Parameter Estimation for a Class of Fractional Stochastic SIRD Models with Random Perturbations

Abstract

The classical SIRD model is extended to the conformable fractional stochastic SIRD model. The differences between the fractional stochastic SIRD model and the integer stochastic SIRD model are analyzed and compared using COVID-19 data from India. The results show that when the order of the fractional stochastic SIRD model is between $[0.93,0.99]$, the root mean square error between the simulated value and the real value of the number of infections is smaller than that of the integer stochastic SIRD model. Then, the maximum likelihood estimation of the parameters of the conformable fractional stochastic SIRD model is carried out, and compared with the maximum likelihood estimation results of the parameters of the integer stochastic SIRD model, It can be seen that the root mean square error of the fractional stochastic SIRD model is smaller when the fractional order is between $[0.93,0.99]$.

Keywords

• COVID-19
• Fractional stochastic SIRD model
• Maximum likelihood estimation

Supporting Institution

This research is partially supported the National Natural Science Foundation of China and the College Students’ innovation and entrepreneurship training program of China

Project Number

No 72031009, No 202210488007

References

1. [1] P. Giles, The mathematical theory of infectious diseases and its applications, J. Oper. Res. Soc., 28(2) (1977), 479-480.
2. [2] W. O. Kermack, A. G. Mckendrick. Contributions to the mathematical theory of epidemics–I 1927, Bull. Math. Biol., 53(1-2) (1991), 33-55.
3. [3] N. Becker, Estimation for an epidemic model, Biom., 32(4) (1976), 769-777.
4. [4] J. Timmer, Parameter estimation in nonlinear stochastic differential equations, Chaos. Soliton. Fract., 11(15) (2000), 2571-2578.
5. [5] E. Buckingham-Jeffery, V. Isham, T. House, Gaussian process approximations for fast inference from infectious disease data, Math. Biosci., 301 (2018), 111-120.
6. [6] K. Senel, M. Ozdinc, S. Ozturkcan, Single parameter estimation approach for robust estimation of SIR model with limited and noisy data: The case for COVID-19, Disaster. Med. Public., 3(15) (2021), E8-E22.
7. [7] M. M. Morato, I. M. L. Pataro, M. V. Americano da Costa, J. E. Normey-Rico, A parametrized nonlinear predictive control strategy for relaxing COVID-19 social distancing measures in Brazil, Isa. T., 124 (2022), 198-214.
8. [8] M. Farman, M. U. Saleem, A. Ahmad, M. O. Ahmad, Analysis and numerical solution of SEIR epidemic model of measles with non-integer time fractional derivatives by using Laplace Adomian Decomposition Method, Ain. Shams. Eng. J., 9(4) (2018), 3391-3397.

9. [9] K. Rajagopal, N. Hasanzaden, F. Parastesh, et al, A fractional-order model for the novel coronavirus (COVID-19) outbreak, Nonlinear. Dynam., 101(1) (2020), 711-718.
10. [10] B. Basti, N. Hammami, I. Berrabah, F. Nouioua, R. Djemiat, N. Benhamidouche, Stability analysis and existence of solutions for a modified SIRD model of COVID-19 with fractional derivatives, Symmetry, 13(8) (2021), 1431.
11. [11] H. Mohammadi, S. Rezapour, A. Jajarmi. On the fractional SIRD mathematical model and control for the transmission of COVID-19: The first and the second waves of the disease in Iran and Japan, Isa. T., 124 (2022), 103-114.
12. [12] S. Fouladi, M. Kohandel, B. Eastman. A comparison and calibration of integer and fractional-order models of COVID-19 with stratified public response, Math. Biosci. Eng., 19(12) (2022), 12792-12813.
13. [13] L. Akinyemi, M Senol, O. S. Iyiola. Exact solutions of the generalized multidimensional mathematical physics models via sub-equation method, Math. Comput. Simulat., 182, (2021), 211-233.
14. [14] F. Ashraf, A. R. Seadawy, S. Rizvi, et al. Multi-wave, M-shaped rational and interaction solutions for fractional nonlinear electrical transmission line equation, JGP., 177, (2022), 104503.
15. [15] Y. X. Kang, S. H. Mao, Y. H. Zhang, Fractional time-varying grey traffic flow model based on viscoelastic fluid and its application, Transport. Res. B-Meth., 157, (2022), 149-174.
16. [16] R. Khalil, M. A. Horani, A. Yousef, et al. A new definition of fractional derivative, J. Comput. Appl. Math., 264(5) (2014), 65-70.
17. [17] D. Fanelli, F. Piazza, Analysis and forecast of COVID-19 spreading in China, Italy and France, Chaos. Sol. Frac., 134 (2020), 109761.
18. [18] X. M. Wang, Applied multivariate analysis. ShangHai: Shanghai University of Finance and Economics Press, (2014).
19. [19] R. Behl, M. Mishra, COVID-19 and India: what next?, Inf. Discov. Deliv., 49(3) (2020), 250-258.
20. [20] Y. Wang, Y. Q. Feng, COVID-19 model and numerical solution based on fractional derivative of Conformable, Complex. Syst. Complex. Sci., 19(03) (2022), 27-32.