Fractional Multi-Step Differential Transformed Method for Approximating a Fractional Stochastic SIS Epidemic Model with Imperfect Vaccination
Abstract
:1. Introduction
2. Fractional Derivative
3. Stochastic SIS Epidemic Model with Imperfect Vaccination
Fractional Stochastic SIS Epidemic Model with Imperfect Vaccination
4. Description of the Method
5. Computational Illustrations
5.1. First Step: Finding FDTM for the First Interval
5.2. Second Step: Apply the MsFDTM for All Intervals
5.3. Case 1: . (Non-Fractional Case)
5.4. Case 2:
5.5. Case 3:
6. Discussion
7. Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
References
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Original Function | Transformed Function |
---|---|
Case | p | e | Comments | ||||||
---|---|---|---|---|---|---|---|---|---|
1 | 1 | stable | |||||||
2 | stable | ||||||||
3 | stable |
t | ||||
---|---|---|---|---|
0. | 0.05 | 0.1 | 0.6 | 0.75 |
1. | 0.105749 | 0.0427621 | 0.545775 | 0.694286 |
2. | 0.109974 | 0.0367964 | 0.587467 | 0.734237 |
3. | 0.113037 | 0.03184 | 0.625454 | 0.770331 |
4. | 0.115206 | 0.0276914 | 0.660073 | 0.802971 |
5. | 0.116686 | 0.0241951 | 0.691627 | 0.832508 |
6. | 0.117631 | 0.0212297 | 0.720388 | 0.859249 |
7. | 0.118162 | 0.0186994 | 0.746603 | 0.883465 |
8. | 0.118374 | 0.0165286 | 0.770498 | 0.9054 |
9. | 0.118339 | 0.0146565 | 0.792275 | 0.92527 |
10. | 0.118115 | 0.0130344 | 0.812122 | 0.943271 |
11. | 0.117749 | 0.0116226 | 0.830208 | 0.95958 |
12. | 0.117278 | 0.0103888 | 0.846687 | 0.974354 |
13. | 0.116731 | 0.00930643 | 0.861702 | 0.987739 |
14. | 0.11613 | 0.0083536 | 0.875381 | 0.999865 |
t | ||||
---|---|---|---|---|
0. | 0.06 | 0.1 | 0.5 | 0.66 |
1. | 0.0846833 | 0.0830972 | 0.513182 | 0.680963 |
2. | 0.102273 | 0.0692909 | 0.52789 | 0.699454 |
3. | 0.114928 | 0.0579974 | 0.543688 | 0.716614 |
4. | 0.12406 | 0.0487393 | 0.560244 | 0.733044 |
5. | 0.130622 | 0.0411297 | 0.5773 | 0.749051 |
6. | 0.135272 | 0.0348562 | 0.594653 | 0.764782 |
7. | 0.138481 | 0.0296671 | 0.612144 | 0.780292 |
8. | 0.140589 | 0.0253599 | 0.629646 | 0.795595 |
9. | 0.141847 | 0.0217719 | 0.647058 | 0.810678 |
10. | 0.142447 | 0.0187717 | 0.664302 | 0.82552 |
11. | 0.142532 | 0.0162536 | 0.681314 | 0.840099 |
12. | 0.142216 | 0.0141322 | 0.698044 | 0.854392 |
13. | 0.141586 | 0.0123381 | 0.714455 | 0.86838 |
14. | 0.140713 | 0.0108151 | 0.730517 | 0.882046 |
15. | 0.139652 | 0.00951743 | 0.746208 | 0.895378 |
16. | 0.138448 | 0.00840763 | 0.761512 | 0.908368 |
17. | 0.137136 | 0.00745506 | 0.776418 | 0.921009 |
18. | 0.135746 | 0.00663451 | 0.790919 | 0.9333 |
19. | 0.134302 | 0.00592521 | 0.805012 | 0.945238 |
20. | 0.132822 | 0.00530997 | 0.818695 | 0.956827 |
t | ||||
---|---|---|---|---|
0 | 0.05 | 0.1 | 0.6 | |
1. | 0.0843185 | 0.0806307 | 0.603477 | 0.768427 |
2. | 0.107201 | 0.0651439 | 0.610994 | 0.783339 |
3. | 0.122691 | 0.0527997 | 0.621487 | 0.796978 |
4. | 0.133229 | 0.0429688 | 0.634175 | 0.810373 |
5. | 0.140346 | 0.0351331 | 0.648474 | 0.823953 |
6. | 0.14504 | 0.0288752 | 0.663941 | 0.837856 |
7. | 0.147983 | 0.0238632 | 0.680235 | 0.852082 |
8. | 0.149641 | 0.0198347 | 0.697094 | 0.86657 |
9. | 0.150343 | 0.016584 | 0.714313 | 0.88124 |
10. | 0.150329 | 0.0139496 | 0.731731 | 0.896009 |
11. | 0.149776 | 0.0118048 | 0.749222 | 0.910803 |
12. | 0.148818 | 0.0100505 | 0.766688 | 0.925557 |
13. | 0.147556 | 0.00860876 | 0.784051 | 0.940216 |
14. | 0.146069 | 0.00741818 | 0.801249 | 0.954737 |
15. | 0.144418 | 0.0064303 | 0.818236 | 0.969084 |
16. | 0.142649 | 0.00560672 | 0.834975 | 0.98323 |
17. | 0.140801 | 0.0049169 | 0.851436 | 0.997154 |
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Abuasad, S.; Yildirim, A.; Hashim, I.; Abdul Karim, S.A.; Gómez-Aguilar, J.F. Fractional Multi-Step Differential Transformed Method for Approximating a Fractional Stochastic SIS Epidemic Model with Imperfect Vaccination. Int. J. Environ. Res. Public Health 2019, 16, 973. https://doi.org/10.3390/ijerph16060973
Abuasad S, Yildirim A, Hashim I, Abdul Karim SA, Gómez-Aguilar JF. Fractional Multi-Step Differential Transformed Method for Approximating a Fractional Stochastic SIS Epidemic Model with Imperfect Vaccination. International Journal of Environmental Research and Public Health. 2019; 16(6):973. https://doi.org/10.3390/ijerph16060973
Chicago/Turabian StyleAbuasad, Salah, Ahmet Yildirim, Ishak Hashim, Samsul Ariffin Abdul Karim, and J.F. Gómez-Aguilar. 2019. "Fractional Multi-Step Differential Transformed Method for Approximating a Fractional Stochastic SIS Epidemic Model with Imperfect Vaccination" International Journal of Environmental Research and Public Health 16, no. 6: 973. https://doi.org/10.3390/ijerph16060973