Modelling of Epidemics with a Generalized Nonlinear Incidence on Complex Networks

In this paper the spreading of epidemic model on complex networks with a generalized nonlinear incidence rate is presented. Firstly the SIS model on homogeneous networks with nonlinear incidence rate is considered, and the existence conditions about the disease-free equilibrium and the endemic equilibrium are given. And then the model on heterogenous scale-free (SF) networks is considered, where the absence of the threshold on SF networks with nonlinear incidence is demonstrated. At last the stability of the disease-free equilibrium on SF networks is obtained. From this paper, it is shown that, while the number of the equilibria is indeed different from the corresponding model with linear incidence rate, the basic reproductive number, which determinate whether the disease is spreading or not, is independent of the functional form of the nonlinear incidence rate.

[1]  J. Yorke,et al.  Recurrent outbreaks of measles, chickenpox and mumps. I. Seasonal variation in contact rates. , 1973, American journal of epidemiology.

[2]  James Watmough,et al.  A simple SIS epidemic model with a backward bifurcation , 2000, Journal of mathematical biology.

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

[4]  A. J. Hall.  Infectious diseases of humans: R. M. Anderson & R. M. May. Oxford etc.: Oxford University Press, 1991. viii + 757 pp. Price £50. ISBN 0-19-854599-1 , 1992 .

[5]  Y. Iwasa,et al.  Influence of nonlinear incidence rates upon the behavior of SIRS epidemiological models , 1986, Journal of mathematical biology.

[6]  Alessandro Vespignani,et al.  Immunization of complex networks. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[7]  Guanrong Chen,et al.  Behaviors of susceptible-infected epidemics on scale-free networks with identical infectivity. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

[8]  R. May,et al.  Infection dynamics on scale-free networks. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[9]  P. Driessche,et al.  A disease transmission model in a nonconstant population , 1993, Journal of mathematical biology.

[10]  Shigui Ruan,et al.  Dynamical behavior of an epidemic model with a nonlinear incidence rate , 2003 .

[11]  Alessandro Vespignani,et al.  Epidemic dynamics and endemic states in complex networks. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[12]  M. E. Alexander,et al.  Periodicity in an epidemic model with a generalized non-linear incidence. , 2004, Mathematical biosciences.

[13]  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.

[14]  Shigui Ruan,et al.  Global analysis of an epidemic model with nonmonotone incidence rate , 2006, Mathematical Biosciences.

[15]  G. Serio,et al.  A generalization of the Kermack-McKendrick deterministic epidemic model☆ , 1978 .

[16]  Albert,et al.  Emergence of scaling in random networks , 1999, Science.

[17]  Yukio Hayashi,et al.  Oscillatory epidemic prevalence in growing scale-free networks. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[18]  S. Levin,et al.  Dynamical behavior of epidemiological models with nonlinear incidence rates , 1987, Journal of mathematical biology.

[19]  Jinshan Wu,et al.  The spread of infectious disease on complex networks with household-structure , 2004 .

[20]  M. Newman.  Spread of epidemic disease on networks. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[21]  J. Yorke,et al.  Recurrent outbreaks of measles, chickenpox and mumps. II. Systematic differences in contact rates and stochastic effects. , 1973, American journal of epidemiology.

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

[23]  Tao Zhou,et al.  Epidemic spreading on heterogeneous networks with identical infectivity , 2007 .

[24]  Murray E Alexander,et al.  Bifurcations of an epidemic model with non-linear incidence and infection-dependent removal rate. , 2006, Mathematical medicine and biology : a journal of the IMA.

[25]  Z. Duan,et al.  An SIS model with infective medium on complex networks , 2008 .