Dynamical analysis of a fractional SIR model with birth and death on heterogeneous complex networks

Abstract In this paper, a fractional SIR model with birth and death rates on heterogeneous complex networks is proposed. Firstly, we obtain a threshold value R 0 based on the existence of endemic equilibrium point E ∗ , which completely determines the dynamics of the model. Secondly, by using Lyapunov function and Kirchhoff’s matrix tree theorem, the globally asymptotical stability of the disease-free equilibrium point E 0 and the endemic equilibrium point E ∗ of the model are investigated. That is, when R 0 1 , the disease-free equilibrium point E 0 is globally asymptotically stable and the disease always dies out; when R 0 > 1 , the disease-free equilibrium point E 0 becomes unstable and in the meantime there exists a unique endemic equilibrium point E ∗ , which is globally asymptotically stable and the disease is uniformly persistent. Finally, the effects of various immunization schemes are studied and compared. Numerical simulations are given to demonstrate the main results.

[1]  Jinde Cao,et al.  A note on global stability of the virose equilibrium for network-based computer viruses epidemics , 2014, Appl. Math. Comput..

[2]  R. Bagley,et al.  On the Appearance of the Fractional Derivative in the Behavior of Real Materials , 1984 .

[3]  R. Bagley,et al.  Fractional order state equations for the control of viscoelasticallydamped structures , 1991 .

[4]  J. Liu,et al.  The spread of disease with birth and death on networks , 2004, q-bio/0402042.

[5]  Mikael Enelund,et al.  Fractional Derivative Viscoelasticity at Large Deformations , 2003 .

[6]  Yongsheng Ding,et al.  A fractional-order differential equation model of HIV infection of CD4+ T-cells , 2009, Math. Comput. Model..

[7]  J. Moon.  Counting labelled trees , 1970 .

[8]  I. Podlubny.  Fractional differential equations : an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications , 1999 .

[9]  Michael Y. Li,et al.  Global-stability problem for coupled systems of differential equations on networks , 2010 .

[10]  C. Vargas‐De‐León,et al.  Volterra-type Lyapunov functions for fractional-order epidemic systems , 2015 .

[11]  Graeme C. Wake,et al.  Lyapunov functions and global stability for SIR, SIRS, and SIS epidemiological models , 2002, Appl. Math. Lett..

[12]  E. Ahmed,et al.  On fractional order differential equations model for nonlocal epidemics , 2007, Physica A: Statistical Mechanics and its Applications.

[13]  Benito M. Chen-Charpentier,et al.  A fractional order epidemic model for the simulation of outbreaks of influenza A(H1N1) , 2014 .

[14]  Elisa F. Long,et al.  HIV epidemic control—a model for optimal allocation of prevention and treatment resources , 2014, Health care management science.

[15]  Hongyong Zhao,et al.  The effect of vaccines on backward bifurcation in a fractional order HIV model , 2015 .

[16]  W. O. Kermack,et al.  A contribution to the mathematical theory of epidemics , 1927 .

[17]  M. Small,et al.  Epidemic dynamics on scale-free networks with piecewise linear infectivity and immunization. , 2008, Physical review. E, Statistical, nonlinear, and soft matter physics.

[18]  S. Islam,et al.  Stability analysis of a general SIR epidemic model , 2013 .

[19]  Marina V. Shitikova,et al.  Application of fractional derivatives to the analysis of damped vibrations of viscoelastic single mass systems , 1997 .

[20]  Maoxing Liu,et al.  Global stability analysis of an SIR epidemic model with demographics and time delay on networks , 2014 .

[21]  C. F. Lorenzo,et al.  Chaos in a fractional order Chua's system , 1995 .

[22]  A. Stanislavsky,et al.  Memory effects and macroscopic manifestation of randomness. , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[23]  Alessandro Vespignani,et al.  Epidemic spreading in scale-free networks. , 2000, Physical review letters.

[24]  Robert J. Plemmons,et al.  Nonnegative Matrices in the Mathematical Sciences , 1979, Classics in Applied Mathematics.

[25]  Ye Yan,et al.  Stability analysis for a fractional differential model of HIV infection of CD4+ T-cells with time delay , 2012, Math. Comput. Simul..

[26]  R. Hilfer.  Applications Of Fractional Calculus In Physics , 2000 .

[27]  Jitao Sun,et al.  Stability analysis of an SIS epidemic model with feedback mechanism on networks , 2014 .