Mathematical Biosciences 279 (2016) 38–42 Contents lists available at ScienceDirect Mathematical Biosciences journal homepage: www.elsevier.com/locate/mbs On extinction time of a generalized endemic chain-binomial model Ozgur Aydogmus Department of Economics, Social Sciences University of Ankara, Ankara, Turkey a r t i c l e i n f o Article history: Received 19 November 2015 Revised 23 May 2016 Accepted 30 June 2016 Available online 9 July 2016 Keywords: Chain-binomial epidemic model Endemic equilibrium Global stability Deterministic approximation Mean extinction time a b s t r a c t We considered a chain-binomial epidemic model not conferring immunity after infection. Mean field dy- namics of the model has been analyzed and conditions for the existence of a stable endemic equilibrium are determined. The behavior of the chain-binomial process is probabilistically linked to the mean field equation. As a result of this link, we were able to show that the mean extinction time of the epidemic increases at least exponentially as the population size grows. We also present simulation results for the process to validate our analytical findings. © 2016 Elsevier Inc. All rights reserved. 1. Introduction A chain binomial epidemic model has been developed at the beginning of 20th century by Reed and Frost. The model is widely used in the literature (see for example [1]) and its simplicity stim- ulated detailed simulation studies [2]. Jacquez [3] criticized the classical formulation of the Reed–Frost model in terms of consis- tency and reasonability of its assumptions. This critique was ini- tiated by the dimensional analysis of Reed–Frost equation. He fol- lowed suggestions of [1] and properly reformulated a more gen- eral epidemic model by using probability generating functions of discrete distributions for the number of contacts per person. Longini [4] modified the classical Reed–Frost process to be able to model diseases such as gonorrhea, rotavirus, meningitis and rhi- novirus in which reinfection take place. The model assumes that there is no removed state so that the sum of the number of in- fected individuals (I ) and of susceptible individuals (R) in the pop- ulation remains constant. If the population consists of N individu- als then the transition probabilities are as follows: Pr (I t +1 = x t +1 |I t = x t ) =  N − x t x t +1  (1 − q xt ) x t+1 q xt (N −xt −x t+1 ) . (1) Here, q is the probability that a susceptible individual escapes from the infection when there is only one infected person in the popu- lation. Since this model assumes that there is no immunity against the disease, it may give rise to the existence of an endemic equi- librium. In fact, the mean dynamics of stochastic model (1) has an E-mail address: ozgur.aydogmus@asbu.edu.tr, aydogmusozgur@gmail.com endemic equilibrium under the condition that the mean number of contact per person is larger than one (see for example [5]). While some probabilistic properties of this model has been given in [4], an analytical study on the extinction time of the pro- cess remains untouched. The very same problem has been high- lighted by Longini [6] as follows: “An interesting analytical question involves the study of the mean stopping time for the endemic process.” Mean extinction time of a birth-death type epidemic model has been studied by Kryscio and Lefévre [7] for large populations and it has been shown that mean extinction time is exponentially in- creasing in population size. In addition, there are other approaches to study mean absorption times for stochastic population models. For instance, a classical result which applies to all discrete time fi- nite Markov chains regarding mean hitting times is given by Norris [8, Theorem 1.3.5]. In ecology literature, Monte-Carlo simulations are used widely (see e.g. [9,10]) due to easiness of implementation. A different approach to obtain analytical results concerning mean extinction times is to write down the master equations and solve them numerically [11,12]. Lastly, diffusion approximations are used in the literature to derive a formula for mean time to extinction [13,14]. We modify the model proposed by Jacquez [3] for infectious diseases not conferring immunity following infection as done by Longini [6] and study extinction times analytically. To be able to work on this problem, we use an unconventional method, namely deterministic approximations. For chain binomial models, deter- ministic approximations have been studied by Weiß and Pollett [15] and Buckley and Pollett [16] as limit theorems. Thus these approximations are valid only for large population sizes. Here we find exponential bounds (decreasing in population size) on http://dx.doi.org/10.1016/j.mbs.2016.06.010 0025-5564/© 2016 Elsevier Inc. All rights reserved.