A stochastic SIS epidemic model with heterogeneous contacts

Abstract

A stochastic model for the spread of an SIS epidemic among a population consisting of N individuals, each having heterogeneous infectiousness and/or susceptibility, is considered and its behavior is analyzed under the practically relevant situation when N is small. The model is formulated as a finite timehomogeneous continuous-time Markov chain X . Based on an appropriate labeling of states, we first construct its infinitesimal rate matrix by using an iterative argument, and we then present an algorithmic procedure for computing steadystate measures, such as the number of infected individuals, the length of an outbreak, the maximum number of infectives, and the number of infections suffered by a marked individual during an outbreak. The time till the epidemic extinction is characterized as a phase-type random variable when there is no external source of infection, and its Laplace-Stieljtes transform and moments are derived in terms of a forward elimination backward substitution solution. The inverse iteration method is applied to the quasi-stationary distribution of X , which provides a good approximation of the process X at a certain time, conditional on non-extinction, after a suitable waiting time. The basic reproduction number R 0 is defined here as a random variable, rather than an expected value.

Stochastic descriptors in an SIR epidemic model for heterogeneous individuals in small networks

“…Since we are interested in the dynamics of the epidemic until no infected individual remains in the population (see Section 3), these are the cases in which our analysis is consistent; we refer the reader to Section 5 for some discussion regarding different situations. In order to analyse X , it is necessary to give some structure to S. It has been recently shown the convenience of grouping states x ∈ S in terms of the number of infected, susceptible or recovered individuals within x [13,39].…”

“…We point out here that the approach followed in this paper complements the work initiated in [13], where the application of a matrix formalism together with the usage of Laplace-Stieltjes transforms, probability generating functions, phase-type distributions [24,Chapter 2], level-dependent quasi-birthand-death processes [24,Chapter 12] and auxiliary absorbing Markov chains are used when analysing a SIS epidemic model. Moreover, the particular application of this approach to the spread of the syndrome Acute coryza among the members of a family is studied in [13] by using empirical data of [9].…”

“…Moreover, the particular application of this approach to the spread of the syndrome Acute coryza among the members of a family is studied in [13] by using empirical data of [9]. The matrix formalism followed in [13] is possible by means of considering a proper order of states within the space of states of the corresponding CTMC, which is similar to the order proposed in [39].…”

“…Moreover, the particular application of this approach to the spread of the syndrome Acute coryza among the members of a family is studied in [13] by using empirical data of [9]. The matrix formalism followed in [13] is possible by means of considering a proper order of states within the space of states of the corresponding CTMC, which is similar to the order proposed in [39]. An alternative order when analysing the dynamics of the SIS epidemic model is the one considered in [42], which yields a particular fractal structure for the infinitesimal generator of the CTMC (see [42, Figures 2 and 3]), for which computational exploitation is an open question [42].…”

“…

Abstract We continue here the work initiated in [13], and analyse an SIR epidemic model for the spread of an epidemic among the members of a small population of N individuals, defined in terms of a continuous-time Markov chain X . We propose a structure by levels and sub-levels of the state space of the process X , and present two different orders, Orders A and B, for states within each sub-level, which are related to a matrix and a scalar formalism, respectively, when developing our analysis.

…”