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Monotonically decreasing behavior of measles epidemic well captured by Atangana–Baleanu–Caputo fractional operator under real measles data of Pakistan

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  • Qureshi, Sania
  • Memon, Zaib-un-Nisa

Abstract

Epidemics have memory characteristics and the deterministic systems in mathematical epidemiology based upon ordinary differential equations are the most suitable tools when modeled under differential operators possessing such memory effects. This paper proposes a new four compartmental based system under the Atangana–Baleanu–Caputo (ABC) fractional order operator for measles epidemic. The mutually exclusive compartments containing susceptible, exposed, infectious and recovered individuals are fractionalyzed with ABC operator while balancing dimension for each parameter of the system. The real measles monthly incidence data of Pakistan from May-December, 2018 reported by WHO is used for parameter estimation thereby getting fitted values for transmission coefficient rate β and the fractional order parameter α via nonlinear least-squares curve fitting technique. The error norm obtained in the ABC case is about 50% smaller than the classical case, that is, when α=1. Fixed point theory proves existence of the unique solution for the fractional order measles system with positive invariant region. Stability analysis of the fractional system shows that both measles-free and endemic equilibria are locally asymptotically stable whenever the basic reproductive number R0 is less and greater than unity, respectively. Later, the least (λ) and the most (α) sensitive parameters are determined using the normalized sensitivity indices for R0. Numerical simulations for the classical and fractional measles epidemiological systems illustrate the better fit for the real data about the infected individuals with the solution obtained via fractional order ABC operator. Various values of the biological parameters are taken to observe their effects on the proposed fractional system thereby suggesting the reduction in the contact rate of measles infected individuals with susceptible population.

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  • Qureshi, Sania & Memon, Zaib-un-Nisa, 2020. "Monotonically decreasing behavior of measles epidemic well captured by Atangana–Baleanu–Caputo fractional operator under real measles data of Pakistan," Chaos, Solitons & Fractals, Elsevier, vol. 131(C).
    • Handle: RePEc:eee:chsofr:v:131:y:2020:i:c:s0960077919304242
    DOI: 10.1016/j.chaos.2019.109478
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    References listed on IDEAS

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    1. Abu Arqub, Omar & Maayah, Banan, 2019. "Modulation of reproducing kernel Hilbert space method for numerical solutions of Riccati and Bernoulli equations in the Atangana-Baleanu fractional sense," Chaos, Solitons & Fractals, Elsevier, vol. 125(C), pages 163-170.
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    4. Altaf Khan, Muhammad & Ullah, Saif & Farooq, Muhammad, 2018. "A new fractional model for tuberculosis with relapse via Atangana–Baleanu derivative," Chaos, Solitons & Fractals, Elsevier, vol. 116(C), pages 227-238.
    5. Qureshi, Sania & Bonyah, Ebenezer & Shaikh, Asif Ali, 2019. "Classical and contemporary fractional operators for modeling diarrhea transmission dynamics under real statistical data," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 535(C).
    6. Arqub, Omar Abu & Maayah, Banan, 2018. "Numerical solutions of integrodifferential equations of Fredholm operator type in the sense of the Atangana–Baleanu fractional operator," Chaos, Solitons & Fractals, Elsevier, vol. 117(C), pages 117-124.
    7. Qureshi, Sania & Yusuf, Abdullahi, 2019. "Modeling chickenpox disease with fractional derivatives: From caputo to atangana-baleanu," Chaos, Solitons & Fractals, Elsevier, vol. 122(C), pages 111-118.
    8. Qureshi, Sania & Yusuf, Abdullahi & Shaikh, Asif Ali & Inc, Mustafa, 2019. "Transmission dynamics of varicella zoster virus modeled by classical and novel fractional operators using real statistical data," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 534(C).
    9. Arqub, Omar Abu & Maayah, Banan, 2019. "Fitted fractional reproducing kernel algorithm for the numerical solutions of ABC – Fractional Volterra integro-differential equations," Chaos, Solitons & Fractals, Elsevier, vol. 126(C), pages 394-402.
    10. Al-Smadi, Mohammed & Arqub, Omar Abu, 2019. "Computational algorithm for solving fredholm time-fractional partial integrodifferential equations of dirichlet functions type with error estimates," Applied Mathematics and Computation, Elsevier, vol. 342(C), pages 280-294.
    11. Qureshi, Sania & Yusuf, Abdullahi, 2019. "Mathematical modeling for the impacts of deforestation on wildlife species using Caputo differential operator," Chaos, Solitons & Fractals, Elsevier, vol. 126(C), pages 32-40.
    12. Qureshi, Sania & Atangana, Abdon, 2019. "Mathematical analysis of dengue fever outbreak by novel fractional operators with field data," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 526(C).
    13. Ullah, Saif & Altaf Khan, Muhammad & Farooq, Muhammad, 2018. "A fractional model for the dynamics of TB virus," Chaos, Solitons & Fractals, Elsevier, vol. 116(C), pages 63-71.
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    Cited by:

    1. Alfifi, H.Y., 2022. "Stability analysis for Schnakenberg reaction-diffusion model with gene expression time delay," Chaos, Solitons & Fractals, Elsevier, vol. 155(C).
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    3. Qureshi, Sania & Jan, Rashid, 2021. "Modeling of measles epidemic with optimized fractional order under Caputo differential operator," Chaos, Solitons & Fractals, Elsevier, vol. 145(C).
    4. Shah, Kamal & Arfan, Muhammad & Ullah, Aman & Al-Mdallal, Qasem & Ansari, Khursheed J. & Abdeljawad, Thabet, 2022. "Computational study on the dynamics of fractional order differential equations with applications," Chaos, Solitons & Fractals, Elsevier, vol. 157(C).
    5. Tuan, Nguyen Huy & Mohammadi, Hakimeh & Rezapour, Shahram, 2020. "A mathematical model for COVID-19 transmission by using the Caputo fractional derivative," Chaos, Solitons & Fractals, Elsevier, vol. 140(C).

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