Controlling disease in migratory bird population: a probable solution through mathematical study

The role of migratory birds in an eco-system cannot be ignored. It becomes more important if they carry a disease. Chatterjee and Chattopadhyay proposed and analysed a three component one season eco-epidemiological model consisting of susceptible migratory birds, infective migratory birds and their predator population (see Chatterjee, S. and Chattopadhyay, J., Role of migratory bird populations in a simple eco-epidemiological model, Mathematical and Computer Modelling of Dynamical systems, in press). They assumed that the recovered infective birds become susceptible again. But, it is observed that in diseases like salmonella and WNV, the recovered birds develop a permanent immunity. Keeping this in mind we modify the model of Chatterjee and Chattopadhyay by adding a recovered class. The main objective of this work is to observe the role of recovery and immunity in such a system. Numerical simulations for a hypothetical set of parameter values are presented to illustrate the analytical findings. It is observed that to obtain a disease free system proper vaccination and proper predation are necessary. The second factor was mentioned in the paper of Chatterjee and Chattopadhyay.

[1]  H. Hethcote,et al.  Disease transmission models with density-dependent demographics , 1992, Journal of mathematical biology.

[2]  J. Velasco-Hernández,et al.  A simple vaccination model with multiple endemic states. , 2000, Mathematical biosciences.

[3]  Joydev Chattopadhyay,et al.  Infection in prey population may act as a biological control in ratio-dependent predator?prey models , 2004 .

[4]  F. R. Gantmakher.  The Theory of Matrices , 1984 .

[5]  Zhien Ma,et al.  A predator-prey model with infected prey. , 2004, Theoretical population biology.

[6]  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.

[7]  J. Rappole,et al.  Migratory birds and spread of West Nile virus in the Western Hemisphere. , 2000, Emerging infectious diseases.

[8]  M. Biendo,et al.  Regional dissemination of Salmonella enterica serovar Enteritidis is season dependent. , 2003, Clinical microbiology and infection : the official publication of the European Society of Clinical Microbiology and Infectious Diseases.

[9]  P. D. N. Srinivasu,et al.  Pelicans at risk in Salton Sea—an eco-epidemiological model-II , 2001 .

[10]  Mark A. McPeek,et al.  Predation, Competition, and Prey Communities: A Review of Field Experiments , 1985 .

[11]  James S. Muldowney,et al.  On Bendixson′s Criterion , 1993 .

[12]  S. Bergström,et al.  Enteropathogenic bacteria in migrating birds arriving in Sweden. , 1997, Scandinavian journal of infectious diseases.

[13]  H. I. Freedman,et al.  Predator-prey populations with parasitic infection , 1989, Journal of mathematical biology.

[14]  V. Deubel,et al.  Introduction of West Nile Virus in the Middle East by Migrating White Storks , 2002, Emerging infectious diseases.

[15]  H. Hethcote,et al.  Four SEI endemic models with periodicity and separatrices. , 1995, Mathematical biosciences.

[16]  A. West,et al.  Recovery and Identification of West Nile Virus from a Hawk in Winter , 2000, Journal of Clinical Microbiology.

[17]  Herbert W. Hethcote,et al.  The Mathematics of Infectious Diseases , 2000, SIAM Rev..

[18]  Kazuo Inoue.  Highly Pathogenic Avian Flu, Japan , 2004, Emerging infectious diseases.

[19]  J. C. Burkill,et al.  Ordinary Differential Equations , 1964 .

[20]  C. Packer,et al.  Keeping the herds healthy and alert: implications of predator control for infectious disease , 2003 .

[21]  Peter Lancaster,et al.  The theory of matrices , 1969 .

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