Extinction of an infectious disease: A large fluctuation in a nonequilibrium system

Physical Review E - Statistical, Nonlinear, and Soft Matter Physics

Volumn 77, Issue 6, 2008, Pages

  Kamenev, Alex ^a   Meerson, Baruch ^a,b  

^a University of Minnesota   (United States)

^b HEBREW UNIVERSITY OF JERUSALEM   (Israel)

Author keywords

[No Author keywords available]

Indexed keywords

COMPUTER NETWORKS; FOOD PRODUCTS; FUNCTION EVALUATION; HEALTH; STOCHASTIC MODELS; TRAJECTORIES;

(E ,2E) THEORY; AMERICAN PHYSICAL SOCIETY (APS); ANALYTICAL RESULTS; BIFURCATION POINTS; EIKONAL APPROXIMATIONS; EPIDEMIOLOGICAL MODELING; EXTINCTION TIME; FIRST-PASSAGE; HAMILTONIAN; HAMILTONIAN SYSTEMS; INFECTIOUS DISEASES; MULTIDIMENSIONAL (MD); NON-EQUILIBRIUM SYSTEMS; OPTIMAL PATHS; PROBABILITY GENERATING FUNCTION (PGF); SPECIES POPULATION; SPIRAL PATH;

HAMILTONIANS;

EID: 44949095232     PISSN: 15393755     EISSN: 15502376     Source Type: Journal    
DOI: 10.1103/PhysRevE.77.061107     Document Type: Article

References (24)

1. Noise in Nonlinear Dynamical Systems, edited by, F. Moss, and, P. V. E. McClintock, (Cambridge University Press, Cambridge, 1989)
2. Fluctuations and Order: The New Synthesis, edited by, M. Milonas, (Springer, New York, 1996).
3. C. W. Gardiner, Handbook of Stochastic Methods (Springer, Berlin, 2004)
4. N. G. van Kampen, Stochastic Processes in Physics and Chemistry (North-Holland, Amsterdam, 2001).
5. B. Gaveau, M. Moreau, and J. Toth, Lett. Math. Phys. 37, 285 (1996) 0377-9017
6. C. R. Doering, K. V. Sargsyan, and L. M. Sander, Multiscale Model. Simul. MMSUBT 1540-3459 10.1137/030602800 3, 283 (2005).
7. V. Elgart and A. Kamenev, Phys. Rev. E PLEEE8 1063-651X 10.1103/PhysRevE.70.041106 70, 041106 (2004).
8. M. Assaf and B. Meerson, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.97.200602 97, 200602 (2006)
9. M. Assaf and B. Meerson, Phys. Rev. E PLEEE8 1063-651X 10.1103/PhysRevE.75.031122 75, 031122 (2007).
10. M. S. Bartlett, Stochastic Population Models in Ecology and Epidemiology (Wiley, New York, 1961)
11. H. Andersson and T. Britton, Stochastic Epidemic Models and Their Statistical Analysis, Lect. Notes Stat., Vol. 151 (Springer, New York, 2000)
12. O. Diekmann and J. A. P. Heesterbeek, Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation (Wiley, Chichester, 2000)
13. D. J. Daley and J. Gani, Epidemic Modelling: An Introduction, Cambridge Studies in Mathematical Biology (Cambridge University Press, Cambridge, 2001).
14. O. A. van Herwaarden and J. Grasman, J. Math. Biol. JMBLAJ 0303-6812 10.1007/BF00298644 33, 581 (1995).
15. I. Nåsell, J. R. Stat. Soc. Ser. B. (Stat. Methodol.) 61, 309 (1999). 0035-9246
16. When the fixed point 4 is a stable focus, the quasistationary state, as described by the stochastic model, will exhibit sustained noisy oscillations, in analogy with other two-species systems [J. P. Aparicio and H. G. Solari, Math. Biosci. MABIAR 0025-5564 10.1016/S0025-5564(00)00050-X 169, 15 (2001)
17. A. J. McKane and T. J. Newman, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.94.218102 94, 218102 (2005)].
18. M. I. Dykman, E. Mori, J. Ross, and P. M. Hunt, J. Chem. Phys. JCPSA6 0021-9606 10.1063/1.467139 100, 5735 (1994).
19. For a single species the equation t G= H G can be analyzed by the spectral theory. This theory, however, has not been extended to multiple species.
20. V. Elgart and A. Kamenev, Phys. Rev. E PLEEE8 1063-651X 10.1103/PhysRevE.74.041101 74, 041101 (2006).
21. M. Dykman, I. B. Schwartz, and A. S. Landsman, e-print arXiv:0801.4902.
22. M. I. Freidlin and A. D. Wentzell, Random Perturbations of Dynamical Systems (Springer-Verlag, New York, 1984); also see the first reference in.
23. B. J. Matkowsky, Z. Schuss, C. Knessl, C. Tier, and M. Mangel, Phys. Rev. A PLRAAN 1050-2947 10.1103/PhysRevA.29.3359 29, 3359 (1984).
24. One can easily compute q2 (t) = (eKδ (t- t0) +1) -1 and p2 (t) =- (e-Kδ (t- t0) +1) -1, where t0 =const.