Fractional Multi-Step Differential Transformed Method for Approximating a Fractional Stochastic SIS Epidemic Model with Imperfect Vaccination
In this paper, we applied a fractional multi-step differential transformed method, which is a generalization of the multi-step differential transformed method, to find approximate solutions to one of the most important epidemiology and mathematical ecology, fractional stochastic SIS epidemic model w...
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| Published in | International journal of environmental research and public health Vol. 16; no. 6; p. 973 |
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| Main Authors | , , , , |
| Format | Journal Article |
| Language | English |
| Published |
Switzerland
MDPI AG
18.03.2019
MDPI |
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| Online Access | Get full text |
| ISSN | 1660-4601 1661-7827 1660-4601 |
| DOI | 10.3390/ijerph16060973 |
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| Abstract | In this paper, we applied a fractional multi-step differential transformed method, which is a generalization of the multi-step differential transformed method, to find approximate solutions to one of the most important epidemiology and mathematical ecology, fractional stochastic SIS epidemic model with imperfect vaccination, subject to appropriate initial conditions. The fractional derivatives are described in the Caputo sense. Numerical results coupled with graphical representations indicate that the proposed method is robust and precise which can give new interpretations for various types of dynamical systems. |
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| AbstractList | In this paper, we applied a fractional multi-step differential transformed method, which is a generalization of the multi-step differential transformed method, to find approximate solutions to one of the most important epidemiology and mathematical ecology, fractional stochastic SIS epidemic model with imperfect vaccination, subject to appropriate initial conditions. The fractional derivatives are described in the Caputo sense. Numerical results coupled with graphical representations indicate that the proposed method is robust and precise which can give new interpretations for various types of dynamical systems. In this paper, we applied a fractional multi-step differential transformed method, which is a generalization of the multi-step differential transformed method, to find approximate solutions to one of the most important epidemiology and mathematical ecology, fractional stochastic SIS epidemic model with imperfect vaccination, subject to appropriate initial conditions. The fractional derivatives are described in the Caputo sense. Numerical results coupled with graphical representations indicate that the proposed method is robust and precise which can give new interpretations for various types of dynamical systems.In this paper, we applied a fractional multi-step differential transformed method, which is a generalization of the multi-step differential transformed method, to find approximate solutions to one of the most important epidemiology and mathematical ecology, fractional stochastic SIS epidemic model with imperfect vaccination, subject to appropriate initial conditions. The fractional derivatives are described in the Caputo sense. Numerical results coupled with graphical representations indicate that the proposed method is robust and precise which can give new interpretations for various types of dynamical systems. Introduction Mathematical modeling of nonlinear systems is a key challenge for contemporary scientists; it is a basic description of physical reality expressed in mathematical terms. Saravanan and Magesh [27] compared two analytical methods: FRDTM vs fractional variational iteration method (FVIM) to find numerical solutions of the linear and nonlinear Fokker-Planck partial differential equations with space and time fractional derivatives, Singh [28] presented FRDTM to compute an alternative approximate solution of initial valued autonomous system of linear and nonlinear fractional partial differential equations and Abuasad et al. The fractional multi-step differential transform method (FMsDTM) is capable of generating approximate solutions of a wide class of linear and nonlinear problems with fractional derivatives that converge quickly to the exact solutions. Dt0αi xi(t)=hi(t,x1,x2,⋯,xn),i=1,2,…,n,t0≤t≤T, with the initial conditions xi(t0)=ci,i=1,2,…,n, where 0<αi≤1 , ci(i=1,2,…,n) are real finite constants, and Dαi is the Caputo fractional derivative of order αi . Before applying the multistep method, we define the fractional differential transform of h(t) as H(k)=1Γ(kα+1)Dtkαh(t)t=t0, then the nth approximate series form solution of fractional initial value problem (FIVP) (5) and (6) can be given by xi=∑k=0NUi(k)(t−t0)kαi,t∈[t0,T], where Ui satisfies the following recurrence relation Ui(K+1)=Γ(kα+1)Γ((k+1)αi+1)Hi(k,U1,U2,⋯,Un),i=1,2,⋯,n, where Hi(k,U1,U2,⋯,Un) denotes the differential transformed function of hi(t,x1,x2,⋯,xn) , subject to the initial conditions Ui(t0)=ci,i=1,2,…,n . Assume that the interval [0,T] is divided into M sub-intervals [ti−1,ti],i=1,2,…,M, of the same step size |
| Author | Gómez-Aguilar, J.F. Yildirim, Ahmet Abuasad, Salah Hashim, Ishak Abdul Karim, Samsul Ariffin |
| AuthorAffiliation | 2 Department of Mathematics, Ege University, 35100 Bornova, Izmir, Turkey; yahmet49ege@gmail.com 4 Fundamental and Applied Sciences Department and Center for Smart Grid Energy Research (CSMER). Institute of Autonomous System, Universiti Teknologi PETRONAS, Bandar Seri Iskandar, 32610 Seri Iskandar, Perak DR, Malaysia; samsul_ariffin@utp.edu.my 1 Preparatory Year Deanship, King Faisal University, 31982 Hofuf, Al-Hasa, Saudi Arabia 5 CONACyT-Tecnológico Nacional de México/CENIDET. Interior Internado Palmira S/N, Col. Palmira, C.P. 62490 Cuernavaca, Morelos, Mexico; jgomez@cenidet.edu.mx 3 School of Mathematical Sciences, Universiti Kebangsaan Malaysia, 43600 UKM Bangi Selangor, Malaysia; ishak_h@ukm.edu.my |
| AuthorAffiliation_xml | – name: 4 Fundamental and Applied Sciences Department and Center for Smart Grid Energy Research (CSMER). Institute of Autonomous System, Universiti Teknologi PETRONAS, Bandar Seri Iskandar, 32610 Seri Iskandar, Perak DR, Malaysia; samsul_ariffin@utp.edu.my – name: 1 Preparatory Year Deanship, King Faisal University, 31982 Hofuf, Al-Hasa, Saudi Arabia – name: 2 Department of Mathematics, Ege University, 35100 Bornova, Izmir, Turkey; yahmet49ege@gmail.com – name: 5 CONACyT-Tecnológico Nacional de México/CENIDET. Interior Internado Palmira S/N, Col. Palmira, C.P. 62490 Cuernavaca, Morelos, Mexico; jgomez@cenidet.edu.mx – name: 3 School of Mathematical Sciences, Universiti Kebangsaan Malaysia, 43600 UKM Bangi Selangor, Malaysia; ishak_h@ukm.edu.my |
| Author_xml | – sequence: 1 givenname: Salah orcidid: 0000-0001-6173-652X surname: Abuasad fullname: Abuasad, Salah – sequence: 2 givenname: Ahmet orcidid: 0000-0001-8989-4271 surname: Yildirim fullname: Yildirim, Ahmet – sequence: 3 givenname: Ishak orcidid: 0000-0003-4237-7140 surname: Hashim fullname: Hashim, Ishak – sequence: 4 givenname: Samsul Ariffin orcidid: 0000-0001-6518-6705 surname: Abdul Karim fullname: Abdul Karim, Samsul Ariffin – sequence: 5 givenname: J.F. orcidid: 0000-0001-9403-3767 surname: Gómez-Aguilar fullname: Gómez-Aguilar, J.F. |
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| CitedBy_id | crossref_primary_10_1155_2019_5703916 crossref_primary_10_1515_phys_2020_0188 crossref_primary_10_37394_232017_2024_15_12 crossref_primary_10_3390_sym15010194 crossref_primary_10_1142_S1793962320410032 crossref_primary_10_3934_mmc_2022015 crossref_primary_10_1063_5_0153122 crossref_primary_10_1155_2022_7374751 crossref_primary_10_3390_math8010096 crossref_primary_10_3390_fractalfract5040168 crossref_primary_10_1016_j_chaos_2019_07_053 crossref_primary_10_3934_mbe_2022201 crossref_primary_10_37394_232021_2023_3_11 crossref_primary_10_1002_jnm_3220 crossref_primary_10_1142_S1793524520500825 crossref_primary_10_1186_s13662_021_03347_3 crossref_primary_10_3390_sym15091721 crossref_primary_10_3390_math8050729 crossref_primary_10_1140_epjp_s13360_020_00856_0 |
| Cites_doi | 10.1016/j.cnsns.2017.01.018 10.1016/j.chaos.2006.09.004 10.1007/978-3-319-17954-4 10.12816/0033742 10.1016/j.jksus.2015.01.003 10.1016/j.apm.2007.09.025 10.1016/S0096-3003(02)00794-4 10.1016/j.amc.2007.07.068 10.1016/j.matcom.2017.06.004 10.1007/s00500-016-2378-5 10.22436/jnsa.010.10.09 10.1016/S0096-3003(03)00581-2 10.1016/j.amc.2015.10.016 10.1007/BFb0067096 10.1016/j.aml.2007.02.022 10.18576/pfda/030404 10.17576/jsm-2018-4711-33 10.1515/IJNSNS.2009.10.6.741 10.1016/j.matcom.2011.07.007 10.1016/S0096-3003(02)00368-5 10.1016/j.amc.2004.10.009 10.12732/ijpam.v95i4.8 |
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| Keywords | fractional calculus imperfect vaccination differential transformed method multi-step differential transformed method stochastic SIS epidemic model |
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| Snippet | In this paper, we applied a fractional multi-step differential transformed method, which is a generalization of the multi-step differential transformed method,... Introduction Mathematical modeling of nonlinear systems is a key challenge for contemporary scientists; it is a basic description of physical reality expressed... |
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| SubjectTerms | Infections Infectious diseases Methods Ordinary differential equations Partial differential equations Population Researchers Vaccines |
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| Title | Fractional Multi-Step Differential Transformed Method for Approximating a Fractional Stochastic SIS Epidemic Model with Imperfect Vaccination |
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