References
- AbdullahiBaba I, Yusuf A, Al-Shomrani M. 2021. A mathematical model for studying rape and its possible mode of control. Results Phys. 22:103917.
- Beretta E, Takeuchi Y. 1995. Global stability of an SIR epidemic model with time delays. J Math Biol. 33:250–260.
- Coddington E, Levinson N. 1955. Theory of ordinary differential equations. New York: Tata McGraw-Hill Education.
- Cooke K. 1979. Stability analysis for a vector disease model. Rocky Mountain J Math. 9:31–42.
- Dalal N, Greenhalgh D, Mao X. 2008. A stochastic model for internal HIV dynamics. J Math Anal Appl. 341:1084–1101.
- Din A, Li Y. 2020. Controlling heroin addiction via age-structured modeling. Adv Diff Equ. 2020:521.
- Din A, Li Y. 2021. Stationary distribution extinction and optimal control for the stochastic hepatitis B epidemic model with partial immunity. Phys Scr. 96:074005.
- Din A, Li Y, Khan T, Anwar K, Zaman G. 2021a. Stochastic dynamics of hepatitis b epidemics. Results Phys. 20:103730.
- Din A, Li Y, Khan T, Zaman G. 2020b. Mathematical analysis of spread and control of the novel corona virus (COVID-19) in china. Chaos Solitons Fractals. 141:110286.
- Din A, Li Y, Liu Q. 2020a. Viral dynamics and control of hepatitis b virus (HBV) using an epidemic model. Alex Eng J. 59(2):667–679.
- Din A, Li Y, Shah MA. 2021b. The complex dynamics of hepatitis b infected in dividuals with optimal control. J Syst Sci Complex. 2021:1–23.
- Din A, Li Y, Yusuf A. 2021c. Stochastic analysis of a delayed hepatitis B epidemic model. Chaos, Solitons Fractals. 146:110839.
- Edmunds W, Medley G, Nokes D. 1996. The transmission dynamics and control of hepatitis b virus in the Gambia. Stat Med. 15:2215–2233.
- Gaff HD, Schaefer E, Lenhart S. 2010. Use of optimal control models to predict treatment time for managing tick-borne disease. J Biol Dyn. 5:517–530.
- Ji C, Jiang D. 2014. Threshold behaviour of a stochastic sir model. Appl Math Model. 39:5067–5079.
- Khan A, Ikram R, Din A, Humphries UW, Akgul A. 2021a. Stochastic COVID-19 SEIQ epidemic model with time-delay. Results Phys. 30:104775.
- Khan A, Zarin R, Hussain G, Usman AH, Humphries UW, Gomez-Aguilar JF. 2021b. Modeling and sensitivity analysis of HBV epidemic model with convex incidence rate. Results Phys. 22:103836.
- Khan T, Jung IH, Zaman G. 2019. A stochastic model for the transmission dynamics of hepatitis b virus. J Biol Dyn. 13(1):328–344.
- Khan T, Khan A, Zaman G. 2018. The extinction and persistence of the stochastic hepatitis B epidemic model. Chaos Solitons Fractals. 108:123–128.
- Khan T, Zaman G, Chohan MI. 2017. The transmission dynamic and optimal control of acute and chronic hepatitis B. J Biol Dyn. 11:172–189.
- Khasminskii R. 2011. Stochastic stability of differential equations. Berlin: Springer Science and Business Media.
- Kiouach D, Sabbar Y. 2019. Modeling the impact of media intervention on controlling the diseases with stochastic perturbations. AIP Conf Proc. 2074(1):20026.
- Kiouach D, Sabbar Y. 2020. Ergodic stationary distribution of a stochastic hepatitisb epidemic model with interval-valued parameters and compensated poisson process. Comput Math Methods Med. 2020:9676501.
- Kuang Y. 1993. Delay differential equations with applications in population dynamics. New York: Academic Press.
- Kumar A, Srivastava PK, Takeuchi Y. 2016. Modeling the role of information and limited optimal treatment on disease prevalence. J Theor Biol. 414:103–119.
- Li F, Zhang S, Meng X. 2019. Dynamics analysis and numerical simulations of a delayed stochastic epidemic model subject to a general response function. Comput Appl Math. 38:95.
- Liu Q, Chen Q. 2015. Analysis of the deterministic and stochastic SIRS epidemic models with nonlinear incidence. Phys A: Stat Mech Appl. 428:140–153.
- Liu Q, Jiang D, Shi N, Hayat T, Alsaedi A. 2017. Stationary distribution and extinction of a stochastic SIRS epidemic model with standard incidence. Phys A Stat Mech Appl. 469:510–517.
- Qi H, Meng X. 2021. Mathematical modeling, analysis and numerical simulation of HIV: the inffuence of stochastic environmental ffuctuations on dynamics. Math Comput Simulat. 187:700–719.
- Rihan FA, Alsakaji HJ, Rajivganthi C. 2020. Stochastic SIRC epidemic model with time-delay for COVID-19. Adv Diff Equ. 502:1–20.
- Russell S. 2004. The economic burden of illness for households in developing countries: a review of studies focusing on malaria, tuberculosis, and human immunodeficiency virus/acquires immunodeficiency syndrome. Am J Trop Med Hyg. 71:147–155.
- Xia W, Kundu S, Maitra S. 2018. Dynamics of a delayed SEIQ epidemic model. Adv Diff Equ 2018:336.
- Xie F, Shan M, Lian X, Weiming Wang. 2017. Periodic solution of a stochastic HBV infection model with logistic hepatocyte growth. Appl Math Comput. 293:630–41.
- Xu C, Li X. 2018. The threshold of a stochastic delayed SIRS epidemic model with temporary immunity and vaccination. Chaos Soliton Fractals. 111:1–8.
- Yuan Y, Allen LJS. 2011. Stochastic models for virus and immune system dynamics. Math Biosci. 234(2):84–94.
- Yusuf A, Acay B, Mustapha UT, Inc M, Baleanu D. 2021. Mathematical modeling of pine wilt disease with caputo fractional operator. Chaos Solitons Fractals. 143:110569.
- Zhang X-B, Huo H-F, Xiang H, Meng X-Y. 2014. Dynamics of the deterministic and stochastic SIQS epidemic model with non-linear incidence. Appl Math Comput. 243:546–588.
- 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 im munity. Phys A: Stat Mech Appl 531:121548.
- Zhao Y, Jiang D. 2014. The threshold of a stochastic SIRS epidemic model with saturated incidence. Appl Math Lett. 34:90–93.