Skip to main content
SpringerLink
Log in

Stability of a Fractional-Order Epidemic Model with Nonlinear Incidences and Treatment Rates

  • Research Paper
  • Published:
Iranian Journal of Science and Technology, Transactions A: Science Aims and scope Submit manuscript

Abstract

In the present study, Caputo derivative-based a new fractional-order epidemic model is presented along with two explicit saturated incidences and saturated treatment rates. For this, a new fearful population compartment is incorporated into the susceptible-infected-recovered compartmental model, which emphasizes to consider two specific incidence rates: one from susceptible individuals’ compartment to infected individuals’ compartment and another from fearful individuals’ compartment to infected individuals’ compartment. The model is analyzed mathematically for disease-free equilibrium (DFE) and endemic equilibrium (EE). The stability of the model’s equilibria is investigated for local as well as global behaviors. It is investigated that DFE is locally asymptotically stable whenever the basic reproduction number \( R_{0} \) is less than one, and EE exists when \( R_{0} \) crosses one. The EE is proved to be locally stable under certain conditions. Further, global stability behavior is investigated for both equilibria using the basic reproduction number \( R_{0} \). Finally, numerical results are presented in support of the analytical findings.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

References

  • Ahmed E, El-Sayed AMA, El-Saka HAA (2006) On some Routh-Hurwitz conditions for fractional order differential equations and their applications in Lorenz, Rossler, Chua and Chen systems. Phys Lett A 358(1):1–4

    MathSciNet  MATH  Google Scholar 

  • Alexander ME, Bowman C, Moghadas SM, Summers R, Gumel AB, Sahai BM (2004) A vaccination model for transmission dynamics of influenza. SIAM J Appl Dyn Syst 3(4):503–524

    MathSciNet  MATH  Google Scholar 

  • Atangana A (2017) Fractal-fractional differentiation and integration: connecting fractal calculus and fractional calculus to predict complex system. Chaos, Solitons Fractals 102:396–406

    MathSciNet  MATH  Google Scholar 

  • Atangana A (2020) Fractional discretization: the African’s tortoise walk. Chaos, Solitons Fractals 130:109399

    MathSciNet  Google Scholar 

  • Capasso V, Serio G (1978) A generalization of the Kermack-McKendrick deterministic epidemic model. Math Biosci 42(1–2):43–61

    MathSciNet  MATH  Google Scholar 

  • Cattani C, Pierro G (2013) On the fractal geometry of DNA by the binary image analysis. Bull Math Biol 75(9):1544–1570

    MathSciNet  MATH  Google Scholar 

  • Diethelm K, Ford NJ, Freed AD (2002) A predictor-corrector approach for the numerical solution of fractional differential equations. Nonlinear Dyn 29:3–22

    MathSciNet  MATH  Google Scholar 

  • Driessche PVD, Watmough J (2002) Reproduction numbers and sub-threshold endemic equilibria for compartment models of disease transmission. Math Biosci 180:29–48

    MathSciNet  MATH  Google Scholar 

  • Dubey B, Patra A, Srivastava PK, Dubey US (2013) Modelling and analysis of a SEIR model with different types of nonlinear treatment rates. J Biol Syst 21(3):1350023

    MATH  Google Scholar 

  • Gao W, Yel G, Baskonus HM, Cattani C (2019) Complex solitons in the conformable (2 + 1)-dimensional Ablowitz-Kaup-Newell-Segur equation. AIMS Math 5(1):507–521

    MathSciNet  Google Scholar 

  • Gao W, Veeresha P, Prakasha DG, Baskonus HM (2020a) Novel dynamic structures of 2019-nCoV with nonlocal operator via powerful computational technique. Biology 9(5):107

    Google Scholar 

  • Gao W, Veeresha P, Prakasha DG, Baskonus HM, Yel G (2020b) New numerical results for the time-fractional phi-four equation using a novel analytical approach. Symmetry 12(3):478

    Google Scholar 

  • Gao W, Veeresha P, Baskonus HM, Prakasha DG, Kumar P (2020c) A new study of unreported cases of 2019-nCOV epidemic outbreaks. Chaos, Solitons Fractals 138:109929

    MathSciNet  Google Scholar 

  • Goel K, Kumar A, Nilam (2020a) A deterministic time-delayed SVIRS epidemic model with incidences and saturated treatment. J Eng Math 121:19–38

    MathSciNet  MATH  Google Scholar 

  • Goel K, Kumar A, Nilam (2020b) Nonlinear dynamics of a time-delayed epidemic model with two explicit aware classes, saturated incidences, and treatment. Nonlinear Dyn. https://doi.org/10.1007/s11071-020-05762-9

    Article  MATH  Google Scholar 

  • Gumel AB, Mccluskey CC, Watmough J (2006) An SVEIR model for assessing the potential impact of an imperfect anti-SARS vaccine. Math Biosci Eng 3:485–494

    MathSciNet  MATH  Google Scholar 

  • Huang G, Takeuchi Y, Ma W, Wei D (2010) Global Stability for delay SIR and SEIR epidemic models with nonlinear incidence rate. Bull Math Biol 72:1192–1207

    MathSciNet  MATH  Google Scholar 

  • Jothimani K, Kaliraj K, Hammouch Z, Ravichandran C (2019) New results on controllability in the framework of fractional integrodifferential equations with nondense domain. Eur Phys J Plus 134(9):441

    Google Scholar 

  • Khan MA, Ullah S, Ullah S, Farhan M (2020) Fractional order SEIR model with generalized incidence rate. AIMS Math 5(4):2843–2857

    MathSciNet  Google Scholar 

  • Kumar A, Nilam (2019) Stability of a delayed SIR epidemic model by introducing two explicit treatment classes along with nonlinear incidence rate and Holling type treatment. Comput Appl Math 38:130

    MathSciNet  MATH  Google Scholar 

  • Kumar A, Nilam, Kishor R (2019) A short study of an sir model with inclusion of an alert class, two explicit nonlinear incidence rates and saturated treatment rate. SeMA J 76(3):505–519

    MathSciNet  MATH  Google Scholar 

  • Kumar A, Goel K, Nilam (2020a) A deterministic time-delayed SIR epidemic model: mathematical modeling and analysis. Theory Biosci 139:67–76

    MATH  Google Scholar 

  • Kumar A, Kumar M, Nilam (2020b) A study on the stability behavior of an epidemic model with ratio-dependent incidence and saturated treatment. Theory Biosci 139:225–234

    Google Scholar 

  • Lu Z, Zhu Y (2018) Comparison principles for fractional differential equations with the Caputo derivatives. Adv Differ Equ. https://doi.org/10.1186/s13662-018-1691-y

    Article  MathSciNet  MATH  Google Scholar 

  • MathWorks (2012) Predictor-corrector PECE method for fractional differential equations. http://www.mathworks.com/matlabcentral/fileexchange/32918

  • Matignon D (1996) Stability results for fractional differential equations with applications to control processing. IEEE-SMC Comput Eng Syst Appl 2:963–968

    Google Scholar 

  • Michael YL, Graef JR, Wang L, Karsai J (1999) Global dynamics of a SEIR model with varying total population size. Math Biosci 160:191–213

    MathSciNet  MATH  Google Scholar 

  • Mouaouine A, Boukhouima A, Hattaf K, Yousfi N (2018) A fractional order SIR epidemic model with nonlinear incidence rate. Adv Differ Equ . https://doi.org/10.1186/s13662-018-1613-z

    Article  MathSciNet  MATH  Google Scholar 

  • Odibat ZM, Shawagfeh NT (2007) Generalized Taylor’s formula. Appl Math Comput 186(1):286–293

    MathSciNet  MATH  Google Scholar 

  • Podlubny I (1999) Fractional differential equations. Academic Press, New York

    MATH  Google Scholar 

  • Rostamy D, Mottaghi E (2016) Stability analysis of a fractional-order epidemics model with multiple equilibriums. Adv Differ Equ. https://doi.org/10.1186/s13662-016-0905-4

    Article  MathSciNet  MATH  Google Scholar 

  • Sánchez YG, Sabir Z, Guirao JLG (2020) Design of a nonlinear SITR fractal model based on the dynamics of a novel coronavirus (Covid-19). Fractals 28(8):2040026

    Google Scholar 

  • Singh J, Kumar D, Hammouch Z, Atangana A (2018) A fractional epidemiological model for computer viruses pertaining to a new fractional derivative. Appl Math Comput 316:504–515

    MathSciNet  MATH  Google Scholar 

  • Valliammal N, Ravichandran C, Hammouch Z, Baskonus HM (2019) A new investigation on fractional-ordered neutral differential systems with state-dependent delay. Int J Nonlinear Sci Numer Simul 20(7–8):803–809

    MathSciNet  MATH  Google Scholar 

  • Wang WD (2002) Global behavior of an SEIRS epidemic model with time delays. Appl Math Lett 15:423–428

    MathSciNet  MATH  Google Scholar 

  • Xu R, Ma Z (2009) Stability of a delayed SIRS epidemic model with a nonlinear incidence rate. Chaos, solut Fractals 41:2319–2325

    MathSciNet  MATH  Google Scholar 

  • Ye X, Xu C (2019) A fractional order epidemic model and simulation for avian influenza dynamics. Math Methods Appl Sci 42(14):4765–4779

    MathSciNet  MATH  Google Scholar 

  • Zhou L, Fan M (2012) Dynamics of a SIR epidemic model with limited medical resources revisited. Nonlinear Anal RWA 13:312–324

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgment

The author thanks the handling editor and anonymous reviewers for their valuable suggestions and comments to improve the quality of the paper.

Author information

Authors and Affiliations

  1. Department of Mathematics, School of Basic Sciences and Research, Sharda University, Greater Noida, 201310, India

    Abhishek Kumar

Authors
  1. Abhishek Kumar
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Abhishek Kumar.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kumar, A. Stability of a Fractional-Order Epidemic Model with Nonlinear Incidences and Treatment Rates. Iran J Sci Technol Trans Sci 44, 1505–1517 (2020). https://doi.org/10.1007/s40995-020-00960-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40995-020-00960-x

Keywords

Search

Navigation