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Journal ArticleDOI

Spatiotemporal dynamics of an ecoepidemiological model with standard incidence

01 Jun 2007-Journal of Mechanics in Medicine and Biology (World Scientific Publishing Company)-Vol. 07, Iss: 02, pp 175-197
TL;DR: To model the gestation lag of the predator species and the spatially heterogeneous characteristics of an ecological population, the concept of diffusionally coupled delay into the system is incorporated and bifurcation behavior of the delayed homogeneous system is studied.
Abstract: We analyze a mathematical model of predator–prey interaction where the prey population is infected with a viral disease. Infection in the prey population is assumed to follow standard incidence. The dynamical behavior of the system is studied in terms of stability aspects. To model the gestation lag of the predator species and the spatially heterogeneous characteristics of an ecological population, we incorporate the concept of diffusionally coupled delay into the system. The bifurcation behavior of the delayed homogeneous system is studied. The existence of traveling wave solutions for the delay–diffusion model is established. Numerical simulations are performed to justify analytical findings.
References
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Journal ArticleDOI
TL;DR: Threshold theorems involving the basic reproduction number, the contact number, and the replacement number $R$ are reviewed for classic SIR epidemic and endemic models and results with new expressions for $R_{0}$ are obtained for MSEIR and SEIR endemic models with either continuous age or age groups.
Abstract: Many models for the spread of infectious diseases in populations have been analyzed mathematically and applied to specific diseases. Threshold theorems involving the basic reproduction number $R_{0}$, the contact number $\sigma$, and the replacement number $R$ are reviewed for the classic SIR epidemic and endemic models. Similar results with new expressions for $R_{0}$ are obtained for MSEIR and SEIR endemic models with either continuous age or age groups. Values of $R_{0}$ and $\sigma$ are estimated for various diseases including measles in Niger and pertussis in the United States. Previous models with age structure, heterogeneity, and spatial structure are surveyed.

5,915 citations

Book
21 Aug 1973
TL;DR: Preface vii Preface to the Second Edition Biology Edition 1.
Abstract: Preface vii Preface to the Second Edition Biology Edition 1. Intoduction 3 2. Mathematical Models and Stability 13 3. Stability versus Complexity in Multispecies Models 4. Models with Few Species: Limit Cycles and Time Delays 79 5. Randomly Fluctuating Environments 109 6. Niche Overlap and Limiting Similarity 139 7. Speculations 172 Appendices 187 Afterthoughts for the Second Edition 211 Bibliography to Afterthoghts 234 Bibliography 241 Author Index 259 Subject Index 263

5,083 citations

Book
02 Feb 2012
TL;DR: Delay Differential Equations as mentioned in this paper are a generalization of delay differential equations and have been used in a variety of applications in population dynamics, such as global stability for single species models and multi-species models.
Abstract: Delay Differential Equations: Introduction. Basic Theory of Delay Differential Equations. Characteristic Equations. Applications in Population Dynamics: Global Stability for Single Species Models. Periodic Solutions, Chaos, Stage Structures, And State Dependent Delays in Single Species Models. Global Stability for Multi-Species Models. Periodic Solutions in Multi-Species Models. Permanence. Neutral Delay Models. References. Appendix. Index.

3,192 citations

Book
01 Jan 1981
TL;DR: The Hopf Bifurcation Theorum has been used in many applications, such as Differential Difference and Integro-differential Equations (by hand).
Abstract: 1. The Hopf Bifurcation Theorum 2. Applications: Ordinary Differential Equations (by hand) 3. Numerical Evaluation of Hopf Bifurcation Formulae 4. Applications: Differential-Difference and Integro-differential Equations (by hand) 5. Applications: Partial Differential Equations (by hand).

2,090 citations