Article Sidebar

Submitted
June 27, 2023
Accepted
September 8, 2024
Published
October 30, 2024
Download
PDF
Statistic
Read Counter : 36 Download : 13

Downloads

Download data is not yet available.

Main Article Content

Abstract

In this paper, we suggest a spatio-temporal epidemic model for coronavirus. Our model will be represented by a system of six partial differential non-linear equations that describe the dynamics of susceptible, exposed, infected, quarantined, removed, and vaccinated individuals. We will start the study of this model by presenting some results of the existence and uniqueness to the solution of our suggested model. By using the method of next-generation matrix, we obtain the basic reproduction number. The model has one disease-free equilibrium point and another endemic steady state. The global stability of these steady states is proved by using some Lyapunouv functions. Finally, different numerical simulations are given to confirm our results given in the theoretical part of the paper.

Keywords

SEIQRV model COVID-19 Reaction-diffusion Global Stability

Article Details

Creative Commons License

This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Author Biographies

Marya Sadki, Laboratory of Mathematics, Computer Science and Applications, Faculty of Sciences and Technologies, University Hassan II of Casablanca, PO Box 146, Mohammedia 20650, Morocco

 

 

Karam Allali

 

 

How to Cite
Yaagoub, Z., Sadki, M., & Allali, K. (2024). GLOBAL STABILITY OF SPATIO-TEMPORAL MODEL WITH QUARANTINE AND VACCINATION. Journal of the Indonesian Mathematical Society, 30(2), 321–337. https://doi.org/10.22342/jims.30.2.1452.321-337
Download Citation
Endnote/Zotero/Mendeley (RIS)
BibTeX
Crossref
Scopus
Google Scholar
Europe PMC

References

  1. Zeb, A., Alzahrani, E., Erturk, V. S., & Zaman, G. (2020). Mathematical model for coronavirus disease 2019 (COVID-19) containing isolation class. BioMed research international, 2020.
  2. Yaagoub, Z., Sadki, M., & Allali, K. (2024). A generalized fractional hepatitis B virus infection model with both cell-to-cell and virus-to-cell transmissions. Nonlinear Dyn (2024). https://doi.org/10.1007/s11071-024-09867-3
  3. Yaagoub, Z. & Allali, K. (2023). Fractional HCV infection model with adaptive immunity and treatment. MATHEMATICAL MODELING AND COMPUTING, 10, 995 –1013.
  4. Allali, K., Tabit, Y., & Harroudi, S. (2017). On HIV model with adaptive immune response, two saturated rates and therapy. Mathematical Modelling of Natural Phenomena, 12(5), 1-14.
  5. Woolliscroft, J. O. (2020). Innovation in response to the COVID-19 pandemic crisis. Academic medicine.
  6. World Health Organization WHO. [Online]; https://covid19.who.int/
  7. Kermack, W. O., & McKendrick, A. G. (1927). A contribution to the mathematical theory of epidemics. Proceedings of the royal society of london. Series A, Containing papers of a mathematical and physical character, 115(772), 700-721.
  8. Khyar, O., & Allali, K. (2020). Global dynamics of a multi-strain SEIR epidemic model with general incidence rates: application to COVID-19 pandemic. Nonlinear dynamics, 102(1), 489-509.
  9. Bentaleb, D., Harroudi, S., Amine, S., & Allali, K. (2020). Analysis and optimal control of a multistrain SEIR epidemic model with saturated incidence rate and treatment. Differential Equations and Dynamical Systems, 1-17.
  10. Rangasamy, M., Chesneau, C., Martin-Barreiro, C., & Leiva, V. (2022). On a novel dynamics of SEIR epidemic models with a potential application to COVID-19. Symmetry, 14(7), 1436.
  11. Kamrujjaman, M., Saha, P., Islam, M. S., & Ghosh, U. (2022). Dynamics of SEIR model: a case study of COVID-19 in Italy. Results in Control and Optimization, 7, 100119.
  12. Chen, Z., Feng, L., Lay Jr, H. A., Furati, K., & Khaliq, A. (2022). SEIR model with unreported infected population and dynamic parameters for the spread of COVID-19. Mathematics and computers in simulation, 198, 31-46.
  13. Mohajan, D., & Mohajan, H. K. (2022). Mathematical analysis of SEIR model to prevent COVID-19 pandemic. Journal of Economic Development, Environment and People, 11(4), 5-30.
  14. Meskaf, A., Khyar, O., Danane, J., & Allali, K. (2020). Global stability analysis of a two-strain epidemic model with non-monotone incidence rates. Chaos, Solitons & Fractals, 133, 109647.
  15. Yaagoub, Z., Danane, J., & Allali, K. (2022). Global Stability Analysis of Two-Strain SEIR Epidemic Model with Quarantine Strategy. In Nonlinear Dynamics and Complexity: Mathematical Modelling of Real-World Problems (pp. 469-493). Cham: Springer International Publishing.
  16. Wagner, C. E., Saad-Roy, C. M., & Grenfell, B. T. (2022). Modelling vaccination strategies for COVID-19. Nature Reviews Immunology, 22(3), 139-141.
  17. Zephaniah, O. C., Nwaugonma, U. I. R., Chioma, I. S., & Adrew, O. (2020). A mathematical model and analysis of an SVEIR model for streptococcus pneumonia with saturated incidence force of infection. Mathematical Modelling and Applications, 5(1), 16-38.
  18. El Hajji, M., & Albargi, A. H. (2022). A mathematical investigation of an “SVEIR” epidemic model for the measles transmission. Math. Biosci. Eng, 19, 2853-2875.
  19. Xu, J. (2022). Global dynamics for an SVEIR epidemic model with diffusion and nonlinear incidence rate. Boundary Value Problems, 2022(1), 80.
  20. Zhu, L., & Wang, X. (2023). Global analysis of a new reaction -diffusion multi-group SVEIR propagation model with time delay. Zeitschrift für angewandte Mathematik und Physik, 74(1), 25.
  21. Nasution, H., Khairani, N., Ahyaningsih, F., & Alamsyah, F. (2022, November). Mathematical modeling of the spread of corona virus disease 19 (COVID-19) with vaccines. In AIP Conference Proceedings (Vol. 2659, No. 1, p. 110009). AIP Publishing LLC.
  22. Zhang, Z., Kundu, S., Tripathi, J. P., & Bugalia, S. (2020). Stability and Hopf bifurcation analysis of an SVEIR epidemic model with vaccination and multiple time delays. Chaos, Solitons & Fractals, 131, 109483.
  23. Al-Darabsah, I. (2021). A time-delayed SVEIR model for imperfect vaccine with a generalized nonmonotone incidence and application to measles. Applied Mathematical Modelling, 91, 74-92.
  24. Nabti, A., & Ghanbari, B. (2021). Global stability analysis of a fractional SVEIR epidemic model. Mathematical Methods in the Applied Sciences, 44(11), 8577-8597.
  25. Sepulveda, G., Arenas, A. J., & González-Parra, G. (2023). Mathematical Modeling of COVID-19 Dynamics under Two Vaccination Doses and Delay Effects. Mathematics, 11(2), 369.
  26. Zhu, L., & Wang, X. (2023). Global analysis of a new reaction -diffusion multi-group SVEIR propagation model with time delay. Zeitschrift für angewandte Mathematik und Physik, 74(1), 25.
  27. Baba, I. A., Kaymakamzade, B., & Hincal, E. (2018). Two-strain epidemic model with two vaccinations. Chaos, Solitons & Fractals, 106, 342-348.
  28. Yaagoub, Z., & Allali, K. (2023). Global Stability of Multi-Strain SEIR Epidemic Model with Vaccination Strategy. Mathematical and Computational Applications, 28(1), 9.
  29. Ojha, R. P., Srivastava, P. K., Sanyal, G., & Gupta, N. (2021). Improved model for the stability analysis of wireless sensor network against malware attacks. Wireless Personal Communications, 116, 2525-2548.
  30. Madhusudanan, V., Srinivas, M. N., & Sridhar, S. (2021). Effect of Noise on Pandemic Structure for Proliferation of Malevolent Nodes in Remote Sensor Network. Wireless Personal Communications, 119, 567-584.
  31. Zou, Y., Yang, W., Lai, J., Hou, J., & Lin, W. (2022). Vaccination and quarantine effect on COVID-19 transmission dynamics incorporating Chinese-spring-festival travel rush: modeling and simulations. Bulletin of Mathematical Biology, 84(2), 30.
  32. Martínez Martínez, I., Florián Quitián, A., Díaz-López, D., Nespoli, P., & Gómez Mármol, F. (2021). MalSEIRS: Forecasting Malware Spread Based on Compartmental Models in Epidemiology. Complexity, 2021, 1-19.
  33. Nwokoye, C. H., & Umeh, I. I. (2017). The SEIQR-V model: On a more accurate analytical characterization of malicious threat defense. International Journal of Information Technology and Computer Science, 9(12), 28-37.
  34. Alshabrawi, M. (2021). Mathematical Model for understanding the spread of COVID-19 in Saudi Arabia with access to vaccination.
  35. Malik, A., Alkholief, M., Aldakheel, F. M., Khan, A. A., Ahmad, Z., Kamal, W., ... & Alshamsan, A. (2022). Sensitivity analysis of COVID-19 with quarantine and vaccination: A fractal-fractional model. Alexandria Engineering Journal, 61(11), 8859-8874.
  36. Pao, C. V. (1982). On nonlinear reaction-diffusion systems. Journal of Mathematical Analysis and Applications, 87(1), 165-198.
  37. Ahmed, N., Tahira, S. S., Rafiq, M., Rehman, M. A., Ali, M., & Ahmad, M. O. (2019). Positivity preserving operator splitting nonstandard finite difference methods for SEIR reaction diffusion model. Open Mathematics, 17(1), 313-330.
  38. Capone, F., De Cataldis, V., & De Luca, R. (2013). On the nonlinear stability of an epidemic SEIR reaction-diffusion model. Ricerche di Matematica, 62, 161-181.
  39. Zhu, L., & Wang, X. (2023). Global analysis of a new reaction -diffusion multi-group SVEIR propagation model with time delay. Zeitschrift für angewandte Mathematik und Physik, 74(1), 25.
  40. Xu, J. (2022). Global dynamics for an SVEIR epidemic model with diffusion and nonlinear incidence rate. Boundary Value Problems, 2022(1), 80.
  41. Zhang, C., Gao, J., Sun, H., & Wang, J. (2019). Dynamics of a reaction -diffusion SVIR model in a spatial heterogeneous environment. Physica A: Statistical Mechanics and its Applications, 533, 122049.
  42. Zheng, W., Zhang, J., & Zhang, W. (2006). Global exponential stability of reaction-diffusion neural networks with both variable time delays and unbounded delay. Lecture notes in computer science, 4113, 377.
  43. Pasha, S. A., Nawaz, Y., & Arif, M. S. (2023). On the nonstandard finite difference method for reaction -diffusion models. Chaos, Solitons & Fractals, 166, 112929.
  44. Wang, Z., & Zhang, Q. (2023). Near-optimal control of a stochastic partial differential equation SEIR epidemic model under economic constraints. European Journal of Control, 69, 100752.
  45. Tu, Y., Hayat, T., Hobiny, A., & Meng, X. (2023). Modeling and multi-objective optimal control of reaction-diffusion COVID-19 system due to vaccination and patient isolation. Applied Mathematical Modelling.
  46. Kouidere A, Elhia M , Balatif O. A spatiotemporal spread of COVID-19 pandemic with vaccination optimal control strategy: A case study in Morocco. MATHEMATICAL MODELING AND COMPUTING, Vol. 10, No. 1, pp. 171 -185 (2023).
  47. Nefedov, N. (2013). Comparison principle for reaction-diffusion-advection problems with boundary and internal layers. In Numerical Analysis and Its Applications: 5th International Conference, NAA 2012, Lozenetz, Bulgaria, June 15-20, 2012, Revised Selected Papers 5 (pp. 62-72). Springer Berlin Heidelberg.
  48. Amann, H. (1978). Existence and stability of solutions for semi-linear parabolic systems, and applications to some diffusion reaction equations. Proceedings of the Royal Society of Edinburgh Section A: Mathematics, 81(1-2), 35-47.
  49. Yaagoub, Z., & Allali, K. (2022). Fractional HBV infection model with both cell-to-cell and virus-to-cell transmissions and adaptive immunity. Chaos, Solitons & Fractals, 165, 112855.
  50. Sadki M, Harroudi S, Allali K. Dynamical analysis of an HCV model with cell-to-cell transmission and cure rate in the presence of adaptive immunity. MATHEMATICAL MODELING AND COMPUTING, Vol. 9, No. 3, pp. 579-593 (2022).
  51. Hattaf, K., & Yousfi, N. (2020). Global stability for fractional diffusion equations in biological systems. Complexity, 2020, 1-6.