Persistence in a discrete-time, stage-structured epidemic model

Salceanu, Paul L.; Smith, Hal L.

doi:10.1080/10236190802400733

Cited by 21 publications

(16 citation statements)

References 8 publications

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“…We use only elementary arguments rather than appealing to the multiplicative ergodic theorem [1,2,4,15]. This extends our earlier work in [11,12,13] which covered only the discrete case.…”

Section: Introductionmentioning

confidence: 48%

Lyapunov Exponents and Uniform Weak Normally Repelling Invariant Sets

Salceanu

Smith

2009

Positive Systems

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Abstract. Let M be a compact invariant set contained in a boundary hyperplane of the positive orthant of R n for a discrete or continuous time dynamical system defined on the positive orthant. Using elementary arguments, we show that M is uniformly weakly repelling in directions normal to the boundary in which M resides provided all normal Lyapunov exponents are positive. This result is useful in establishing uniform persistence of the dynamics.

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“…We use only elementary arguments rather than appealing to the multiplicative ergodic theorem [1,2,4,15]. This extends our earlier work in [11,12,13] which covered only the discrete case.…”

Section: Introductionmentioning

confidence: 48%

Lyapunov Exponents and Uniform Weak Normally Repelling Invariant Sets

Salceanu

Smith

2009

Positive Systems

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“…We mention that our next three results (i.e., Lemma 2.4 and Theorems 2.5 and 2.6) are similar, respectively, to those given in [10], Lemma 3.1, Theorems 4.3 and 4.4; thus we encourage the reader to find the analogous proofs in the above mentioned reference. The relationship between K (see (3)) and r(A K ), K = S, I , is given in the following Lemma.…”

Section: Resultsmentioning

confidence: 68%

“…By neglecting the larval stage of the full model, which we did here, we were able to obtain somewhat better sufficient conditions for this global convergence to hold in Theorem 2.3 than in the corresponding result in [10]. Strictly speaking, the results are not comparable but here we were not forced to assume either that c JS vanishes or that the system is monotone [12].…”

Section: Discussionmentioning

confidence: 74%

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Persistence in a discrete-time stage-structured fungal disease model

Salceanu

Smith

2009

Journal of Biological Dynamics

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A discrete-time susceptible and infected (SI) epidemic model, with less than 100% vertical disease transmission, for the spread of a fungal disease in a structured amphibian host population, is analysed. Criteria for persistence of the population as well as the disease are established. Stability results for host extinction and for the disease-free equilibrium are presented. Bifurcation theory is used to establish existence of an endemic equilibrium.

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“…The models will be useful to further study populations in areas where they are threatened or endangered. A variety of mathematical models have been developed to study the impact of the environment or disease on small or declining populations of amphibians [3,7,15,16,24,25,30,31]. We will summarize these models and discuss how they relate to our modelling efforts.…”

Section: Introductionmentioning

confidence: 99%

Stability and permanence in gender- and stage-structured models for the boreal toad

Mallawaarachchi¹,

Allen²,

Carey³

2011

Journal of Biological Dynamics

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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 depends on its life cycle and breeding strategies. In addition, an epizootic model for the spread of Bd is formulated. Analysis of these models provides two thresholds. The first threshold, the basic reproduction number for the population, ℛ(0), determines whether the population persists and the second threshold, the basic reproduction number for the fungal disease, ℛ(F), determines whether the disease persists. If ℛ(0)>1 and ℛ(F)<1, then the population becomes disease-free. However, if both thresholds are greater than one, the population levels are severely reduced by the fungal pathogen.

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Persistence in a discrete-time, stage-structured epidemic model

Cited by 21 publications

References 8 publications

Lyapunov Exponents and Uniform Weak Normally Repelling Invariant Sets

Lyapunov Exponents and Uniform Weak Normally Repelling Invariant Sets

Persistence in a discrete-time stage-structured fungal disease model

Stability and permanence in gender- and stage-structured models for the boreal toad

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