochastic perturbations. Math. Comput. Simul. 45(3-4), 269-277 (1998) MathSciNetView ArticleMATHGoogle Scholar
- Yu, J, Jiang, D, Shi, N: Global stability of two-group SIR model with random perturbation. J. Math. Anal. Appl. 360(1), 235-244 (2009) MathSciNetView ArticleMATHGoogle Scholar
- Ji, C, Jiang, D: Threshold behaviour of a stochastic SIR model. Appl. Math. Model. 38(21-22), 5067-5079 (2014) MathSciNetView ArticleGoogle Scholar
- Feng, T, Meng, X, Liu, L, Gao, S: Application of inequalities technique to dynamics analysis of a stochastic eco-epidemiology model. J. Inequal. Appl. 2016(1), 327 (2016) MathSciNetView ArticleMATHGoogle Scholar
- Ma, H, Jia, Y: Stability analysis for stochastic differential equations with infinite Markovian switchings. J. Math. Anal. Appl. 435(1), 593-605 (2016) MathSciNetView ArticleMATHGoogle Scholar
- Zhao, W, Li, J, Zhang, T, Meng, X, Zhang, T: Persistence and ergodicity of plant disease model with Markov conversion and impulsive toxicant input. Commun. Nonlinear Sci. Numer. Simul. 48, 70-84 (2017) MathSciNetView ArticleGoogle Scholar
- Tuckwell, HC, Williams, RJ: Some properties of a simple stochastic epidemic model of SIR type. Math. Biosci. 208(1), 76-97 (2007) MathSciNetView ArticleMATHGoogle Scholar
- Cai, Y, Kang, Y, Banerjee, M, Wang, W: A stochastic SIRS epidemic model with infectious force under intervention strategies. J. Differ. Equ. 259(12), 7463-7502 (2015) MathSciNetView ArticleMATHGoogle Scholar
- Gray, A, Greenhalgh, D, Mao, X, Pan, J: The SIS epidemic model with Markovian switching. J. Math. Anal. Appl. 394(2), 496-516 (2012) MathSciNetView ArticleMATHGoogle Scholar
- Zhang, X, Jiang, D, Alsaedi, A, Hayat, T: Stationary distribution of stochastic SIS epidemic model with vaccination under regime switching. Appl. Math. Lett. 59, 87-93 (2016) MathSciNetView ArticleMATHGoogle Scholar
- Meng, X, Zhao, S, Feng, T, Zhang, T: Dynamics of a novel nonlinear stochastic SIS epidemic model with double epidemic hypothesis. J. Math. Anal. Appl. 433(1), 227-242 (2016) MathSciNetView ArticleMATHGoogle Scholar
- Tornatore, E, Buccellato, SM, Vetro, P: Stability of a stochastic sir system. Phys. A, Stat. Mech. Appl. 354, 111-126 (2005) View ArticleGoogle Scholar
- Chang, Z, Meng, X, Lu, X: Analysis of a novel stochastic SIRS epidemic model with two different saturated incidence rates. Phys. A, Stat. Mech. Appl. 472, 103-116 (2017) MathSciNetView ArticleGoogle Scholar
- Gray, A, Greenhalgh, D, Hu, L, Mao, X, Pan, J: A stochastic differential equation SIS epidemic model. SIAM J. Appl. Math. 71(3), 876-902 (2011) MathSciNetView ArticleMATHGoogle Scholar
- Lin, Y, Jiang, D: Long-time behaviour of a perturbed SIR model by white noise. Discrete Contin. Dyn. Syst., Ser. B 18(7), 1873-1887 (2013) MathSciNetView ArticleMATHGoogle Scholar
- Schurz, H, Tosun, K: Stochastic asymptotic stability of SIR model with variable diffusion rates. J. Dyn. Differ. Equ. 27(1), 69-82 (2015) MathSciNetView ArticleMATHGoogle Scholar
- Lu, Q: Stability of SIRS system with random perturbations. Phys. A, Stat. Mech. Appl. 388(18), 3677-3686 (2009) MathSciNetView ArticleGoogle Scholar
- Wei, F, Liu, J: Long-time behavior of a stochastic epidemic model with varying population size. Phys. A, Stat. Mech. Appl. 470, 146-153 (2017) MathSciNetView ArticleGoogle Scholar
- Dalal, N, Greenhalgh, D, Mao, X: A stochastic model of AIDS and condom use. J. Math. Anal. Appl. 325(1), 36-53 (2007) MathSciNetView ArticleMATHGoogle Scholar
- Xu, C: Global threshold dynamics of a stochastic differential equation SIS model. J. Math. Anal. Appl. 447(2), 736-757 (2017) MathSciNetView ArticleMATHGoogle Scholar
- Lahrouz, A, Settati, A: Necessary and sufficient condition for extinction and persistence of SIRS system with random perturbation. Appl. Math. Comput. 233, 10-19 (2014) MathSciNetMATHGoogle Scholar
- Lahrouz, A, Settati, A: Qualitative study of a nonlinear stochastic SIRS epidemic system. Stoch. Anal. Appl. 32(6), 992-1008 (2014) MathSciNetView ArticleMATHGoogle Scholar
- Zhao, D, Zhang, T, Yuan, S: The threshold of a stochastic SIVS epidemic model with nonlinear saturated incidence. Phys. A, Stat. Mech. Appl. 443, 372-379 (2016) MathSciNetView ArticleGoogle Scholar
- Zhao, Y, Lin, Y, Jiang, D, Mao, X, Li, Y: Stationary distribution of stochastic SIRS epidemic model with standard incidence. Discrete Contin. Dyn. Syst., Ser. B 21(7), 2363-2378 (2016) MathSciNetView ArticleMATHGoogle Scholar
- Miao, A, Zhang, J, Zhang, T, Sampath Aruna Pradeep, BG: Threshold dynamics of a stochastic SIR model with vertical transmission and vaccination. Comput. Math. Methods Med. 2017, Article ID 4820183 (2017) MathSciNetView ArticleGoogle Scholar
- Lahrouz, A, Settati, A, Akharif, A: Effects of stochastic perturbation on the SIS epidemic system. J. Math. Biol. 74(1), 469-498 (2017) MathSciNetView ArticleMATHGoogle Scholar
- Liu, Q, Jiang, D, Shi, N, Hayat, T, Alsaedi, A: Stationary distribution and extinction of a stochastic SIRS epidemic model with standard incidence. Phys. A, Stat. Mech. Appl. 469, 510-517 (2017) MathSciNetView ArticleGoogle Scholar
- Dieu, NT, Nguyen, DH, Du, NH, Yin, G: Classification of asymptotic behavior in a stochastic SIR model. SIAM J. Appl. Dyn. Syst. 15(2), 1062-1084 (2016) MathSciNetView ArticleMATHGoogle Scholar
- Zhao, Y, Jiang, D: The threshold of a stochastic SIS epidemic model with vaccination. Appl. Math. Comput. 243, 718-727 (2014) MathSciNetMATHGoogle Scholar
- Jiang, D, Liu, Q, Shi, N, Hayat, T, Alsaedi, A, Xia, P: Dynamics of a stochastic HIV-1 infection model with logistic growth. Phys. A, Stat. Mech. Appl. 469, 706-717 (2017) MathSciNetView ArticleGoogle Scholar
- Cai, Y, Kang, Y, Banerjee, M, Wang, W: A stochastic epidemic model incorporating media coverage. Commun. Math. Sci. 14(4), 893-910 (2016) MathSciNetView ArticleMATHGoogle Scholar
- Du, NH, Nhu, NN: Permanence and extinction of certain stochastic SIR models perturbed by a complex type of noises. Appl. Math. Lett. 64, 223-230 (2017) MathSciNetView ArticleMATHGoogle Scholar
- Liu, Q, Chen, Q: Analysis of the deterministic and stochastic SIRS epidemic models with nonlinear incidence. Phys. A, Stat. Mech. Appl. 428, 140-153 (2015) MathSciNetView ArticleGoogle Scholar
- Zhang, X, Jiang, D, Hayat, T, Ahmad, B: Dynamics of a stochastic SIS model with double epidemic diseases driven by Lévy jumps. Phys. A, Stat. Mech. Appl. 471, 767-777 (2017) View ArticleGoogle Scholar
- Li, C, Pei, Y, Liang, X, Fang, D: A stochastic toxoplasmosis spread model between cat and oocyst with jumps process. Commun. Math. Biol. Neurosci. 2016, 18 (2016) Google Scholar
- Zhou, Y, Yuan, S, Zhao, D: Threshold behavior of a stochastic SIS model with jumps. Appl. Math. Comput. 275, 255-267 (2016) MathSciNetGoogle Scholar
- Jiang, D, Ji, C, Shi, N, Yu, J: The long time behavior of DI SIR epidemic model with stochastic perturbation. J. Math. Anal. Appl. 372(1), 162-180 (2010) MathSciNetView ArticleMATHGoogle Scholar
- Liu, M, Bai, C, Wang, K: Asymptotic stability of a two-group stochastic SEIR model with infinite delays. Commun. Nonlinear Sci. Numer. Simul., 19(10), 3444-3453 (2014) MathSciNetView ArticleGoogle Scholar
- Liu, Q, Jiang, D, Shi, N, Hayat, T, Alsaedi, A: Asymptotic behavior of a stochastic delayed SEIR epidemic model with nonlinear incidence. Phys. A, Stat. Mech. Appl. 462, 870-882 (2016) MathSciNetView ArticleGoogle Scholar
- Kloeden, PE, Platen, E: Higher-order implicit strong numerical schemes for stochastic differential equations. J. Stat. Phys. 66(1), 283-314 (1992) MathSciNetView ArticleMATHGoogle Scholar