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Appl. Math."],"published-print":{"date-parts":[[2021,1]]},"abstract":"<jats:p>Stochastic discrete-time susceptible--infected--susceptible (SIS) and susceptible--infected--recovered (SIR) models of endemic diseases are introduced and analyzed. For the deterministic, mean-field model, the basic reproductive number $R_0$ determines their global dynamics. If $R_0\\le 1$, then the frequency of infected individuals asymptotically converges to zero. If $R_0&gt;1$, then the infectious class uniformly persists for all time; conditions for a globally stable, endemic equilibrium are given. In contrast, the infection goes extinct in finite time with a probability of 1 in the stochastic models for all $R_0$ values. To understand the length of the transient prior to extinction as well as the behavior of the transients, the quasi-stationary distributions and the associated mean time to extinction are analyzed using large deviation methods. When $R_0&gt;1$, these mean times to extinction are shown to increase exponentially with the population size $N$. Moreover, as $N$ approaches $\\infty$, the quasi-stationary distributions are supported by a compact set bounded away from extinction; sufficient conditions for convergence to a Dirac measure at the endemic equilibrium of the deterministic model are also given. In contrast, when $R_0&lt;1$, the mean times to extinction are bounded above $1\/(1-\\alpha)$, where $\\alpha&lt;1$ is the geometric rate of decrease of the infection when rare; as $N$ approaches $\\infty$, the quasi-stationary distributions converge to a Dirac measure at the disease-free equilibrium for the deterministic model. For several special cases, explicit formulas for approximating the quasi-stationary distribution and the associated mean extinction are given. These formulas illustrate how for arbitrarily small $R_0$ values, the mean time to extinction can be arbitrarily large, and how for arbitrarily large $R_0$ values, the mean time to extinction can be arbitrarily large.<\/jats:p>","DOI":"10.1137\/20m1339015","type":"journal-article","created":{"date-parts":[[2021,10,5]],"date-time":"2021-10-05T11:55:47Z","timestamp":1633434947000},"page":"2195-2217","source":"Crossref","is-referenced-by-count":17,"title":["Extinction and Quasi-Stationarity for Discrete-Time, Endemic SIS and SIR Models"],"prefix":"10.1137","volume":"81","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-5481-4822","authenticated-orcid":true,"given":"Sebastian J.","family":"Schreiber","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Shuo","family":"Huang","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jifa","family":"Jiang","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Hao","family":"Wang","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2021,10,5]]},"reference":[{"key":"atypb1","doi-asserted-by":"publisher","DOI":"10.1017\/S026996480000084X"},{"key":"atypb2","doi-asserted-by":"publisher","DOI":"10.1016\/0025-5564(94)90025-6"},{"key":"atypb3","doi-asserted-by":"publisher","DOI":"10.1016\/S0025-5564(99)00047-4"},{"key":"atypb4","doi-asserted-by":"publisher","DOI":"10.1080\/10236190802332308"},{"key":"atypb5","doi-asserted-by":"publisher","DOI":"10.1038\/280361a0"},{"key":"atypb6","doi-asserted-by":"crossref","unstructured":"R. M. Anderson and R. M. May,\n                      Infectious Diseases of Humans: Dynamics and Control\n                      , Oxford University Press, New York, 1991.","DOI":"10.1093\/oso\/9780198545996.001.0001"},{"key":"atypb7","doi-asserted-by":"publisher","DOI":"10.1007\/s002850000060"},{"key":"atypb8","doi-asserted-by":"publisher","DOI":"10.1007\/s00285-010-0336-x"},{"key":"atypb9","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.2013182117"},{"key":"atypb10","doi-asserted-by":"publisher","DOI":"10.1093\/biomet\/62.2.477"},{"key":"atypb11","unstructured":"M. S. Bartlett,\n                      An Introduction to Stochastic Processes With Special Reference to Methods and Applications\n                      , Cambridge University Press, London-New York, 1966."},{"key":"atypb12","doi-asserted-by":"publisher","DOI":"10.1016\/S0362-546X(01)00587-9"},{"key":"atypb13","doi-asserted-by":"publisher","DOI":"10.1007\/s11009-018-9621-8"},{"key":"atypb14","doi-asserted-by":"publisher","DOI":"10.2307\/3211876"},{"key":"atypb15","doi-asserted-by":"publisher","DOI":"10.2307\/3212311"},{"key":"atypb16","doi-asserted-by":"crossref","unstructured":"O. Diekmann, H. Heesterbeek, and T. Britton,\n                      Mathematical Tools for Understanding Infectious Disease Dynamics\n                      , in Princetion Ser. Theor. Comput. Biol. 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