Extinction and Ergodic Stationary Distribution of COVID-19 Epidemic Model with Vaccination Effects
Abstract
1. Introduction
2. Models Formulation
3. The Qualitative Analysis for the Positive Solution
4. Extinction
5. Extinction
6. The Stationary Distribution of the Disease
Stationary Distribution
- 1.
- In both sides, open input U and in its neighbor, the smallest eigenvalue of A(t) has bounds that are separate.
- 2.
- If the average time τ (at which a curve starts from x going to the set U) is of finiteness, and for every compact subset . Next, if is an integrating function having measurement π, then
7. Numerical Simulations
8. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
- Huang, C.; Wang, Y.; Li, X.; Ren, L.; Zhao, J.; Hu, Y.; Zhang, L. Clinical features of patients infected with 2019 novel coronavirus in Wuhan, China. Lancet 2020, 395, 497–506. [Google Scholar] [CrossRef] [PubMed]
- Ding, Y.; Jiao, J.; Zhang, Q.; Zhang, Y.; Ren, X. Stationary Distribution and Extinction in a Stochastic SIQR Epidemic Model Incorporating Media Coverage and Markovian Switching. Symmetry 2021, 13, 1122. [Google Scholar] [CrossRef]
- Atangana, A. Modelling the spread of COVID-19 with new fractal-fractional operators: Can the lockdown save mankind before vaccination? Chaos Solitons Fractals 2020, 136, 109860. [Google Scholar] [CrossRef] [PubMed]
- Tul, A.Q.; Anjum, N.; Zeb, A.; Djilali, S.; Khan, Z.A. On the analysis of Caputo fractional order dynamics of Middle East Lungs Coronavirus (MERS-CoV) model. Alex. Eng. J. 2022, 61, 5123–5131. [Google Scholar]
- Ali, I.; Khan, S.U. Threshold of Stochastic SIRS Epidemic Model from Infectious to Susceptible Class with Saturated Incidence Rate Using Spectral Method. Symmetry 2022, 14, 1838. [Google Scholar] [CrossRef]
- Anwarud, D.; Li, Y.; Shah, M.A. The complex dynamics of hepatitis B infected individuals with optimal control. J. Syst. Sci. Complex. 2021, 34, 1301–1323. [Google Scholar]
- Allen, L.J.S. An introduction to stochastic epidemic models. In Mathematical Epidemiology; Springer: Berlin/Heidelberg, Germany, 2008; pp. 81–130. [Google Scholar]
- Lei, Q.; Yang, Z. Dynamical behaviours of a stochastic SIRI epidemic model. Appl. Anal. 2016, 96, 1–13. [Google Scholar]
- Özdemir, N.; Agrawal, O.P.; Karadeniz, D.; İskender, B.B. Fractional optimal control problem of an axis-symmetric diffusion-wave propagation. Phys. Scr. 2009, T136, 014024. [Google Scholar] [CrossRef]
- Din, A.; Yassine, S. Long-term bifurcation and stochastic optimal control of a triple-delayed Ebola illness model with vaccination and quarantine strategies. Fractal Fract. 2022, 6, 578. [Google Scholar] [CrossRef]
- Khan, A.; Yassine, S. Stochastic modeling of the Monkeypox 2022 epidemic with cross-infection hypothesis in a highly disturbed environment. Math. Biosci. Eng. 2022, 19, 13560–13581. [Google Scholar] [CrossRef] [PubMed]
- Din, A.; Li, Y.; Tahir, K.; Gul, Z. Mathematical analysis of spread and control of the novel corona virus (COVID-19) in China. Chaos Solitons Fractals 2020, 141, 110286. [Google Scholar] [CrossRef] [PubMed]
- Din, A.; Li, Y.; Abdullahi, Y. Delayed hepatitis B epidemic model with stochastic analysis. Chaos Solitons Fractals 2021, 146, 110839. [Google Scholar] [CrossRef]
- Yassine, S.; Khan, A.; Din, A.; Kiouach, D.; Rajasekar, S.P. Determining the global threshold of an epidemic model with general interference function and high-order perturbation. AIMS Math. 2022, 7, 19865–19890. [Google Scholar]
- Alzahrani, E.O.; Khan, M.A. Modeling the dynamics of Hepatitis E with optimal control. Chaos Solitons Fractals 2018, 116, 287–301. [Google Scholar] [CrossRef]
- Olver, P.J. Applications of Lie Groups to Differential Equations; Springer: Berlin/Heidelberg, Germany, 2000. [Google Scholar]
- Olver, P.J. Equivalence, Invariants, and Symmetry; Cambridge University Press: Cambridge, UK, 1995. [Google Scholar]
- Zhang, T.; Ding, T.; Gao, N.; Song, Y. Dynamical behavior of a stochastic SIRC model for influenza A. Symmetry 2020, 12, 745. [Google Scholar] [CrossRef]
- Gaeta, G.; Rodriguez-Quintero, N. Lie-point symmetries and stochastic differential equations. J. Phys. A Math. Gen. 1999, 32, 8485–8505. [Google Scholar] [CrossRef]
- Gaeta, G. Lie-point symmetries and stochastic differential equations: II. J. Phys. A Math. Gen. 2000, 33, 4883–4902. [Google Scholar] [CrossRef]
- Zhang, X.-B.; Wang, X.-D.; Huo, H. Extinction and stationary distribution of a stochastic SIRS epidemic model with standard incidence rate and partial immunity. Phys. Stat. Mech. Appl. 2019, 531, 121548. [Google Scholar] [CrossRef]
- Din, A.; Saida, A.; Amina, A. A stochastically perturbed co-infection epidemic model for COVID-19 and hepatitis B virus. Nonlinear Dyn. 2022, 28, 1–25. [Google Scholar] [CrossRef] [PubMed]
- Khasminskii, R. Stochastic Stability of Differential Equations; Springer Science & Business Media: Berlin/Heidelberg, Germany, 2011; Volume 66. [Google Scholar]
Parameter | Description |
---|---|
Rate of recruitment. | |
Rate of infection effectively | |
Vaccinated population in percentage. | |
effect of Vaccination | |
Rate of natural death | |
Rate of sign reported by lab | |
Recovery rate from | |
COVID-19 death rate | |
Transferred rate from to to | |
COVID-19 death rate | |
recovered rate of |
Parameters | Source | ||
---|---|---|---|
1.50 | 3.50 | assumed | |
0.02 | 0.30 | assumed | |
0.07 | 0.03 | assumed | |
0.01 | 0.03 | assumed | |
0.01 | 0.05 | assumed | |
0.02 | 0.04 | assumed | |
0.05 | 0.10 | assumed | |
0.35 | 0.20 | assumed | |
0.05 | 0.30 | assumed | |
0.55 | 0.40 | assumed | |
0.15 | 0.50 | assumed | |
50.0 | 4.00 | assumed | |
20.0 | 3.00 | assumed | |
30.0 | 1.00 | assumed | |
40.0 | 2.00 | assumed | |
40.0 | 2.00 | assumed | |
10.0 | 1.00 | assumed | |
1.25 | 1.20 | assumed | |
1.23 | 1.25 | assumed | |
1.35 | 1.15 | assumed | |
1.20 | 1.05 | assumed | |
1.15 | 1.22 | assumed | |
1.10 | 1.15 | assumed |
Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |
© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).
Share and Cite
Batool, H.; Li, W.; Sun, Z. Extinction and Ergodic Stationary Distribution of COVID-19 Epidemic Model with Vaccination Effects. Symmetry 2023, 15, 285. https://doi.org/10.3390/sym15020285
Batool H, Li W, Sun Z. Extinction and Ergodic Stationary Distribution of COVID-19 Epidemic Model with Vaccination Effects. Symmetry. 2023; 15(2):285. https://doi.org/10.3390/sym15020285
Chicago/Turabian StyleBatool, Humera, Weiyu Li, and Zhonggui Sun. 2023. "Extinction and Ergodic Stationary Distribution of COVID-19 Epidemic Model with Vaccination Effects" Symmetry 15, no. 2: 285. https://doi.org/10.3390/sym15020285
APA StyleBatool, H., Li, W., & Sun, Z. (2023). Extinction and Ergodic Stationary Distribution of COVID-19 Epidemic Model with Vaccination Effects. Symmetry, 15(2), 285. https://doi.org/10.3390/sym15020285