Qualitative analysis of a fractional-order two-strain epidemic model with vaccination and general non-monotonic incidence rate
Tóm tắt
In this paper, a fractional-order two-strain epidemic model with vaccination and general non-monotonic incidence rate is analyzed. The studied problem is formulated using susceptible, infectious and recovered compartmental model. A Caputo fractional operator is incorporated in each compartment to describe the memory effect related to an epidemic evolution. First, the global existence, positivity and boundedness of solutions of the proposed model are proved. The basic reproduction numbers associated with studied problem are calculated. Four steady states are given, namely the disease-free equilibrium, the strain 1 endemic equilibrium, the strain 2 endemic equilibrium, and the endemic equilibrium associated with both strains. By considering appropriate Lyapunov functions, the global stability of the equilibrium points is proven according to the model parameters. Our modeling approach using a generalized non-monotonic incidence functions encloses a variety of fractional-order epidemic models existing in the literature. Finally, the theoretical findings are illustrated using numerical simulations.
Tài liệu tham khảo
Podlubny I (1999) Fractional differential equations. Maths Sci. Eng, San Diego
citation_journal_title=Nonlinear Anal Real World Appl; citation_title=The effect of vaccines on backward bifurcation in a fractional order HIV model; citation_author=J Huo, H Zhao, L Zhu; citation_volume=26; citation_publication_date=2015; citation_pages=289-305; citation_doi=10.1016/j.nonrwa.2015.05.014; citation_id=CR2
citation_journal_title=Alex Eng J; citation_title=Modelling the dynamics of novel coronavirus (2019-nCov) with fractional derivative; citation_author=MA Khan, A Atagana; citation_volume=59; citation_issue=4; citation_publication_date=2020; citation_pages=2379-2389; citation_doi=10.1016/j.aej.2020.02.033; citation_id=CR3
citation_title=Fractional-order nonlinear systems: modeling, analysis and simulation; citation_publication_date=2011; citation_id=CR4; citation_author=I Petráš; citation_publisher=Springer Science & Business Media
citation_journal_title=Phys Lett A; citation_title=On some Routh–Hurwitz conditions for fractional order differential equations and their applications in Lorenz, Rössler, Chua and Chen systems; citation_author=E Ahmed, A El-Sayed, HA El-Saka; citation_volume=358; citation_issue=1; citation_publication_date=2006; citation_pages=1-4; citation_doi=10.1016/j.physleta.2006.0.087; citation_id=CR5
citation_journal_title=Phys Rev E; citation_title=Memory effects on epidemic evolution: the susceptible-infected-recovered epidemic model; citation_author=M Saeedian, M Khalighi, N Azimi-Tafreshi, G Jafari, M Ausloos; citation_publication_date=2017; citation_doi=10.1103/PhysRevE.95.022409; citation_id=CR6
citation_journal_title=Results Phys; citation_title=Mathematical modeling and optimal control of the COVID-19 dynamics; citation_author=ZH Shen, YM Chu, MA Khan, S Muhammad, OA Al-Hartomy, M Higazy; citation_volume=31; citation_publication_date=2021; citation_doi=10.1016/j.rinp.2021.105028; citation_id=CR7
citation_journal_title=Results Phys; citation_title=Modeling and simulation of the novel coronavirus in Caputo derivative; citation_author=M Awais, FS Alshammari, S Ullah, MA Khan, S Islam; citation_volume=19; citation_publication_date=2020; citation_doi=10.1016/j.rinp.2020.103588; citation_id=CR8
citation_journal_title=Num Methods Partial Differ Equ; citation_title=Mathematical modeling for novel coronavirus (COVID-19) and control; citation_author=MS Alqarni, M Alghamdi, T Muhammad, AS Alshomrani, MA Khan; citation_volume=38; citation_issue=4; citation_publication_date=2022; citation_pages=760-776; citation_doi=10.1002/num.22695; citation_id=CR9
citation_journal_title=Commun Nonlinear Sci Numer Simul; citation_title=Optimal control strategies of Zika virus model with mutant; citation_author=EO Alzahrani, W Ahmad, MA Khan, SJ Malebary; citation_volume=93; citation_publication_date=2021; citation_doi=10.1016/j.cnsns.2020.105532; citation_id=CR10
citation_journal_title=Math Probl Eng; citation_title=On fractional order dengue epidemic model; citation_author=H Al-Sulami, M El-Shahed, JJ Nieto, W Shammakh; citation_publication_date=2014; citation_doi=10.1155/2014/456537; citation_id=CR11
citation_journal_title=Adv Differ Equ; citation_title=On a fractional order Ebola epidemic model; citation_author=I Area, H Batarf, J Losada, JJ Nieto, W Shammakh, Á Torres; citation_volume=1; citation_publication_date=2015; citation_pages=278; citation_doi=10.1186/s13662-015-0613-5; citation_id=CR12
citation_journal_title=Nonlinear Dyn; citation_title=A fractional calculus based model for the simulation of an outbreak of dengue fever; citation_author=K Diethelm; citation_volume=71; citation_issue=4; citation_publication_date=2013; citation_pages=613-619; citation_doi=10.1007/s11071-012-0475-2; citation_id=CR13
El-Saka H (2013) The fractional-order sir and sirs epidemic models with variable population size. Math Sci Lett 2(3):195.
https://doi.org/10.12785/msl/020308
citation_journal_title=Proc Series Brazil Soc Comput Appl Math; citation_title=Global stability of fractional sir epidemic model; citation_author=JPC Santos, E Monteiro, GB Vieira; citation_publication_date=2017; citation_doi=10.5540/03.2017.005.01.0019; citation_id=CR15
citation_journal_title=Pathogens; citation_title=Evidence for SARS-COV-2 infection of animal hosts; citation_author=A Abdel-Moneim, ES Abdelwhab; citation_volume=9; citation_publication_date=2020; citation_pages=529; citation_doi=10.3390/pathogens9070529; citation_id=CR16
citation_journal_title=Chaos Solitons Fract; citation_title=Fractional order mathematical modeling of COVID-19 transmission; citation_author=S Ahmad, A Ullah, QM Al-Mdallal, H Khan, K Shah, A Khan; citation_publication_date=2020; citation_doi=10.1016/j.chaos.2020.110256; citation_id=CR17
citation_journal_title=Results Phys; citation_title=Dynamics of fractional order COVID-19 model with a case study of Saudi Arabia; citation_author=YM Chu, A Ali, MA Khan, S Islam, S Ullah; citation_publication_date=2021; citation_doi=10.1016/j.rinp.2020.103787; citation_id=CR18
citation_journal_title=Model Earth Syst Environ; citation_title=The influence of awareness campaigns on the spread of an infectious disease: a qualitative analysis of a fractional epidemic model; citation_author=K Akdim, A Ez-Zetouni, M Zahid; citation_volume=8; citation_publication_date=2022; citation_pages=1311-1319; citation_doi=10.1007/s40808-021-01158-9; citation_id=CR19
citation_journal_title=AIMS Math; citation_title=Mathematical analysis of a fractional-order epidemic model with nonlinear incidence function; citation_author=S Djillali, A Atangana, A Zeb, C Park; citation_volume=7; citation_issue=2; citation_publication_date=2022; citation_pages=2160-2175; citation_doi=10.3934/math.2022123; citation_id=CR20
citation_journal_title=Math Biosci; citation_title=Global analysis of an epidemic model with nonmonotone incidence rate; citation_author=D Xiao, S Ruan; citation_volume=208; citation_issue=2; citation_publication_date=2007; citation_pages=419-29; citation_doi=10.1016/j.mbs.2006.09.025; citation_id=CR21
citation_journal_title=Commun Nonlinear Sci Numer Simul; citation_title=Dynamics of a delayed epidemic model with non-monotonic incidence rate; citation_author=HF Huo, ZP Ma; citation_volume=15; citation_publication_date=2010; citation_pages=459-468; citation_doi=10.1016/j.cnsns.2009.04.018; citation_id=CR22
citation_journal_title=J Public Affairs; citation_title=Dynamical analysis of novel COVID-19 epidemic model with non-monotonic incidence function; citation_author=R Prem Kumar, S Basu, D Ghosh, PK Santra, GS Mahapatra; citation_publication_date=2021; citation_doi=10.1002/pa.2754; citation_id=CR23
citation_journal_title=J Math Res; citation_title=Modeling and analysis of an epidemic model with non-monotonic incidence rate under treatment; citation_author=TK Kar, A Batabyal; citation_publication_date=2010; citation_doi=10.5539/jmr.v2n1p103; citation_id=CR24
citation_journal_title=Eur Phys J Plus; citation_title=Modeling the impact of the vaccine on the COVID-19 epidemic transmission via fractional derivative; citation_author=S Arshad, S Khalid, S Javed, N Amin, F Nawaz; citation_volume=137; citation_issue=7; citation_publication_date=2022; citation_pages=802; citation_doi=10.1140/epjp/s13360-022-02988-x; citation_id=CR25
citation_journal_title=PLoS ONE; citation_title=Modelling and optimal control of multi strain epidemics, with application to COVID-19; citation_author=EF Arruda, SS Das, CM Dias, DH Pastore; citation_volume=16; citation_issue=9; citation_publication_date=2021; citation_doi=10.1371/journal.pone.0257512; citation_id=CR26
citation_journal_title=J Theor Biol; citation_title=A multi-strain epidemic model for COVID-19 with infected and asymptomatic cases: application to French data; citation_author=M Massard, R Eftimie, A Perasso, B Saussereau; citation_volume=545; citation_publication_date=2022; citation_doi=10.1016/j.jtbi.2022.111117; citation_id=CR27
citation_journal_title=AIP Conf Proc; citation_title=A fractional-order two-strain epidemic model with two vaccinations; citation_author=B Kaymakamzade, E Hincal, D Amilo; citation_volume=2325; citation_publication_date=2021; citation_doi=10.1063/5.0040309; citation_id=CR28
citation_journal_title=Int J Biomath; citation_title=A stochastic vaccinated epidemic model incorporating Lévy processes with a general awareness-induced incidence; citation_author=K Akdim, A Ez-Zetouni, M Zahid; citation_publication_date=2021; citation_doi=10.1142/S1793524521500443; citation_id=CR29
citation_journal_title=Results Control Optim; citation_title=A fractal fractional order vaccination model of COVID-19 pandemic using Adam’s Moulton analysis; citation_author=GM Vijayalakshmi; citation_volume=8; citation_publication_date=2022; citation_doi=10.1016/j.rico.2022.100144; citation_id=CR30
Baba IA, Hincal E (2017) Global stability analysis of two-strain epidemic model with bilinear and non-monotone incidence rates. Eur Phys J Plus 132:208.
https://doi.org/10.1140/epjp/i2017-11476-x
Ez-Zetouni A, Khyar O, Allali K, Akdim K, Zahid M (2022) Stochastic and deterministic analysis of a COVID-19 pandemic model under vaccination strategy: real cases application, 07 March 2022, PREPRINT (Version 1) available at Research Square [
https://doi.org/10.21203/rs.3.rs-1346960/v1
]
citation_journal_title=Phys A; citation_title=A fractional-order model with different strains of COVID-19; citation_author=IA Baba, FA Rihan; citation_volume=603; citation_publication_date=2022; citation_doi=10.1016/j.physa.2022.127813; citation_id=CR33
citation_journal_title=Appl Math Sci; citation_title=An epidemic model with modified non-monotonic incidence rate under treatment; citation_author=G Ujjainkar, VK Gupta, B Singh, R Khandelwal, N Trivedi; citation_volume=6; citation_issue=21–24; citation_publication_date=2012; citation_pages=1159-1171; citation_id=CR34
citation_journal_title=Iranian J Sci Technol Trans A Sci; citation_title=Global Dynamics of a diffusive two-strain epidemic model with non-monotone incidence rate; citation_author=A Khatua, D Pal, TK Kar; citation_volume=13; citation_publication_date=2022; citation_pages=1; citation_id=CR35
citation_journal_title=J Differ Eqs; citation_title=A stochastic SIRS epidemic model with infectious force under intervention strategies; citation_author=Y Cai, Y Kang, M Banerjee, W Wang; citation_volume=59; citation_issue=12; citation_publication_date=2015; citation_pages=7463-7502; citation_doi=10.1016/j.jde.2015.08.024; citation_id=CR36
citation_journal_title=Proc R Soc Lond B; citation_title=Modelling strategies for controlling SARS outbreaks; citation_author=AB Gumel; citation_volume=271; citation_publication_date=2004; citation_pages=22-23; citation_doi=10.1098/rspb.2004.2800; citation_id=CR37
citation_journal_title=Chaos Solitons Fract; citation_title=A fractional model for the dynamics of TB virus; citation_author=S Ullah, MA Khan, M Farooq; citation_volume=1; citation_issue=116; citation_publication_date=2018; citation_pages=63-71; citation_doi=10.1016/j.chaos.2018.09.001; citation_id=CR38
citation_journal_title=Nonlinear Dyn; citation_title=Stability analysis of Caputo fractional-order nonlinear systems revisited; citation_author=H Delavari, D Baleanu, J Sadati; citation_volume=67; citation_issue=4; citation_publication_date=2012; citation_pages=2433-9; citation_doi=10.1007/s11071-011-0157-5; citation_id=CR39
citation_journal_title=Appl Math Comput; citation_title=Generalized Taylors formula; citation_author=ZM Odibat, NT Shawagfeh; citation_volume=186; citation_issue=1; citation_publication_date=2007; citation_pages=286-93; citation_id=CR40
citation_journal_title=J Math Anal Appl; citation_title=Global existence theory and chaos control of fractional differential equations; citation_author=W Lin; citation_volume=332; citation_publication_date=2007; citation_pages=709-726; citation_doi=10.1016/j.jmaa.2006.10.040; citation_id=CR41
citation_journal_title=Math Biosci; citation_title=Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission; citation_author=P Driessche, J Watmough; citation_volume=180; citation_publication_date=2002; citation_pages=29-48; citation_doi=10.1016/S0025-5564(02)00108-6; citation_id=CR42
citation_journal_title=Commun Nonlinear Sci Numer Simul; citation_title=Application of generalized diferential transform method to multi-order fractional diferential equations; citation_author=VS Erturk, S Momani, Z Odibat; citation_volume=13; citation_issue=8; citation_publication_date=2008; citation_pages=1642-1654; citation_doi=10.1016/j.cnsns.2007.02.006; citation_id=CR43
citation_journal_title=Chaos Solitons Fract; citation_title=A fractional-order epidemic model with time-delay and nonlinear incidence rate; citation_author=FA Rihan, QM Al-Mdallal, HJ AlSakaji, A Hashish; citation_volume=126; citation_publication_date=2019; citation_pages=97-105; citation_doi=10.1016/j.chaos.2019.05.039; citation_id=CR44
citation_journal_title=Chaos Solitons Fract; citation_title=Solutions of a disease model with fractional white noise; citation_author=MA Akinlar, M Inc, JF Gómez-Aguilar, B Boutarfa; citation_publication_date=2020; citation_doi=10.1016/j.chaos.2020.109840; citation_id=CR45
citation_journal_title=Phys A; citation_title=Stochastic viral infection model with lytic and nonlytic immune responses driven by Levy noise; citation_author=K Akdim, A Ez-zetouni, J Danane, K Allali; citation_volume=549; citation_publication_date=2020; citation_doi=10.1016/j.physa.2020.124367; citation_id=CR46
Sabbar Y, Kiouach D, Rajasekar SP (2022) Acute threshold dynamics of an epidemic system with quarantine strategy driven by correlated white noises and Lévy jumps associated with infinite measure. Int J Dyn Control 1–14