Global stability of a fractional-order logistic growth model with infectious disease

Hasan S. Panigoro, Emli Rahmi

Abstract


Infectious disease has an influence on the density of a population. In this paper, a fractional-order logistic growth model with infectious disease is formulated. The population grows logistically and divided into two compartments i.e. susceptible and infected populations. We start by investigating the existence, uniqueness, non-negativity, and boundedness of solutions. Furthermore, we show that the model has three equilibrium points namely the population extinction point, the disease-free point, and the endemic point. The population extinction point is always a saddle point while others are conditionally asymptotically stable. For the non-trivial equilibrium points, we successfully show that the local and global asymptotic stability have the similar properties. Especially, when the endemic point exists, it is always globally asymptotically stable. We also show the existence of forward bifurcation in our model. We portray some numerical simulations consist of the phase portraits, time series, and a bifurcation diagram to validate the analytical findings.

Keywords


Dynamics, Fractional-Order, Logistic Growth, Infectious Disease

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References


P.-F. Verhulst, “Notice sur la loi que la population poursuit dans son accroissement,” Correspondance mathématique et physique, vol. 10, pp. 113–121, 1838.

T. R. Malthus, An essay on the principle ofpopulation. (Reprinted from 1798 edition, Johnson, London), reprinted ed. Michigan, USA: University of Michigan, 1798.

F. J. Richards, “A Flexible growth function for empirical use,” Journal ofExperimental Botany, vol. 10, no. 29, pp. 290–300, 1959.

A. A. Blumberg, “Logistic growth rate functions,” Journal of Theoretical Biology, vol. 21, no. 1, pp. 42–44, 1968.

A. Tsoularis and J. Wallace, “Analysis of logistic growth models,” Mathematical Biosciences, vol. 179, no. 1, pp. 21–55, 2002.

Y. Kang and O. Udiani, “Dynamics of a single species evolutionary model with Allee effects,” Journal of Mathematical Analysis and Applications, vol. 418, no. 1, pp. 492–515, 2014.

S. Abbas, M. Banerjee, and S. Momani, “Dynamical analysis of fractional-order modified logistic model,” Computers and Mathematics with Applications, vol. 62, no. 3, pp. 1098–1104, 2011.

J. Leonel Rocha and A. K. Taha, “Allee’s effect bifurcation in generalized logistic maps,” International Journal ofBifurcation and Chaos, vol. 29, no. 3, pp. 1–19, 2019.

Q. Lin, “Stability analysis of a single species logistic model with Allee effect and feedback control,” Advances in Difference Equations, vol. 2018, no. 1, pp. 1–13, 2018.

M. T. Hoang and O. F. Egbelowo, “Numerical dynamics of nonstandard finite difference schemes for a Logistics model with feedback control,” Annali dell’Universita di Ferrara, vol. 66, no. 1, pp. 51–65, 2020.

H. S. Panigoro, A. Suryanto, W. M. Kusumahwinahyu, and I. Darti, “Dynamics of a fractional-order predator-prey model with infectious diseases in prey,” Communication in Biomathematical Sciences, vol. 2, no. 2, p. 105, 2019.

A. Suryanto, I. Darti, H. S. Panigoro, and A. Kilicman, “A fractional-order predator-prey model with ratio-dependent functional response and linear harvesting,” Mathematics, vol. 7, no. 11, pp. 1–13, 2019.

H. S. Panigoro, A. Suryanto, W. M. Kusumawinahyu, and I. Darti, “Continuous threshold harvesting in a gause-type predator-prey model with fractional-order,” AIP Conference Proceedings, vol. 2264, no. 1, p. 040001, 2020.

H. S. Panigoro, E. Rahmi, N. Achmad, and S. L. Mahmud, “The influence of additive Allee effect and periodic harvesting to the dynamics of Leslie-Gower predator-prey model,” Jambura Journal ofMathematics, vol. 2, no. 2, pp. 87–96, 2020.

D. Mukherjee and C. Maji, “Bifurcation analysis of a Holling type II predator-prey model with refuge,” Chinese Journal of Physics, vol. 65, no. February, pp. 153–162, 2020.

L. K. Beay and M. Saija, “A stage-structure Rosenzweig-MacArthur model with effect of prey refuge,” Jambura Journal of Biomathematics (JJBM), vol. 1, no. 1, pp. 1–7, 2020.

B. Malavika, S. Marimuthu, M. Joy, A. Nadaraj, E. S. Asirvatham, and L. Jeyaseelan, “Forecasting COVID-19 epidemic in India and high incidence states using SIR and logistic growth models,” Clinical Epidemiology and Global Health, no. May, pp. 1–8, 2020.

P. Wang, X. Zheng, J. Li, and B. Zhu, “Prediction of epidemic trends in COVID-19 with logistic model and machine learning technics,” Chaos, Solitons and Fractals, vol. 139, p. 110058, 2020.

N. Nuraini, K. Khairudin, and M. Apri, “Modeling simulation of COVID-19 in Indonesia based on early endemic data,” Communication in Biomathematical Sciences, vol. 3, no. 1, pp. 1–8, 2020.

C. Y. Shen, “Logistic growth modelling of COVID-19 proliferation in China and its international implications,” International Journal ofInfectious Diseases, vol. 96, pp. 582–589, 2020.

M. Rayungsari, M. Aufin, and N. Imamah, “Parameters estimation of generalized Richards model for COVID-19 cases in Indonesia using genetic algorithm,” Jambura Journal ofBiomathematics (JJBM), vol. 1, no. 1, pp. 25–30, 2020.

M. A. Abdelaziz, A. I. Ismail, F. A. Abdullah, and M. H. Mohd, “Bifurcations and chaos in a discrete SI epidemic model with fractional order,” Advances in Difference Equations, vol. 2018, no. 1, 2018.

S. Iwami, Y. Takeuchi, and X. Liu, “Avian–human influenza epidemic model,” Mathematical Biosciences, vol. 207, no. 1, pp. 1–25, 2007.

F. M. Hilker, M. Langlais, and H. Malchow, “The Allee effect and infectious diseases: Extinction, multistability, and the (dis-)appearance of oscillations,” American Naturalist, vol. 173, no. 1, pp. 72–88, 2009.

N. Liu, J. Fang, W. Deng, and J. wei Sun, “Stability analysis of a fractional-order SIS model on complex networks with linear treatment function,” Advances in Difference Equations, vol. 2019, no. 1, 2019.

F. W. Zhang and L. F. Nie, “Dynamics of SIS epidemic model with varying total population and multivaccination control strategies,” Studies in Applied Mathematics, vol. 139, no. 4, pp. 533–550, 2017.

E. Okyere, J. Ackora-Prah, and F. T. Oduro, “A Caputo based SIRS and SIS fractional order models with standard incidence rate and varying population,” pp. 1–25, 2019.

H. El-Saka, “The fractional-order SIS epidemic model with variable population size,” Journal ofthe Egyptian Mathematical Society, vol. 22, no. 1, pp. 50–54, 2014.

M. Hassouna, A. Ouhadan, and E. H. El Kinani, “On the solution of fractional order SIS epidemic model,” Chaos, Solitons and Fractals, vol. 117, pp. 168–174, 2018.

A. Suryanto, I. Darti, and S. Anam, “Stability analysis of a fractional order modified Leslie-Gower model with additive Allee effect,” International Journal ofMathematics and Mathematical Sciences, vol. 2017, no. 0, pp. 1–9, 2017.

H. S. Panigoro and D. Savitri, “Bifurkasi Hopf pada Model Lotka-Volterra orde-fraksional dengan efek Allee aditif pada predator,” Jambura Journal of Biomathematics, vol. 1, no. 1, pp. 16–24, 2020.

H. S. Panigoro, A. Suryanto, W. M. Kusumawinahyu, and I. Darti, “A Rosenzweig–MacArthur model with continuous threshold harvesting in predator involving fractional derivatives with power Law and Mittag–Leffler kernel,” Axioms, vol. 9, no. 4, p. 122, 2020.

M. Moustafa, M. H. Mohd, A. I. Ismail, and F. A. Abdullah, “Dynamical analysis of a fractional-order eco- epidemiological model with disease in prey population,” Advances in Difference Equations, vol. 2020, no. 1, p. 48, 2020.

S. Othman and S. Othman, “Global stability of a fractional order sir epidemic model with double epidemic hypothesis and nonlinear incidence rate,” pp. 1–15, 2020.

R. Mondal, D. Kesh, and D. Mukherjee, “Dynamical analysis of a fractional order Rosenzweig-MacArthur model incorporating herbivore induced plant volatile,” Chinese Journal ofPhysics, vol. 68, pp. 258–269, 2020.

I. Petras, Fractional-order nonlinear systems: modeling, analysis and simulation. Beijing: Springer London, 2011, vol. 1. 201 [37] Y. Li, Y. Q. Chen, and I. Podlubny, “Stability of fractional-order nonlinear dynamic systems: Lyapunov direct method and generalized Mittag–Leffler stability,” Computers & Mathematics with Applications, vol. 59, no. 5, pp. 1810–1821, 2010.

H.-L. Li, L. Zhang, C. Hu, Y.-l. L. Jiang, and Z. Teng, “Dynamical analysis of a fractional-order predator-prey model incorporating a prey refuge,” Journal ofApplied Mathematics and Computing, vol. 54, no. 1-2, pp. 435–449, 2017.

C. Vargas-De-León, “Volterra-type Lyapunov functions for fractional-order epidemic systems,” Communications in Nonlinear Science and Numerical Simulation, vol. 24, no. 1-3, pp. 75–85, 2015.

J. Huo, H. Zhao, and L. Zhu, “The effect of vaccines on backward bifurcation in a fractional order HIV model,” Nonlinear Analysis: Real World Applications, vol. 26, pp. 289–305, 2015.

E. Ahmed, A. M. A. El-Sayed, and H. A. A. El-Saka, “On some Routh–Hurwitz conditions for fractional order differential equations and their applications in Lorenz, Rössler, Chua and Chen systems,” Physics Letters A, vol. 358, no. 1, pp. 1–4, 2006.

K. Diethelm, N. J. Ford, and A. D. Freed, “A predictor-corrector approach for the numerical solution of fractional differential equations,” Nonlinear Dynamics, vol. 29, no. 1-4, pp. 3–22, 2002.




DOI: https://doi.org/10.34312/jjbm.v1i2.8135

Copyright (c) 2020 Hasan S. Panigoro; Emli Rahmi

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