JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org. This content downloaded from 165.123.34.86 on Wed, 17 Jun 2015 18:34:14 UTC All use subject to JSTOR Terms and Conditions Editor's Note Invasions of new territory-by plants, animals, or genes-are an old topic in ecology, but hardly obsolete: the spread of pests such as the gypsy moth, exotic plants, recurrent and emerging infectious diseases, and genetically engineered organisms are important contemporary problems for ecology. For nearly half a . . ~~century, reaction-diffusion models have been the main analytic framework for Emphasizing9 modeling spatial spread, in part because of the well-developed mathematical theory new ideas that tells us how to compute things like the long-term rate of spread and the conditions for spatial pattern formation. In this paper, we are given the tools to to study the rate of spread for invading organisms in a very different kind of model, integrodifference equations. Unlike diffusion equations, these models can accomstimulate research modate leptokurtic (broad-tailed) dispersal patterns, and in such cases they can exhibit the accelerating rates of spread that have been observed in some invasions.in ecology Of course this does not mean that we should stop using diffusion models, but it gives us an alternative with a different set of assumptions that is likely to be more accurate when the dispersal pattern of individuals is far from the Gaussian distribution implicit in a diffusion model. AbstractModels that describe the spread of invading organisms often assume that the dispersal distances of propagules are normally distributed. In contrast, measured dispersal curves are typically leptokurtic, not normal. In this paper, we consider a class of models, integrodifference equations, that directly incorporate detailed dispersal data as well as population growth dynamics. We provide explicit formulas for the speed of invasion for compensatory growth and for different choices of the propagule redistribution kernel and apply these formulas to the spread of D. pseudoobscura. We observe that: (1) the speed of invasion of a spreading population is extremely sensitive to the precise shape of the redistribution kernel and, in particular, to the tail of the distribution; (2) fat-tailed kernels can generate accelerating invasions rather than constant-speed travelling waves; (3) normal redistribution kernels (and by inference, many reaction-diffusion models) may grossly underestimate rates of spread of invading populations in comparison with models that incorporate more realistic leptokurtic distributions; and (4) the relative superiority of different redistribution kernels depends, in general, on the precise magnitude of the net reproductive rate. The addition of...
This primer article focuses on the basic reproduction number, normalℛ0, for infectious diseases, and other reproduction numbers related to normalℛ0 that are useful in guiding control strategies. Beginning with a simple population model, the concept is developed for a threshold value of normalℛ0 determining whether or not the disease dies out. The next generation matrix method of calculating normalℛ0 in a compartmental model is described and illustrated. To address control strategies, type and target reproduction numbers are defined, as well as sensitivity and elasticity indices. These theoretical ideas are then applied to models that are formulated for West Nile virus in birds (a vector-borne disease), cholera in humans (a disease with two transmission pathways), anthrax in animals (a disease that can be spread by dead carcasses and spores), and Zika in humans (spread by mosquitoes and sexual contacts). Some parameter values from literature data are used to illustrate the results. Finally, references for other ways to calculate normalℛ0 are given. These are useful for more complicated models that, for example, take account of variations in environmental fluctuation or stochasticity.
Two systematic methods are presented to guide the construction of Lyapunov functions for general infectious disease models and are thus applicable to establish their global dynamics. Specifically, a matrix-theoretic method using the Perron eigenvector is applied to prove the global stability of the disease-free equilibrium, while a graph-theoretic method based on Kirchhoff's matrix tree theorem and two new combinatorial identities are used to prove the global stability of the endemic equilibrium. Several disease models in the literature and two new cholera models are used to demonstrate the applications of these methods.
Severe acute respiratory syndrome (SARS), a new, highly contagious, viral disease, emerged in China late in 2002 and quickly spread to 32 countries and regions causing in excess of 774 deaths and 8098 infections worldwide. In the absence of a rapid diagnostic test, therapy or vaccine, isolation of individuals diagnosed with SARS and quarantine of individuals feared exposed to SARS virus were used to control the spread of infection. We examine mathematically the impact of isolation and quarantine on the control of SARS during the outbreaks in Toronto, Hong Kong, Singapore and Beijing using a deterministic model that closely mimics the data for cumulative infected cases and SARS-related deaths in the first three regions but not in Beijing until mid-April, when China started to report data more accurately. The results reveal that achieving a reduction in the contact rate between susceptible and diseased individuals by isolating the latter is a critically important strategy that can control SARS outbreaks with or without quarantine. An optimal isolation programme entails timely implementation under stringent hygienic precautions defined by a critical threshold value. Values below this threshold lead to control, but those above are associated with the incidence of new community outbreaks or nosocomial infections, a known cause for the spread of SARS in each region. Allocation of resources to implement optimal isolation is more effective than to implement sub-optimal isolation and quarantine together. A community-wide eradication of SARS is feasible if optimal isolation is combined with a highly effective screening programme at the points of entry.
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