Bulletin of Mathematical Biology

Xiao, Yanni; Xiang, Changcheng; Cheke, Robert A.; Tang, Sanyi

doi: 10.1007/s11538-020-00736-9pmid: 32390107

There are many challenges to coupling the macroscale to the microscale in temporal or spatial contexts. In order to examine effects of an individual movement and spatial control measures on a disease outbreak, we developed a multiscale model and extended the semi-stochastic simulation method by linking individual movements to pathogen’s diffusion, linking the slow dynamics for disease transmission at the population level to the fast dynamics for pathogen shedding/excretion at the individual level. Numerical simulations indicate that during a disease outbreak individuals with the same infection status show the property of clustering and, in particular, individuals’ rapid movements lead to an increase in the average reproduction number R0\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$R_0$$\end{document}, the final size and the peak value of the outbreak. It is interesting that a high level of aggregation the individuals’ movement results in low new infections and a small final size of the infected population. Further, we obtained that either high diffusion rate of the pathogen or frequent environmental clearance lead to a decline in the total number of infected individuals, indicating the need for control measures such as improving air circulation or environmental hygiene. We found that the level of spatial heterogeneity when implementing control greatly affects the control efficacy, and in particular, an uniform isolation strategy leads to low a final size and small peak, compared with local measures, indicating that a large-scale isolation strategy with frequent clearance of the environment is beneficial for disease control.

Aikens, Melissa L.

doi: 10.1007/s11538-020-00739-6pmid: 32399760

Over the last 25 years, reforms in undergraduate biology education have transformed the way biology is taught at many institutions of higher education. This has been fueled in part by a burgeoning discipline-based education research community, which has advocated for evidence-based instructional practices based on findings from research. This perspective will review some of the changes to undergraduate biology education that have gained or are currently gaining momentum, becoming increasingly common in undergraduate biology classrooms. However, there are still areas in need of improvement. Although more underrepresented minority students are enrolling in and graduating from biology programs than in the past, there is a need to understand the experiences and broaden participation of other underserved groups in biology and ensure biology classroom learning environments are inclusive. Additionally, although understanding biology relies on understanding concepts from the physical sciences and mathematics, students still rarely connect the concepts they learn from other STEM disciplines to biology. Integrating concepts and practices across the STEM disciplines will be critical for biology graduates as they tackle the biological problems of the twenty-first century.

Hiller, Rebecca M.; von Kügelgen, Julius; Bao, Huan; Van Hoa, Franck Duong; Cytrynbaum, Eric N.

doi: 10.1007/s11538-020-00737-8pmid: 32415547

The MalFGK2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$_2$$\end{document} transporter regulates the movement of maltose across the inner membrane of E. coli and serves as a model system for bacterial ATP binding cassette (ABC) importers. Despite the wealth of biochemical and structural data available, a general model describing the various translocation pathways is still lacking. In this study, we formulate a mathematical model with the goal of determining the transporter reaction pathway, specifically looking at the order of binding events and conformation changes by which transport proceeds. Fitting our mathematical model to equilibrium binding data, we estimate the unknown equilibrium parameters of the system, several of which are key determinants of the transport process. Using these estimates along with steady-state ATPase rate data, we determine which of several possible reaction pathways is dominant, as a function of five underdetermined kinetic parameter values. Because neither experimental measurements nor estimates of certain kinetic rate constants are available, the problem of deciding which of the reaction pathways is responsible for transport remains unsolved. However, using the mathematical framework developed here, a firmer conclusion regarding the dominant reaction pathway as a function of MalE and maltose concentration could be drawn once these unknown kinetic parameters are determined.

Bressoud, David M.

doi: 10.1007/s11538-020-00738-7pmid: 32410076

This is a survey of the developments in the first two years of undergraduate mathematics, beginning in the early 1950s and continuing up to the present. It documents the repeated efforts at making this instruction relevant to the partner disciplines, especially Biology, and describes the challenges for the future.

Yuan, Sanling; Wu, Dongmei; Lan, Guijie; Wang, Hao

doi: 10.1007/s11538-020-00733-ypmid: 32350614

Stoichiometric producer–grazer models are nonsmooth due to the Liebig’s Law of Minimum and can generate new dynamics such as bistability for producer–grazer interactions. Environmental noises can be extremely important and change dynamical behaviors of a stoichiometric producer–grazer model. In this paper, we consider a stochastically forced producer–grazer model and study the phenomena of noise-induced state switching between two stochastic attractors in the bistable zone. Namely, there is a frequent random hopping of phase trajectories between attracting basins of the attractors. In addition, by applying the stochastic sensitivity function technique, we construct the confidence ellipse and confidence band to find the configurational arrangement of equilibria and a limit cycle, respectively.

Lin, Guo; Niu, Yibin; Pan, Shuxia; Ruan, Shigui

doi: 10.1007/s11538-020-00725-ypmid: 32314098

In this paper, we study the spreading speed in an integrodifference system which models invasion of predators into the habitat of the prey. Without the requirement of comparison principle, we construct several auxiliary integrodifference equations and use the results of monotone scalar equations to estimate the spreading speed of the invading predators. We also present some numerical simulations to support our theoretical results and demonstrate that the integrodifference predator–prey system exhibits very complex dynamics. Our theory and numerical results imply that the invasion of predators may have a rough constant speed. Moreover, our numerical simulations indicate that the spatial contact of individuals and the overcompensatory phenomenon of the prey may be conducive to the persistence of nonmonotone biological systems and lead to instability of the predator-free state.

Lange, Alexander

doi: 10.1007/s11538-020-00730-1pmid: 32350621

Despite being similar in structure, functioning, and size, viral pathogens enjoy very different, usually well-defined ways of life. They occupy their hosts for a few days (influenza), for a few weeks (measles), or even lifelong (HCV), which manifests in acute or chronic infections. The various transmission routes (airborne, via direct physical contact, etc.), degrees of infectiousness (referring to the viral load required for transmission), antigenic variation/immune escape and virulence define further aspects of pathogenic lifestyles. To survive, pathogens must infect new hosts; the success determines their fitness. Infection happens with a certain likelihood during contact of hosts, where contact can also be mediated by vectors. Besides structural aspects of the host-contact network, three parameters appear to be key: the contact rate and the infectiousness during contact, which encode the mode of transmission, and third the immunity of susceptible hosts. On these grounds, what can be said about the reproductive success of viral pathogens? This is the biological question addressed in this paper. The answer extends earlier results of the author and makes explicit connection to another basic work on the evolution of pathogens. A mathematical framework is presented that models intra- and inter-host dynamics in a minimalistic but unified fashion covering a broad spectrum of viral pathogens, including those that cause flu-like infections, childhood diseases, and sexually transmitted infections. These pathogens turn out as local maxima of numerically simulated fitness landscapes. The models involve differential and integral equations, agent-based simulation, networks, and probability.

Chen, Mingshu; Wu, Hong; Wang, Yuanshi

doi: 10.1007/s11538-020-00735-wpmid: 32385574

This paper considers plant–pollinator–herbivore systems where the plant produces food for the pollinator, the pollinator provides pollination service for the plant in return, while the herbivore consumes both the food and the plant itself without providing pollination service. Based on these resource–consumer interactions, we form a plant–pollinator–herbivore model which includes the intermediary food. Using qualitative method and Kuznetsov theorem, we show global dynamics of the subsystems, uniform persistence of the whole system and periodic oscillation by Hopf bifurcation. Rigorous analysis on the system demonstrates mechanisms by which varying parameters could make the system transition between extinction of herbivore, coexistence of the three species at steady states, coexistence in periodic oscillations and extinction of pollinator. It is shown that (i) in plant–pollinator interactions, the plant would produce food; (ii) in plant–herbivore interactions, the plant would produce toxin; (iii) in the presence of both pollinator and herbivore, the plant would produce both food and toxin, and intermediate productions are analytically given by which the plant can reach its maximal density; and (iv) an appropriate toxin production could drive the herbivore into extinction, an unappropriate one would drive the pollinator into extinction, while too much toxin production will drive the plant itself into extinction. The analysis leads to explanations for experimental observations and provides new insights.

Volkening, A.; Abbott, M. R.; Chandra, N.; Dubois, B.; Lim, F.; Sexton, D.; Sandstede, B.

doi: 10.1007/s11538-020-00731-0pmid: 32356149

As zebrafish develop, black and gold stripes form across their skin due to the interactions of brightly colored pigment cells. These characteristic patterns emerge on the growing fish body, as well as on the anal and caudal fins. While wild-type stripes form parallel to a horizontal marker on the body, patterns on the tailfin gradually extend distally outward. Interestingly, several mutations lead to altered body patterns without affecting fin stripes. Through an exploratory modeling approach, our goal is to help better understand these differences between body and fin patterns. By adapting a prior agent-based model of cell interactions on the fish body, we present an in silico study of stripe development on tailfins. Our main result is a demonstration that two cell types can produce stripes on the caudal fin. We highlight several ways that bone rays, growth, and the body–fin interface may be involved in patterning, and we raise questions for future work related to pattern robustness.

Greer, Meredith L.; Akman, Olcay; Comar, Timothy D.; Hrozencik, Daniel; Rubin, Jonathan E.

doi: 10.1007/s11538-020-00728-9pmid: 32399614

Mathematical biology education provides key foundational underpinnings for the scholarly work of mathematical biology. Professional societies support such education efforts via funding, public speaking opportunities, Web presence, publishing, workshops, prizes, opportunities to discuss curriculum design, and support of mentorship and other means of sustained communication among communities of scholars. Such programs have been critical to the broad expansion of the range and visibility of research and educational activities in mathematical biology. We review these efforts, past and present, across multiple societies—the Society for Mathematical Biology (SMB), the Symposium on Biomathematics and Ecology Education and Research (BEER), the Mathematical Association of America (MAA), and the Society for Industrial and Applied Mathematics (SIAM). We then proceed to suggest ways that professional societies can serve as advocates and community builders for mathematical biologists at all levels, noting that education continues throughout a career and also emphasizing the value of educating new generations of students. Our suggestions include collecting and disseminating data related to biomath education; developing and maintaining mentoring systems and research communities; and providing incentives and visibility for educational efforts within mathematical biology.