New Method to Investigate the Impact of Independent Quadratic <i>α</i>-Stable Poisson Jumps on the Dynamics of a Disease under Vaccination Strategy
Long-run bifurcation analysis aims to describe the asymptotic behavior of a dynamical system. One of the main objectives of mathematical epidemiology is to determine the acute threshold between an infection’s persistence and its elimination. In this study, we use a more comprehensive SVIR epidemic m...
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| Format: | Article |
| Language: | English |
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MDPI AG
2023-03-01
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| Series: | Fractal and Fractional |
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| Online Access: | https://www.mdpi.com/2504-3110/7/3/226 |
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| author | Yassine Sabbar Asad Khan Anwarud Din Mouhcine Tilioua |
| author_facet | Yassine Sabbar Asad Khan Anwarud Din Mouhcine Tilioua |
| author_sort | Yassine Sabbar |
| collection | DOAJ |
| description | Long-run bifurcation analysis aims to describe the asymptotic behavior of a dynamical system. One of the main objectives of mathematical epidemiology is to determine the acute threshold between an infection’s persistence and its elimination. In this study, we use a more comprehensive SVIR epidemic model with large jumps to tackle this and related challenging problems in epidemiology. The huge discontinuities arising from the complexity of the problem are modelled by four independent, tempered, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>α</mi></semantics></math></inline-formula>-stable quadratic Lévy processes. A new analytical method is used and for the proposed stochastic model, the critical value <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi mathvariant="fraktur">R</mi><mn>0</mn><mo>🟉</mo></msubsup></semantics></math></inline-formula> is calculated. For strictly positive value of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi mathvariant="fraktur">R</mi><mn>0</mn><mo>🟉</mo></msubsup></semantics></math></inline-formula>, the stationary and ergodic properties of the perturbed model are verified (continuation scenario). However, for a strictly negative value of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi mathvariant="fraktur">R</mi><mn>0</mn><mo>🟉</mo></msubsup></semantics></math></inline-formula>, the model predicts that the infection will vanish exponentially (disappearance scenario). The current study incorporates a large number of earlier works and provides a novel analytical method that can successfully handle numerous stochastic models. This innovative approach can successfully handle a variety of stochastic models in a wide range of applications. For the tempered <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>α</mi></semantics></math></inline-formula>-stable processes, the Rosinski (2007) algorithm with a specific Lévy measure is implemented as a numerical application. It is concluded that both noise intensities and parameter <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>α</mi></semantics></math></inline-formula> have a great influence on the dynamical transition of the model as well as on the shape of its associated probability density function. |
| first_indexed | 2024-03-11T06:30:52Z |
| format | Article |
| id | doaj.art-4789c2ef70d14a3bb48e1542eacb6a12 |
| institution | Directory Open Access Journal |
| issn | 2504-3110 |
| language | English |
| last_indexed | 2024-03-11T06:30:52Z |
| publishDate | 2023-03-01 |
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| series | Fractal and Fractional |
| spelling | doaj.art-4789c2ef70d14a3bb48e1542eacb6a122023-11-17T11:12:05ZengMDPI AGFractal and Fractional2504-31102023-03-017322610.3390/fractalfract7030226New Method to Investigate the Impact of Independent Quadratic <i>α</i>-Stable Poisson Jumps on the Dynamics of a Disease under Vaccination StrategyYassine Sabbar0Asad Khan1Anwarud Din2Mouhcine Tilioua3MAIS Laboratory, MAMCS Group, FST Errachidia, Moulay Ismail University of Meknes, P.O. Box 509, Errachidia 52000, MoroccoSchool of Computer Science and Cyber Engineering, Guangzhou University, Guangzhou 510006, ChinaDepartment of Mathematics, Sun Yat-sen University, Guangzhou 510006, ChinaMAIS Laboratory, MAMCS Group, FST Errachidia, Moulay Ismail University of Meknes, P.O. Box 509, Errachidia 52000, MoroccoLong-run bifurcation analysis aims to describe the asymptotic behavior of a dynamical system. One of the main objectives of mathematical epidemiology is to determine the acute threshold between an infection’s persistence and its elimination. In this study, we use a more comprehensive SVIR epidemic model with large jumps to tackle this and related challenging problems in epidemiology. The huge discontinuities arising from the complexity of the problem are modelled by four independent, tempered, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>α</mi></semantics></math></inline-formula>-stable quadratic Lévy processes. A new analytical method is used and for the proposed stochastic model, the critical value <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi mathvariant="fraktur">R</mi><mn>0</mn><mo>🟉</mo></msubsup></semantics></math></inline-formula> is calculated. For strictly positive value of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi mathvariant="fraktur">R</mi><mn>0</mn><mo>🟉</mo></msubsup></semantics></math></inline-formula>, the stationary and ergodic properties of the perturbed model are verified (continuation scenario). However, for a strictly negative value of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi mathvariant="fraktur">R</mi><mn>0</mn><mo>🟉</mo></msubsup></semantics></math></inline-formula>, the model predicts that the infection will vanish exponentially (disappearance scenario). The current study incorporates a large number of earlier works and provides a novel analytical method that can successfully handle numerous stochastic models. This innovative approach can successfully handle a variety of stochastic models in a wide range of applications. For the tempered <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>α</mi></semantics></math></inline-formula>-stable processes, the Rosinski (2007) algorithm with a specific Lévy measure is implemented as a numerical application. It is concluded that both noise intensities and parameter <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>α</mi></semantics></math></inline-formula> have a great influence on the dynamical transition of the model as well as on the shape of its associated probability density function.https://www.mdpi.com/2504-3110/7/3/226dynamical systemnoisebifurcationergodicitylévy processesjumps |
| spellingShingle | Yassine Sabbar Asad Khan Anwarud Din Mouhcine Tilioua New Method to Investigate the Impact of Independent Quadratic <i>α</i>-Stable Poisson Jumps on the Dynamics of a Disease under Vaccination Strategy Fractal and Fractional dynamical system noise bifurcation ergodicity lévy processes jumps |
| title | New Method to Investigate the Impact of Independent Quadratic <i>α</i>-Stable Poisson Jumps on the Dynamics of a Disease under Vaccination Strategy |
| title_full | New Method to Investigate the Impact of Independent Quadratic <i>α</i>-Stable Poisson Jumps on the Dynamics of a Disease under Vaccination Strategy |
| title_fullStr | New Method to Investigate the Impact of Independent Quadratic <i>α</i>-Stable Poisson Jumps on the Dynamics of a Disease under Vaccination Strategy |
| title_full_unstemmed | New Method to Investigate the Impact of Independent Quadratic <i>α</i>-Stable Poisson Jumps on the Dynamics of a Disease under Vaccination Strategy |
| title_short | New Method to Investigate the Impact of Independent Quadratic <i>α</i>-Stable Poisson Jumps on the Dynamics of a Disease under Vaccination Strategy |
| title_sort | new method to investigate the impact of independent quadratic i α i stable poisson jumps on the dynamics of a disease under vaccination strategy |
| topic | dynamical system noise bifurcation ergodicity lévy processes jumps |
| url | https://www.mdpi.com/2504-3110/7/3/226 |
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