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Global stability analysis of a general nonlinear scabies dynamics model

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  • AlShamrani, N.H.
  • Elaiw, A.M.
  • Batarfi, H.
  • Hobiny, A.D.
  • Dutta, H.

Abstract

In this work we present a general nonlinear model of scabies infection dynamics. The dynamics is described by a five-dimensional system of ordinary differential equations that expresses the transmissions between susceptible, infectious/infective individuals and adult scabiei mites. The intrinsic growth rate for susceptible individuals, the infection rates as well as the removal and transmission rates of infected individuals and adult mites are modeled by general nonlinear functions. Based on a set of conditions on these general functions we investigate and analyze our model. Nonnegativity and boundedness of solutions of the model are conducted. A basic reproduction number, ℜ0M, is calculated for the model which ensures the existence and stability of all corresponding equilibria. Using candidate Lyapunov functions, it is shown that whenever the basic reproduction number is less than or equal unity, the model has an associated disease-free equilibrium, Q0M, that is globally asymptotically stable. In addition, when the threshold exceeds unity the model has a globally asthmatically stable endemic equilibrium, Q¯M. Finally, using some parameter values related to the scabies infection dynamics, numerical simulation results are demonstrated to clarify the strength of our main theoretical results. Sensitivity analysis of the endemic equilibrium has been performed.

Suggested Citation

  • AlShamrani, N.H. & Elaiw, A.M. & Batarfi, H. & Hobiny, A.D. & Dutta, H., 2020. "Global stability analysis of a general nonlinear scabies dynamics model," Chaos, Solitons & Fractals, Elsevier, vol. 138(C).
    • Handle: RePEc:eee:chsofr:v:138:y:2020:i:c:s0960077920305294
    DOI: 10.1016/j.chaos.2020.110133
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    References listed on IDEAS

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    1. Naomi van der Linden & Kees van Gool & Karen Gardner & Helen Dickinson & Jason Agostino & David G Regan & Michelle Dowden & Rosalie Viney, 2019. "A systematic review of scabies transmission models and data to evaluate the cost-effectiveness of scabies interventions," PLOS Neglected Tropical Diseases, Public Library of Science, vol. 13(3), pages 1-18, March.
    2. Vargas-De-León, Cruz, 2011. "On the global stability of SIS, SIR and SIRS epidemic models with standard incidence," Chaos, Solitons & Fractals, Elsevier, vol. 44(12), pages 1106-1110.
    3. Stephen J Gilmore, 2011. "Control Strategies for Endemic Childhood Scabies," PLOS ONE, Public Library of Science, vol. 6(1), pages 1-14, January.
    4. Zhu, Min & Xu, Yong, 2019. "A time-periodic dengue fever model in a heterogeneous environment," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 155(C), pages 115-129.
    5. Dantas, Eber & Tosin, Michel & Cunha Jr, Americo, 2018. "Calibration of a SEIR–SEI epidemic model to describe the Zika virus outbreak in Brazil," Applied Mathematics and Computation, Elsevier, vol. 338(C), pages 249-259.
    6. Timothy Kinyanjui & Jo Middleton & Stefan Güttel & Jackie Cassell & Joshua Ross & Thomas House, 2018. "Scabies in residential care homes: Modelling, inference and interventions for well-connected population sub-units," PLOS Computational Biology, Public Library of Science, vol. 14(3), pages 1-24, March.
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