Extinction and Quasi-Stationarity in the Stochastic Logistic SIS Model

Frontmatter

Chapter 1. Introduction

Abstract

This monograph is devoted to an analysis of a classical mathematical model in population biology, known as the stochastic logistic SIS model. It serves as a model both for the spread of an infection that gives no immunity and for density dependent population growth, and it also appears as an important special case of a contact process that accounts for spatial influences.

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Chapter 2. Model Formulation

Abstract

The mathematical model dealt with in this monograph is referred to as a stochastic logistic SIS model. In this chapter we show that there are many logistic models, and that the SIS model is just one of them, and probably the simplest one.

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Chapter 3. Stochastic Process Background

Abstract

This chapter contains some stochastic process background that is of value for the study of the SIS model. In the first seven sections of the chapter we give results for a much larger class of stochastic processes than the specific SIS model, while the last two sections deal with the SIS model. The results in the first seven sections are expected to be of value in the study of quasi-stationarity for other stochastic logistic models than the SIS model that is the main object of study throughout the monograph.

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Chapter 4. The SIS Model: First Approximations of the Quasi-stationary Distribution

Abstract

The preceding chapter has given a number of results for a class of stochastic models containing the SIS model. We shall use these results in the chapters that follow, where we gradually derive results that lead up to approximations of the quasi-stationary distribution.

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Chapter 5. Some Approximations Involving the Normal Distribution

Abstract

Approximations play a central role in this monograph. The subject area of asymptotic analysis is therefore important. It contains a number of powerful results and ideas. A classic reference to this field is the book by Olver [54].

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Chapter 6. Preparations for the Study of the Stationary Distribution p(1) of the SIS Model

Abstract

We devote the present chapter to derivations of results that are needed in the next chapter, where we deal with approximations of the stationary distribution p ⁽¹⁾ of the auxiliary process \(\{X^{(1)}(t)\}\) of the SIS model.

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Chapter 7. Approximation of the Stationary Distribution p(1) of the SIS Model

Abstract

We use this chapter to derive approximations of the stationary distribution p ⁽¹⁾, while we deal with the stationary distribution p ⁽⁰⁾ in Chap. 9. The reason for dealing with p ⁽¹⁾ first is that it is the easy one, and that we actually use some of the results derived for p ⁽¹⁾ in the derivation of approximations for p ⁽⁰⁾.

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Chapter 8. Preparations for the Study of the Stationary Distribution p(0) of the SIS Model

Abstract

This chapter prepares for the approximations of the stationary distribution p ⁽⁰⁾ of the SIS model that are derived in Chap. 9.

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Chapter 9. Approximation of the Stationary Distribution p(0) of the SIS Model

Abstract

The stationary distribution p⁽⁰⁾ is by (3.7) determined from the

$$ p_n^{(0)}=\frac{\pi n}{{\sum_{n=1}^N}\pi_n},\,\,\,n=1,2,\ldots,N. $$

(9.1)

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Chapter 10. Approximation of Some Images Under Ψ for the SIS Model

Abstract

It was mentioned in Chap. 3, Sect. 3.6, that Ferrari et al. [27] have proved an important result concerning the map \( \Psi \) namely that if \( \nu \) is an arbitrary discrete distribution with the state space \( \{1,2,\ldots,N \}\), then the sequence \( \nu,\Psi(\nu),\Psi^{2}(\nu),\ldots,\)converges to the quasi-stationary distribution q.

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Chapter 11. Approximation of the Quasi-stationary Distribution q of the SIS Model

Abstract

The quasi-stationary distribution q has a central position in our study of the stochastic SIS model. We give approximations of this distribution in each of the three parameter regions in this chapter. As has been mentioned before, we actually work with two kinds of approximation of quite different type. The first type of approximation is a stationary distribution of some related process.

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Chapter 12. Approximation of the Time to Extinction for the SIS Model

Abstract

We noted in Sect. 3.4, (3.42), that the time to extinction \( \tau_{\rm q}\) from quasi-stationarity for a birth–death process with finite state space and with an absorbing state at the origin has an exponential distribution with expectation equal to \( \rm E{\tau}_{Q}= 1 /(\mu_{1}\rm q_{1}).\)

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Chapter 13. Uniform Approximations for the SIS Model

Abstract

Our main concern in this monograph is with the quasi-stationary distribution and the time to extinction for the stochastic logistic SIS model. Explicit expressions are not available for these quantities. Furthermore, they behave in qualitatively different ways in three different parameter regions.

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Chapter 14. Thresholds for the SIS Model

Abstract

As mentioned in the introduction, threshold concepts are the most powerful results that mathematical modelling can contribute to theoretical epidemiology. They lead to threshold functions that partition the parameter space into subsets in which the model solutions behave in qualitatively different ways. Early threshold results are all based on deterministic models. For the deterministic version of the SIS model, one studies the proportion of infected individuals as time approaches infinity.

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Chapter 15. Concluding Comments

Abstract

The story about the the extinction time and the quasi-stationary distribution of the SIS model has come to an end. The final results are given in terms of uniform approximations of the quasi-stationary distribution in Theorem 13.3, and uniform approximations of the expected times to extinction from the state one and from quasi-stationarity in Theorem 13.4. We remind the reader that these final approximations are uniform across the three parameter regions in which qualitatively different behaviors are observed.

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