Workshop Details

Dynamic SEIR (Susceptible, Exposed, Infectious, Removed) compartmental models provide a tool for predicting the size and duration of unfettered outbreaks, and exploring the impacts of interventions on epidemics (e.g., isolation of infectious individuals, vaccination of susceptible individuals). The performance of these models as adequate representations of reality (Getz et al, 2018) depends on the extent to which the assumptions that: i) the same disease progression processes and rates apply equally to all individuals in the population, and ii) the population itself is well-mixed. Variability in individual's behavior, exposure history, immunological and coinfection states, requires a vast amount of data compared with a single time series of incidence data that is used to fit SEIR models to specific epidemics. Incidence data, however, typically have a spatial element to them because these data often come tagged with a location where they were collected (e.g. clinic, town, or city). Here we explore ways to make use of incidence data aggregated at levels of spatial resolution where the assumption that the outbreak consists of a network of process and spatially homogeneous subpopulations to which SEIVD (naturally Vaccinated, Dead) models can be locally applied. Our continuous and discrete-time, deterministic and stochastic versions of these models are constructed with the Numerus Model Builder platform (Getz and Dougherty, 2018; Getz et al, in press) , which permits direct extension to metapopulation versions and one-click generation of web browser implementations. We apply these models to incidence data from the 2014 West African Ebola outbreak, focusing on a comparison of the disease process parameters that locally apply and the degree to which it may be reasonable to apply average parameters to making predications at a country level.

References:

Getz, W. M., C. R. Marshall, C. J. Carlson1, L. Giuggioli, S. J. Ryan, C. Boettiger, S. D. Chamberlain,L. Larsen, P. D’Odorico, D. O’Sullivan, S. S. Romañach, 2018a. Making ecosystem models adequate. Ecology Letters, 21 (2): 153-166.

Getz, W. M. and E. J. Dougherty. Discrete Stochastic Analogs of Erlang Epidemic Models, 2018. Journal of Biological Dynamics,12 (1), 16-38

Getz, W. M., R. Salter, O. Muellerklein, H. S. Yoon, K. Tallam, in press. An Epidemic Modeling Primer and Nova Model Builder Implementation. Epidemics.

Recently, there has been considerable interest in both inference and predictions for compartmental epidemic models on multiple physical scales, like, for instance, a single host and a population of hosts. Both viral invasions and global pandemics are often described by similar mathematical constructs known as SIR models. In this talk I will review some basic concepts related to such models across scales and present a simple unifying framework that allows to conceptually connect both deterministic (e.g., population level) and stochastic (e.g., molecular level) SIR models with the help of tools of statistical theory of survival analysis.

We use a general autonomous discrete-time infectious disease model to extend the next generation matrix approach for calculating the basic reproduction number, Râ‚€, to account for populations with locally asymptotically stable period k cycles in the disease-free systems, where k≥1.When Râ‚€<1 and the demographic equation (in the absence of the disease) has a locally asymptotically stable period k population cycle, we prove the local asymptotic stability of the disease-free period k cycle. That is, the disease goes extinct whenever Râ‚€<1. Under the same period k demographic assumption but with Râ‚€>1, we prove that the disease-free period k population cycle is unstable and the disease persists. Using the Ricker recruitment function, we apply our results to discrete-time infectious disease models that are formulated for SIR infections with and without vaccination, and ISAv infections in a salmon population. When Râ‚€>1, our simulations show that the disease-free period k cycle dynamics drives the SIR disease dynamics, but not the ISAv disease dynamics (This is joint work with P. van den Driessche). 

Infrastructure - such as roads, bridges, railways - is the backbone of a functional and healthy community. When parts of this infrastructure are threatened, it is critical for society to respond to that threat or risk loss of life and property. One of the most significant threats to our infrastructure in recent U.S. history has been as the result of hurricanes, most memorably Hurricane Katrina, which devastated New Orleans, and Hurricane Sandy, from which the infrastructure of New York City is still recovering from.  Likewise, the power grid failure in the U.S. Northeast and Canada in 2003, exhibited an underlying and poorly understood weakness in a vital system that we depend on daily.  In this talk we will discuss ongoing work, on several fronts, using mathematics and computing to model and plan for extreme events such as those described above.

The analysis of energy systems works on many time scales, spanning fractions of a second to decades.  In this talk I will describe a series of models that span seconds to a year.  The first, SMART-Invest, is policy model that optimizes investments in renewables and storage using a yearly model that captures wind and solar variations in hourly increments.  SMART-Invest can model the full dispatch stack, but does not capture the grid, and does not schedule individual generators.  Next, SMART-ISO is a highly detailed simulator of the stochastic unit commitment process, simulating week-long periods in 5-minute increments using a process that closely matches the nested planning processes used by PJM.  SMART-ISO has a full model of the grid, and schedules fast and slow generators using actual notification times, while capturing the dynamics and uncertainty of renewables.  SMART-Storage uses the unit-commitment decisions from SMART-ISO, but co-optimizes generators and grid level storage over a period of a day to a week, in five-minute increments.  Finally, our newest model co-optimizes a battery being used simultaneously for frequency regulation (responding to signals every 2 seconds) and energy/peak shifting which requires horizons of a day or more.  All of these models represent a challenging form of stochastic optimization, since the models have to balance making robust decisions in the presence of uncertainty, while modeling the complex dynamics of a physical system.  We present a modeling framework that blends four classes of policies that integrate all the fields of stochastic optimization, and show how each time scale draws on the capabilities of different classes of policies. 

In this talk, I will report on projects related to Mathematics of Planet Earth that were done by undergraduate students at Georgetown University under my direction during the last four years. These projects typically last two to six months and involve analytical and computational work together with visualization, and sometimes statistical analysis. Topics include an analysis of ridership patterns in an urban bikeshare system, a study of the well-known Lorenz'96 model for a one-dimensional planet, an exploration of a model for vegetation patterns in drylands, and work on extrapolation and interpolation of climate extreme indices that were obtained from ensembles of general circulation models. I shall report on characteristics of such short projects, choices of topics, and typical student trajectories leading up to the project and beyond.     

Eight modules demonstrating the role of the mathematical sciences in issues of sustainability were developed as part of the original MPE2013 project. These modules were designed for undergraduate mathematics at the first and second year level. In May 2016, a joint conference was held at Catawba Valley Community College in North Carolina and Northampton Community College in Pennsylvania to discuss how to introduce MPE to community colleges. A recommendation of the community college participants was to rework the modules to better fit within the community college classroom environment. This talk presents the efforts made to edit and revise the MPE modules so they reflect the needs of the community college classroom.

I will begin with several examples of ecological systems that raise important issues of management approaches: invasive species and coral reef ecosystems.  I will then present general approaches that can be used to address management problems for these systems as well as other systems.  Themes will include the role played by simple models on the one hand, and, (1) the importance of recognizing nonlinearities, (2) the possibility of tipping points, (3) importance of transients (4) large stochastic influences (5) conflicting goals and (6) ecological limitations.

In this talk, we present an interactive web-based real-time decision support suite, RealOpt©. ÂŒe system integrates visualization, information and cognitive analytics, and dynamic large-scale computational modeling and optimization tools that allow public health emergency preparedness coordinators to determine optimal response facilities and locations, resource needs and supply-routes, and population ƒow in real time. With design flexibility and future system expansion, RealOpt© is designed in modular format allowing direct linkage to multiple functional modules. Currently, the system has twelve modules covering emergency response preparedness and operations including biological, chemical, radiological/ nuclear incidents, biosurveillance, epidemiology, and decontainment models, operations logistics and networks, and real-time crowd sourcing data feed, and evacuation. RealOpt© has been used for biodefense and H1N1 regional planning and operations, regional flood and hurricane responses, 2010 Haiti earthquake disaster relief, 2011 Japan Fukushima disasters, 2014-2015 Ebola containment assistance and after-event public health preparedness training in West Africa, and 2016-17 Zika response and containment. ÂŒe fast solution engines enable its real-time use for rapid decision and scenario analysis, since it requires only a CPU minute to determine an optimal network of facilities and resource needs to serve a population of over 10 million.

Bacteria remineralize dissolved organic carbon in the ocean, thereby influencing the cycles of carbon and other nutrients. To utilize organic substrates, bacteria must perform extracellular degradation of polymers, exploiting either dissolved substrates in the bulk environment or using motility to colonize particles and gels. Here we develop a theoretical model to study the co-evolution of motility and degradation traits in marine heterotrophic bacteria. With this model, we link evolutionarily stable trait combinations to properties of the organic matter pool. We find that the diversity of organic matter, i.e. the number of distinct compounds present in the organic polymer pool is a critical parameter controlling the ecology and kinetics of organic matter consumption. As diversity progressively increases, there is an evolutionary bifurcation of the population from small, non-motile bacteria which degrade polymers primarily in the bulk, to large, motile bacteria which colonize and particulate matter and gels. We show that when monomers are modeled as a leaky public good, there should be coexistence between non-motile and motile bacteria, with the latter responsible for degradation. We hypothesize that the ecological role of motility is for bacteria to colonize persistent gel particles, and that this also explains the correlations between motility and other copiotrophic traits in marine bacteria.
To further explore the influence of the gel-decomposition paradigm on organic matter, we use polymer physics to study the kinetics of gel formation and dispersion as a function of polymer length, and make predictions about the age-size spectrum of dissolved organic polymers.

Many populations are spatially-structured and exist as metapopulations, with distinct patches connected through movement and exchange of individuals. Understanding the dynamics of metapopulations and how they persist is necessary for managing natural resources but particularly challenging in the marine environment, where much of the connectivity takes place through tiny larvae. A metapopulation can persist either by individual patches retaining enough of their own offspring to exist in isolation – termed self-persistence – or through exchange among patches that provides sufficient recruits for a patch to persist within the network – called network persistence. Though there has been much theoretical development of persistence mechanisms, they have not been well tested in empirical systems.
We use a model system of yellowtail clownfish (Amphiprion clarkii) in the Philippines to understand connectivity and persistence in an empirical marine metapopulation. Clownfish are particularly well-suited to metapopulation studies because they have limited movement as adults, a relatively short pelagic larval duration, and distinct habitat patches at a range of scales in our study site. Using mark-recapture and genetic parentage methods, we seek to estimate survival, reproduction, and dispersal within this system to understand and characterize persistence mechanisms.
The 17 subpopulations at our study site have persisted throughout seven sampling years with stable abundance and similar size distributions throughout that time, suggesting the overall metapopulation is persistent. Our preliminary findings support the potential for self-persistence for some of the larger subpopulations, but network persistence is likely required for overall metapopulation persistence in this system. By understanding persistence in this system in detail, we can determine what lessons and methods might be useful in tackling more complicated systems, like many populations of fished species.

The adaptive potential of metapopulations under climate change will depend on both ecological and evolutionary processes that occur on complex networks. However, most work has assessed the population-level effects of these processes in isolation within a single patch or along relatively simple landscapes. We focus on corals as our study system, where populations can be linked across patches that are tens to thousands of kilometers apart through hydrodynamically driven larval dispersal. Here, we use an eco-evolutionary framework to quantify the relative contribution of dispersal and directional selection to coral adaptive potential across different types of coral reef seascapes. Within each patch, we modeled the competitive dynamics of three functional groups: a fast-growing coral, a stress-tolerant coral and macroalgae. We then simulated reef metacommunity dynamics under different temperature increase scenarios and quantified coral persistence within networks that can be classified as regular, random or small-world. We find that evolutionary capacity is the main driver of coral population persistence across all network types, such that coral populations are unable to persist without some genetic variation. We also show that larval production and seascape configuration are secondary drivers and thus modulate the effects of evolution on the ability of coral populations to persist under projected future warming. Though we specifically modeled coral reefs, our work broadly applies to other communities composed of a network of patches, especially those that are demographically linked by complex dispersal patterns.

Voluntary vaccination coexists with mandatory vaccination on a global and even national scale. For example, most states in the US use voluntary vaccination for MMR (a combined vaccine against Measles, Mumps and Rubella); but in Mississippi, West Virginia and California, MMR is a mandatory requirement for school entry and only medical exemptions of MMR are allowed. This project tries to compare these two strategies from the perspective of epidemiology and economic cost. Specifically, with voluntary vaccination, individual decisionmaking depends on the perceived payoffs of disease and vaccines rather than the real ones. Population structure on which information and diseases spread and the ability of processing information are two factors leading to the difference between perceived and real payoffs. Here, network and prospect theory are used to capture features of these two factors, respectively. For well-mixed populations, the voluntary strategy gives rise to an endemic, non-vaccinator equilibrium, and the mandatory strategy takes more time to reach the stable state with endemic disease and high vaccination coverage. Even though the voluntary strategy costs less than the mandatory one when populations stabilize, the former leads to much more cumulative cost during the evolutionary process. In addition, people can move easily and freely, and regions with these two opposite strategies are not isolated. The coupling of these two strategies by migration of people in both well-mixed and structured populations is an ongoing study.

Land conversion and the consequential habitat fragmentation has arguably been the single largest contributor to disease emergence within the past century. As the world population grows, and the needs and demands for goods and services continue to increase, so does the current necessity for land conversion; however, an unintended consequence of this process is zoonotic disease spillover to humans and  domesticated species. This research builds theory about the role that habitat fragmentation has on the spread of infectious diseases via edge effects. It combines metrics in landscape ecology with theoretical epidemiological models to understand how the shape and size of a habitat fragment on a landscape affects zoonotic disease transmission. Equilibrium stability and bifurcation analysis of the general model are conducted, and numerical simulations are performed to make implications for control strategies.

It is easy to lose confidence in the capacity for human social and political systems to respond effectively to the challenges from rising average global temperature and associated climate change. A working group of diverse researchers with backgrounds in mathematical modeling, climate science, psychology, sociology, geography and ecology has addressed the question as to whether there is any rational basis to expect that human behavioral changes can sufficiently impact climate to significantly reduce future mean global temperatures. Climate models can easily make assumptions about reductions in future greenhouse gas emissions and project the implications, but they do this with no rational basis for human responses. We have built this rational basis by developing a model based on a set of standard assumptions from the psychology literature in the theory of planned behavior, linking these to extreme events obtained from a climate model, and then allowing feedback to global emissions in a climate model on a yearly basis as affected by human behavior. The key result from this is that there is indeed some rational basis for hope, whereby a meaningful reduction of future average temperature occurs under circumstances in which mitigation arises from infrastructure changes which allow for cumulative impacts.

Interest in models that acknowledge the two-way coupling between human and environmental systems has grown steadily over the past few decades, and has accelerated, especially in recent years. Nevertheless, we still face many catastrophes that, in theory, may be avoidable. I will discuss simple mathematical models of human and environmental systems with and without two-way coupling to highlight this dilemma. In particular, I will highlight the vital roles of social learning and social norms as well as economic and non-economic factors, interdependently. I will conclude with a few words about future directions in this area. This presentation is based largely on joint work of the Anand and Bauch research groups.

Humans have had and are having major impacts on their environments, and one cannot consider management of natural systems without considering the coupling with socio-economic systems (Arrow, K.J.; Ehrlich 2014).  Global environmental problems in general present problems of the commons, whose solutions require “mutual coercion, mutually agreed upon.”(Hardin 1968)  However, as Ostrom showed (Ostrom 1990), the most effective approaches to this are typically bottom-up, involving social norms, insurance arrangements and other collective dynamics. (Nyborg et al. 2016; Levin 2014).   These pose unique mathematical problems with broad importance, and will be the focus of the lecture.

Arrow, K.J.; Ehrlich, P.R. ; and S.A. Levin. 2014. “Some Perspectives on Linked Ecosystems and Socio-Economic Systems.” In Environment and Development Economics: Essays in Honor of Sir Partha Dasgupta, Ed. S. Barrett et Al., Springer-Verlag, 95–116.

Hardin, G. 1968. “The Tragedy of the Commons.” Sciences 162: 1243–48.

Levin, S A. 2014. “Public Goods in Relation to Competition, Cooperation, and Spite.” Proceedings of the National Academy of Sciences of the United States of America 111: 10838–45.

Nyborg, Karine, John M. Anderies, Astrid Dannenberg, Therese Lindahl, Caroline Schill, Maja Schlüter, W. Neil Adger, et al. 2016. “Social Norms as Solutions.” Science. doi:10.1126/science.aaf8317.

Ostrom, E. 1990. Governing the Commons: The Evolution of Institutions for Collective Action. Cambridge: Cambridge University Press.

The agricultural establishment has made significant progress in its efforts to improve agricultural productivity and efficiency. Yet, with about one billion people hungry, two billion people with insufficient nutrients, and over two billion people already overweight or obese, undernutrition and malnutrition are affecting more than half the world's population. Clearly, when enough food is produced but sizable fractions of the population suffer from malnutrition or are overweight, we need to get a better understanding of the global food system.

In this talk, we give an overview of recent efforts at modeling food systems using concepts from the theory of complex systems and highlight research issues for MPE.

The numerical modeling of the Earth system comprising the components of atmosphere, ocean, biosphere and cryosphere, is based on mathematical formulations which arise from fundamental principles in physics, chemistry, and dynamics. The principles eschew the governance of the conservation laws of energy, momentum, and mass. The components have significant differences in their characteristic timescales such that the interactions across the components lead to a host of important phenomena ranging from the weather to seasonal to annual to decadal and longer timescales. A particularly important element is the natural variations of the coupled system superposed on which are key factors forcing changes in the variables. An example is the anthropogenic climate change e.g., increases in greenhouse gases. The combination of variability and the trend imposed by forced climate change has given rise to the concept of non-stationarity of the Earth system, with anomalously aperiodic excursions in variables across timescales signifying the occurrence of extremes that can be of serious societal concern (e.g., droughts, heat waves, hurricanes). An additional challenge in mathematical representations in the model and quantification of climate state is the uncertainties in the processes that stem from gaps in basic knowledge e.g., convection, clouds, and aerosol pollution. How these uncertainties affect the confidence in the computational simulations is becoming an important aspect of understanding the Earth’s climate sensitivity and future climates.    

Climate change is known to significantly affect the dynamics of vector-borne diseases, such as malaria.  In particular, the species involved in the transmission dynamics of malaria are affected by various abiotic conditions, such as temperature, precipitation, humidity and vapor pressure . A number of models, typically statistical (using data and statistical approaches to correlate some climate variables with malaria incidence) or mechanistic (accounting for the detailed dynamic nonlinear processes involved in disease transmission), have been employed to assess the likely impact of anthropogenic climate change on malaria transmission dynamics and control.  These models have (generally) reached divergent conclusions, with some predicting a large expansion in the continental land area suitable for transmission and in the number of people at risk of malaria, while others predict only modest poleward (and altitudinal) shifts in the burden of disease, with little net effect , and the issue remains unresolved thus far.  This talk addresses some of the key challenges (and some recent results) associated with the mathematical modeling of the effect of climate change on malaria transmission dynamics.

Forecasting the impacts of climate change on Aedes-borne viruses—especially dengue, chikungunya, and Zika—is a key component of public health preparedness, and provides a framework for thinking of a suite of vector borne diseases in a world of changing climate. We applied an empirically parameterized Bayesian model of Aedes-borne viruses as a function of temperature to predict current cumulative monthly global transmission risk, and projected risk in 2050 and 2070 based on general circulation models (GCMs). Our results show that shifting suitability will track optimal temperatures for transmission (26-29 °C), potentially leading to poleward shifts. Furthermore, especially for Ae. albopictus, extreme temperatures are likely to limit transmission risk in current zones of endemicity, especially the tropics. Due to the uncertainty and variation in both mechanistic and climate models, the range of potential risk outcome is in the order of a billion people. Importantly, the geographic shift in suitable temperatures as a result of climate change predictions allows for people currently at risk to entirely escape transmission, while new individuals in new areas are put at risk. We present synoptic descriptions of this range of predictions, to facilitate communication of the ranges created by climate model input variations.