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Computational and Mathematical Methods in Medicine
Volume 2017, Article ID 4820183, 10 pages
https://doi.org/10.1155/2017/4820183
Research Article

Threshold Dynamics of a Stochastic Model with Vertical Transmission and Vaccination

1College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, China
2State Key Laboratory of Mining Disaster Prevention and Control Co-Founded by Shandong Province and the Ministry of Science and Technology, Shandong University of Science and Technology, Qingdao 266590, China
3Department of Mathematics, University of Ruhuna, 81000 Matara, Sri Lanka

Correspondence should be addressed to Tongqian Zhang; nc.ude.tsuds@naiqgnotgnahz

Received 25 March 2017; Accepted 4 June 2017; Published 6 July 2017

Academic Editor: Gul Zaman

Copyright © 2017 Anqi Miao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

A stochastic model with vertical transmission and vaccination is proposed and investigated in this paper. The threshold dynamics are explored when the noise is small. The conditions for the extinction or persistence of infectious diseases are deduced. Our results show that large noise can lead to the extinction of infectious diseases which is conducive to epidemic diseases control.

1. Introduction

The history of mankind is filled with struggle with diseases. Infectious diseases such as smallpox, cholera, plague of leprosy, diphtheria, syphilis, typhus fever, malaria, rabies, and tuberculosis have threatened the health of human beings. People have realized the importance of quantitative studies on the spread of infectious diseases to predict and to control them. It can be known from referring to the literature [14] that, with the aid of the establishment of infectious disease models, people can understand the crucial laws of infectious diseases and provide reliable and enough information to predict and control infectious diseases. For example, as early as 1760, Bernoulli and Blower [5] proposed the first mathematical model in epidemiology for studying the spread and inoculation of smallpox. Further, in 1927, Kermack and McKendrick [6] proposed the concept of the so-called “compartmental model,” in which all the population was classified into three compartments: susceptible compartment , infected compartment , and removed compartment It is assumed in the model that the susceptible class can transform into the infective class through contact with infected individuals, and the infectives can recover through treatment so that they have permanent immunity. Therefore, it is now well known that many scholars have paid attention to models; as a result, it can be seen in the literature that a large number of mathematical models of ordinary differential equations, delay differential equations, and partial differential equations have been constructed to study the spread of infectious diseases (see, e.g., [723]). In the last decades, we observed that scholars published few papers in scientific journals related to mathematics considering infectious diseases with vertical transmission which are transmitted from parents to their offspring (e.g., [1, 2426]). Although scholars neglect the effect of vertical transmission, it is very important to study the real situation of the transmission of infectious diseases. The current diseases affecting humanity such as AIDS [2731], Chagas’ disease [3234], hepatitis B [35, 36], and hepatitis C [37] are vertically transmitted. From this, it can be clearly seen that mathematical modeling including vertical transmission, horizontal transmission, and vaccination [38, 39] is more realistic than without them. Therefore, in this study, we have focused our attention on this and an epidemic model involving vertical transmission and vaccination was proposed as follows [1, 24] (see Figure 1):where , , and represent the members of the susceptible, the infectious, and the removed or the recovered members from infection, respectively. is the birth and death rate of and is the birth and death rate of is the contact rate, and is the vaccination proportion to the newborn from and . Then, constant is the proportion of the offspring of infective parents that are susceptible individuals and . is the recovery rate of the infective individuals. Obviously, the total population size is normalized to one, and the basic reproductive number of system (1) is By constructing a Lyapunov function and using the LaSalle invariance principle, we can show that if , the infection-free equilibrium is globally asymptotically stable, while if , the infection-free equilibrium is unstable and the endemic equilibrium is globally asymptotically stable.

Figure 1: The compartmental diagram for the model with vertical transmission and vaccination.

In fact, the spread of diseases is inevitably disturbed by the influence of random factors; the stochastic epidemic system is more in line with the actual situation. Therefore, epidemic systems described by stochastic differential equations have been paid extensive attention in recent years (see, e.g., [4046]). Various stochastic perturbation approaches have been introduced into epidemic systems and excellent results have been obtained. In this study, our main objective is to introduce four approaches. The first one is to analyze epidemic systems including the environment noise by using the method of time Markov chain (see, e.g., [4751]). The second one is to consider the parameters’ perturbation (see, e.g., [5272]). The third one is to introduce Lévy jump noise into the system [7375]. The fourth one is to investigate stochastic perturbation around the positive equilibria of deterministic systems (see, e.g., [41, 42, 7678]).

Parameter perturbation induced by white noises is an important and common form to describe the effect of stochasticity. In this paper, we adopt the perturbation with white noises, that is, , where is a standard Brownian motion with intensity Then, the resultant system transforms into the following form:

This paper is organized as follows. In Section 3, we will discuss the extinction of infectious diseases and explore the conditions leading to the extinction of infectious diseases. In Section 4, we will deduce the condition for a disease in order to be persistent.

2. Preliminaries

Throughout this paper, we let be the -dimensional Euclidean space. , that is, the positive cone.

Let be a one-dimensional Brownian motion defined on the complete probability space adapted to the filtration Let denote the family of all -valued measurable -adapted processes such that Let denote the family of all real-valued functions defined on such that they are continuously twice differentiable in and once in . We set Clearly, when , we have , . Then, we have the following.

Lemma 1 (one-dimensional Itô’s formula [40, 79, 80]). Let be an Itô process on with the stochastic differential where and . Let . Then, is again an Itô process with the stochastic differential given by almost surely.

By using the methods from Lahrouz and Omari [81], we can prove the following lemma.

Lemma 2. For any initial value , there exists a unique solution to system (2) on , and the solution will remain in with probability one, namely.

Lemma 3. On the basis of Lemma 2, if , then , almost surely. Thus, the region is a positively invariant set of system (2).

3. Extinction

In this section, we deduce the condition which will cause a disease to die out.

Definition 4. For system (2), the infected individual is said to be extinctive if , almost surely.

Let us introduce for convenience; then, we have the following results that we have mentioned in the following theorem.

Theorem 5. If or and , then the infected individual of system (2) goes to extinction almost surely.

Proof. Let be a solution of system (2) with initial value . Applying Itô’s formula to the second equation of system (2) leads toIntegrating both sides of (8) from 0 to giveswhere and is the local continuous martingale with Next, we have two cases to be discussed, depending on whether
If , we can easily see from (9) thatDividing both sides of (10) by , we haveSince almost surely, by the large number theorem for martingales (see, e.g., [53]), one can obtain that Then, taking the limit superior on both sides of (11) leads to when , which implies
If , similarly, one can have thatDividing both sides of (14) by , we haveBy taking the superior limit on both sides of (15), one can have that Then, when , we obtain which implies This completes the proof of Theorem 5.

Remark 6. Theorem 5 shows that when , the infectious disease of system (2) goes to extinction almost surely; namely, large white noise stochastic disturbance is conducive to control infectious diseases. When the white noise is not large and , the infectious disease of system (2) also goes to extinction almost surely; then, is the threshold associated with the extinction of infectious diseases.

4. Persistence in Mean

Definition 7. For system (2), the infected individual is said to be permanent in mean if , almost surely, where is defined as

Let us denote for convenience; then, we have the following results that we have mentioned in the following theorem.

Theorem 8. If , then the infected individual is persistent in mean; moreover, satisfies almost surely.

Proof. Integrating from 0 to and dividing by on both sides of the third equation of system (2) yield Note that ; then, one can get Applying Itô’s formula givesIntegrating from 0 to and dividing by on both sides of (22) yieldFrom (23), we obtainSince both and , then one has , , and . Note that ; by taking the inferior limit of both sides of (24), we have This completes the proof of Theorem 8.

Remark 9. Theorems 5 and 8 show that the condition for the disease to die out or persist depends on the intensity of white noise disturbances strongly. And small white noise disturbances will be beneficial for long-term prevalence of the disease; conversely, large white noise disturbances may cause the epidemic disease to die out.

5. Conclusion and Numerical Simulation

In this paper, a stochastic system with vertical transmission and vaccination is proposed. The threshold dynamics depending on the stochastic perturbation are deduced by using the theory of stochastic differential equation and inequality technique. Our results show that the dynamics of the stochastic system are different with the deterministic case due to the effect of stochastic perturbation, and the persistent diseases in the deterministic system may be eliminated under the stochastic perturbation.

In the following, by employing the Euler Maruyama (EM) method [40], we perform some numerical simulations to illustrate the extinction and persistence of the diseases in the stochastic system and corresponding deterministic system for comparison.

For numerical simulations, we set parameters as , , , , , and in system (1). A simple computation shows that , and then system (1) has a stable infection-free equilibrium , which implies that the disease of system (1) will be eliminated ultimately (see Figure 2(a)). If we change to , in this case, , and then system (1) has a stable infection equilibrium , which implies that the disease of system (1) will be persistent ultimately (see Figure 2(b)).

Figure 2: Illustration for the deterministic system where , , , , and .

Next, we consider the effect of stochastic white noise based on the persistent system. Let , and obviously, ; by Theorem 5, the disease dies out under a large white noise disturbance (see Figure 3). If we change to , in this case, and ; then, by Theorem 5, the disease dies out (see Figure 4). If we reduce the intensity of noise to , obviously, ; by Theorem 8, the disease is persistent (see Figure 5).

Figure 3: Comparison of the deterministic system and stochastic system, where , , , , , , , and
Figure 4: Comparison of the deterministic system and stochastic system, where , , , , , , , , and
Figure 5: Comparison of the deterministic system and stochastic system, where , , , , , , , , and

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This work is supported by Shandong Provincial Natural Science Foundation of China (no. ZR2015AQ001), the National Natural Science Foundation of China (no. 11371230), and Research Funds for Joint Innovative Center for Safe and Effective Mining Technology and Equipment of Coal Resources by Shandong Province and SDUST (2014TDJH102).

References

  1. R. Anderson and R. May, Infectious Diseases in Humans: Dynamics and Control, Oxford University Press, Oxford, UK, 1992.
  2. C. V. De-León, “On the global stability of SIS, SIR and SIRS epidemic models with standard incidence,” Chaos, Solitons & Fractals, vol. 44, no. 12, pp. 1106–1110, 2011. View at Publisher · View at Google Scholar · View at Scopus
  3. F. Brauer, Epidemic Models in Populations of Varying Size, Springer, Berlin, Germany, 1989.
  4. H. W. Hethcote, “The mathematics of infectious diseases,” SIAM Review, vol. 42, no. 4, pp. 599–653, 2000. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  5. D. Bernoulli and S. Blower, “An attempt at a new analysis of the mortality caused by smallpox and of the advantages of inoculation to prevent it,” Reviews in Medical Virology, vol. 14, no. 5, pp. 275–288, 2004. View at Publisher · View at Google Scholar · View at Scopus
  6. W. O. Kermack and A. G. McKendrick, “A contribution to the mathematical theory of epidemics,” Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, vol. 115, no. 772, pp. 700–721, 1927. View at Publisher · View at Google Scholar
  7. G. Huang and Y. Takeuchi, “Global analysis on delay epidemiological dynamic models with nonlinear incidence,” Journal of Mathematical Biology, vol. 63, no. 1, pp. 125–139, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  8. G. Huang, Y. Takeuchi, W. Ma, and D. Wei, “Global stability for delay SIR and SEIR epidemic models with nonlinear incidence rate,” Bulletin of Mathematical Biology, vol. 72, no. 5, pp. 1192–1207, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  9. Z. Hu, W. Ma, and S. Ruan, “Analysis of SIR epidemic models with nonlinear incidence rate and treatment,” Mathematical Biosciences, vol. 238, no. 1, pp. 12–20, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  10. B. G. Pradeep and W. Ma, “Global stability of a delayed mosquito-transmitted disease model with stage structure,” Electronic Journal of Differential Equations, vol. 10, pp. 1–19, 2015. View at Google Scholar · View at MathSciNet
  11. Z. Jiang and W. Ma, “Permanence of a delayed SIR epidemic model with general nonlinear incidence rate,” Mathematical Methods in the Applied Sciences, vol. 38, no. 3, pp. 505–516, 2015. View at Publisher · View at Google Scholar · View at MathSciNet
  12. T. Zhang, X. Meng, T. Zhang, and Y. Song, “Global dynamics for a new high-dimensional SIR model with distributed delay,” Applied Mathematics and Computation, vol. 218, no. 24, pp. 11806–11819, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  13. W. Zhao, T. Zhang, Z. Chang, X. Meng, and Y. Liu, “Dynamical analysis of SIR epidemic models with distributed delay,” Journal of Applied Mathematics, vol. 2013, Article ID 154387, 15 pages, 2013. View at Publisher · View at Google Scholar · View at Scopus
  14. K. I. Kim, Z. Lin, and L. Zhang, “Avian-human influenza epidemic model with diffusion,” Nonlinear Analysis: Real World Applications, vol. 11, no. 1, pp. 313–322, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  15. G. Zaman, Y. Han Kang, and I. H. Jung, “Stability analysis and optimal vaccination of an SIR epidemic model,” BioSystems, vol. 93, no. 3, pp. 240–249, 2008. View at Publisher · View at Google Scholar · View at Scopus
  16. L. Zhang, Z.-C. Wang, and X.-Q. Zhao, “Threshold dynamics of a time periodic reaction-diffusion epidemic model with latent period,” Journal of Differential Equations, vol. 258, no. 9, pp. 3011–3036, 2015. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  17. W. Wang and X.-Q. Zhao, “An epidemic model with population dispersal and infection period,” SIAM Journal on Applied Mathematics, vol. 66, no. 4, pp. 1454–1472, 2006. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  18. W. Wang, W. Ma, and X. Lai, “Repulsion effect on superinfecting virions by infected cells for virus infection dynamic model with absorption effect and chemotaxis,” Nonlinear Analysis: Real World Applications, vol. 33, pp. 253–283, 2017. View at Publisher · View at Google Scholar · View at MathSciNet
  19. W. Wang, W. Ma, and X. Lai, “A diffusive virus infection dynamic model with nonlinear functional response, absorption effect and chemotaxis,” Communications in Nonlinear Science and Numerical Simulation, vol. 42, pp. 585–606, 2017. View at Publisher · View at Google Scholar · View at MathSciNet
  20. G. Zaman, Y. H. Kang, and I. H. Jung, “Optimal treatment of an SIR epidemic model with time delay,” BioSystems, vol. 98, no. 1, pp. 43–50, 2009. View at Publisher · View at Google Scholar · View at Scopus
  21. X. Meng, Y. Yang, and S. Zhao, “Adaptive evolution of virulence-related traits in a susceptible-infected model with treatment,” Abstract and Applied Analysis, vol. 2014, Article ID 891401, 10 pages, 2014. View at Publisher · View at Google Scholar · View at MathSciNet
  22. W. Zhao, J. Li, and X. Meng, “Dynamical analysis of SIR epidemic model with nonlinear pulse vaccination and lifelong immunity,” Discrete Dynamics in Nature and Society, vol. 2015, Article ID 848623, 10 pages, 2015. View at Publisher · View at Google Scholar · View at MathSciNet
  23. S. Guo and W. Ma, “Global behavior of delay differential equations model of HIV infection with apoptosis,” Discrete and Continuous Dynamical Systems. Series B, vol. 21, no. 1, pp. 103–119, 2016. View at Publisher · View at Google Scholar · View at MathSciNet
  24. X. Meng and L. Chen, “The dynamics of a new SIR epidemic model concerning pulse vaccination strategy,” Applied Mathematics and Computation, vol. 197, no. 2, pp. 582–597, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
  25. Z. Lu, X. Chi, and L. Chen, “The effect of constant and pulse vaccination on SIR epidemic model with horizontal and vertical transmission,” Mathematical and Computer Modelling, vol. 36, no. 9-10, pp. 1039–1057, 2002. View at Publisher · View at Google Scholar · View at MathSciNet
  26. Y. Chen, J. Evans, and M. Feldlaufer, “Horizontal and vertical transmission of viruses in the honey bee, Apis mellifera,” Journal of Invertebrate Pathology, vol. 92, no. 3, pp. 152–159, 2006. View at Publisher · View at Google Scholar · View at Scopus
  27. S. Sprecher, G. Soumenkoff, F. Puissant, and M. Degueldre, “Vertical transmission of hiv in 15-week fetus,” The Lancet, vol. 328, no. 8501, pp. 288-289, 1986. View at Publisher · View at Google Scholar · View at Scopus
  28. A. B. Van't Wout, N. A. Kootstra, G. A. Mulder-Kampinga et al., “Macrophage-tropic variants initiate human immunodeficiency virus type 1 infection after sexual, parenteral, and vertical transmission,” Journal of Clinical Investigation, vol. 94, no. 5, pp. 2060–2067, 1994. View at Publisher · View at Google Scholar · View at Scopus
  29. T. Zhang, X. Meng, and T. Zhang, “Global dynamics of a virus dynamical model with cell-to-cell transmission and cure rate,” Computational and Mathematical Methods in Medicine, vol. 2015, Article ID 758362, 8 pages, 2015. View at Publisher · View at Google Scholar · View at MathSciNet
  30. T. Zhang, X. Meng, and T. Zhang, “Global analysis for a delayed SIV model with direct and environmental transmissions,” The Journal of Applied Analysis and Computation, vol. 6, no. 2, pp. 479–491, 2016. View at Google Scholar · View at MathSciNet
  31. K. Hattaf, N. Yousfi, and A. Tridane, “Mathematical analysis of a virus dynamics model with general incidence rate and cure rate,” Nonlinear Analysis: Real World Applications, vol. 13, no. 4, pp. 1866–1872, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  32. J. R. Coura and P. A. Vĩas, “Chagas disease: a new worldwide challenge,” Nature, vol. 465, no. 7301, pp. S6–S7, 2010. View at Publisher · View at Google Scholar · View at Scopus
  33. J. A. Pérez-Molina, A. M. Perez, F. F. Norman, B. Monge-Maillo, and R. López-Vélez, “Old and new challenges in chagas disease,” The Lancet Infectious Diseases, vol. 15, no. 11, pp. 1347–1356, 2015. View at Publisher · View at Google Scholar · View at Scopus
  34. A. L. Ribeiro, M. P. Nunes, M. M. Teixeira, and M. O. Rocha, “Diagnosis and management of Chagas disease and cardiomyopathy,” Nature Reviews Cardiology, vol. 9, no. 10, pp. 576–589, 2012. View at Publisher · View at Google Scholar · View at Scopus
  35. R. P. Beasley, C. Trepo, C. E. Stevens, and W. Szmuness, “The e antigen and vertical transmission of hepatitis B surface antigen,” The American Journal of Epidemiology, vol. 105, no. 2, pp. 94–98, 1977. View at Publisher · View at Google Scholar · View at Scopus
  36. K. Hattaf and N. Yousfi, “A generalized HBV model with diffusion and two delays,” Computers and Mathematics with Applications, vol. 69, no. 1, pp. 31–40, 2015. View at Publisher · View at Google Scholar · View at Scopus
  37. M. M. Thaler, D. W. Wara, G. Veereman-Wauters et al., “Vertical transmission of hepatitis C virus,” The Lancet, vol. 338, no. 8758, pp. 17-18, 1991. View at Publisher · View at Google Scholar · View at Scopus
  38. X. Meng, L. Chen, and B. Wu, “A delay SIR epidemic model with pulse vaccination and incubation times,” Nonlinear Analysis: Real World Applications, vol. 11, no. 1, pp. 88–98, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  39. T. Zhang, X. Meng, and T. Zhang, “SVEIRS: a new epidemic disease model with time delays and impulsive effects,” Abstract and Applied Analysis, vol. 2014, Article ID 542154, 15 pages, 2014. View at Publisher · View at Google Scholar · View at MathSciNet
  40. X. Mao, Stochastic Differential Equations and Applications, Horwood Publishing, Chichester, UK, 2nd edition, 1997. View at Publisher · View at Google Scholar · View at MathSciNet
  41. E. Beretta, V. Kolmanovskii, and L. Shaikhet, “Stability of epidemic model with time delays influenced by stochastic perturbations,” Mathematics and Computers in Simulation, vol. 45, no. 3-4, pp. 269–277, 1998. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  42. J. Yu, D. Jiang, and N. Shi, “Global stability of two-group SIR model with random perturbation,” Journal of Mathematical Analysis and Applications, vol. 360, no. 1, pp. 235–244, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  43. N. Bacaër, “On the stochastic SIS epidemic model in a periodic environment,” Journal of Mathematical Biology, vol. 71, no. 2, pp. 491–511, 2015. View at Publisher · View at Google Scholar · View at MathSciNet
  44. C. Ji and D. Jiang, “Threshold behaviour of a stochastic SIR model,” Applied Mathematical Modelling, vol. 38, no. 21-22, pp. 5067–5079, 2014. View at Publisher · View at Google Scholar · View at MathSciNet
  45. T. Feng, X. Meng, L. Liu, and S. Gao, “Application of inequalities technique to dynamics analysis of a stochastic eco-epidemiology model,” Journal of Inequalities and Applications, vol. 1, article 327, 2016. View at Publisher · View at Google Scholar · View at MathSciNet
  46. X. Meng, “Stability of a novel stochastic epidemic model with double epidemic hypothesis,” Applied Mathematics and Computation, vol. 217, no. 2, pp. 506–515, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  47. A. Gray, D. Greenhalgh, X. Mao, and J. Pan, “The SIS epidemic model with Markovian switching,” Journal of Mathematical Analysis and Applications, vol. 394, no. 2, pp. 496–516, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  48. H. C. Tuckwell and R. J. Williams, “Some properties of a simple stochastic epidemic model of SIR type,” Mathematical Biosciences, vol. 208, no. 1, pp. 76–97, 2007. View at Publisher · View at Google Scholar · View at MathSciNet
  49. X. Zhang, D. Jiang, A. Alsaedi, and T. Hayat, “Stationary distribution of stochastic SIS epidemic model with vaccination under regime switching,” Applied Mathematics Letters, vol. 59, pp. 87–93, 2016. View at Publisher · View at Google Scholar · View at MathSciNet
  50. Y. Cai, Y. Kang, M. Banerjee, and W. Wang, “A stochastic SIRS epidemic model with infectious force under intervention strategies,” Journal of Differential Equations, vol. 259, no. 12, pp. 7463–7502, 2015. View at Publisher · View at Google Scholar · View at MathSciNet
  51. W. Zhao, J. Li, T. Zhang, X. Meng, and T. Zhang, “Persistence and ergodicity of plant disease model with Markov conversion and impulsive toxicant input,” Communications in Nonlinear Science and Numerical Simulation, vol. 48, pp. 70–84, 2017. View at Publisher · View at Google Scholar · View at MathSciNet
  52. E. Tornatore, S. M. Buccellato, and P. Vetro, “Stability of a stochastic SIR system,” Physica A: Statistical Mechanics and Its Applications, vol. 354, no. 1-4, pp. 111–126, 2005. View at Publisher · View at Google Scholar · View at Scopus
  53. A. Gray, D. Greenhalgh, L. Hu, X. Mao, and J. Pan, “A stochastic differential equation SIS epidemic model,” SIAM Journal on Applied Mathematics, vol. 71, no. 3, pp. 876–902, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  54. Y. Zhao and D. Jiang, “The threshold of a stochastic SIS epidemic model with vaccination,” Applied Mathematics and Computation, vol. 243, pp. 718–727, 2014. View at Publisher · View at Google Scholar · View at MathSciNet
  55. Y. Lin and D. Jiang, “Long-time behaviour of a perturbed SIR model by white noise,” Discrete and Continuous Dynamical Systems. Series B, vol. 18, no. 7, pp. 1873–1887, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  56. H. Schurz and K. Tosun, “Stochastic asymptotic stability of SIR model with variable diffusion rates,” Journal of Dynamics and Differential Equations, vol. 27, no. 1, pp. 69–82, 2015. View at Publisher · View at Google Scholar · View at MathSciNet
  57. Q. Lu, “Stability of SIRS system with random perturbations,” Physica A. Statistical Mechanics and Its Applications, vol. 388, no. 18, pp. 3677–3686, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  58. X. Meng, S. Zhao, T. Feng, and T. Zhang, “Dynamics of a novel nonlinear stochastic SIS epidemic model with double epidemic hypothesis,” Journal of Mathematical Analysis and Applications, vol. 433, no. 1, pp. 227–242, 2016. View at Publisher · View at Google Scholar · View at MathSciNet
  59. F. Wei and J. Liu, “Long-time behavior of a stochastic epidemic model with varying population size,” Physica A: Statistical Mechanics and Its Applications, vol. 470, pp. 146–153, 2017. View at Publisher · View at Google Scholar · View at MathSciNet
  60. N. Dalal, D. Greenhalgh, and X. Mao, “A stochastic model of AIDS and condom use,” Journal of Mathematical Analysis and Applications, vol. 325, no. 1, pp. 36–53, 2007. View at Publisher · View at Google Scholar · View at MathSciNet
  61. C. Xu, “Global threshold dynamics of a stochastic differential equation SIS model,” Journal of Mathematical Analysis and Applications, vol. 447, no. 2, pp. 736–757, 2017. View at Publisher · View at Google Scholar · View at MathSciNet
  62. A. Lahrouz and A. Settati, “Necessary and sufficient condition for extinction and persistence of SIRS system with random perturbation,” Applied Mathematics and Computation, vol. 233, pp. 10–19, 2014. View at Publisher · View at Google Scholar · View at MathSciNet
  63. A. Lahrouz and A. Settati, “Qualitative study of a nonlinear stochastic SIRS epidemic system,” Stochastic Analysis and Applications, vol. 32, no. 6, pp. 992–1008, 2014. View at Publisher · View at Google Scholar · View at MathSciNet
  64. D. Zhao, T. Zhang, and S. Yuan, “The threshold of a stochastic SIVS epidemic model with nonlinear saturated incidence,” Physica A. Statistical Mechanics and Its Applications, vol. 443, pp. 372–379, 2016. View at Publisher · View at Google Scholar · View at MathSciNet
  65. Y. Zhao, Y. Lin, D. Jiang, X. Mao, and Y. Li, “Stationary distribution of stochastic SIRS epidemic model with standard incidence,” Discrete and Continuous Dynamical Systems. Series B, vol. 21, no. 7, pp. 2363–2378, 2016. View at Publisher · View at Google Scholar · View at MathSciNet
  66. Q. Liu, D. Jiang, N. Shi, T. Hayat, and A. Alsaedi, “Stationary distribution and extinction of a stochastic SIRS epidemic model with standard incidence,” Physica A. Statistical Mechanics and Its Applications, vol. 469, pp. 510–517, 2017. View at Publisher · View at Google Scholar · View at MathSciNet
  67. D. Jiang, Q. Liu, N. Shi, T. Hayat, A. Alsaedi, and P. Xia, “Dynamics of a stochastic HIV-1 infection model with logistic growth,” Physica A. Statistical Mechanics and Its Applications, vol. 469, pp. 706–717, 2017. View at Publisher · View at Google Scholar · View at MathSciNet
  68. Y. Cai, Y. Kang, M. Banerjee, and W. Wang, “A stochastic epidemic model incorporating media coverage,” Communications in Mathematical Sciences, vol. 14, no. 4, pp. 893–910, 2016. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  69. N. T. Dieu, D. H. Nguyen, N. H. Du, and G. Yin, “Classification of asymptotic behavior in a stochastic SIR model,” SIAM Journal on Applied Dynamical Systems, vol. 15, no. 2, pp. 1062–1084, 2016. View at Publisher · View at Google Scholar · View at MathSciNet
  70. N. H. Du and N. N. Nhu, “Permanence and extinction of certain stochastic SIR models perturbed by a complex type of noises,” Applied Mathematics Letters, vol. 64, pp. 223–230, 2017. View at Publisher · View at Google Scholar · View at MathSciNet
  71. Q. Liu and Q. Chen, “Analysis of the deterministic and stochastic SIRS epidemic models with nonlinear incidence,” Physica A. Statistical Mechanics and Its Applications, vol. 428, pp. 140–153, 2015. View at Publisher · View at Google Scholar · View at MathSciNet
  72. Z. Chang, X. Meng, and X. Lu, “Analysis of a novel stochastic SIRS epidemic model with two different saturated incidence rates,” Physica A. Statistical Mechanics and Its Applications, vol. 472, pp. 103–116, 2017. View at Publisher · View at Google Scholar · View at MathSciNet
  73. X. Zhang, D. Jiang, T. Hayat, and B. Ahmad, “Dynamics of a stochastic SIS model with double epidemic diseases driven by Lévy jumps,” Physica A. Statistical Mechanics and Its Applications, vol. 471, pp. 767–777, 2017. View at Publisher · View at Google Scholar · View at MathSciNet
  74. Y. Zhou, S. Yuan, and D. Zhao, “Threshold behavior of a stochastic SIS model with Lévy jumps,” Applied Mathematics and Computation, vol. 275, pp. 255–267, 2016. View at Publisher · View at Google Scholar · View at MathSciNet
  75. C. Li, Y. Pei, X. Liang, and D. Fang, “A stochastic toxoplasmosis spread model between cat and oocyst with jumps process,” Communications in Mathematical Biology and Neuroscience, 2016. View at Google Scholar
  76. D. Jiang, C. Ji, N. Shi, and J. Yu, “The long time behavior of DI SIR epidemic model with stochastic perturbation,” Journal of Mathematical Analysis and Applications, vol. 372, no. 1, pp. 162–180, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  77. M. Liu, C. Bai, and K. Wang, “Asymptotic stability of a two-group stochastic SEIR model with infinite delays,” Communications in Nonlinear Science and Numerical Simulation, vol. 19, no. 10, pp. 3444–3453, 2014. View at Publisher · View at Google Scholar · View at MathSciNet
  78. Q. Liu, D. Jiang, N. Shi, T. Hayat, and A. Alsaedi, “Asymptotic behavior of a stochastic delayed SEIR epidemic model with nonlinear incidence,” Physica A. Statistical Mechanics and Its Applications, vol. 462, pp. 870–882, 2016. View at Publisher · View at Google Scholar · View at MathSciNet
  79. X. Li, X. Lin, and Y. Lin, “Lyapunov-type conditions and stochastic differential equations driven by G-brownian motion,” Journal of Mathematical Analysis and Applications, vol. 439, no. 1, pp. 235–255, 2016. View at Publisher · View at Google Scholar · View at MathSciNet
  80. H. Ma and Y. Jia, “Stability analysis for stochastic differential equations with infinite Markovian switchings,” Journal of Mathematical Analysis and Applications, vol. 435, no. 1, pp. 593–605, 2016. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  81. A. Lahrouz and L. Omari, “Extinction and stationary distribution of a stochastic SIRS epidemic model with non-linear incidence,” Statistics and Probability Letters, vol. 83, no. 4, pp. 960–968, 2013. View at Publisher · View at Google Scholar · View at Scopus