Deterministic and Stochastic Nonlinear Model for Transmission Dynamics of COVID-19 with Vaccinations Following Bayesian-Type Procedure
Abstract
:1. Introduction
Model Formulation
2. Fundamental Analysis
3. Uniqueness and Existence
4. Mathematical Analysis of Deterministic Model
4.1. Disease-Free Equilibrium Points
4.2. Endemic Equilibrium
4.3. Expression for
- (i)
- If or/and , then there is no positive equilibrium.
- (ii)
- If or/and There exists a unique positive equilibrium called the endemic equilibrium.
5. Stability Analysis
5.1. Local Stability
5.2. Global Stability
- 1.
- A unique single variant of the delta infection equilibrium if and only if .
- 2.
- A unique single variant of the omicron infection equilibrium if and only if .
- 3.
- A double infection equilibrium when .
6. Second-Order Derivative Model
6.1. Strength Number
6.2. Mathematical Analysis
7. Numerical Results and Discussion for Deterministic Model
8. Stochastic form of the Proposed Model
Existence and Uniqueness of (23)
- (i)
- (ii)
- then
9. Basic Reproduction Number for Stochastic Model
10. Stability Analysis of Stochastic Model
11. Stochastic Model and Numerical Interpretation
12. Conclusions
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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Parameters | Physical Meaning and Representation |
---|---|
Susceptible compartment | |
Vaccinated compartment | |
Delta virus infected compartment | |
Omicron virus infected compartment | |
Recovered compartment | |
Birth rate | |
Disease death rate of delta virus | |
Natural death rate | |
Disease death rate of omicron virus | |
Vaccination rate | |
k | Contact rate |
Transmission of delta virus | |
Transmission of omicron virus | |
Delta virus vaccine efficacy | |
Loss of immunity | |
Vaccination rate | |
Recovery from delta virus | |
Recovery of omicron virus |
Parameters | Approximate Values [31,32] |
---|---|
0.0016 | |
0.0000683 | |
0.0032 | |
0.0032 | |
0.011 | |
k | 0.024 |
0.127 | |
0.127 | |
0.16 | |
0.01111 | |
0.09 | |
0.09 | |
0.09 |
Error Analysis | ||
---|---|---|
Compartment | Error of Approximation for Deterministic Model | Error of Approximation for Stochastic Model |
0.05432 | 0.00345 | |
0.03214 | 0.008765 | |
0.01236 | 0.001129 | |
0.06754 | 0.003214 | |
0.00769 | 0.000341 |
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Jeelani, M.B.; Din, R.U.; Alhamzi, G.; Hleili, M.; Alrabaiah, H. Deterministic and Stochastic Nonlinear Model for Transmission Dynamics of COVID-19 with Vaccinations Following Bayesian-Type Procedure. Mathematics 2024, 12, 1662. https://doi.org/10.3390/math12111662
Jeelani MB, Din RU, Alhamzi G, Hleili M, Alrabaiah H. Deterministic and Stochastic Nonlinear Model for Transmission Dynamics of COVID-19 with Vaccinations Following Bayesian-Type Procedure. Mathematics. 2024; 12(11):1662. https://doi.org/10.3390/math12111662
Chicago/Turabian StyleJeelani, Mohammadi Begum, Rahim Ud Din, Ghaliah Alhamzi, Manel Hleili, and Hussam Alrabaiah. 2024. "Deterministic and Stochastic Nonlinear Model for Transmission Dynamics of COVID-19 with Vaccinations Following Bayesian-Type Procedure" Mathematics 12, no. 11: 1662. https://doi.org/10.3390/math12111662
APA StyleJeelani, M. B., Din, R. U., Alhamzi, G., Hleili, M., & Alrabaiah, H. (2024). Deterministic and Stochastic Nonlinear Model for Transmission Dynamics of COVID-19 with Vaccinations Following Bayesian-Type Procedure. Mathematics, 12(11), 1662. https://doi.org/10.3390/math12111662