TY - JOUR

T1 - Comparison of deterministic and stochastic SIS and SIR models in discrete time

AU - Allen, Linda J.S.

AU - Burgin, Amy M.

N1 - Funding Information:
We acknowledge the support of the NSF grant DMS-9626417. We thank anonymous referees for their helpful comments and suggestions.

PY - 2000/1

Y1 - 2000/1

N2 - The dynamics of deterministic and stochastic discrete-time epidemic models are analyzed and compared. The discrete-time stochastic models are Markov chains, approximations to the continuous-time models. Models of SIS and SIR type with constant population size and general force of infection are analyzed, then a more general SIS model with variable population size is analyzed. In the deterministic models, the value of the basic reproductive number R0 determines persistence or extinction of the disease. If R0 < 1, the disease is eliminated, whereas if R0 > 1, the disease persists in the population. Since all stochastic models considered in this paper have finite state spaces with at least one absorbing state, ultimate disease extinction is certain regardless of the value of R0. However, in some cases, the time until disease extinction may be very long. In these cases, if the probability distribution is conditioned on non-extinction, then when R0 > 1, there exists a quasi-stationary probability distribution whose mean agrees with deterministic endemic equilibrium. The expected duration of the epidemic is investigated numerically.

AB - The dynamics of deterministic and stochastic discrete-time epidemic models are analyzed and compared. The discrete-time stochastic models are Markov chains, approximations to the continuous-time models. Models of SIS and SIR type with constant population size and general force of infection are analyzed, then a more general SIS model with variable population size is analyzed. In the deterministic models, the value of the basic reproductive number R0 determines persistence or extinction of the disease. If R0 < 1, the disease is eliminated, whereas if R0 > 1, the disease persists in the population. Since all stochastic models considered in this paper have finite state spaces with at least one absorbing state, ultimate disease extinction is certain regardless of the value of R0. However, in some cases, the time until disease extinction may be very long. In these cases, if the probability distribution is conditioned on non-extinction, then when R0 > 1, there exists a quasi-stationary probability distribution whose mean agrees with deterministic endemic equilibrium. The expected duration of the epidemic is investigated numerically.

KW - Epidemic

KW - Markov process

KW - Quasi-stationary

KW - Stochastic

UR - http://www.scopus.com/inward/record.url?scp=0033979874&partnerID=8YFLogxK

U2 - 10.1016/S0025-5564(99)00047-4

DO - 10.1016/S0025-5564(99)00047-4

M3 - Article

C2 - 10652843

AN - SCOPUS:0033979874

VL - 163

SP - 1

EP - 33

JO - Mathematical Biosciences

JF - Mathematical Biosciences

SN - 0025-5564

IS - 1

ER -