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Journal of Mathematical Biology

Timothy Chumley, Ozgur Aydogmus, Anastasios Matzavinos, Alexander Roitershtein
We study stochastic evolutionary game dynamics in a population of finite size. Individuals in the population are divided into two dynamically evolving groups. The structure of the population is formally described by a Wright-Fisher type Markov chain with a frequency dependent fitness. In a strong selection regime that favors one of the two groups, we obtain qualitatively matching lower and upper bounds for the fixation probability of the advantageous population. In the infinite population limit we obtain an exact result showing that a single advantageous mutant can invade an infinite population with a positive probability...
May 16, 2017: Journal of Mathematical Biology
Sze-Bi Hsu, King-Yeung Lam, Feng-Bin Wang
This paper presents a PDE system modeling the growth of a single species population consuming inorganic carbon that is stored internally in a poorly mixed habitat. Inorganic carbon takes the forms of "CO2" (dissolved CO2 and carbonic acid) and "CARB" (bicarbonate and carbonate ions), which are substitutable in their effects on algal growth. We first establish a threshold type result on the extinction/persistence of the species in terms of the sign of a principal eigenvalue associated with a nonlinear eigenvalue problem...
May 11, 2017: Journal of Mathematical Biology
Fabio A C C Chalub, Max O Souza
This work is a systematic study of discrete Markov chains that are used to describe the evolution of a two-types population. Motivated by results valid for the well-known Moran (M) and Wright-Fisher (WF) processes, we define a general class of Markov chains models which we term the Kimura class. It comprises the majority of the models used in population genetics, and we show that many well-known results valid for M and WF processes are still valid in this class. In all Kimura processes, a mutant gene will either fixate or become extinct, and we present a necessary and sufficient condition for such processes to have the probability of fixation strictly increasing in the initial frequency of mutants...
May 10, 2017: Journal of Mathematical Biology
U A Rozikov
We define a DNA as a sequence of [Formula: see text]'s and embed it on a path of Cayley tree. Using group representation of the Cayley tree, we give a hierarchy of a countable set of DNAs each of which 'lives' on the same Cayley tree. This hierarchy has property that each vertex of the Cayley tree belongs only to one of DNA. Then we give a model (energy, Hamiltonian) of this set of DNAs by an analogue of Ising model with three spin values (considered as DNA base pairs) on a set of admissible configurations...
May 8, 2017: Journal of Mathematical Biology
Brittany Stephenson, Cristina Lanzas, Suzanne Lenhart, Judy Day
The spore-forming, gram-negative bacteria Clostridium difficile can cause severe intestinal illness. A striking increase in the number of cases of C. difficile infection (CDI) among hospitals has highlighted the need to better understand how to prevent the spread of CDI. In our paper, we modify and update a compartmental model of nosocomial C. difficile transmission to include vaccination. We then apply optimal control theory to determine the time-varying optimal vaccination rate that minimizes a combination of disease prevalence and spread in the hospital population as well as cost, in terms of time and money, associated with vaccination...
May 8, 2017: Journal of Mathematical Biology
Gabriel Cardona, Joan Carles Pons
Phylogenetic networks have gained attention from the scientific community due to the evidence of the existence of evolutionary events that cannot be represented using trees. A variant of phylogenetic networks, called LGT networks, models specifically lateral gene transfer events, which cannot be properly represented with generic phylogenetic networks. In this paper we treat the problem of the reconstruction of LGT networks from substructures induced by three leaves, which we call tri-LGT-nets. We first restrict ourselves to a class of LGT networks that are both mathematically treatable and biologically significant, called BAN-LGT networks...
April 27, 2017: Journal of Mathematical Biology
Pablo G Barrientos, J Ángel Rodríguez, Alfonso Ruiz-Herrera
We prove analytically the existence of chaotic dynamics in the forced SIR model. Although numerical experiments have already suggested that this model can exhibit chaotic dynamics, a rigorous proof (without computer-aided) was not given before. Under seasonality in the transmission rate, the coexistence of low birth and mortality rates with high recovery and transmission rates produces infinitely many periodic and aperiodic patterns together with sensitive dependence on the initial conditions.
April 22, 2017: Journal of Mathematical Biology
Jeremy G Sumner, Amelia Taylor, Barbara R Holland, Peter D Jarvis
Recently there has been renewed interest in phylogenetic inference methods based on phylogenetic invariants, alongside the related Markov invariants. Broadly speaking, both these approaches give rise to polynomial functions of sequence site patterns that, in expectation value, either vanish for particular evolutionary trees (in the case of phylogenetic invariants) or have well understood transformation properties (in the case of Markov invariants). While both approaches have been valued for their intrinsic mathematical interest, it is not clear how they relate to each other, and to what extent they can be used as practical tools for inference of phylogenetic trees...
April 22, 2017: Journal of Mathematical Biology
F M G Magpantay
A general model of an imperfect vaccine for a childhood disease is presented and the effects of different types of vaccine failure on transmission were investigated using models that consider both homogeneous and age-specific mixing. The models are extensions of the standard SEIR equations with an additional vaccinated component that allows for five different vaccine parameters: three types of vaccine failure in decreasing susceptibility to infection via failure in degree ("leakiness"), take ("all-or-nothingness") and duration (waning of vaccine-derived immunity); one parameter reflecting the relative reduction in infectiousness of vaccinated individuals who get infected; and one parameter that reflects the relative reduction in reporting probability of vaccinated individuals due to a possible reduction in severity of symptoms...
April 17, 2017: Journal of Mathematical Biology
Robert R Wilkinson, Frank G Ball, Kieran J Sharkey
We consider a very general stochastic model for an SIR epidemic on a network which allows an individual's infectious period, and the time it takes to contact each of its neighbours after becoming infected, to be correlated. We write down the message passing system of equations for this model and prove, for the first time, that it has a unique feasible solution. We also generalise an earlier result by proving that this solution provides a rigorous upper bound for the expected epidemic size (cumulative number of infection events) at any fixed time [Formula: see text]...
April 13, 2017: Journal of Mathematical Biology
Pia Domschke, Dumitru Trucu, Alf Gerisch, Mark A J Chaplain
The dynamic interplay between collective cell movement and the various molecules involved in the accompanying cell signalling mechanisms plays a crucial role in many biological processes including normal tissue development and pathological scenarios such as wound healing and cancer. Information about the various structures embedded within these processes allows a detailed exploration of the binding of molecular species to cell-surface receptors within the evolving cell population. In this paper we establish a general spatio-temporal-structural framework that enables the description of molecular binding to cell membranes coupled with the cell population dynamics...
April 12, 2017: Journal of Mathematical Biology
Xiao Zhao, Stephan Noack, Wolfgang Wiechert, Eric von Lieres
Dynamic flux balance analysis (DFBA) extends flux balance analysis and enables the combined simulation of both intracellular and extracellular environments of microbial cultivation processes. A DFBA model contains two coupled parts, a dynamic part at the upper level (extracellular environment) and an optimization part at the lower level (intracellular environment). Both parts are coupled through substrate uptake and product secretion rates. This work proposes a Karush-Kuhn-Tucker condition based solution approach for DFBA models, which have a nonlinear objective function in the lower-level part...
April 11, 2017: Journal of Mathematical Biology
Xiuli Cen, Zhilan Feng, Yiqiang Zheng, Yulin Zhao
Antibiotic-resistant bacteria have posed a grave threat to public health by causing a number of nosocomial infections in hospitals. Mathematical models have been used to study transmission dynamics of antibiotic-resistant bacteria within a hospital and the measures to control antibiotic resistance in nosocomial pathogens. Studies presented in Lipstich et al. (Proc Natl Acad Sci 97(4):1938-1943, 2000) and Lipstich and Bergstrom (Infection control in the ICU environment. Kluwer, Boston, 2002) have provided valuable insights in understanding the transmission of antibiotic-resistant bacteria in a hospital...
April 10, 2017: Journal of Mathematical Biology
Gabriel P Langlois, Morgan Craig, Antony R Humphries, Michael C Mackey, Joseph M Mahaffy, Jacques Bélair, Thibault Moulin, Sean R Sinclair, Liangliang Wang
We develop a mathematical model of platelet, megakaryocyte, and thrombopoietin dynamics in humans. We show that there is a single stationary solution that can undergo a Hopf bifurcation, and use this information to investigate both normal and pathological platelet production, specifically cyclic thrombocytopenia. Carefully estimating model parameters from laboratory and clinical data, we then argue that a subset of parameters are involved in the genesis of cyclic thrombocytopenia based on clinical information...
April 8, 2017: Journal of Mathematical Biology
Zhigui Lin, Huaiping Zhu
In this paper, a reaction-diffusion system is proposed to model the spatial spreading of West Nile virus in vector mosquitoes and host birds in North America. Transmission dynamics are based on a simplified model involving mosquitoes and birds, and the free boundary is introduced to model and explore the expanding front of the infected region. The spatial-temporal risk index [Formula: see text], which involves regional characteristic and time, is defined for the simplified reaction-diffusion model with the free boundary to compare with other related threshold values, including the usual basic reproduction number [Formula: see text]...
April 4, 2017: Journal of Mathematical Biology
Sophie Hautphenne, Melanie Massaro, Peter Taylor
In this paper, we use a finite-state continuous-time Markov chain with one absorbing state to model an individual's lifetime. Under this model, the time of death follows a phase-type distribution, and the transient states of the Markov chain are known as phases. We then attempt to provide an answer to the simple question "What is the conditional age distribution of the individual, given its current phase"? We show that the answer depends on how we interpret the question, and in particular, on the phase observation scheme under consideration...
April 3, 2017: Journal of Mathematical Biology
Laurent Tournier, Anne Goelzer, Vincent Fromion
Central to the functioning of any living cell, the metabolic network is a complex network of biochemical reactions. It may also be viewed as an elaborate production system, integrating a diversity of internal and external signals in order to efficiently produce the energy and the biochemical precursors to ensure all cellular functions. Even in simple organisms like bacteria, it shows a striking level of coordination, adapting to very different growth media. Constraint-based models constitute an efficient mathematical framework to compute optimal metabolic configurations, at the scale of a whole genome...
March 30, 2017: Journal of Mathematical Biology
Christine Sample, Benjamin Allen
Evolutionary game theory is a mathematical approach to studying how social behaviors evolve. In many recent works, evolutionary competition between strategies is modeled as a stochastic process in a finite population. In this context, two limits are both mathematically convenient and biologically relevant: weak selection and large population size. These limits can be combined in different ways, leading to potentially different results. We consider two orderings: the [Formula: see text] limit, in which weak selection is applied before the large population limit, and the [Formula: see text] limit, in which the order is reversed...
March 28, 2017: Journal of Mathematical Biology
Renaud Dessalles, Vincent Fromion, Philippe Robert
This paper analyzes, in the context of a prokaryotic cell, the stochastic variability of the number of proteins when there is a control of gene expression by an autoregulation scheme. The goal of this work is to estimate the efficiency of the regulation to limit the fluctuations of the number of copies of a given protein. The autoregulation considered in this paper relies mainly on a negative feedback: the proteins are repressors of their own gene expression. The efficiency of a production process without feedback control is compared to a production process with an autoregulation of the gene expression assuming that both of them produce the same average number of proteins...
March 13, 2017: Journal of Mathematical Biology
Veronika Hajnová, Lenka Přibylová
The structured population LPA model is studied. The model describes flour beetle (Tribolium) population dynamics of four stage populations: eggs, larvae, pupae and adults with cannibalism between these stages. We concentrate on the case of non-zero cannibalistic rates of adults on eggs and adults on pupae and no cannibalism of larvae on eggs, but the results can be numerically continued to non-zero cannibalism of larvae on eggs. In this article two-parameter bifurcations in LPA model are analysed. Various stable and unstable invariant sets are found, different types of hysteresis are presented and abrupt changes in dynamics are simulated to explain the complicated way the system behaves near two-parameter bifurcation manifolds...
March 10, 2017: Journal of Mathematical Biology
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