An epidemic in a dynamic population with importation of infectives
Consider a large uniformly mixing dynamic population, which has constant birth rate and exponentially distributed lifetimes, with mean population size $n$. A Markovian SIR (susceptible $\to$ infective $\to$ recovered) infectious disease, having importation of infectives, taking place in this popula...
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| Format: | Article |
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Institute of Mathematical Statistics
2017
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| Online Access: | https://eprints.nottingham.ac.uk/34284/ |
| _version_ | 1848794816490504192 |
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| author | Ball, Frank Britton, Tom Trapman, Pieter |
| author_facet | Ball, Frank Britton, Tom Trapman, Pieter |
| author_sort | Ball, Frank |
| building | Nottingham Research Data Repository |
| collection | Online Access |
| description | Consider a large uniformly mixing dynamic population, which has constant birth rate and exponentially distributed lifetimes, with mean population size $n$. A Markovian SIR (susceptible $\to$ infective $\to$ recovered) infectious disease, having importation of infectives, taking place in this population is analysed. The main situation treated is where $n\to\infty$, keeping the basic reproduction number $R_0$ as well as the importation rate of infectives fixed, but assuming that the quotient of the average infectious period and the average lifetime tends to 0 faster than $1/\log n$. It is shown that, as $ n \to \infty$, the behaviour of the 3-dimensional process describing the evolution of the fraction of the population that are susceptible, infective and recovered, is encapsulated in a 1-dimensional regenerative process $S=\{ S(t);t\ge 0\}$ describing the limiting fraction of the population that are susceptible. The process $S$ grows deterministically, except at one random time point per regenerative cycle, where it jumps down by a size that is completely determined by the waiting time since the previous jump. Properties of the process $S$, including the jump size and stationary distributions, are determined. |
| first_indexed | 2025-11-14T19:22:12Z |
| format | Article |
| id | nottingham-34284 |
| institution | University of Nottingham Malaysia Campus |
| institution_category | Local University |
| last_indexed | 2025-11-14T19:22:12Z |
| publishDate | 2017 |
| publisher | Institute of Mathematical Statistics |
| recordtype | eprints |
| repository_type | Digital Repository |
| spelling | nottingham-342842020-05-04T18:36:48Z https://eprints.nottingham.ac.uk/34284/ An epidemic in a dynamic population with importation of infectives Ball, Frank Britton, Tom Trapman, Pieter Consider a large uniformly mixing dynamic population, which has constant birth rate and exponentially distributed lifetimes, with mean population size $n$. A Markovian SIR (susceptible $\to$ infective $\to$ recovered) infectious disease, having importation of infectives, taking place in this population is analysed. The main situation treated is where $n\to\infty$, keeping the basic reproduction number $R_0$ as well as the importation rate of infectives fixed, but assuming that the quotient of the average infectious period and the average lifetime tends to 0 faster than $1/\log n$. It is shown that, as $ n \to \infty$, the behaviour of the 3-dimensional process describing the evolution of the fraction of the population that are susceptible, infective and recovered, is encapsulated in a 1-dimensional regenerative process $S=\{ S(t);t\ge 0\}$ describing the limiting fraction of the population that are susceptible. The process $S$ grows deterministically, except at one random time point per regenerative cycle, where it jumps down by a size that is completely determined by the waiting time since the previous jump. Properties of the process $S$, including the jump size and stationary distributions, are determined. Institute of Mathematical Statistics 2017-03-06 Article PeerReviewed Ball, Frank, Britton, Tom and Trapman, Pieter (2017) An epidemic in a dynamic population with importation of infectives. Annals of Applied Probability, 27 (1). pp. 242-274. ISSN 1050-5164 Branching process Regenerative process SIR epidemic Skorohod metric Weak convergence http://projecteuclid.org/euclid.aoap/1488790828 doi:10.1214/16-AAP1203 doi:10.1214/16-AAP1203 |
| spellingShingle | Branching process Regenerative process SIR epidemic Skorohod metric Weak convergence Ball, Frank Britton, Tom Trapman, Pieter An epidemic in a dynamic population with importation of infectives |
| title | An epidemic in a dynamic population with importation of infectives |
| title_full | An epidemic in a dynamic population with importation of infectives |
| title_fullStr | An epidemic in a dynamic population with importation of infectives |
| title_full_unstemmed | An epidemic in a dynamic population with importation of infectives |
| title_short | An epidemic in a dynamic population with importation of infectives |
| title_sort | epidemic in a dynamic population with importation of infectives |
| topic | Branching process Regenerative process SIR epidemic Skorohod metric Weak convergence |
| url | https://eprints.nottingham.ac.uk/34284/ https://eprints.nottingham.ac.uk/34284/ https://eprints.nottingham.ac.uk/34284/ |