Efficiency of vaccines for COVID-19 and stability analysis with fractional derivative

Document Type : Research Paper


1 Department of Mathematics, Faulty of Basic Science, Bu-Ali Sina University, Hamedan 65178-38695, Iran.

2 Department of Mathematics, Hamedan University of Technology, Hamedan, Iran.

3 Gofa Camp, Near Gofa Industrial College and German Adebabay, Nifas Silk-Lafto, 26649 Addis Ababa, Ethiopia.

4 Institute of Mathematical Sciences, Faculty of Science, University of Malaya, Kuala Lumpur 50603, Malaysia.

5 Department of Pediatrics, Hamadan University of Medical Science, Hamadan, Iran.

6 Department of Mathematics and General Sciences, Prince Sultan University, Riyadh, Saudi Arabia.

7 Department of Industrial Engineering, OST˙IM Technical University, 06374 Ankara, Turkey.

8 Department of Applied Mathematics and Statistics, Technological University of Cartagena, Cartagena 30203, Spain.


The objectives of this study are to develop the SEIR model for COVID-19 and evaluate its main parameters such as therapeutic vaccines, vaccination rate, and effectiveness of prophylactic. Global and local stability of the model and numerical simulation are examined. The local stability of equilibrium points was classified. A Lyapunov function is constructed to analyze the global stability of the disease-free equilibrium. The simulation part is based on two situations, including the USA and Iran. Our results provide a good contribution to the current research on this topic.


Main Subjects

  •  [1] P. Agarwal, J. J. Nieto, M. Ruzhansky, and D. F. M. Torres, Analysis of Infectious Disease Problems (COVID-19) and Their Global Impact, Queen Mary University of London, London, UK, 2021.
  • [2] F. Ahmed, N. Ahmed, C. Pissarides, and J. Stiglitz, Why inequality could spread COVID-19, The Lancet Regional Health, 5(5) (2020), E240.
  • [3] M. Altaf Khan and A. Atangana, Modeling the dynamics of novel coronavirus (2019-nCov) with fractional derivative, Alexandria Engineering Journal, 59(4) (2020), 2379–2389.
  • [4] M. Amdouni, J. Alzabut, M. E. Samei, W. Sudsutad, and C. Thaiprayoon, A generalized approach of the GilpinAyala model with fractional derivatives under numerical simulation, Mathematics, 10(19) (2022), 3655.
  • [5] S. M. Aydogan, D. Baleanu, H. Mohammadi, and S. Rezapour, On the mathematical model of Rabies by using the fractional Caputo-Fabrizio derivative, Advances in Difference Equations, 2020 (2020), 382.
  • [6] I. A. Baba, U. W. Humphries, F. A. Rihan, and J. E. N´apoles Vald´es, Fractional–order modeling and control of COVID-19 with shedding effect, Axioms, 12 (2023), 321.
  • [7] M. Caputo and M. Fabrizio, A new definition of fractional derivative without singular kernel, Progress in Fractional Differentiation and Application, 1(2) (2015), 73–85.
  • [8] K. Diethelm, The Analysis of Fractional Differential Equations, Springer, Berlin, Heidelberg, 2010.
  • [9] E. F. Doungmo Goufo, Chaotic processes using the two-parameter derivative with non-singular and non-local kernel: Basic theory and applications, Chaos, 26(8) (2016), 084305.
  • [10] E. F. Doungmo Goufo and A. Atangana, Analytical and numerical schemes for a derivative with filtering property and no singular kernel with applications to diffusion, The European Physical Journal Plus, 131 (2016), 269.
  • [11] A. K. Golmankhaneh, I. Tejado, H. Sevli, and J. E. N´apoles Vald´es, On initial value problems of fractal delay equations, Applied Mathematics and Computation, 449 (2023), 127980.
  • [12] T. Guo, Q. Shen, W. Guo, W. He, J. Li, Y. Zhang, Y. Wang, Z. Zhou, D. Deng, X. Ouyang, Z. Xiang, W. Jiang, H. Luo, P. Chen, and H. Peng, Clinical characteristics of Elderly patients with COVID-19 in Hunan province, China: a multicenter, retrospective study, Gerontology, 66(5) (2020), 467–475.
  • [13] Z. Hu, C. Song, C. Xu, G. Jin, Y. Chen, X. Xu, H. Ma, W. Chen, Y. Lin, Y. Zheng, J. Wang, Z. Hu, Y. Yi, and H. Shen, Clinical characteristics of 24 asymptomatic infections with COVID-19 screened among close contacts in Nanjing, China, Sci. China Life Sci., 63(5) (2020), 706–711.
  • [14] C. Huang, Y. Wang, X. Li, L. Ren, J. Zhao, Y. Hu, L. Zhang, G. Fan, J. Xu, X. Gu, Z. Cheng, T. Yu, J. Xia, W. Wu, X. Xie, W. Yin, H. Li, M. Liu, Y. Xiao, H. Gao, L. Guo, J. Xie, G. Wang, R. Jiang, Z. Gao, Q. Jin, J. Wang, and B. Cao, Clinical features of patients infected with 2019 novel coronavirus in Wuhan, China, Lancet, 395(10223) (2020), 497–506.
  • [15] D. S. Hui, E. I. Azhar, T. A. Madani, F. Ntoumi, R. Kock, O. Dar, G. Ippolito, T. D. Mchugh, Z. A. Memish, C. Drosten, A. Zumla, and E. Petersen, The continuing 2019-nCoV epidemic threat of novel coronaviruses to global health - The latest 2019 novel coronavirus outbreak in Wuhan, China, International journal of infectious diseases, 91 (2020), 264–266.
  • [16] M. Jakovljevic, S. Bjedov, N. Jaksic, and I. Jakovljevic, COVID-19 pandemia and public and global mental health from the perspective of global health securit, Psychiatr Danub, 32(1) (2020), 6–14.
  •  [17] M. K. A. Kaabar, V. Kalvandi, N. Eghbali, M. E. Samei, Z. Siri, and F. Mart´ınez, Generalized Mittag-LefflerHyers-Ulam stability of a quadratic fractional integral equation, Nonlinear Engineering, 10 (2021), 414–427.
  • [18] A. A. Kilbas, H. M.Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 204, Elsevier, 2006.
  • [19] M. Lipsitch, D. L. Swerdlow, and L. Finelli, Defining the epidemiology of Covid-19 studies needed, New England journal of medicine, 382(13) (2020), 1194–1196.
  • [20] H. Mohammadi, S. Kumar, S. Rezapour, and S. Etemad, A theoretical study of the Caputo-Fabrizio fractional modeling for hearing loss due to Mumps virus with optimal control, Chaos Solitons & Fractals, 144 (2020), 110668.
  • [21] H. Nishiura, T. Kobayashi, T. Miyama, A. Suzuki, S. M. Jung, K. Hayashi, R. Kinoshita, Y. Yang, B. Yuan, A. R. Akhmetzhanov, and N. M. Linton, Estimation of the asymptomatic ratio of novel coronavirus infections (COVID-19), International journal of infectious diseases, 94 (2020), 154–155.
  • [22] R. Qesmi and A. Hammoumi, Lifting lockdown control measure assessment: From finite to infinite-dimensional epidemic models for COVID-19, arXiv, 2021 (2021).
  • [23] S. Rezapour and H. Mohammadi, A study on the AH1N1/09 influenza transmission model with the fractional Caputo-Fabrizio derivative, Advances in Difference Equations, 2020 (2020), 488.
  • [24] S. Rezapour, H. Mohammadi, and M. E. Samei, SEIR epidemic model for Covid-19 transmission by caputo derivative of fractional order, Advances in Difference Equations, 2020 (2020), 490.
  • [25] M. E. Samei, R. Ghaffari, S. W. Yao, M. K. A. Kaabar, F. Mart´ınez, and M. Inc, Existence of solutions for a singular fractional q−differential equations under Riemann–Liouville integral boundary condition, Symmetry, 13 (2021), 135.
  • [26] M. Vajdy, Induction and maintenance of long-term immunological memory following infection or vaccination, Frontiers in immunology, 10 (2019), 2658.
  • [27] D. Wang, B. Hu, C. Hu, F. Zhu, X. Liu, J. Zhang, B. Wang, H. Xiang, Z. Cheng, Y. Xiong, Y. Zhao, Y. Li, X. Wang, and Z. Peng, Clinical characteristics of 138 hospitalized patients with 2019 novel coronavirus–infected pneumonia in wuhan, china, JAMA Network, 323(11) (2020), 1061–1069.
  • [28] X. Wang, A. Berhail, N. Tabouche, M. M. Matar, M. E. Samei, M. K. A. Kaabar, and X. G. Yue, A novel investigation of non-periodic snap bvp in the G-caputo sense, Axioms, 11 (2022) , 390.
  • [29] P. Wintachai and K. Prathom, Stability analysis of SEIR model related to efficiency of vaccines for Covid-19 situation, Heliyon, 7(4) (2021) , e06812.
  • [30] H. Zhou, J. Alzabut, S. Rezapour, M. E. Samei, Uniform persistence and almost periodic solutions of a nonautonomous patch occupancy model, Advances in Difference Equations, 2020 (2020), 143.
  • [31] N. Zhu, D. Zhang, W. Wang, X. Li, B. Yang, J. Song, X. Zhao, B. Huang, W. Shi, R. Lu, P. Niu, F. Zhan, X. Ma, D. Wang, W. Xu, G. Wu, G. F. Gao, and W. Tan, A novel coronavirus from patients with pneumonia in China 2019, New England Journal of Medicine, 382(8) (2020), 727–733.