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- L J Allen, A M Burgin
- Mathematical biosciences
- 2000

The dynamics of deterministic and stochastic discrete-time epidemic models are analyzed and compared. The discrete-time stochastic models are Markov chains, approximations to the continuous-time models. Models of SIS and SIR type with constant population size and general force of infection are analyzed, then a more general SIS model with variable population… (More)

- L J Allen
- Mathematical biosciences
- 1994

Discrete-time models, or difference equations, of some well-known SI, SIR, and SIS epidemic models are considered. The discrete-time SI and SIR models give rise to systems of nonlinear difference equations that are similar in behavior to their continuous analogues under the natural restriction that solutions to the discrete-time models be positive. It is… (More)

- Linda J S Allen, Michel Langlais, Carleton J Phillips
- Mathematical biosciences
- 2003

An SI epidemic model for a host with two viral infections circulating within the population is developed, analyzed, and numerically simulated. The model is a system of four differential equations which includes a state for susceptible individuals, two states for individuals infected with a single virus, one which is vertically transmitted and the other… (More)

- Linda J S Allen, Glenn E Lahodny
- Journal of biological dynamics
- 2012

The basic reproduction number, ℛ(0), one of the most well-known thresholds in deterministic epidemic theory, predicts a disease outbreak if ℛ(0)>1. In stochastic epidemic theory, there are also thresholds that predict a major outbreak. In the case of a single infectious group, if ℛ(0)>1 and i infectious individuals are introduced into a susceptible… (More)

- L J S Allen, P van den Driessche
- Mathematical biosciences
- 2013

Thresholds for disease extinction provide essential information for control, eradication or management of diseases. Through relations between branching process theory and the corresponding deterministic model, it is shown that the deterministic and stochastic thresholds are in agreement for discrete-time and continuous-time infectious disease models with… (More)

- D K Mallawaarachchi, Linda J S Allen, Cynthia Carey
- Journal of biological dynamics
- 2011

The boreal toad Bufo boreas boreas, once common in the western USA, is listed as an endangered species in Colorado and New Mexico, and protected in Wyoming. Populations have dramatically declined due to the presence of the fungal pathogen Batrachochytrium dendrobatidis (Bd). A gender- and stage-structured model for the boreal toad is formulated which… (More)

- Nadarajah Kirupaharan, Linda J S Allen
- Bulletin of mathematical biology
- 2004

Stochastic differential equations that model an SIS epidemic with multiple pathogen strains are derived from a system of ordinary differential equations. The stochastic model assumes there is demographic variability. The dynamics of the deterministic model are summarized. Then the dynamics of the stochastic model are compared to the deterministic model. In… (More)

- L J Allen, M A Jones, C F Martin
- Mathematical biosciences
- 1991

A discrete-time, age-independent SIR-type epidemic model is formulated and analyzed. The effects of vaccination are also included in the model. Three mathematically important properties are verified for the model: solutions are nonnegative, the population size is time-invariant, and the epidemic concludes with all individuals either remaining susceptible or… (More)

- Curtis L Wesley, Linda J S Allen
- Journal of biological dynamics
- 2009

Patterns of contact in social behaviour and seasonality due to environmental influences often affect the spread and persistence of diseases. Models of epidemics with seasonality and patterns in the contact rate include time-periodic coefficients, making the systems nonautonomous. No general method exists for calculating the basic reproduction number, the… (More)

- Linda J S Allen, P van den Driessche
- Mathematical biosciences and engineering : MBE
- 2006

Two new stochastic epidemic models, a continuous-time Markov chain model and a stochastic differential equation model, are formulated. These are based on a deterministic model that includes vaccination and is applicable to pertussis. For some parameter values, the deterministic model exhibits a backward bifurcation if the vaccine is imperfect. Thus a region… (More)