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2018 | Book

Understanding Complex Biological Systems with Mathematics


About this book

This volume examines a variety of biological and medical problems using mathematical models to understand complex system dynamics. Featured topics include autism spectrum disorder, ectoparasites and allogrooming, argasid ticks dynamics, super-fast nematocyst firing, cancer-immune population dynamics, and the spread of disease through populations. Applications are investigated with mathematical models using a variety of techniques in ordinary and partial differential equations, difference equations, Markov-chain models, Monte-Carlo simulations, network theory, image analysis, and immersed boundary method. Each article offers a thorough explanation of the methodologies used and numerous tables and color illustrations to explain key results. This volume is suitable for graduate students and researchers interested in current applications of mathematical models in the biosciences.

The research featured in this volume began among newly-formed collaborative groups at the 2017 Women Advancing Mathematical Biology Workshop that took place at the Mathematical Biosciences Institute in Columbus, Ohio. The groups spent one intensive week working at MBI and continued their collaborations after the workshop, resulting in the work presented in this volume.

Table of Contents

Searching for Superspreaders: Identifying Epidemic Patterns Associated with Superspreading Events in Stochastic Models
The importance of host transmissibility in disease emergence has been demonstrated in historical and recent pandemics that involve infectious individuals, known as superspreaders, who are capable of transmitting the infection to a large number of susceptible individuals. To investigate the impact of superspreaders on epidemic dynamics, we formulate deterministic and stochastic models that incorporate differences in superspreaders versus nonsuperspreaders. In particular, continuous-time Markov chain models are used to investigate epidemic features associated with the presence of superspreaders in a population. We parameterize the models for two case studies, Middle East respiratory syndrome (MERS) and Ebola. Through mathematical analysis and numerical simulations, we find that the probability of outbreaks increases and time to outbreaks decreases as the prevalence of superspreaders increases in the population. In particular, as disease outbreaks occur more rapidly and more frequently when initiated by superspreaders, our results emphasize the need for expeditious public health interventions.
Christina J. Edholm, Blessing O. Emerenini, Anarina L. Murillo, Omar Saucedo, Nika Shakiba, Xueying Wang, Linda J. S. Allen, Angela Peace
How Disease Risks Can Impact the Evolution of Social Behaviors and Emergent Population Organization
Individuals living in social groups are susceptible to disease spread through their social networks. The network’s structure including group stability, clustering, and an individual’s behavior and affiliation choice all have some impact on the effect of disease spread. Moreover, under certain scenarios, a social group may change its own structure to suppress the transmission of infectious disease. While many studies have focused on how different network structures shape the disease dynamics, relatively few have directly considered the equally important evolutionary question of how disease dynamics shape the success of social systems. In this paper, we summarize the relevant mathematical and biological literature on evolutionary theory and population network structure to discuss what is known about how the synergistic effects of network-based epidemiology of infection and social behavior can shape the evolution of social behaviors and the population structures that emerge from them. We close by discussing open questions, including how these insights may shift when instead considering macro-parasites as the infection spreading on the network.
Nakeya D. Williams, Heather Z. Brooks, Maryann E. Hohn, Candice R. Price, Ami E. Radunskaya, Suzanne S. Sindi, Shelby N. Wilson, Nina H. Fefferman
Mathematical Analysis of the Impact of Social Structure on Ectoparasite Load in Allogrooming Populations
In many social species, there exist a few highly connected individuals living among a larger majority of poorly connected individuals. Previous studies have shown that, although this social network structure may facilitate some aspects of group-living (e.g., collective decision-making), these highly connected individuals can act as super-spreaders of circulating infectious pathogens. We build on this literature to instead consider the impact of this type of network structure on the circulation of ectoparasitic infections in a population. We consider two ODE models that each approximate a simplified network model; one with uniform social contacts, and one with a few highly connected individuals. We find that, rather than increasing risk, the inclusion of highly connected individuals increases the probability that a population will be able to eradicate ectoparasitic infection through social grooming.
Heather Z. Brooks, Maryann E. Hohn, Candice R. Price, Ami E. Radunskaya, Suzanne S. Sindi, Nakeya D. Williams, Shelby N. Wilson, Nina H. Fefferman
Modeling the Argasid Tick (Ornithodoros moubata) Life Cycle
The first mathematical models for an argasid tick are developed to explore the dynamics and identify knowledge gaps of these poorly studied ticks. These models focus on Ornithodoros moubata, an important tick species throughout Africa and Europe. Ornithodoros moubata is a known vector for African swine fever (ASF), a catastrophically fatal disease for domesticated pigs in Africa and Europe. In the absence of any previous models for soft-bodied ticks, we propose two mathematical models of the life cycle of O. moubata. One is a continuous-time differential equation model that simplifies the tick life cycle to two stages, and the second is a discrete-time difference equation model that uses four stages. Both models use two host types: small hosts and large hosts, and both models find that either host type alone could support the tick population and that the final tick density is a function of host density. While both models predict similar tick equilibrium values, we observe significant differences in the time to equilibrium. The results demonstrate the likely establishment of these ticks if introduced into a new area even if there is only one type of host. These models provide the basis for developing future models that include disease states to explore infection dynamics and possible management of ASF.
Sara M. Clifton, Courtney L. Davis, Samantha Erwin, Gabriela Hamerlinck, Amy Veprauskas, Yangyang Wang, Wenjing Zhang, Holly Gaff
A Mathematical Model for Tumor–Immune Dynamics in Multiple Myeloma
We propose a mathematical model that describes the dynamics of multiple myeloma and three distinct populations of the innate and adaptive immune system: cytotoxic T cells, natural killer cells, and regulatory T cells. The model includes significant biologically- and therapeutically-relevant pathways for inhibitory and stimulatory interactions between these populations. Due to the model complexity, we propose a reduced version that captures the principal biological aspects for advanced disease, while still including potential targets for therapeutic interventions. Analysis of the reduced two-dimensional model revealed details about long-term model behavior. In particular, theoretical results describing equilibria and their associated stability are described in detail. Consistent with the theoretical analysis, numerical results reveal parameter regions for which bistability exits. The two stable states in these cases may correspond to long-term disease control or a higher level of disease burden. This initial analysis of the dynamical system provides a foundation for later work, which will consider combination therapies, their expected outcomes, and optimization of regimens.
Jill Gallaher, Kamila Larripa, Urszula Ledzewicz, Marissa Renardy, Blerta Shtylla, Nessy Tania, Diana White, Karen Wood, Li Zhu, Chaitali Passey, Michael Robbins, Natalie Bezman, Suresh Shelat, Hearn Jay Cho, Helen Moore
Fluid Dynamics of Nematocyst Prey Capture
A nematocyst is a specialized organelle within cells of jellyfish and other Cnidarians that sting. Nematocysts are also present in some single-celled protists. They contain a barbed, venomous thread that accelerates faster than almost anything else in the animal kingdom. Here we simulate the fluid–structure interaction of the barbed thread accelerating through water to puncture its prey using the 2D immersed boundary method. For simplicity, our model describes the discharge of a single barb harpooning a single-celled organism, as in the case of dinoflagellates. One aspect of this project that is particularly interesting is that the micron-sized barbed thread reaches Reynolds numbers above one, where inertial effects become important. At this scale, even small changes in speed and shape can have dramatic effects on the local flow field. This suggests that the large variety of sizes and shapes of nematocysts may have important fluid dynamic consequences. We find that reaching the inertial regime is critical for hitting prey over short distances since the large boundary layers surrounding the barb characteristic of viscous dominated flows effectively push the prey out of the way.
Wanda Strychalski, Sarah Bryant, Baasansuren Jadamba, Eirini Kilikian, Xiulan Lai, Leili Shahriyari, Rebecca Segal, Ning Wei, Laura A. Miller
Simulations of the Vascular Network Growth Process for Studying Placenta Structure and Function Associated with Autism
Placenta chorionic surface vascular networks differ in individuals at-risk for autism compared to controls in terms of longer, straighter, thicker vessels; less branching; smaller changes in flow directions; and better coverage to the placental boundary. What mechanism(s) could drive these differences and how these mechanisms would impact blood transport has not been widely investigated. We used a Monte-Carlo simulation to mimic three mechanisms for controlling vascular growth: vessels grow faster and longer, terminate more frequently before branching, and flow directions are more tightly controlled in the at-risk simulations. For each mechanism, we analyzed simulated vascular networks based on structural properties and blood flow, assuming Poiseuille’s law and distensible vessels. Our simulations showed that none of these mechanisms alone could reproduce all structural properties of vascular networks in placentas identified as at-risk for autism. Terminating vessels more frequently or growing longer vessels could each reproduce longer vessels and less branching, but not greater boundary coverage or smaller changes in flow directions. As for their influence on blood flow, longer vessels and less branching have large, opposing effects on network function. Networks with longer vessels are less efficient in terms of slower flow rates and higher total network volume; in contrast, networks with less branching are more efficient. Our results suggest either these mechanisms work together to drive observed differences in vascular networks of at-risk individuals by balancing their impacts on network function; or another mechanism not considered here might drive these differences.
Catalina Anghel, Kellie Archer, Jen-Mei Chang, Amy Cochran, Anca Radulescu, Carolyn M. Salafia, Rebecca Turner, Yacoubou Djima Karamatou, Lan Zhong
Placental Vessel Extraction with Shearlets, Laplacian Eigenmaps, and a Conditional Generative Adversarial Network
The placenta is the key organ of maternal–fetal interactions, where nutrient, oxygen, and waste transfer take place. Differences in the morphology of the placental chorionic surface vascular network (PCSVN) have been associated with developmental disorders such as autism, hinting that the PCSVN could potentially serve as a biomarker for early diagnosis and treatment of autism. Studying PCSVN features in large cohorts requires a reliable and automated mechanism to extract the vascular networks. This paper presents two distinct methods for PCSVN enhancement and extraction. Our first algorithm, which builds upon a directional multiscale mathematical framework based on a combination of shearlets and Laplacian eigenmaps, is able to intensify the appearance of vessels with high success in high-contrast images such as those produced in CT scans. Our second algorithm, which applies a conditional generative adversarial neural network (cGAN), was trained to simulate a human-traced PCSVN given a digital photograph of the placental chorionic surface. This method surpasses any existing automated PCSVN extraction methods reported on digital photographs of placentas. We hypothesize that a suitable combination of the two methods could further improve PCSVN extraction results and should be studied in the future.
Catalina Anghel, Kellie Archer, Jen-Mei Chang, Amy Cochran, Anca Radulescu, Carolyn M. Salafia, Rebecca Turner, Karamatou Yacoubou Djima, Lan Zhong
Understanding Complex Biological Systems with Mathematics
Ami Radunskaya
Rebecca Segal
Blerta Shtylla
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