The probability of extinction in a bovine respiratory syncytial virus epidemic model

Mathematical Biosciences

Volumn 231, Issue 2, 2011, Pages 144-158

  Griffiths, Martin ^a,b   Greenhalgh, David ^a  

^a UNIVERSITY OF STRATHCLYDE   (United Kingdom)

^b UNIVERSITY OF MANCHESTER   (United Kingdom)

Author keywords

Backward bifurcation; Branching process; Deterministic model; Probabilities of extinction; Stochastic model; Stochastic simulation

Indexed keywords

BACKWARD BIFURCATION; BRANCHING PROCESS; DEGREE OF COMPLEXITY; DETERMINISTIC EPIDEMIC MODELS; DETERMINISTIC MODEL; DETERMINISTIC MODELS; EPIDEMIC MODELS; RESPIRATORY SYNCYTIAL VIRUS; STOCHASTIC FORMULATION; STOCHASTIC SIMULATIONS;

BIFURCATION (MATHEMATICS); COMPUTER SIMULATION; EPIDEMIOLOGY; PROBABILITY DISTRIBUTIONS; STOCHASTIC SYSTEMS; VIRUSES;

STOCHASTIC MODELS;

BIFURCATION; EPIDEMIC; NUMERICAL MODEL; PROBABILITY; STOCHASTICITY;

ANALYTIC METHOD; ARTICLE; BOVINE RESPIRATORY SYNCYTIAL VIRUS INFECTION; DISEASE PREDISPOSITION; EPIDEMIC; HUMAN; MATHEMATICAL COMPUTING; NONHUMAN; POPULATION SIZE; POPULATION STRUCTURE; PROBABILITY; RESPIRATORY SYNCYTIAL PNEUMOVIRUS; SIMULATION; STOCHASTIC MODEL;

ANIMALS; CATTLE; CATTLE DISEASES; COMPUTER SIMULATION; DISEASE OUTBREAKS; MODELS, BIOLOGICAL; RESPIRATORY SYNCYTIAL VIRUS INFECTIONS; RESPIRATORY SYNCYTIAL VIRUS, BOVINE;

BOVINE RESPIRATORY SYNCYTIAL VIRUS;

EID: 79954677689     PISSN: 00255564     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.mbs.2011.02.011     Document Type: Article

References (27)

1. Allen L.J.S., van den Driessche P. Stochastic epidemic models with a backward bifurcation. Math. Biosci. Eng. 2006, 3:445.
2. Anderson R.M., May R.M. Population biology of infectious diseases: Part 1. Nature 1979, 280:361.
3. Bailey N.T.J. The Mathematical Theory of Infectious Diseases 1975, Charles Griffin. second ed.
4. Bean N.G., Bright L., Latouche G., Pearce C.E.M., Pollett P.K., Taylor P.G. The quasi-stationary behaviour of quasi-birth-and-death processes. Ann. Appl. Probab. 1997, 7(1):134.
5. Castillo-Chavez C., Cooke K., Huang W., Levin S.A. Results on the dynamics for models for the sexual transmission of the human immunodeficiency virus. Appl. Math. Lett. 1989, 2:327.
6. Castillo-Chavez C., Cooke K., Huang W., Levin S.A. The role of long incubation periods in the dynamics of HIV/AIDS. Part 2: multiple group models. Mathematical and statistical approaches to AIDS epidemiology. Lecture Notes in Biomathematics 1989, vol. 83:200. Springer-Verlag.
7. Clancy D., O'Neill P.D., Pollett P.K. Approximations for the long-term behaviour of an open-population epidemic model. Methodol. Comput. Appl. Probab. 2001, 3:75.
8. de Jong M.C.M., van der Poel W.H.M., Kramps J.A., Brand A., van Oirschot J.T. Persistence and recurrent outbreaks of bovine respiratory syncytial virus on dairy farms. Am. J. Vet. Res. 1996, 57:628.
9. Dickman R., Vidigal R. Quasi-stationary distributions for stochastic processes with an absorbing state. J. Phys. A: Math. Gen. 2002, 35:1147.
10. Diekmann O., Heesterbeek J.A.P., Metz J.A.J. On the definition and computation of the basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations. J. Math. Biol. 1990, 28:365.
11. Dushoff J. Incorporating immunological ideas in epidemiological models. J. Theor. Biol. 1996, 180:181.
12. Greenhalgh D., Diekmann O., de Jong M.C.M. Subcritical endemic steady states in mathematical models for animal infections with incomplete immunity. Math. Biosci. 2000, 165:1.
13. Greenhalgh D., Griffiths M., bifurcation Backward equilibrium and stability phenomena in a three-stage extended BRSV epidemic model. J. Math. Biol. 2009, 59:1.
14. Greenhalgh D., Griffiths M. Dynamic phenomena arising from an extended Core Group model. Math. Biosci. 2009, 221:136.
15. M. Griffiths, Backward Bifurcation and Associated Phenomena in Epidemic Models, Ph.D. Thesis, University of Strathclyde, 2007.
16. Grimmett G., Stirzaker D. Probability and Random Processes 2001, Oxford University.
17. Huang W., Cooke K.L., Castillo-Chavez C. Stability and bifurcation for a multiple-group model for the dynamics of HIV/AIDS transmission. SIAM J. Appl. Math. 1992, 52:835.
18. Karlin S., Taylor H.M. A First Course in Stochastic Processes 1975, Academic Press. second ed.
19. Kermack W.O., McKendrick A.G. A contribution to the mathematical theory of epidemics. Proc. R. Soc. A 1927, 115:700.
20. Kurtz T.G. Solutions of ordinary differential equations as limits of pure jump Markov processes. J. Appl. Probab. 1970, 7:49.
21. Kurtz T.G. Limit theorems for sequences of jump Markov processes approximating ordinary differential processes. J. Appl. Probab. 1971, 8:344.
22. Nasell I. Stochastic models of some endemic infections. Math. Biosci. 2002, 179:1.
23. Nisbet R.M., Gurney W.S.C. Modelling Fluctuating Populations 1982, Wiley.
24. Renshaw E. Modelling Biological Populations in Space and Time 1991, Cambridge University.
25. van Kampen N.G. Stochastic Processes in Physics and Chemistry 1992, Elsevier. second ed.
26. Whittle P. The outcome of a stochastic epidemic - a note on Bailey's paper. Biometrika 1955, 42:116.
27. Wiggins S. Introduction to Applied Nonlinear Dynamical Systems and Chaos 1990, Springer-Verlag.