Stochastic models of some endemic infections. (English) Zbl 0991.92026

Summary: Stochastic models are established and studied for several endemic infections with demography. Approximations of quasi-stationary distributions and of times to extinction are derived for stochastic versions of SI, SIS, SIR, and SIRS models. The approximations are valid for sufficiently large population sizes. Conditions for validity of the approximations are given for each of the models. These are also conditions for validity of the corresponding deterministic models. It is noted that some deterministic models are unacceptable approximations of the stochastic models for a large range of realistic parameter values.


92D30 Epidemiology
60J70 Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.)
Full Text: DOI


[1] Nåsell, I., The threshold concept in stochastic epidemic and endemic models, (Mollison, D., Epidemic Models: Their Structure and Relation to Data (1995), Cambridge University: Cambridge University Cambridge) · Zbl 0850.92047
[2] Nåsell, I., On the time to extinction in recurrent epidemics, J. Roy. Statist. Soc. B, Part 2, 61, 309 (1999) · Zbl 0917.92023
[3] Bailey, N. T.J., The Mathematical Theory of Infectious Diseases and Its Applications (1975), Griffin: Griffin London · Zbl 0115.37202
[4] Hethcote, H. W., The mathematics of infectious diseases, SIAM Rev., 42, 4, 599 (2001) · Zbl 0993.92033
[5] Mena-Lorca, J.; Hethcote, H. W., Dynamic models of infectious diseases as regulators of population sizes, J. Math. Biol., 30, 693 (1992) · Zbl 0748.92012
[6] Hethcote, H. W., Qualitative analysis of communicable disease models, Math. Biosci., 28, 335 (1976) · Zbl 0326.92017
[7] Jacquez, J. A.; Simon, C. A., The stochastic SI model with recruitment and deaths. I. Comparison with the closed SIS model, Math. Biosci., 117, 77 (1993) · Zbl 0785.92025
[8] Nåsell, I., The quasi-stationary distribution of the closed endemic SIS model, Adv. Appl. Prob., 28, 895 (1996) · Zbl 0854.92020
[9] Nåsell, I., On the quasi-stationary distribution of the stochastic logistic epidemic, Math. Biosci., 156, 21 (1999) · Zbl 0954.92024
[11] Andersson, H.; Britton, T., Stochastic epidemics in dynamic populations: quasi-stationarity and extinction, J. Math. Biol., 41, 6, 559 (2000) · Zbl 1002.92018
[12] Allen, L. J.S.; Burgin, A. M., Comparison of deterministic and stochastic SIS and SIR models in discrete time, Math. Biosci., 163, 11 (2000)
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.