Modeling and Analysis of Fractional Leptospirosis Model Using Atangana–Baleanu Derivative

  • Saif Ullah
  • Muhammad Altaf KhanEmail author
Part of the Studies in Systems, Decision and Control book series (SSDC, volume 194)


In this chapter, a fractional epidemic model for the leptospirosis disease with Atangana–Baleanu (AB) derivative is formulated. Initially, we present the model equilibria and basic reproduction number. The local stability of disease free equilibrium point is proved using fractional Routh Harwitz criteria. The Picard–Lindelof method is applied to show the existence and uniqueness of solutions for the model. A numerical scheme using Adams–Bashforth method for solving the proposed fractional model involving the AB derivative is presented. Finally, numerical simulations are performed in order to validate the importance of the arbitrary order derivative. The numerical result shows that the fractional order plays an important role to better understand the dynamics of disease.


Fractional calculus Atangana–Baleanu fractional derivative Leptospirosis model 


  1. 1.
    Marr, J.S., Cathey, J.T.: New hypothesis for cause of epidemic among native 236 americans, new england, 1616–1619. Emerg. Infect. Dis. 16(2), 1–281 (2010)Google Scholar
  2. 2.
    Victoriano, A.F.B., Smythe, L.D., Barzaga, N.G., Cavinta, L.L., Kasai, T., Limpakarnjanarat, K., Ong, B.L., Gongal, G., Hall, J., Coulombe, C.A.: Leptospirosis in the asia pacific region. BMC Infect. Dis. 9(1), 1–147 (2009)Google Scholar
  3. 3.
    Holt, J., Davis, S., Leirs, H.: A model of leptospirosis infection in an African rodent to determine risk to humans: seasonal fluctuations and the impact of rodent control. Acta Trop. 99(2–3), 218–225 (2006)Google Scholar
  4. 4.
    Okosun, K.O., Mukamuri, M., Makinde, D.O.: Global stability analysis and control of leptospirosis. Open Math. 14(1), 567–585 (2016)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Baca-Carrasco, D., Olmos, D., Barradas, I.: A mathematical model for human and animal leptospirosis. J. Biol. Syst. 23(01), 55–65 (2015)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Chadsuthi, S., Modchang, C., Lenbury, Y., Iamsirithaworn, S., Triampo, W.: Modeling seasonal leptospirosis transmission and its association with rainfall and temperature in Thailand using time-series and ARIMAX analyses. Asian Pac. J. Trop. Med. 5(7), 539–546 (2012)Google Scholar
  7. 7.
    Khan, M.A., Saddiq, S.F., Islam, S., Khan, I., Shafie, S.: Dynamic behavior of leptospirosis disease with saturated incidence rate. Int. J. Appl. Comput. Math. 2(4), 435–452 (2016)MathSciNetGoogle Scholar
  8. 8.
    Sadiq, S.F., Khan, M.A., Islam, S., Zaman, G., Jung, H., Khan, S.A.: Optimal control of an epidemic model of leptospirosis with nonlinear saturated incidences. Annu. Res. Rev. Biol. 4(3), 560–576 (2014)Google Scholar
  9. 9.
    Khan, M.A., Zaman, G., Islam, S., Chohan, M.I.: Optimal campaign in leptospirosis epidemic by multiple control variables. Appl. Math. 3(11), 1655–1663 (2012)Google Scholar
  10. 10.
    Khan, M.A., Islam, S., Khan, S.A., Khan, I., Shafie, S., Gul, T.: Prevention of Leptospirosis infected vector and human population by multiple control variables. Abstr. Appl. Anal. (Hindawi) 1, 1–14 (2014)Google Scholar
  11. 11.
    Sadiq, S.F., Khan, M.A., Islam, S., Zaman, G., Jung, H., Khan, S.A.: Optimal control of an epidemic model of leptospirosis with nonlinear saturated incidences. Annu. Res. Rev. Biol. 4(3), 560–576 (2014)Google Scholar
  12. 12.
    Khan, M.A., Islam, S., Khan, S.A.: Mathematical modeling towards the dynamical interaction of leptospirosis. Appl. Math. Inf. Sci. 8(3), 1–8 (2014)Google Scholar
  13. 13.
    Podlubny, I.: Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications. Academic, San Diego (1999)zbMATHGoogle Scholar
  14. 14.
    Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach, Yverdon (1993)zbMATHGoogle Scholar
  15. 15.
    Caputo, M., Fabrizio, M.: A new definition of fractional derivative without singular kernel. Prog. Fract. Differ. Appl. 1(2), 73–85 (2015)Google Scholar
  16. 16.
    Atangana, A., Baleanu, D.: New fractional derivatives with non-local and non-singular kernel: theory and application to heat transfer model. Therm. Sci. 20(2), 763–769 (2016)Google Scholar
  17. 17.
    Atangana, A.: Non validity of index law in fractional calculus: a fractional differential operator with Markovian and non-Markovian properties. Phys. A Stat. Mech. Appl. 505, 688–706 (2018)MathSciNetGoogle Scholar
  18. 18.
    Atangana, A., Gómez-Aguilar, J.F.: Fractional derivatives with no-index law property: application to chaos and statistics. Chaos Solitons Fractals 114, 516–535 (2018)MathSciNetGoogle Scholar
  19. 19.
    Atangana, A., Gómez-Aguilar, J.F.: Decolonisation of fractional calculus rules: breaking commutativity and associativity to capture more natural phenomena. Eur. Phys. J. Plus 133, 1–22 (2018)Google Scholar
  20. 20.
    Atangana, A., Koca, I.: Chaos in a simple nonlinear system with Atangana-Baleanu derivatives with fractional order. Chaos Solitons Fractals 89, 447–454 (2016)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Atangana, A., Gómez-Aguilar, J.F.: A new derivative with normal distribution kernel: theory, methods and applications. Phys. A Stat. Mech. Appl. 476, 1–14 (2017)MathSciNetGoogle Scholar
  22. 22.
    Gómez-Aguilar, J.F., Dumitru, B.: Fractional transmission line with losses. Zeitschrift für Naturforschung A 69(10–11), 539–546 (2014)Google Scholar
  23. 23.
    Gómez-Aguilar, J.F., Torres, L., Yépez-Martínez, H., Baleanu, D., Reyes, J.M., Sosa, I.O.: Fractional Liénard type model of a pipeline within the fractional derivative without singular kernel. Adv. Differ. Equ. 2016(1), 1–17 (2016)zbMATHGoogle Scholar
  24. 24.
    Yépez-Martínez, H., Gómez-Aguilar, J.F., Sosa, I.O., Reyes, J.M., Torres-Jiménez, J.: The Feng’s first integral method applied to the nonlinear mKdV space-time fractional partial differential equation. Rev. Mex. Fís 62(4), 310–316 (2016)MathSciNetGoogle Scholar
  25. 25.
    Saad, K.M., Gómez-Aguilar, J.F.: Analysis of reaction-diffusion system via a new fractional derivative with non-singular kernel. Phys. A Stat. Mech. Appl. 509, 703–716 (2018)MathSciNetGoogle Scholar
  26. 26.
    Gómez-Aguilar, J.F., Escobar-Jiménez, R.F., López-López, M.G., Alvarado-Martínez, V.M.: Atangana-Baleanu fractional derivative applied to electromagnetic waves in dielectric media. J. Electromagn. Waves Appl. 30(15), 1937–1952 (2016)Google Scholar
  27. 27.
    Coronel-Escamilla, A., Gómez-Aguilar, J.F., Baleanu, D., Córdova-Fraga, T., Escobar-Jiménez, R.F., Olivares-Peregrino, V.H., Qurashi, M.M.A.l.: Bateman-Feshbach tikochinsky and Caldirola–Kanai oscillators with new fractional differentiation. Entropy 19(2), 1–21 (2017)Google Scholar
  28. 28.
    Gómez-Aguilar, J.F., Yépez-Martínez, H., Escobar-Jiménez, R.F., Astorga-Zaragoza, C.M., Morales-Mendoza, L.J., González-Lee, M.: Universal character of the fractional space-time electromagnetic waves in dielectric media. J. Electromagn. Waves Appl. 29(6), 727–740 (2015)Google Scholar
  29. 29.
    Saad, K.M., Gómez-Aguilar, J.F.: Coupled reaction-diffusion waves in a chemical system via fractional derivatives in Liouville-Caputo sense. Rev. Mex. Fís. 64(5), 539–547 (2018)MathSciNetGoogle Scholar
  30. 30.
    Coronel-Escamilla, A., Gómez-Aguilar, J.F., López-López, M.G., Alvarado-Martínez, V.M., Guerrero-Ramírez, G.V.: Triple pendulum model involving fractional derivatives with different kernels. Chaos Solitons Fractals 91, 248–261 (2016)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Gómez-Aguilar, J.F., Atangana, A.: New insight in fractional differentiation: power, exponential decay and Mittag-Leffler laws and applications. Eur. Phys. J. Plus 132(1), 1–13 (2017)Google Scholar
  32. 32.
    Alkahtani, B.S.T.: Chua’s circuit model with Atangana-Baleanu derivative with fractional order. Chaos Solitons Fractals 89, 547–551 (2016)MathSciNetzbMATHGoogle Scholar
  33. 33.
    Gómez-Aguilar, J.F., Yépez-Martínez, H., Escobar-Jiménez, R.F., Astorga-Zaragoza, C.M., Reyes-Reyes, J.: Analytical and numerical solutions of electrical circuits described by fractional derivatives. Appl. Math. Model. 40(21–22), 9079–9094 (2016)MathSciNetGoogle Scholar
  34. 34.
    Toufik, M., Atangana, A.: New numerical approximation of fractional derivative with non-local and non-singular kernel: application to chaotic models. Eur. Phys. J. Plus 132, 444 (2017)Google Scholar
  35. 35.
    Algahtani, O.J.J.: Comparing the Atangana-Baleanu and Caputo-Fabrizio derivative with fractional order: Allen Cahn model. Chaos Solitons Fractals 89, 552–559 (2016)MathSciNetzbMATHGoogle Scholar
  36. 36.
    Gómez-Aguilar, J.F.: Analytical and numerical solutions of a nonlinear alcoholism model via variable-order fractional differential equations. Phys. A Stat. Mech. Appl. 494, 52–75 (2018)MathSciNetGoogle Scholar
  37. 37.
    Coronel-Escamilla, A., Gómez-Aguilar, J.F., Torres, L., Valtierra-Rodríguez, M., Escobar-Jiménez, R.F.: Design of a state observer to approximate signals by using the concept of fractional variable-order derivative. Digit. Signal Process. 69, 127–139 (2017)Google Scholar
  38. 38.
    Solís-Pérez, J.E., Gómez-Aguilar, J.F., Atangana, A.: Novel numerical method for solving variable-order fractional differential equations with power, exponential and Mittag-Leffler laws. Chaos Solitons Fractals 114, 175–185 (2018)MathSciNetGoogle Scholar
  39. 39.
    Alkahtani, B.S.T., Atangana, A., Koca, I.: Novel analysis of the fractional Zika model using the Adams type predictor-corrector rule for non-singular and non-local fractional operators. J. Nonlinear Sci. Appl. 10, 3191–3200 (2017)MathSciNetGoogle Scholar
  40. 40.
    Koca, I.: Modelling the spread of Ebola virus with Atangana-Baleanu fractional operators. Eur. Phys. J. Plus 133(3), 1–16 (2018)Google Scholar
  41. 41.
    Morales-Delgado, V.F., Gómez-Aguilar, J.F., Taneco-Hernándeza, M.A., Escobar-Jiménez, R.F., Olivares-Peregrino, V.H.: Mathematical modeling of the smoking dynamics using fractional differential equations with local and nonlocal kernel. J. Nonlinear Sci. Appl. 11, 994–1014 (2018)MathSciNetGoogle Scholar
  42. 42.
    Khan, M.A., Saddiq, S.F., Islam, S., Khan, I., Ching, D.L.C.: Epidemic model of leptospirosis containing fractional order. Abstr. Appl. Anal. (Hindawi) 1, 1–13 (2014)Google Scholar
  43. 43.
    El-Shahed, M.: Fractional order model for the spread of leptospirosis. Int. J. Math. Anal. 8(54), 2651–2667 (2014)Google Scholar
  44. 44.
    Zaman, G., Khan, M.A., Islam, S., Chohan, M.I., Jung, I.H.: Modeling dynamical interactions between leptospirosis infected vector and human population. Appl. Math. Sci. 6(26), 1287–1302 (2012)zbMATHGoogle Scholar
  45. 45.
    Van den Driessche, P., Watmough, J.: Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math. Biosci. 180(1–2), 29–48 (2002)MathSciNetzbMATHGoogle Scholar
  46. 46.
    Ahmed, E., El-Sayed, A.M.A., El-Saka, H.A.: On some Routh-Hurwitz conditions for fractional order differential equations and their applications in Lorenz, Rössler, Chua and Chen systems. Phys. Lett. A 358(1), 1–4 (2006)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of PeshawarPeshawarPakistan
  2. 2.Department of MathematicsCity University of Science and Information TechnologyPeshawarPakistan

Personalised recommendations