Bulletin of Mathematical Biology

Lof, Marjolein; Etienne, Rampal; Powell, James; Gee, Maarten; Hemerik, Lia

doi: 10.1007/s11538-008-9327-0pmid: 18780001

Animal aggregation is a general phenomenon in ecological systems. Aggregations are generally considered as an evolutionary advantageous state in which members derive the benefits of protection and mate choice, balanced by the costs of limiting resources and competition. In insects, chemical information conveyance plays an important role in finding conspecifics and forming aggregations. In this study, we describe a spatio-temporal simulation model designed to explore and quantify the effects of these infochemicals, i.e., food odors and an aggregation pheromone, on the spatial distribution of a fruit fly (Drosophila melanogaster) population, where the lower and upper limit of local population size are controlled by an Allee effect and competition. We found that during the spatial expansion and strong growth of the population, the use of infochemicals had a positive effect on population size. The positive effects of reduced mortality at low population numbers outweighed the negative effects of increased mortality due to competition. At low resource densities, attraction toward infochemicals also had a positive effect on population size during recolonization of an area after a local population crash, by decreasing the mortality due to the Allee effect. However, when the whole area was colonized and the population was large, the negative effects of competition on population size were larger than the positive effects of the reduction in mortality due to the Allee effect. The use of infochemicals thus has mainly positive effects on population size and population persistence when the population is small and during the colonization of an area.

Gee, Maarten; Lof, Marjolein; Hemerik, Lia

doi: 10.1007/s11538-008-9329-ypmid: 18780000

In a companion paper (Lof et al., in Bull. Math. Biol., 2008), we describe a spatio-temporal model for insect behavior. This model includes chemical information for finding resources and conspecifics. As a model species, we used Drosophila melanogaster, because its behavior is documented comparatively well.

Brauer, Fred

doi: 10.1007/s11538-008-9326-1pmid: 18663538

We consider a two-group epidemic model with treatment and establish a final size relation that gives the extent of the epidemic. This relation can be established with arbitrary mixing between the groups even though it may not be feasible to determine the reproduction number for the model. If the mixing of the two groups is proportionate, there is an explicit expression for the reproductive number and the final size relation is expressible in terms of the components of the reproduction number. We also extend the results to a two-group influenza model with proportionate mixing. Some numerical simulations suggest that (i) the assumption of no disease deaths is a good approximation if the disease death rate is small and (ii) a one-group model is a close approximation to a two-group model but a two-group model is necessary for comparing targeted treatment strategies.

Zhao, Ruijun; Milner, Fabio

doi: 10.1007/s11538-008-9330-5pmid: 18668296

We describe and analyze a mathematical model for schistosomiasis in which infected snails are distinguished from susceptible through increased mortality and no reproduction. We based the model on the same derivation as Anderson and May (J. Anim. Ecol. 47:219–247, 1978), Feng and Milner (A New Mathematical Model of Schistosomiasis, Mathematical Models in Medical and Health Science, Nashville, TN, 1997. Innov. Appl. Math., Vanderbilt Univ. Press, Nashville, pp. 117–128, 1998), and May and Anderson (J. Anim. Ecol. 47:249–267, 1978), but used logistic growth both in human and snail hosts. We introduce a parameter r, the effective coverage of medical treatment/prevention to control the infection. We determine a reproductive number for the disease directly related to its persistence and extinction. Finally, we obtain a critical value for r that indicates the minimum treatment effort needed in order to clear out the disease from the population.

Grünewald, S.; Huber, K.; Wu, Q.

doi: 10.1007/s11538-008-9331-4pmid: 18665426

A contemporary and fundamental problem faced by many evolutionary biologists is how to puzzle together a collection ℘ of partial trees (leaf-labeled trees whose leaves are bijectively labeled by species or, more generally, taxa, each supported by, e.g., a gene) into an overall parental structure that displays all trees in ℘. This already difficult problem is complicated by the fact that the trees in ℘ regularly support conflicting phylogenetic relationships and are not on the same but only overlapping taxa sets. A desirable requirement on the sought after parental structure, therefore, is that it can accommodate the observed conflicts. Phylogenetic networks are a popular tool capable of doing precisely this. However, not much is known about how to construct such networks from partial trees, a notable exception being the Z-closure super-network approach, which is based on the Z-closure rule, and the Q-imputation approach. Although attractive approaches, they both suffer from the fact that the generated networks tend to be multidimensional making it necessary to apply some kind of filter to reduce their complexity.

Ortega, Neli R. S.; Santos, Fabiano S.; Zanetta, Dirce M. T.; Massad, Eduardo

doi: 10.1007/s11538-008-9332-3pmid: 18663537

In this paper, we present a fuzzy approach to the Reed–Frost model for epidemic spreading taking into account uncertainties in the diagnostic of the infection. The heterogeneities in the infected group is based on the clinical signals of the individuals (symptoms, laboratorial exams, medical findings, etc.), which are incorporated into the dynamic of the epidemic. The infectivity level is time-varying and the classification of the individuals is performed through fuzzy relations. Simulations considering a real problem with data of the viral epidemic in a children daycare are performed and the results are compared with a stochastic Reed–Frost generalization.

Campos, Daniel; Méndez, Vicenç; Ortega-Cejas, Vicente

doi: 10.1007/s11538-008-9333-2pmid: 18696165

We analyze traveling front solutions for a class of reaction-transport Lattice Models (LMs) in order to claim their interest on the description of biological invasions. As lattice models are spatially discrete models, we address here the problem of invasions trough patchy habitats, where every node in the lattice represents a different patch. Distributed generation times for the individuals are considered, so that different temporal patterns can be studied. Specifically, we explore some examples of seasonal and nonseasonal patterns which may be of ecological interest. The main advantage of the LMs described here is that a direct correspondence between these discrete models and a mesoscopic description of Continuous-Time Random Walks (CTRW) can be found. This point is of great importance, since many times one needs analytical expressions to support or validate numerical results, or vice versa. Finally, that correspondence allows us to provide a discussion about some general aspects of reaction-dispersal models.

Morishita, Yoshihiro; Iwasa, Yoh

doi: 10.1007/s11538-008-9334-1pmid: 18668295

Many genes and their regulatory relationships are involved in developmental phenomena. However, by chemical information alone, we cannot fully understand changing organ morphologies through tissue growth because deformation and growth of the organ are essentially mechanical processes. Here, we develop a mathematical model to describe the change of organ morphologies through cell proliferation. Our basic idea is that the proper specification of localized volume source (e.g., cell proliferation) is able to guide organ morphogenesis, and that the specification is given by chemical gradients. We call this idea “growth-based morphogenesis.” We find that this morphogenetic mechanism works if the tissue is elastic for small deformation and plastic for large deformation. To illustrate our concept, we study the development of vertebrate limb buds, in which a limb bud protrudes from a flat lateral plate and extends distally in a self-organized manner. We show how the proportion of limb bud shape depends on different parameters and also show the conditions needed for normal morphogenesis, which can explain abnormal morphology of some mutants. We believe that the ideas shown in the present paper are useful for the morphogenesis of other organs.

Bertram, Richard; Rhoads, Joseph; Cimbora, Wendy

doi: 10.1007/s11538-008-9335-0pmid: 18648884

We describe a novel dynamic mechanism for episodic or compound bursting oscillations, in which bursts of electrical impulses are clustered together into episodes, separated by long silent phases. We demonstrate the mechanism for episodic bursting using a minimal mathematical model for “phantom bursting.” Depending on the location in parameter space, this model can produce fast, medium, or slow bursting, or in the present case, fast, slow, and episodic bursting. The episodic bursting is modestly robust to noise and to parameter variation, and the effect that noise has on the episodic bursting pattern is quite different from that of an alternate episodic burst mechanism in which the slow envelope is produced by metabolic oscillations. This mechanism could account for episodic bursting produced in endocrine cells or neurons, such as pancreatic islets or gonadotropin releasing neurons of the hypothalamus.

Kim, Peter; Lee, Peter; Levy, Doron

doi: 10.1007/s11538-008-9336-zpmid: 18663536

We derive a model for describing the dynamics of imatinib-treated chronic myelogenous leukemia (CML). This model is a continuous extension of the agent-based CML model of Roeder et al. (Nat. Med. 12(10), 1181–1184, 2006) and of its recent formulation as a system of difference equations (Kim et al. in Bull. Math. Biol. 70(3), 728–744, 2008). The new model is formulated as a system of partial differential equations that describe various stages of differentiation and maturation of normal hematopoietic cells and of leukemic cells.