# zbMATH — the first resource for mathematics

On the number of recovered individuals in the $$SIS$$ and $$SIR$$ stochastic epidemic models. (English) Zbl 1200.92035
Summary: The basic models of infectious disease dynamics (the $$SIS$$ and $$SIR$$ models) are considered. Particular attention is paid to the number of infected individuals that recovered and its relationship with the final epidemic size. We investigate this descriptor both until the extinction of the epidemic and in the transient regime. Simple and efficient methods to obtain the distribution of the number of recovered individuals and its moments are proposed and discussed with respect to previous work. The methodology could also be extended to other stochastic epidemic models. The theory is illustrated by numerical experiments, which demonstrate that the proposed computational methods can be applied efficiently. In particular, we use the distribution of the number of individuals removed in the $$SIR$$ model in conjunction with data of outbreaks of $$ESBL$$ observed in the intensive care unit of a Spanish hospital.

##### MSC:
 92D30 Epidemiology 65C50 Other computational problems in probability (MSC2010) 65C20 Probabilistic models, generic numerical methods in probability and statistics
MCQueue
Full Text:
##### References:
 [1] Abate, J.; Whitt, W., Numerical inversion of Laplace transforms of probability distributions, ORSA J. comput., 7, 36, (1995) · Zbl 0821.65085 [2] Allen, L.J.S., An introduction to stochastic processes with applications to biology, (2003), Prentice-Hall New Jersey [3] Andersson, H.; Britton, T., Stochastic epidemic models and their statistical analysis, Lecture notes in biostatistics, vol. 151, (2000), Springer-Verlag New York · Zbl 0951.92021 [4] Artalejo, J.R.; Economou, A.; Lopez-Herrero, M.J., Evaluating growth measures in an immigration process subject to binomial and geometric catastrophes, Math. biosci. eng., 4, 573, (2007) · Zbl 1143.92030 [5] Artalejo, J.R.; Economou, A.; Lopez-Herrero, M.J., The maximum number of infected individuals in SIS epidemic models: computational techniques and quasi-stationary distributions, J. comput. appl. math., 233, 2563, (2010) · Zbl 1180.92073 [6] Artalejo, J.R.; Gomez-Corral, A., A state-dependent Markov-modulated mechanism for generating events and stochastic models, Math. methods appl. sci., 33, 1342, (2010) · Zbl 1211.60035 [7] Bailey, N.T.J., The total size of a general stochastic epidemic, Biometrika, 40, 177, (1953) · Zbl 0050.36601 [8] Bailey, N.T.J., The mathematical theory of infectious diseases and its applications, (1975), Charles Griffin & Company Ltd. London · Zbl 0115.37202 [9] Bailey, N.T.J., The elements of stochastic processes with applications to the natural sciences, (1990), John Wiley & Sons New York · Zbl 0127.11203 [10] Ball, F., A unified approach to the distribution of total size and total area under the trajectory of infectives in epidemic models, Adv. appl. prob., 18, 289, (1986) · Zbl 0606.92018 [11] Ball, F.; Nåsell, I., The shape of the size distribution of an epidemic in a finite population, Math. biosci., 123, 167, (1994) · Zbl 0805.92023 [12] Ball, F.; Neal, P., Network epidemic models with two levels of mixing, Math. biosci., 212, 69, (2008) · Zbl 1132.92020 [13] Britton, T., Stochastic epidemic models: a survey, Math. biosci., 225, 24, (2010) · Zbl 1188.92031 [14] Chernick, M.R., Boostrap methods: A guide for practitioners and researchers, (2008), Wiley Hoboken · Zbl 1136.62029 [15] Clancy, D., A stochastic SIS infection model incorporating indirect transmission, J. appl. prob., 42, 726, (2005) · Zbl 1082.60065 [16] Cohen, A.M., Numerical methods for Laplace transform inversion, (2007), Springer New York · Zbl 1127.65094 [17] Coolen-Schrijner, P.; van Doorn, E.A., Quasi-stationary distributions for a class of discrete-time Markov chains, Methodol. comput. appl. probab., 8, 449, (2006) · Zbl 1106.60064 [18] Daley, D.J.; Gani, J., Epidemic modelling: an introduction, Cambridge studies in mathematical biology, vol. 15, (1999), Cambridge University Press Cambridge · Zbl 0922.92022 [19] Demeris, N.; O’Neill, P.D., Computation of final outcome probabilities for the generalised stochastic epidemic, Stat. comput., 16, 309, (2006) [20] Isham, V.; Harden, S.; Nekovee, M., Stochastic epidemics and rumours on finite random networks, Physica A, 389, 561, (2010) [21] Keeling, M.J.; Ross, J.V., On methods for studying stochastic disease dynamics, J.R. soc. interface, 5, 171, (2008) [22] Lindholm, M., On the time to extinction for a two-type version of bartlett’s epidemic model, Math. biosci., 212, 99, (2008) · Zbl 1132.92022 [23] Lucet, J.C.; Decré, D.; Fichelle, A.; Joly-Guillou, M.L.; Pernet, M.; Deblangy, C.; Kosmann, M.J.; Régnier, B.J.C., Control of a prolonged outbreak of extended-spectrum beta-lactamase-producing enterobacteriaceae in a university hospital, Clin. infect. dis., 29, 1411, (1999) [24] National Committee for Clinical Laboratory Standards, Performance standards for antimicrobial susceptibility testing, in: 12th Informational Supplement. Approved standard M100-S12. NCCLS, Wayne, Pa, 2002. [25] Neal, P., Coupling of two SIR epidemic models with variable susceptibles and infectives, J. appl. prob., 44, 41, (2007) [26] Neuts, M.F.; Li, J.M., An algorithmic study of S-I-R stochastic epidemic models, (), 295 · Zbl 0857.92013 [27] Paterson, D.L.; Bonomo, R.A., Extended-spectrum beta-lactamases: clinical update, Clin. microbiol. rev., 18, 657, (2005) [28] Riggs, T.; Koopman, J.S., Maximizing statistical power in group-randomized vaccine trials, Epidemiol. infect., 133, 993, (2005) [29] Stone, P.; Wilkinson-Herbots, H.; Isham, V., A stochastic model for head lice infections, J. math. biol., 56, 743, (2008) · Zbl 1146.92324 [30] Tijms, H.C., A first course in stochastic models, (2003), Wiley Chichester · Zbl 1088.60002 [31] van Doorn, E.A.; Pollett, P.K., Survival in a quasi-death process, Linear algebra appl., 429, 776, (2008) · Zbl 1148.60056 [32] Whittle, P., The outcome of a stochastic epidemic – a note on bailey’s paper, Biometrika, 42, 116, (1955) · Zbl 0064.39103 [33] Wierman, J.C.; Marchette, D.J., Modeling computer virus prevalence with a susceptible-infected-susceptible model with reintroduction, Comput. stat. data ann., 45, 3, (2004) · Zbl 1429.68037 [34] Xu, Y.; Allen, L.J.S.; Perelson, A.S., Stochastic model of an influenza epidemic with drug resistance, J. theor. biol., 248, 179, (2007)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.