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Mathematical Biosciences and Engineering (MBE)
 

An application of queuing theory to SIS and SEIS epidemic models
Pages: 809 - 823, Volume 7, Issue 4, October 2010

doi:10.3934/mbe.2010.7.809      Abstract        References        Full text (405.0K)                  Related Articles

Carlos M. Hernández-Suárez - Facultad de Ciencias, Universidad de Colima, Apdo. Postal 25, Colima, Colima, Mexico (email)
Carlos Castillo-Chavez - Mathematics, Computational and Modeling Sciences Center, Arizona State University PO Box 871904, Tempe, AZ, 85287, United States (email)
Osval Montesinos López - Facultad de Ciencias, Universidad de Colima, Apdo. Postal 25, Colima, Colima, Mexico (email)
Karla Hernández-Cuevas - Facultad de Ciencias, Universidad de Colima, Apdo. Postal 25, Colima, Colima, Mexico (email)

1 S. Ross, "Introduction to Probability Models," Academic Press, 2007.
2 D. Kendall, Some problems in the theory of queues, Journal of the Royal Statistical Society Series B (Methodological), 13 (1951) 151-185.
3 D. Kendall, Stochastic processes occurring in the theory of queues and their analysis by the method of the imbedded markov chain, The Annals of Mathematical Statistics, 24 (1953), 338-354.
4 M. Kitaev, The M/G/1 processor-sharing model: Transient behavior, Queueing Systems, 14 (1993), 239-273.
5 H. Andersson and T. Britton, "Stochastic Epidemic Models and Their Statistical Analysis," Springer Verlag, 2000.
6 T. Sellke, On the asymptotic distribution of the size of a stochastic epidemic, J. Appl. Probab., 20 (1983), 390-394.       
7 F. Ball and P. Donnelly, Strong approximations for epidemic models, Stochastic Processes and Their Applications, 55 (1995), 1-21.
8 P. Trapman and M. Bootsma, A useful relationship between epidemiology and queueing theory: The distribution of the number of infectives at the moment of the first detection, Mathematical Biosciences, 219 (2009), 15-22.
9 H. Andersson and B. Djehiche, A threshold limit theorem for the stochastic logistic epidemic, J. Appl. Probab., 35 (1998), 662-670, http://projecteuclid.org/getRecord?id=euclid.jap/1032265214.       
10 H. Andersson and T. Britton, Stochastic epidemics in dynamic populations: Quasi-stationarity and extinction, J. Math. Biol., 41 (2000), 559-580.
11 J. N. Darroch and E. Seneta, On quasi-stationary distributions in absorbing continuous-time finite Markov chains, J. Appl. Probability, 4 (1967), 192-196.       
12 J. Cavender, Quasi-stationary distributions of birth-and-death processes, Advances in Applied Probability, 10 (1978), 570-586.
13 R. J. Kryscio and C. Lefèvre, On the extinction of the S-I-S stochastic logistic epidemic, J. Appl. Probab., 26 (1989), 685-694.       
14 W. Kermack and A. McKendrick, Contributions to the mathematical theory of epidemics-iii. Further studies of the problem of endemicity, Bulletin of Mathematical Biology, 53 (1991), 89-118.
15 R. H. Norden, On the distribution of the time to extinction in the stochastic logistic population model, Adv. in Appl. Probab., 14 (1982), 687-708.       
16 I. Nåsell, On the quasi-stationary distribution of the stochastic logistic epidemic, Math. Biosci., 156 (1999), 21-40, epidemiology, cellular automata and evolution (Sofia, 1997).       
17 G. Weiss and M. Dishon, On the asymptotic behavior of the stochastic and deterministic models of an epidemic, Math. Biosci., 11 (1971), 261-265.
18 D. J. Bartholomew, Continuous time diffusion models with random duration of interest, J. Mathematical Sociology, 4 (1976), 187-199.       
19 O. Ovaskainen, The quasistationary distribution of the stochastic logistic model, J. Appl. Probab., 38 (2001), 898-907.       
20 I. Nåsell, On the quasi-stationary distribution of the Ross malaria model, Mathematical Biosciences, 107 (1991), 187.
21 I. Nåsell, Extinction and quasi-stationarity in the Verhulst logistic model, Journal of Theoretical Biology, 211 (2001), 11-27.
22 I. Nåsell, Stochastic models of some endemic infections, Math. Biosci., 179 (2002), 1-19.       
23 C. Hernández-Suárez and C. Castillo-Chavez, A basic result on the integral for birth-death Markov processes, Mathematical Biosciences, 161 (1999), 95-104.
24 V. T. Stefanov and S. Wang, A note on integrals for birth-death processes, Math. Biosci., 168 (2000), 161-165.
25 F. Ball and V. T. Stefanov, Further approaches to computing fundamental characteristics of birth-death processes, J. Appl. Probab., 38 (2001), 995-1005.       
26 M. VanHoorn, Algorithms and approximations for queueing systems, CWI Tract No. 8, CWI, Amsterdam, 1984.
27 D. Cox, "Renewal Theory," Monographs on Applied Probability and Statistics, Methuen and Co., 1962.

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