On expected durations of birth-death processes with applications to branching processes and SIS epidemics
We study continuous-time birth–death type processes, where individuals have independent and identically distributed lifetimes, according to a random variable Q, with E[Q] = 1, and where the birth rate if the population is currently in state (has size) n is α(n). We focus on two important examples, n...
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| Format: | Article |
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Applied Probability Trust
2016
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| Online Access: | https://eprints.nottingham.ac.uk/34191/ |
| _version_ | 1848794795016716288 |
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| author | Ball, Frank Britton, Tom Neal, Peter |
| author_facet | Ball, Frank Britton, Tom Neal, Peter |
| author_sort | Ball, Frank |
| building | Nottingham Research Data Repository |
| collection | Online Access |
| description | We study continuous-time birth–death type processes, where individuals have independent and identically distributed lifetimes, according to a random variable Q, with E[Q] = 1, and where the birth rate if the population is currently in state (has size) n is α(n). We focus on two important examples, namely α(n) = λn being a branching process, and α(n) = λn(N −n)/N which corresponds to an SIS (susceptible → infective → susceptible) epidemic model in a homogeneously mixing community of fixed size N. The processes are assumed to start with a single individual, i.e. in state 1. Let T , An, C, and S denote the (random) time to extinction, the total time spent in state n, the total number of individuals ever alive, and the sum of the lifetimes of all individuals in the birth–death process, respectively. We give expressions for the expectation of all these quantities and show that these expectations are insensitive to the distribution of Q. We also derive an asymptotic expression for the expected time to extinction of the SIS epidemic, but now starting at the endemic state, which is not independent of the distribution of Q. The results are also applied to the household SIS epidemic, showing that, in contrast to the household SIR (susceptible → infective → recovered) epidemic, its threshold parameter R∗ is insensitive to the distribution of Q. |
| first_indexed | 2025-11-14T19:21:52Z |
| format | Article |
| id | nottingham-34191 |
| institution | University of Nottingham Malaysia Campus |
| institution_category | Local University |
| last_indexed | 2025-11-14T19:21:52Z |
| publishDate | 2016 |
| publisher | Applied Probability Trust |
| recordtype | eprints |
| repository_type | Digital Repository |
| spelling | nottingham-341912020-05-04T17:40:38Z https://eprints.nottingham.ac.uk/34191/ On expected durations of birth-death processes with applications to branching processes and SIS epidemics Ball, Frank Britton, Tom Neal, Peter We study continuous-time birth–death type processes, where individuals have independent and identically distributed lifetimes, according to a random variable Q, with E[Q] = 1, and where the birth rate if the population is currently in state (has size) n is α(n). We focus on two important examples, namely α(n) = λn being a branching process, and α(n) = λn(N −n)/N which corresponds to an SIS (susceptible → infective → susceptible) epidemic model in a homogeneously mixing community of fixed size N. The processes are assumed to start with a single individual, i.e. in state 1. Let T , An, C, and S denote the (random) time to extinction, the total time spent in state n, the total number of individuals ever alive, and the sum of the lifetimes of all individuals in the birth–death process, respectively. We give expressions for the expectation of all these quantities and show that these expectations are insensitive to the distribution of Q. We also derive an asymptotic expression for the expected time to extinction of the SIS epidemic, but now starting at the endemic state, which is not independent of the distribution of Q. The results are also applied to the household SIS epidemic, showing that, in contrast to the household SIR (susceptible → infective → recovered) epidemic, its threshold parameter R∗ is insensitive to the distribution of Q. Applied Probability Trust 2016-03-24 Article PeerReviewed Ball, Frank, Britton, Tom and Neal, Peter (2016) On expected durations of birth-death processes with applications to branching processes and SIS epidemics. Journal of Applied Probability, 53 (1). pp. 203-215. ISSN 0021-9002 Birth–death process; branching processes; SIS epidemics; insensitivity results http://journals.cambridge.org/abstract_S0021900215000194 doi:10.1017/jpr.2015.19 doi:10.1017/jpr.2015.19 |
| spellingShingle | Birth–death process; branching processes; SIS epidemics; insensitivity results Ball, Frank Britton, Tom Neal, Peter On expected durations of birth-death processes with applications to branching processes and SIS epidemics |
| title | On expected durations of birth-death processes with applications to branching processes and SIS epidemics |
| title_full | On expected durations of birth-death processes with applications to branching processes and SIS epidemics |
| title_fullStr | On expected durations of birth-death processes with applications to branching processes and SIS epidemics |
| title_full_unstemmed | On expected durations of birth-death processes with applications to branching processes and SIS epidemics |
| title_short | On expected durations of birth-death processes with applications to branching processes and SIS epidemics |
| title_sort | on expected durations of birth-death processes with applications to branching processes and sis epidemics |
| topic | Birth–death process; branching processes; SIS epidemics; insensitivity results |
| url | https://eprints.nottingham.ac.uk/34191/ https://eprints.nottingham.ac.uk/34191/ https://eprints.nottingham.ac.uk/34191/ |