A fractional order epidemic model for the simulation of outbreaks of Ebola
Abstract The Ebola outbreak in 2014 caused many infections and deaths. Some literature works have proposed some models to study Ebola virus, such as SIR, SIS, SEIR, etc. It is proved that the fractional order model can describe epidemic dynamics better than the integer order model. In this paper, we...
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| Format: | Article |
| Language: | English |
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SpringerOpen
2021-03-01
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| Series: | Advances in Difference Equations |
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| Online Access: | https://doi.org/10.1186/s13662-021-03272-5 |
| _version_ | 1818649168828170240 |
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| author | Weiqiu Pan Tianzeng Li Safdar Ali |
| author_facet | Weiqiu Pan Tianzeng Li Safdar Ali |
| author_sort | Weiqiu Pan |
| collection | DOAJ |
| description | Abstract The Ebola outbreak in 2014 caused many infections and deaths. Some literature works have proposed some models to study Ebola virus, such as SIR, SIS, SEIR, etc. It is proved that the fractional order model can describe epidemic dynamics better than the integer order model. In this paper, we propose a fractional order Ebola system and analyze the nonnegative solution, the basic reproduction number R 0 $R_{0}$ , and the stabilities of equilibrium points for the system firstly. In many studies, the numerical solutions of some models cannot fit very well with the real data. Thus, to show the dynamics of the Ebola epidemic, the Gorenflo–Mainardi–Moretti–Paradisi scheme (GMMP) is taken to get the numerical solution of the SEIR fractional order Ebola system and the modified grid approximation method (MGAM) is used to acquire the parameters of the SEIR fractional order Ebola system. We consider that the GMMP method may lead to absurd numerical solutions, so its stability and convergence are given. Then, the new fractional orders, parameters, and the root-mean-square relative error g ( U ∗ ) = 0.4146 $g(U^{*})=0.4146$ are obtained. With the new fractional orders and parameters, the numerical solution of the SEIR fractional order Ebola system is closer to the real data than those models in other literature works. Meanwhile, we find that most of the fractional order Ebola systems have the same order. Hence, the fractional order Ebola system with different orders using the Caputo derivatives is also studied. We also adopt the MGAM algorithm to obtain the new orders, parameters, and the root-mean-square relative error which is g ( U ∗ ) = 0.2744 $g(U^{*})=0.2744$ . With the new parameters and orders, the fractional order Ebola systems with different orders fit very well with the real data. |
| first_indexed | 2024-12-17T01:30:02Z |
| format | Article |
| id | doaj.art-b6103390455d43d3a1dc69047f0a4ba7 |
| institution | Directory Open Access Journal |
| issn | 1687-1847 |
| language | English |
| last_indexed | 2024-12-17T01:30:02Z |
| publishDate | 2021-03-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Advances in Difference Equations |
| spelling | doaj.art-b6103390455d43d3a1dc69047f0a4ba72022-12-21T22:08:36ZengSpringerOpenAdvances in Difference Equations1687-18472021-03-012021112110.1186/s13662-021-03272-5A fractional order epidemic model for the simulation of outbreaks of EbolaWeiqiu Pan0Tianzeng Li1Safdar Ali2School of Mathematics and Statistics, Sichuan University of Science and EngineeringSchool of Mathematics and Statistics, Sichuan University of Science and EngineeringSchool of Mathematics and Statistics, Sichuan University of Science and EngineeringAbstract The Ebola outbreak in 2014 caused many infections and deaths. Some literature works have proposed some models to study Ebola virus, such as SIR, SIS, SEIR, etc. It is proved that the fractional order model can describe epidemic dynamics better than the integer order model. In this paper, we propose a fractional order Ebola system and analyze the nonnegative solution, the basic reproduction number R 0 $R_{0}$ , and the stabilities of equilibrium points for the system firstly. In many studies, the numerical solutions of some models cannot fit very well with the real data. Thus, to show the dynamics of the Ebola epidemic, the Gorenflo–Mainardi–Moretti–Paradisi scheme (GMMP) is taken to get the numerical solution of the SEIR fractional order Ebola system and the modified grid approximation method (MGAM) is used to acquire the parameters of the SEIR fractional order Ebola system. We consider that the GMMP method may lead to absurd numerical solutions, so its stability and convergence are given. Then, the new fractional orders, parameters, and the root-mean-square relative error g ( U ∗ ) = 0.4146 $g(U^{*})=0.4146$ are obtained. With the new fractional orders and parameters, the numerical solution of the SEIR fractional order Ebola system is closer to the real data than those models in other literature works. Meanwhile, we find that most of the fractional order Ebola systems have the same order. Hence, the fractional order Ebola system with different orders using the Caputo derivatives is also studied. We also adopt the MGAM algorithm to obtain the new orders, parameters, and the root-mean-square relative error which is g ( U ∗ ) = 0.2744 $g(U^{*})=0.2744$ . With the new parameters and orders, the fractional order Ebola systems with different orders fit very well with the real data.https://doi.org/10.1186/s13662-021-03272-5Epidemic modelsFractional order modelEbolaModified grid approximation method |
| spellingShingle | Weiqiu Pan Tianzeng Li Safdar Ali A fractional order epidemic model for the simulation of outbreaks of Ebola Advances in Difference Equations Epidemic models Fractional order model Ebola Modified grid approximation method |
| title | A fractional order epidemic model for the simulation of outbreaks of Ebola |
| title_full | A fractional order epidemic model for the simulation of outbreaks of Ebola |
| title_fullStr | A fractional order epidemic model for the simulation of outbreaks of Ebola |
| title_full_unstemmed | A fractional order epidemic model for the simulation of outbreaks of Ebola |
| title_short | A fractional order epidemic model for the simulation of outbreaks of Ebola |
| title_sort | fractional order epidemic model for the simulation of outbreaks of ebola |
| topic | Epidemic models Fractional order model Ebola Modified grid approximation method |
| url | https://doi.org/10.1186/s13662-021-03272-5 |
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