559 research outputs found
Epidemiological models with parametric heterogeneity: Deterministic theory for closed populations
We present a unified mathematical approach to epidemiological models with
parametric heterogeneity, i.e., to the models that describe individuals in the
population as having specific parameter (trait) values that vary from one
individuals to another. This is a natural framework to model, e.g.,
heterogeneity in susceptibility or infectivity of individuals. We review, along
with the necessary theory, the results obtained using the discussed approach.
In particular, we formulate and analyze an SIR model with distributed
susceptibility and infectivity, showing that the epidemiological models for
closed populations are well suited to the suggested framework. A number of
known results from the literature is derived, including the final epidemic size
equation for an SIR model with distributed susceptibility. It is proved that
the bottom up approach of the theory of heterogeneous populations with
parametric heterogeneity allows to infer the population level description,
which was previously used without a firm mechanistic basis; in particular, the
power law transmission function is shown to be a consequence of the initial
gamma distributed susceptibility and infectivity. We discuss how the general
theory can be applied to the modeling goals to include the heterogeneous
contact population structure and provide analysis of an SI model with
heterogeneous contacts. We conclude with a number of open questions and
promising directions, where the theory of heterogeneous populations can lead to
important simplifications and generalizations.Comment: 26 pages, 6 figures, submitted to Mathematical Modelling of Natural
Phenomen
Adaptive Fitness Landscape for Replicator Systems: To Maximize or not to Maximize
Sewall Wright's adaptive landscape metaphor penetrates a significant part of
evolutionary thinking. Supplemented with Fisher's fundamental theorem of
natural selection and Kimura's maximum principle, it provides a unifying and
intuitive representation of the evolutionary process under the influence of
natural selection as the hill climbing on the surface of mean population
fitness. On the other hand, it is also well known that for many more or less
realistic mathematical models this picture is a sever misrepresentation of what
actually occurs. Therefore, we are faced with two questions. First, it is
important to identify the cases in which adaptive landscape metaphor actually
holds exactly in the models, that is, to identify the conditions under which
system's dynamics coincides with the process of searching for a (local) fitness
maximum. Second, even if the mean fitness is not maximized in the process of
evolution, it is still important to understand the structure of the mean
fitness manifold and see the implications of this structure on the system's
dynamics. Using as a basic model the classical replicator equation, in this
note we attempt to answer these two questions and illustrate our results with
simple well studied systems.Comment: 13 pages, 4 figure
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