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Influence of fractal–fractional differentiation and independent quadratic Lévy jumps on the dynamics of a general epidemic model with vaccination strategy

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  • Sabbar, Yassine
  • Din, Anwarud
  • Kiouach, Driss

Abstract

The emergence and reemergence of various infectious diseases have adversely affected human life both from a social and economic point of view, and thus the study of epidemic diseases has diverted the attention of scientists from different fields. Particularly, to understand the dynamics and control mechanisms of the novel infections, one must focus on mathematical modeling, which is assumed to be the most important tool for analyzing the new epidemics. In this sense, the most common strategy used in the literature is to characterize the dynamics of infections via compartmental models. In this work, we formulated a general mathematical model along with a vaccination strategy to reveal the novel findings on the asymptotic properties of the underlying model. The complicated rapid fluctuations of the problem are taken into consideration, and second-order quadratic jumps with four independent compensated Poisson processes are studied. The model was further enhanced by including the two general interference functions. The key threshold determinant R0⋆ is calculated and it is proved that for R0⋆>1, the model has the properties of stationarity and ergodicity (i.e., permanence scenario). However, if R0⋆ is less than one, the infection will disappear exponentially (i.e., disappearance scenario). These results exhibit that noise sources play a dominant role in explaining the asymptotic behavior of any biological phenomenon. Numerically, we confirm the above-mentioned illustrations, and the simulation suggests the following results: (a) the mean-time of the solution depends on the noise intensities (b) the second-order Poisson jumps produce a negative effect on the time required for the survival of the infection. The study was further extended by considering non-integer differential equations, and the underlying model was studied both from fractional and fractal dimensions. This combination tries to give a physical explanation of infectious pathways. The stability of fractional ordered model was studied by using the Hyers–Ulam (HU) approach. To find the numerical solution of the fractional model, we presented and developed a fractal–fractional numerical scheme by employing the Adams Bashforth’s method. We find that studying the models from a fractal and fractional point of view has a major impact on the infection’s development. The dynamics of the times-series and chaotic behaviors vary by merely modifying the fractal or fractional orders of the model. More generally, the theoretical and numerical results of this paper provide excellent insight into the long-run behavior of the epidemic under a vaccination strategy.

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  • Sabbar, Yassine & Din, Anwarud & Kiouach, Driss, 2023. "Influence of fractal–fractional differentiation and independent quadratic Lévy jumps on the dynamics of a general epidemic model with vaccination strategy," Chaos, Solitons & Fractals, Elsevier, vol. 171(C).
  • Handle: RePEc:eee:chsofr:v:171:y:2023:i:c:s0960077923003351
    DOI: 10.1016/j.chaos.2023.113434
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    References listed on IDEAS

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    1. Avila-Vales, Eric & Pérez, Ángel G.C., 2019. "Dynamics of a time-delayed SIR epidemic model with logistic growth and saturated treatment," Chaos, Solitons & Fractals, Elsevier, vol. 127(C), pages 55-69.
    2. Atangana, Abdon & Qureshi, Sania, 2019. "Modeling attractors of chaotic dynamical systems with fractal–fractional operators," Chaos, Solitons & Fractals, Elsevier, vol. 123(C), pages 320-337.
    3. Liu, Qun & Jiang, Daqing & Hayat, Tasawar & Ahmad, Bashir, 2018. "Stationary distribution and extinction of a stochastic predator–prey model with additional food and nonlinear perturbation," Applied Mathematics and Computation, Elsevier, vol. 320(C), pages 226-239.
    4. Atangana, Abdon, 2017. "Fractal-fractional differentiation and integration: Connecting fractal calculus and fractional calculus to predict complex system," Chaos, Solitons & Fractals, Elsevier, vol. 102(C), pages 396-406.
    5. Tong, Jinying & Zhang, Zhenzhong & Bao, Jianhai, 2013. "The stationary distribution of the facultative population model with a degenerate noise," Statistics & Probability Letters, Elsevier, vol. 83(2), pages 655-664.
    6. Atangana, Abdon & İğret Araz, Seda, 2021. "New concept in calculus: Piecewise differential and integral operators," Chaos, Solitons & Fractals, Elsevier, vol. 145(C).
    7. Liu, Qun & Jiang, Daqing & Hayat, Tasawar & Ahmad, Bashir, 2017. "Periodic solution and stationary distribution of stochastic SIR epidemic models with higher order perturbation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 482(C), pages 209-217.
    8. Liu, Qun & Jiang, Daqing & Shi, Ningzhong & Hayat, Tasawar & Alsaedi, Ahmed, 2017. "Stationary distribution and extinction of a stochastic SIRS epidemic model with standard incidence," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 469(C), pages 510-517.
    9. Lv, Xuejin & Meng, Xinzhu & Wang, Xinzeng, 2018. "Extinction and stationary distribution of an impulsive stochastic chemostat model with nonlinear perturbation," Chaos, Solitons & Fractals, Elsevier, vol. 110(C), pages 273-279.
    10. Li, Xiao-Ping & Din, Anwarud & Zeb, Anwar & Kumar, Sunil & Saeed, Tareq, 2022. "The impact of Lévy noise on a stochastic and fractal-fractional Atangana–Baleanu order hepatitis B model under real statistical data," Chaos, Solitons & Fractals, Elsevier, vol. 154(C).
    11. Peng, Shige & Zhu, Xuehong, 2006. "Necessary and sufficient condition for comparison theorem of 1-dimensional stochastic differential equations," Stochastic Processes and their Applications, Elsevier, vol. 116(3), pages 370-380, March.
    12. Yanan Zhao & Daqing Jiang, 2014. "The Behavior of an SVIR Epidemic Model with Stochastic Perturbation," Abstract and Applied Analysis, Hindawi, vol. 2014, pages 1-7, June.
    13. Sabbar, Yassine & Kiouach, Driss & Rajasekar, S.P. & El-idrissi, Salim El Azami, 2022. "The influence of quadratic Lévy noise on the dynamic of an SIC contagious illness model: New framework, critical comparison and an application to COVID-19 (SARS-CoV-2) case," Chaos, Solitons & Fractals, Elsevier, vol. 159(C).
    14. Đorđević, J. & Papić, I. & Šuvak, N., 2021. "A two diffusion stochastic model for the spread of the new corona virus SARS-CoV-2," Chaos, Solitons & Fractals, Elsevier, vol. 148(C).
    15. Bashkirtseva, Irina & Ryashko, Lev & Ryazanova, Tatyana, 2020. "Analysis of regular and chaotic dynamics in a stochastic eco-epidemiological model," Chaos, Solitons & Fractals, Elsevier, vol. 131(C).
    16. Yassine Sabbar & Mehmet Yavuz & Fatma Özköse, 2022. "Infection Eradication Criterion in a General Epidemic Model with Logistic Growth, Quarantine Strategy, Media Intrusion, and Quadratic Perturbation," Mathematics, MDPI, vol. 10(22), pages 1-16, November.
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