Pattern formation of an epidemic model with diffusion

One subject of spatial epidemiology is spatial variation in disease risk or incidence. The spread of epidemics can result in strong spatial patterns of such risk or incidence: for example, pathogen dispersal might be highly localized, vectors or reservoirs for pathogens might be spatially restricted, or susceptible hosts might be clumped. Here, spatial pattern of an epidemic model with nonlinear incidence rates is investigated. The conditions for Hopf bifurcation and Turing bifurcation are gained and, in particular, exact Turing domain is found in the two parameters space. Furthermore, numerical results show that force of infection, namely β, plays an important role in the spatial pattern. More specifically, different patterns emerge as β increases. The mathematical analysis and numerical results well extend the finding of pattern formation in the epidemic models and may well explain the field observed in some areas.

[1]  Zhen Jin,et al.  CROSS-DISCIPLINARY PHYSICS AND RELATED AREAS OF SCIENCE AND TECHNOLOGY: Emergence of Strange Spatial Pattern in a Spatial Epidemic Model , 2008 .

[2]  T. Geisel,et al.  Forecast and control of epidemics in a globalized world. , 2004, Proceedings of the National Academy of Sciences of the United States of America.

[3]  J M Hughes,et al.  Emerging infectious diseases: public health issues for the 21st century. , 1999, Science.

[4]  Zhen Jin,et al.  SPATIAL PATTERN IN AN EPIDEMIC SYSTEM WITH CROSS-DIFFUSION OF THE SUSCEPTIBLE , 2009 .

[5]  Alan Hastings,et al.  Spatial heterogeneity and ecological models , 1990 .

[6]  Zhen Jin,et al.  Self-organized wave pattern in a predator-prey model , 2010 .

[7]  R. May,et al.  Modelling vaccination strategies against foot-and-mouth disease , 2003, Nature.

[8]  Zhen Jin,et al.  Predator cannibalism can give rise to regular spatial pattern in a predator–prey system , 2009 .

[9]  S. Cornell,et al.  Dynamics of the 2001 UK Foot and Mouth Epidemic: Stochastic Dispersal in a Heterogeneous Landscape , 2001, Science.

[10]  R. May,et al.  Stability and Complexity in Model Ecosystems , 1976, IEEE Transactions on Systems, Man, and Cybernetics.

[11]  S. Riley,et al.  Smallpox transmission and control: Spatial dynamics in Great Britain , 2006, Proceedings of the National Academy of Sciences.

[12]  P. Driessche,et al.  Dispersal data and the spread of invading organisms. , 1996 .

[13]  N. Rashevsky,et al.  Mathematical biology , 1961, Connecticut medicine.

[14]  Li Li,et al.  Pattern dynamics of a spatial predator–prey model with noise , 2012 .

[15]  S. Blower,et al.  Mixing ecology and epidemiology , 1991, Proceedings of the Royal Society of London. Series B: Biological Sciences.

[16]  R. Ostfeld,et al.  Spatial epidemiology: an emerging (or re-emerging) discipline. , 2005, Trends in ecology & evolution.

[17]  Zhen Jin,et al.  Effect of noise on the pattern formation in an epidemic model , 2010 .

[18]  A. McMichael,et al.  Environmental and social influences on emerging infectious diseases: past, present and future. , 2004, Philosophical transactions of the Royal Society of London. Series B, Biological sciences.

[19]  J. Noble,et al.  Geographic and temporal development of plagues , 1974, Nature.

[20]  N. Ferguson,et al.  Planning for smallpox outbreaks , 2003, Nature.

[21]  D. Rand,et al.  Correlation models for childhood epidemics , 1997, Proceedings of the Royal Society of London. Series B: Biological Sciences.

[22]  Steve Leach,et al.  Transmission potential of smallpox in contemporary populations , 2001, Nature.

[23]  Marjorie J. Wonham,et al.  An epidemiological model for West Nile virus: invasion analysis and control applications , 2004, Proceedings of the Royal Society of London. Series B: Biological Sciences.

[24]  Noble Jv,et al.  Geographic and temporal development of plagues , 1974 .

[25]  Zhen Jin,et al.  Pattern formation in a spatial S–I model with non-linear incidence rates , 2007 .

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

[27]  O. Bjørnstad,et al.  Travelling waves and spatial hierarchies in measles epidemics , 2001, Nature.

[28]  Ilkka Hanski,et al.  Coexistence of Competitors in Patchy Environment , 1983 .

[29]  David L Smith,et al.  Predicting the spatial dynamics of rabies epidemics on heterogeneous landscapes , 2002, Proceedings of the National Academy of Sciences of the United States of America.

[30]  C. Dye,et al.  Modeling the SARS Epidemic , 2003, Science.

[31]  Matt J Keeling,et al.  Using conservation of pattern to estimate spatial parameters from a single snapshot , 2004, Proceedings of the National Academy of Sciences of the United States of America.

[32]  Jim M Cushing,et al.  Chaos in Ecology: Experimental Nonlinear Dynamics , 2002 .

[33]  Zhen Jin,et al.  Chaos induced by breakup of waves in a spatial epidemic model with nonlinear incidence rate , 2008 .

[34]  Zhen Jin,et al.  Influence of infection rate and migration on extinction of disease in spatial epidemics. , 2010, Journal of theoretical biology.

[35]  W. M. Lonsdale,et al.  GLOBAL PATTERNS OF PLANT INVASIONS AND THE CONCEPT OF INVASIBILITY , 1999 .

[36]  李莉,et al.  Emergence of Strange Spatial Pattern in a Spatial Epidemic Model , 2008 .

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