Quasi-stationary and ratio of expectations distributions: A comparative study

Abstract

Abstract. Many stochastic systems, including biological applications, use Markov chains in which there is a set of absorbing states. It is then needed to consider analogues of the stationary distribution of an irreducible chain. In this context, quasi-stationary distributions play a fundamental role to describe the long-term behavior of the system. The rationale for using quasi-stationary distribution is well established in the abundant existing literature. The aim of this study is to reformulate the ratio of means approach Seneta, 1965, 1967) which provides a simple alternative. We have a two-fold objective. The first objective is viewing quasi-stationarity and ratio of expectations as two different approaches for understanding the dynamics of the system before absorption. At this point, we remark that the quasi-stationary distribution and a ratio of means distribution may give or not give similar information. In this way, we arrive to the second objective; namely, to investigate the possibility of using the ratio of expectations distribution as an approximation to the quasistationary distribution. This second objective is explored by comparing both distributions in some selected scenarios, which are mainly inspired in stochastic epidemic models. Previously, the rate of convergence to the quasi-stationary regime is taking into account in order to make meaningful the comparison.

On SIR-models with Markov-modulated events: Length of an outbreak, total size of the epidemic and number of secondary infections

“…In recent years, there has been a rapidly growing interest in SIR-models and their applications; a comprehensive review of the existing literature and some results can be found in the survey papers by Britton [21], and Isham [34,35]. In the stochastic setting, three important quantities of the SIR-model include the final size distribution, the expected duration of an epidemic, and the basic reproduction number R 0 , which are analyzed by using a variety of tools in applied probability; see, for example, the articles by Allen [3], Allen and Burgin [4], Artalejo et al [9], and Artalejo and López-Herrero [12], among others. Gani and Purdue [27] outline a matrix-geometric method for determining the total size distribution in a recursive manner, and El Maroufy et al [25] explore how this method can be used to study SIR-models with a generalized mechanism of infection.…”

“…The concepts of local infectious clump and local susceptibility set are used by Ball and Neal [18] to develop a unified approach to the threshold behavior of SIR epidemics with two levels of mixing. The objective of Artalejo and López-Herrero [12] is to investigate quasi-stationarity and the ratio of expectations as two conceptually different approaches for understanding the dynamics of the SIR-model and its variant with demography before the extinction of an epidemic. New descriptors, including the time to reach specific numbers of infectives and susceptible individuals [15,Section 3.1], and the time to reach a critical number of infections [15,Section 3.2] are also investigated by using efficient numerical tools; for a related work, see the paper [5] where the interest is in the time to reach a critical number of infections in the SIR-model with infective and susceptible immigrants.…”

The stochastic SEIR model before extinction: Computational approaches

“…Another possibility for the quantification of an epidemic before its extinction is the use of the so-called Ratio-of-Expectations (RE) distribution and its variants [8,11,18]. In a sense, the RE-distribution gives the 'mean' fractions of time that a population spends in the various states till the extinction of the epidemic.…”

“…The stationary distribution of this 'returned' process that has been called also pseudo-transient distribution (see Ewens [22]) is also related to the RE-distribution. For a comparative study for the quasi-stationary and RE-distributions, we refer to the paper by Artalejo and Lopez-Herrero [8].…”

A model for the spread of an SIS epidemic in a human population

“…When the absorbing set A contains more than one state, they can be amalgamated into one single absorbing state and we can set q A = 0 and q A j = 0 for j ∈ C. Consequently, the above procedure can be applied to derive a quasi-equilibrium distribution of the Markov chain (see for example [40,13] and [121,Section 4.7]). If the state space is multi-dimensional, then we can use an appropriate bijection, as discussed in the following paragraph, which makes Q C a square matrix.…”