References
- Maltezou, H. C., Wicker, S., Borg, M., Heininger, U., Puro, V., Theodoridou, M., Poland, G. A. (2011). Vaccination policies for health-care workers in acute health-care facilities in Europe. Vaccine. 29(51):9557–9562.
- Buonomo, B., D'Onofrio, A., Lacitignola, D. (2008). Global stability of an SIR epidemic model with information dependent vaccination. Math. Biosci. 216(1):9–16. DOI: https://doi.org/10.1016/j.mbs.2008.07.011.
- Cao, B., Huo, H. F., Xiang, H. (2017). Global stability of an age-structure epidemic model with imperfect vaccination and relapse. Physica A. 486(15):638–655. DOI: https://doi.org/10.1016/j.physa.2017.05.056.
- Parsamanesh, M., Erfanian, M. (2018). Global dynamics of an epidemic model with standard incidence rate and vaccination strategy. Chaos Solitons Fract. 117:192–199. DOI: https://doi.org/10.1016/j.chaos.2018.10.022.
- Li, J., Yang, Y., Zhou, Y. (2011). Global stability of an epidemic model with latent stage and vaccination. Nonlinear Anal Real World Appl. 12(4):2163–2173. DOI: https://doi.org/10.1016/j.nonrwa.2010.12.030.
- Li, J. Q., Ma, Z. E. (2004). Stability analysis for SIS epidemic models with vaccination and constant population size. Discrete Continuous Dyn Syst. B. 4(3):635–642. DOI: https://doi.org/10.3934/dcdsb.2004.4.635.
- Li, J. Q., Ma, Z. N. (2004). Global analysis of SIS epidemic models with variable total population size. Math. Comput. Model. 39:1231–1242.
- May, R. (1973). Stability and Complexity in Model Ecosystems. Princeton. NJ: Princeton University.
- Mao, X. (2011). Stationary distribution of stochastic population systems. Syst. Contr. Lett. 60(6):398–405. DOI: https://doi.org/10.1016/j.sysconle.2011.02.013.
- Spencer, S. (2008). Stochastic epidemic models for emerging diseases. Ph.D. dissertation. University of Nottingham, Nottingham, UK.
- Beddington, J. R., May, R. M. (1977). Harvesting natural populations in a randomly fluctuating environment. Science. 197(4302):463–465. DOI: https://doi.org/10.1126/science.197.4302.463.
- Durrett, R. (1999). Stochastic spatial models. SIAM Rev. 41(4):677–718. DOI: https://doi.org/10.1137/S0036144599354707.
- Dennis, B. (2002). Allee effects in stochastic populations. Oikos. 96(3):389–401. DOI: https://doi.org/10.1034/j.1600-0706.2002.960301.x.
- Britton, T. (2010). Stochastic epidemic models: A survey. Math. Biosci. 225(1):24–35. DOI: https://doi.org/10.1016/j.mbs.2010.01.006.
- Zhu, C., Yin, G. (2009). On competitive Lotka-Volterra model in random environments. J. Math. Anal. Appl. 357(1):154–170. DOI: https://doi.org/10.1016/j.jmaa.2009.03.066.
- Imhof, L., Walcher, S. (2005). Exclusion and persistence in deterministic and stochastic chemostat models. J. Differ. Equat. 217(1):26–53. DOI: https://doi.org/10.1016/j.jde.2005.06.017.
- Mao, X., Marion, G., Renshaw, E. (2002). Environmental Brownian noise suppresses explosions in population dynamics. Stochast. Process. Appl. 97(1):95–110. DOI: https://doi.org/10.1016/S0304-4149(01)00126-0.
- Lahrouz, A., Omari, L., Kiouach, D. (2013). Extinction and stationary distribution of a stochastic SIRS epidemic model with non-linear incidence. Stat. Prob. Lett. 83(4):960–968. DOI: https://doi.org/10.1016/j.spl.2012.12.021.
- Lin, Y., Jiang, D., Wang, S. (2014). Stationary distribution of a stochastic SIS epidemic model with vaccination. Phys. A. 394:187–197. DOI: https://doi.org/10.1016/j.physa.2013.10.006.
- Zhang, X. B., Wang, X. D., Huo, H. F. (2019). Extinction and stationary distribution of a stochastic SIRS epidemic model with standard incidence rate and partial immunity. Physica A. 531:121548. DOI: https://doi.org/10.1016/j.physa.2019.121548.
- Ji, C., Jiang, D., Shi, N. (2011). Stability of epidemic model with time delays influenced by stochastic perturbations. Math. Comput. Simul. 45:1747–1762.
- Liu, Q. (2016). The threshold of a stochastic Susceptible-Infective epidemic model under regime switching. Nonlinear Anal. Hybrid Syst. 21:49–58. DOI: https://doi.org/10.1016/j.nahs.2016.01.002.
- Zhang, X. B., Chang, S. Q., Huo, H. F. (2019). Dynamic behavior of a stochastic SIR epidemic model with vertical transmission. Electron. J. Differ. Equat. 2019(125):1–20.
- Cai, S. Y., Cai, Y. M., Mao, X. R. (2019). A stochastic differential equation SIS epidemic model with two correlated Brownian motions. Nonlinear Dyn. 97(4):2175–2187. DOI: https://doi.org/10.1007/s11071-019-05114-2.
- Zhang, X. H., Jiang, D. Q., Alsaedi, A., Hayat, T. (2016). Stationary distribution of stochastic SIS epidemic model with vaccination under regime switching. Appl. Math. Lett. 59:87–93. DOI: https://doi.org/10.1016/j.aml.2016.03.010.
- Cao, B. Q., Shan, M. J., Zhang, Q. M., Wang, W. M. (2017). A stochastic SIS epidemic model with vaccination. Physica A. 486(15):127–143. DOI: https://doi.org/10.1016/j.physa.2017.05.083.
- Lin, Y. G., Jiang, D. Q., Wang, S. (2014). Stationary distribution of a stochastic SIS epidemic model with vaccination. Physica A. 394(15):187–197. DOI: https://doi.org/10.1016/j.physa.2013.10.006.
- Farnoosh, R., Parsamanesh, M. (2017). Stochastic differential equation systems for an SIS epidemic model with vaccination and immigration. Commun. Stat.: Theory Methods. 46(17):8723–8736. DOI: https://doi.org/10.1080/03610926.2016.1189571.
- Zhao, Y., Jiang, D. Q. (2014). The threshold of a stochastic SIS epidemic model with vaccination. Appl. Math. Comput. 243(15):718–727. DOI: https://doi.org/10.1016/j.amc.2014.05.124.
- Zhao, Y., Jiang, D., Mao, X., Gray, A. (2015). The threshold of a stochastic SIRS epidemic model in a population with varying size. DCDS-B. 20(4):1289–1307. DOI: https://doi.org/10.3934/dcdsb.2015.20.1289.
- Liu, Q., Chen, Q., Jiang, D. (2016). The threshold of a stochastic delayed SIR epidemic model with temporary immunity. Physica A. 450:115–125. DOI: https://doi.org/10.1016/j.physa.2015.12.056.
- Zhao, Y., Zhang, L., Yuan, S. (2018). The effect of media coverage on threshold dynamics for a stochastic SIS epidemic model. Physica A. 512:248–260. DOI: https://doi.org/10.1016/j.physa.2018.08.113.
- Cai, Y., Kang, Y., Banerjee, M., Wang, W. (2015). A stochastic SIRS epidemic model with infectious force under intervention strategies. J. Differ. Equat. 259(12):7463–7502. DOI: https://doi.org/10.1016/j.jde.2015.08.024.
- Gray, A., Greenhalgh, D., Hu, L., Mao, X., Pan, J. (2011). A stochastic differential equation SIS epidemic model. SIAM J. Appl. Math. 71(3):876–902. DOI: https://doi.org/10.1137/10081856X.
- Liu, Q., Jiang, D. Q., Hayat, T., Alsaedi, A. (2019). Threshold behavior in a stochastic delayed SIS epidemic model with vaccination and double diseases. J. Franklin Inst. 356(13):7466–7485. DOI: https://doi.org/10.1016/j.jfranklin.2018.11.055.
- Mao, X. R. (2011). Stationary distribution of stochastic population systems. Syst. Control Lett. 60(6):398–405. DOI: https://doi.org/10.1016/j.sysconle.2011.02.013.
- Freedman, H. I., Ruan, S. (1995). Uniform persistence in functional differential equations. J. Differ. Equat. 115(1):173–192. DOI: https://doi.org/10.1006/jdeq.1995.1011.
- Gopalsamy, K. (1992). Stability and Oscillations in Delay Differential Equations of Population Dynamics. Dordrecht: Kluwer Academic.
- He, X., Gopalsamy, K. (1997). Persistence, attractivity, and delay in facultative mutualism. J. Math. Anal. Appl. 215(1):154–173. DOI: https://doi.org/10.1006/jmaa.1997.5632.
- Teng, Z., Yu, Y. (2000). Some new results of nonautonomous Lotka-Volterra competitive systems with delays. J. Math. Anal. Appl. 241(2):254–275. DOI: https://doi.org/10.1006/jmaa.1999.6643.
- Zaman, G., Kang, Y. H., Jung, H. (2008). Stability analysis and optimal vaccination of an SIR epidemic model. Biosystems. 93(3):240–249. DOI: https://doi.org/10.1016/j.biosystems.2008.05.004.
- Zhang, X. B., Chang, S., Shi, Q., Huo, H. F. (2018). Qualitative study of a stochastic sis epidemic model with vertical transmission. Physica A. 505:805–817. DOI: https://doi.org/10.1016/j.physa.2018.04.022.
- Zhao, Y., Jiang, D. (2014). The threshold of a stochastic SIS epidemic model with vaccination. Appl. Math. Comput. 243:718–727. DOI: https://doi.org/10.1016/j.amc.2014.05.124.
- Zhang, X., Yuan, R. (2019). The existence of stationary distribution of a stochastic delayed chemostat model. Appl. Math. Lett. 93:15–21. DOI: https://doi.org/10.1016/j.aml.2019.01.034.
- Kinnally, M. (2009). Stationary Distributions for Stochastic Delay Differential Equations with Non-Negativity Constraints. San Diego, CA: University of California.
- National Bureau of Statistics of China, Annual Data (2019). Available from: http://http://data.stats.gov.cn/easyquery.htm?cn=C01.
- Chinese Center for Disease Control, Prevention, Significant Effect of Hepatitis B Control in China (2013). Available at: http://www.nhc.gov.cn/jkj/s3582/201307/518216575e544109b2caca07fca3b430.shtml
- National Health, Family Planning Commission, Transcript of media conference on World Hepatitis Day (2015). Available at: http://www.nhc.gov.cn/jkj/s3582/201507/280845c41c764f968aaf5678f43f8208.shtml
- Higham, D. J. (2001). An algorithmic introduction to numerical simulation of stochastic differential equations. Siam Rev. 43(3):525–546.