Structural instability and linear allocation control in generalized models of substance use disorder.

Substance use disorder (SUD) is a complex disease involving nontrivial biological, psychological, environmental, and social factors. While many mathematical studies have proposed compartmental models for SUD, almost all of these exclusively model new cases as the result of an infectious process, neglecting any SUD that was primarily developed in social isolation. While these decisions were likely made to facilitate mathematical analysis, isolated SUD development is critical for the most common substances of abuse today, including opioid use disorder developed through prescription use and alcoholism developed primarily due to genetic factors or stress, depression, and other psychological factors. In this paper we will demonstrate that even a simple infectious disease model is structurally unstable with respect to a linear perturbation in the infection term - precisely the sort of term necessary to model SUD development in isolation. This implies that models of SUD which exclusively treat problematic substance use as an infectious disease will have misleading dynamics whenever a non-trivial rate of isolated SUD development exists in actuality. As we will show, linearly perturbed SUD models do not have a use disorder-free equilibrium. To investigate management strategies, we implement optimal control techniques with the goal of minimizing the number of SUD cases over time.

Generalized Stressors on Hive and Forager Bee Colonies.

Hive-forming bees play an integral role in promoting agricultural sustainability and ecosystem preservation. The recent worldwide decline of several species of bees, and in particular, the honeybee in the United States, highlights the value in understanding possible causes. Over the past decade, numerous mathematical models and empirical experiments have worked to understand the causes of colony stress, with a particular focus on colony collapse disorder. We integrate and enhance major mathematical models of the past decade to create a single, analytically tractable model using a traditional disease modeling framework that incorporates both lethal and sublethal stressors. On top of this synthesis, a major innovation of our model is the generalization of stressor attributes including their transmissibility, impairment level, lethality, duration, and temporal-occurrence. Our model is validated against numerous emergent, biological characteristics and demonstrates that precocious foraging and labor destabilization can produce colony collapse disorder. The thresholds for these phenomena to occur depend on the characteristics and timing of the stressor, thus motivating further empirical and theoretical studies into stressor characteristics.

Prioritization of the concepts and skills in quantitative education for graduate students in biomedical science.

Substantial guidance is available on undergraduate quantitative training for biologists, including reports focused on biomedical science. Far less attention has been paid to the graduate curriculum and the particular challenges of the diversity of specialization within the life sciences. We propose an innovative approach to quantitative education that goes beyond recommendations of a course or set of courses or activities, derived from analysis of the expectations for students in particular programs. Due to the plethora of quantitative methods, it is infeasible to expect that biomedical PhD students can be exposed to more than a minority of the quantitative concepts and techniques employed in modern biology. We collected key recent papers suggested by the faculty in biomedical science programs, chosen to include important scientific contributions that the faculty consider appropriate for all students in the program to be able to read with confidence. The quantitative concepts and methods inherent in these papers were then analyzed and categorized to provide a rational basis for prioritization of those concepts to be emphasized in the education program. This novel approach to prioritization of quantitative skills and concepts provides an effective method to drive curricular focus based upon program-specific faculty input for science programs of all types. The results of our particular application to biomedical science training highlight the disconnect between typical undergraduate quantitative education for life science students, focused on continuous mathematics, and the concepts and skills in graphics, statistics, and discrete mathematics that arise from priorities established by biomedical science faculty. There was little reference in the key recent papers chosen by faculty to classic mathematical areas such as calculus which make up a large component of the formal undergraduate mathematics training of graduate students in biomedical areas.

Multiscale flow between the branches and polyps of gorgonians.

Gorgonians, including sea fans, are soft corals well known for their elaborate branching structure and how they sway in the ocean. This branching structure can modify environmental flows to be beneficial for feeding in a particular range of velocities and, presumably, for a particular size of prey. As water moves through the elaborate branches, it is slowed, and recirculation zones can form downstream of the colony. At the smaller scale, individual polyps that emerge from the branches expand their tentacles, further slowing the flow. At the smallest scale, the tentacles are covered in tiny pinnules where exchange occurs. In this paper, we quantified the gap to diameter ratios for various gorgonians at the scale of the branches, the polyp tentacles and the pinnules. We then used computational fluid dynamics to determine the flow patterns at all three levels of branching. We quantified the leakiness between the branches, tentacles and pinnules over the biologically relevant range of Reynolds numbers and gap-to-diameter ratios, and found that the branches and tentacles can act as either leaky rakes or solid plates depending upon these dimensionless parameters. The pinnules, in contrast, mostly impede the flow. Using an agent-based modeling framework, we quantified plankton capture as a function of the gap-to-diameter ratio of the branches and the Reynolds number. We found that the capture rate depends critically on both morphology and Reynolds number. The results of the study have implications for how gorgonians modify ambient flows for efficient feeding and exchange.

Modeling the effects of size-dependent harvesting strategies on the population dynamics of tropical trees.

Several forest plant species are harvested both lethally for their timber and non-lethally for their non-timber forest products by the local people for cultural and economic reasons. To maximize yield, harvesters target various life stages of these species including both adults and juveniles particularly when the number of harvestable adults decline. The demographic consequences of harvesting various plant sizes differ based on what life stage is targeted. In this paper, we develop a size-structured, seasonal system of difference equations and corresponding matrix model with time-varying harvest to model the effects of size-dependent harvesting strategies on the population dynamics of tropical trees. We illustrate numerically our work specifically on African mahogany, Khaya senegalensis, a tropical tree in Benin. Novel applications and combinations of previously established matrix compression algorithms are presented to determine certain rates in our model, with other rates coming from the use of generalized linear modeling and ordinary least squares estimation incorporating observed population data. Harvesting rates for two types of populations are estimated, one with simulated harvest and the other experiencing natural harvest. Eigenvalue analysis suggests that for the populations in our study, harvesting may not have a drastic effect on the long-term persistence of the population. However, this should be taken with caution given that our model does not account for stochastic environmental variations that can interactively reduce population growth rates.

A Data-Driven Mathematical Model of the Heroin and Fentanyl Epidemic in Tennessee.

Opioid addiction represents a major national health issue spanning decades. In recent years, prescription opioid use disorder has increasingly led to heroin and fentanyl use, with subsequent increases in mortality rates due to overdose. In this paper, we present a mechanistic, epidemic model for prescription opioid addiction and illicit heroin or fentanyl addiction which aims to better understand and predict the dynamics between these two stages of opioid use disorder. Our model aims to be both parsimonious and robust: as a system of five differential equations it is appropriate for use in theory advancement and yet it remains powerful enough to capture state-level data from Tennessee for the period 2013-2018. A key finding from our data-driven analysis is that, in the face of changing policy around prescription opioids, heroin and fentanyl are now the driving force behind the Tennessee opioid epidemic. Model projections suggest that both addictions and overdoses related to heroin and fentanyl will continue to increase in the next few years (2020-2022), even as addiction to prescription drugs continues to fall. Finally, management strategy analysis suggests that in the changing face of the epidemic, the most successful approach will target availability of treatment with subsequent monitoring of stably recovered individuals to see that they do not relapse, coincident with direct efforts to decrease opioid overdose fatalities (e.g., further availability of Naloxone).

A semi-automated finite difference mesh creation method for use with immersed boundary software IB2d and IBAMR.

Numerous fluid-structure interaction problems in biology have been investigated using the immersed boundary method. The advantage of this method is that complex geometries, e.g., internal or external morphology, can easily be handled without the need to generate matching grids for both the fluid and the structure. Consequently, the difficulty of modeling the structure lies often in discretizing the boundary of the complex geometry (morphology). Both commercial and open source mesh generators for finite element methods have long been established; however, the traditional immersed boundary method is based on a finite difference discretization of the structure. Here we present a software library for obtaining finite difference discretizations of boundaries for direct use in the 2D immersed boundary method. This library provides tools for extracting such boundaries as discrete mesh points from digital images. We give several examples of how the method can be applied that include passing flow through the veins of insect wings, within lymphatic capillaries, and around starfish using open-source immersed boundary software.

Agent-based and continuous models of hopper bands for the Australian plague locust: How resource consumption mediates pulse formation and geometry.

Locusts are significant agricultural pests. Under favorable environmental conditions flightless juveniles may aggregate into coherent, aligned swarms referred to as hopper bands. These bands are often observed as a propagating wave having a dense front with rapidly decreasing density in the wake. A tantalizing and common observation is that these fronts slow and steepen in the presence of green vegetation. This suggests the collective motion of the band is mediated by resource consumption. Our goal is to model and quantify this effect. We focus on the Australian plague locust, for which excellent field and experimental data is available. Exploiting the alignment of locusts in hopper bands, we concentrate solely on the density variation perpendicular to the front. We develop two models in tandem; an agent-based model that tracks the position of individuals and a partial differential equation model that describes locust density. In both these models, locust are either stationary (and feeding) or moving. Resources decrease with feeding. The rate at which locusts transition between moving and stationary (and vice versa) is enhanced (diminished) by resource abundance. This effect proves essential to the formation, shape, and speed of locust hopper bands in our models. From the biological literature we estimate ranges for the ten input parameters of our models. Sobol sensitivity analysis yields insight into how the band's collective characteristics vary with changes in the input parameters. By examining 4.4 million parameter combinations, we identify biologically consistent parameters that reproduce field observations. We thus demonstrate that resource-dependent behavior can explain the density distribution observed in locust hopper bands. This work suggests that feeding behaviors should be an intrinsic part of future modeling efforts.

The total dispersal kernel: a review and future directions.

The distribution and abundance of plants across the world depends in part on their ability to move, which is commonly characterized by a dispersal kernel. For seeds, the total dispersal kernel (TDK) describes the combined influence of all primary, secondary and higher-order dispersal vectors on the overall dispersal kernel for a plant individual, population, species or community. Understanding the role of each vector within the TDK, and their combined influence on the TDK, is critically important for being able to predict plant responses to a changing biotic or abiotic environment. In addition, fully characterizing the TDK by including all vectors may affect predictions of population spread. Here, we review existing research on the TDK and discuss advances in empirical, conceptual modelling and statistical approaches that will facilitate broader application. The concept is simple, but few examples of well-characterized TDKs exist. We find that significant empirical challenges exist, as many studies do not account for all dispersal vectors (e.g. gravity, higher-order dispersal vectors), inadequately measure or estimate long-distance dispersal resulting from multiple vectors and/or neglect spatial heterogeneity and context dependence. Existing mathematical and conceptual modelling approaches and statistical methods allow fitting individual dispersal kernels and combining them to form a TDK; these will perform best if robust prior information is available. We recommend a modelling cycle to parameterize TDKs, where empirical data inform models, which in turn inform additional data collection. Finally, we recommend that the TDK concept be extended to account for not only where seeds land, but also how that location affects the likelihood of establishing and producing a reproductive adult, i.e. the total effective dispersal kernel.

Modeling the Prescription Opioid Epidemic.

Opioid addiction has become a global epidemic and a national health crisis in recent years, with the number of opioid overdose fatalities steadily increasing since the 1990s. In contrast to the dynamics of a typical illicit drug or disease epidemic, opioid addiction has its roots in legal, prescription medication-a fact which greatly increases the exposed population and provides additional drug accessibility for addicts. In this paper, we present a mathematical model for prescription drug addiction and treatment with parameters and validation based on data from the opioid epidemic. Key dynamics considered include addiction through prescription, addiction from illicit sources, and treatment. Through mathematical analysis, we show that no addiction-free equilibrium can exist without stringent control over how opioids are administered and prescribed, in which case we estimate that the epidemic would cease to be self-sustaining. Numerical sensitivity analysis suggests that relatively low states of endemic addiction can be obtained by primarily focusing on medical prevention followed by aggressive treatment of remaining cases-even when the probability of relapse from treatment remains high. Further empirical study focused on understanding the rate of illicit drug dependence versus overdose risk, along with the current and changing rates of opioid prescription and treatment, would shed significant light on optimal control efforts and feasible outcomes for this epidemic and drug epidemics in general.

IB2d: a Python and MATLAB implementation of the immersed boundary method.

The development of fluid-structure interaction (FSI) software involves trade-offs between ease of use, generality, performance, and cost. Typically there are large learning curves when using low-level software to model the interaction of an elastic structure immersed in a uniform density fluid. Many existing codes are not publicly available, and the commercial software that exists usually requires expensive licenses and may not be as robust or allow the necessary flexibility that in house codes can provide. We present an open source immersed boundary software package, IB2d, with full implementations in both MATLAB and Python, that is capable of running a vast range of biomechanics models and is accessible to scientists who have experience in high-level programming environments. IB2d contains multiple options for constructing material properties of the fiber structure, as well as the advection-diffusion of a chemical gradient, muscle mechanics models, and artificial forcing to drive boundaries with a preferred motion.