An application of queuing theory to SIS and SEIS epidemic models

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2010, 7(4): 809-823. doi: 10.3934/mbe.2010.7.809

An application of queuing theory to SIS and SEIS epidemic models

Carlos M. Hernández-Suárez ^1, , Carlos Castillo-Chavez ^2, , Osval Montesinos López ^1, and Karla Hernández-Cuevas ^1,

1.	Facultad de Ciencias, Universidad de Colima, Apdo. Postal 25, Colima, Colima, Mexico, Mexico, Mexico
2.	Mathematics, Computational and Modeling Sciences Center, Arizona State University PO Box 871904, Tempe, AZ, 85287, United States

Received February 2010 Revised May 2010 Published October 2010

Abstract
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Under a Creative Commons license

In this work we consider every individual of a population to be a server whose state can be either busy (infected) or idle (susceptible). This server approach allows to consider a general distribution for the duration of the infectious state, instead of being restricted to exponential distributions. In order to achieve this we first derive new approximations to quasistationary distribution (QSD) of SIS (Susceptible- Infected- Susceptible) and SEIS (Susceptible- Latent- Infected- Susceptible) stochastic epidemic models. We give an expression that relates the basic reproductive number, $R_0$ and the server utilization, $\rho$.

Keywords: SIS; SEIS; Queuing theory; $R_{0}$; basic reproductive number; stochastic epidemic models..

Mathematics Subject Classification: Primary: 92B05; Secondary: 62J2.

Citation: Carlos M. Hernández-Suárez, Carlos Castillo-Chavez, Osval Montesinos López, Karla Hernández-Cuevas. An application of queuing theory to SIS and SEIS epidemic models. Mathematical Biosciences & Engineering, 2010, 7 (4) : 809-823. doi: 10.3934/mbe.2010.7.809

References:

[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. doi: doi:10.1214/aoms/1177728975.
[4]	M. Kitaev, The M/G/1 processor-sharing model: Transient behavior, Queueing Systems, 14 (1993), 239-273. doi: doi:10.1007/BF01158868.
[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. doi: doi:10.2307/3213811.
[7]	F. Ball and P. Donnelly, Strong approximations for epidemic models, Stochastic Processes and Their Applications, 55 (1995), 1-21. doi: doi:10.1016/0304-4149(94)00034-Q.
[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. doi: doi:10.1016/j.mbs.2009.02.001.
[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. doi: doi:10.1007/s002850000060.
[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. doi: doi:10.2307/3212311.
[12]	J. Cavender, Quasi-stationary distributions of birth-and-death processes, Advances in Applied Probability, 10 (1978), 570-586. doi: doi:10.2307/1426635.
[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. doi: doi:10.2307/3214374.
[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. doi: doi:10.2307/1427019.
[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). doi: doi:10.1016/S0025-5564(98)10059-7.
[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. doi: doi:10.1016/0025-5564(71)90087-3.
[18]	D. J. Bartholomew, Continuous time diffusion models with random duration of interest, J. Mathematical Sociology, 4 (1976), 187-199. doi: doi:10.1080/0022250X.1976.9989853.
[19]	O. Ovaskainen, The quasistationary distribution of the stochastic logistic model, J. Appl. Probab., 38 (2001), 898-907. doi: doi:10.1239/jap/1011994180.
[20]	I. Nåsell, On the quasi-stationary distribution of the Ross malaria model, Mathematical Biosciences, 107 (1991), 187. doi: doi:10.1016/0025-5564(91)90004-3.
[21]	I. Nåsell, Extinction and quasi-stationarity in the Verhulst logistic model, Journal of Theoretical Biology, 211 (2001), 11-27. doi: doi:10.1006/jtbi.2001.2328.
[22]	I. Nåsell, Stochastic models of some endemic infections, Math. Biosci., 179 (2002), 1-19. doi: doi:10.1016/S0025-5564(02)00098-6.
[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. doi: doi:10.1016/S0025-5564(99)00034-6.
[24]	V. T. Stefanov and S. Wang, A note on integrals for birth-death processes, Math. Biosci., 168 (2000), 161-165. doi: doi:10.1016/S0025-5564(00)00046-8.
[25]	F. Ball and V. T. Stefanov, Further approaches to computing fundamental characteristics of birth-death processes, J. Appl. Probab., 38 (2001), 995-1005. doi: doi:10.1239/jap/1011994187.
[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.

show all references

References:

[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. doi: doi:10.1214/aoms/1177728975.
[4]	M. Kitaev, The M/G/1 processor-sharing model: Transient behavior, Queueing Systems, 14 (1993), 239-273. doi: doi:10.1007/BF01158868.
[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. doi: doi:10.2307/3213811.
[7]	F. Ball and P. Donnelly, Strong approximations for epidemic models, Stochastic Processes and Their Applications, 55 (1995), 1-21. doi: doi:10.1016/0304-4149(94)00034-Q.
[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. doi: doi:10.1016/j.mbs.2009.02.001.
[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. doi: doi:10.1007/s002850000060.
[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. doi: doi:10.2307/3212311.
[12]	J. Cavender, Quasi-stationary distributions of birth-and-death processes, Advances in Applied Probability, 10 (1978), 570-586. doi: doi:10.2307/1426635.
[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. doi: doi:10.2307/3214374.
[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. doi: doi:10.2307/1427019.
[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). doi: doi:10.1016/S0025-5564(98)10059-7.
[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. doi: doi:10.1016/0025-5564(71)90087-3.
[18]	D. J. Bartholomew, Continuous time diffusion models with random duration of interest, J. Mathematical Sociology, 4 (1976), 187-199. doi: doi:10.1080/0022250X.1976.9989853.
[19]	O. Ovaskainen, The quasistationary distribution of the stochastic logistic model, J. Appl. Probab., 38 (2001), 898-907. doi: doi:10.1239/jap/1011994180.
[20]	I. Nåsell, On the quasi-stationary distribution of the Ross malaria model, Mathematical Biosciences, 107 (1991), 187. doi: doi:10.1016/0025-5564(91)90004-3.
[21]	I. Nåsell, Extinction and quasi-stationarity in the Verhulst logistic model, Journal of Theoretical Biology, 211 (2001), 11-27. doi: doi:10.1006/jtbi.2001.2328.
[22]	I. Nåsell, Stochastic models of some endemic infections, Math. Biosci., 179 (2002), 1-19. doi: doi:10.1016/S0025-5564(02)00098-6.
[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. doi: doi:10.1016/S0025-5564(99)00034-6.
[24]	V. T. Stefanov and S. Wang, A note on integrals for birth-death processes, Math. Biosci., 168 (2000), 161-165. doi: doi:10.1016/S0025-5564(00)00046-8.
[25]	F. Ball and V. T. Stefanov, Further approaches to computing fundamental characteristics of birth-death processes, J. Appl. Probab., 38 (2001), 995-1005. doi: doi:10.1239/jap/1011994187.
[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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