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## Workshop 4: Multiscale Dynamics of Infections

### Organizers

Rebecca Garabed
Veterinary Preventive Medicine, The Ohio State University
Juan B. Gutierrez
Mathematics, University of Georgia
Grzegorz Rempala
Biostatistics, OSU

Modern methods of genomic data collection reveal a great heterogeneity and diversity among strains of pathogens with different genotypes infecting hosts with significant differences in virulence, immunogenicity, and antigenic variation on a micro-scale (Workshop 1). The presence/persistence of specific different sub-groups of such pathogens depends heavily on macro-scale interactions of various human and animal hosts, travel patterns, environment, and intervention strategies (Workshop 2-3). In order to properly understand the main drivers of transmission of infectious disease, these ecological, molecular, and immunological factors need to be analyzed together, and their joint correct characterization requires a comprehensive interdisciplinary and multi-scale modeling approach. We expect that different infection outcomes are the result of the interplay of events at organ tissue cellular and molecular as well as ecosystem scales over the time course of minutes to years. A recent example is the Zika virus epidemic where a 3-5 days infection of a pregnant woman can result in birth defects lasting for the newborn’s lifetime. In order to study such problems, the unifying modeling framework for simultaneously analyzing multiple scales of empirical information from the level of single cells to organs, organisms, population and ecosystems appears necessary.

However, integrating the information from the micro- and macro-scales presents a great scientific challenge as the data on different scales is generated by different mechanisms and comes with different characteristics and uncertainties. In fact, integration across scales can be seen as the fundamental challenge for biology in the 21st century. The mathematical challenges raised are truly substantial, as the integration requires dealing with multiple temporal and spatial scales, as well as organizational scales. If scales in time and space are well separated, there exist, at least for deterministic systems, methods based on perturbation approaches that can be used. However, even these approaches are difficult to apply to systems with the kinds of underlying heterogeneities that will be found in systems with diseases. There are methods for averaging over heterogeneities but these methods as well have limitations. And, the different time scales of observations and the different framework that models at different scales use present substantial challenges.

This workshop will bring together scientists from multiple disciplines, to exchange ideas about new perspectives for the quantification of within-host dynamics and between-host transmission of infectious disease as well as integration over multiple temporal and spatial scales. Approaches that consider how to integrate approaches at different scales that range from deterministic to stochastic to computational will be a central theme, including how to incorporate data and how to deal with situations with limited data. It is assumed that a representative subset of participants in Workshops 1-3 will also participate. In the context of some of the problems Participants will discuss novel molecular and ecological data that has become available ('omic, clinical, entomological, and epidemiological data), and will discuss modeling perspectives that will allow their integration beyond traditional epidemiological models of transmission with the goals of improving public health practice and policies.

### Daily Themes

 Monday Molecular Scale Tuesday Geographic Scale Wednesday Physiologic Scale Thursday Aggregations Across Scales and Multi Scale Friday Roundtable Discussion

### Accepted Speakers

Julien Arino
Mathematics, University of Manitoba
Neda Bagheri
Chemical & Biological Engineering, Northwestern University
Caleb Bastian
Princeton University
Lindsay Beck-Johnson
Department of Biology, Colorado State University
Lauren Childs
Mathematics, Virginia Polytechnic Institute and State University
Winston Garira
Department of Math. & Applied Math. C/o Modelling Health and Environmental Linkages Research Group (MHELRG), University of Venda - South Africa
Hayriye Gulbudak
Mathematics, University of Louisiana at Lafayette
Juan B. Gutierrez
Mathematics, University of Georgia
James Hyman
Mathematics, Tulane University
Renata Ivanek
Population Medicine and Diagnostic Sciences, Cornell University
Anna Jolles
College of Veterinary Medicine / Department of Integrative Biology, Oregon State University
Eben Kenah
College of Public Health, The Ohio State University
Michael Kirby
Cristina Lanzas
Population Health and Pathobiology, North Carolina State University
Daniel Linder
Biostatistics, Georgia Health Sciences University
Tom Lindström
IFM,
Maia Martcheva
Mathematics, University of Florida
Sean Moore
Biological Sciences, University of Notre Dame
Todd Parsons
Laboratoire de Probabilités, Statistique et Modélisation, Sorbonne Université
Laura Pomeroy
Veterinary Preventive Medicine, Ohio State University
Grzegorz Rempala
Biostatistics, OSU
Eberhard Voit
Dept. of Biomedical Engineering, Georgia Tech and Emory University
Monday, April 23, 2018
Time Session
08:45 AM

Shuttle to MBI

09:00 AM
10:00 AM

Breakfast

10:00 AM
10:30 AM

Welcome Overview and Introduction

10:30 AM
11:15 AM
Eberhard Voit - Dynamic Models of Malaria

Malaria is a complex disease that afflicts 200 million individuals worldwide and causes about 500,000 deaths per year. It is transmitted by Anopheles mosquitoes and caused by one of several Plasmodium species, which have a complicated life cycle that is distributed between the mosquito and the human host. The molecular events, host-pathogen interactions, physiological host responses, and the global reach of the disease create a truly multi-scale system. I will present five vignettes describing mathematical models of molecular and cellular events during malaria infections in non-human primates, which offer excellent models of the human disease processes. Three vignettes will focus on different aspects related to the dynamics of red blood cells and reticulocytes, whose disappearance is the dominant driver of malarial anemia, one of the most severe hallmarks of the disease. The other two vignettes will demonstrate how insights into the meaning of transcriptomic changes during the disease can be gained through models of concomitant processes at the metabolic and physiological levels.

This work is joint work with Luis L. Fonseca and Anuj Gupta.

11:15 AM
12:00 PM
Caleb Bastian - Transition Dynamics Convey Key Programs in Cellular Populations

The epithelial-mesenchymal transition (EMT) is a key cellular program of growth and development and of mature cells, such as wound healing. Evidence for its role in pathology, such as cancer invasion, has been accumulating following from intensive study, yet little is known about the dynamics of the transition per se. Using a multidisciplinary approach, this talk explores the dynamics of the transition and its role in conveying distinct gene programs. The global spatiotemporal distribution of metastatic (dysregulatory) burden is found to be critically sensitive to key non-linear dynamics.

12:00 PM
02:30 PM

Lunch Break

02:30 PM
03:15 PM
Neda Bagheri - Hybrid models predict emergent dynamics of multiscale cell populations

Computational models are essential tools that can be used to simultaneously explain and guide biological intuition. With increasingly high-resolution, high-throughput, and dynamic experimental data, computational biologists are better equipped to develop informed models that to characterize complex cellular responses and direct experimental design. I introduce an agent-based model as an intuitive, modular, and flexible framework to study emergence of heterogeneous cell populations. We use this framework to interrogate the inherent multiscale nature of cells—reinforcing how “the whole is greater than the sum of its parts”—and to predict cell population dynamics from the composition of simpler biological modules. Elucidating the compositionality problem is fundamental to advancing our understanding of basic science; to promoting the impact of synthetic biology; and to designing precise dynamic therapeutic strategies.

03:15 PM
04:00 PM
Todd Parsons - The population genentics of pathogen virulence

Life history theory provides a powerful framework to understand the evolution of pathogens in both epidemic and endemic situations. This framework, however, relies on the assumption that pathogen populations are very large and that one can neglect the effects of demographic stochasticity. In my talk, I will present an alternative approach, based in population genetics, which will explore the effects of finite population size on the evolution of pathogen virulence and transmission. I will show that demographic stochasticity introduces additional evolutionary forces that can affect qualitatively the dynamics and the evolutionary outcome. In particular, I will discuss scenarios where finite population size can either select for lower or higher virulence.

04:00 PM
04:45 PM

Panel Discussion

05:00 PM
07:00 PM

Reception and Poster session in MBI Lounge

Tuesday, April 24, 2018
Time Session
08:00 AM

Shuttle to MBI

08:15 AM
09:00 AM

Breakfast

09:00 AM
09:45 AM
Tom Lindström - Challenges when building a continental scale livestock disease spread model: Animal movements, farm locations, and computation

Applied epidemiological modeling offers powerful tools to inform policy decisions regarding control actions, identify spatial hotspots, or predict the course of an outbreak. However, modeling is often challenged by limited information about the system. For instance, in the USA, there are no federal databases with animal movements that can be used to model disease spread via these contacts. I will present some of our ongoing work to circumvent this issue by developing the United States Animal Movement Model (USAMM), which is parameterized based on interstate certificates of veterinary inspection and can be used to predict movement across the USA, including within state shipments.

In an international collaboration with researchers from Linkping University (Sweden), Colorado State University (USA), United States Department of Agriculture, and University of Warwick (UK), we connect USAMM predictions to a disease spread model (United States Disease Outbreak Simulator, USDOS), which also accounts for local spread, to simulate outbreak of livestock diseases. This faces another challenge: where are the farms? As with animal movements, there are no federal databases with farm locations, and the most detailed information available is census data from the National Agricultural Statistics Service (NASS) at the county level. Using predictions based on the Farm Location and Animal Population Simulator (FLAPS), I will present some of our work focused on determining whether fine-scale farm distribution matters when modeling disease outbreaks on the US farm population, using Foot and Mouth Diseases as an example. For this purpose, we use both analytical methods and stochastic simulations. The latter involves simulations among nearly 850000 farms, and efficient computation is crucial. I will also present an approach to speed up simulations for spatially explicit disease simulations.

09:45 AM
10:30 AM
Lindsay Beck-Johnson - Exploring the role of geographic scale on foot and mouth disease outbreaks

Foot and mouth disease (FMD) is a viral infection of cloven-hoofed animals, including livestock species such as cattle, pigs, and sheep. FMD is highly transmissible and outbreaks in non-endemic countries are very expensive. Therefore, there is interest from policy makers in understanding potential outbreaks and identifying ways to reduce losses. The United States Disease Outbreak Simulation (USDOS) is a simulation model that allows for exploration of national-scale infection patterns and control of potential U.S. FMD outbreaks in cattle. USDOS is a premises-level model, incorporating information on both local spread and long-distance transmission events, which are informed by cattle shipment networks. In this presentation, I will discuss how the national scale outcomes of potential FMD outbreaks in the U.S. are influenced by local, regional, and national factors and how these may be altered by control actions.

Different FMD outbreak models incorporate geographic scale in different ways and in systems, like FMD, where there are multiple quality models, it can be difficult for policy makers to select a single model on which to base high-stakes decisions. Here I will present an international collaborative project on the development of a multi-model ensemble that includes four different FMD models that are commonly used in policy. Ensemble modeling methods provide a standardized and transparent way of producing a single, interpretable projection from multiple model outputs. Here we use the multi-model ensemble and data from the initial weeks of the 2001 U.K. FMD outbreak to explore whether the method could improve the accuracy of the model predictions early in an epidemic before the outcome of the outbreak is known.

10:30 AM
10:45 AM

Break

10:45 AM
11:30 AM
Laura Pomeroy - Modeling endemic foot-and-mouth disease in Cameroon

Foot-and-mouth disease virus (FMDV) threatens animal health and leads to considerable economic losses worldwide. Endemic disease dynamics occur throughout Asia and Africa; however, most models of infectious disease dynamics represent locations throughout North America, Europe, and Australia that experience epidemic dynamics or have been FMD-free for decades. This dichotomy has resulted in a void in our understanding of natural FMD dynamics. Therefore, we investigated disease dynamics in the Far North Region, Cameroon, where FMDV is endemic and five serotypes have been detected in cattle. First, we characterized and quantified serotype-specific transmission and immunodynamics in cattle. Second, we investigated if pastoral mobility plays a role in endemic maintenance. This work underscores the importance of using empirical data in models to understand processes that influence the transmission of infectious diseases at multiple scales.

11:30 AM
12:15 PM
Julien Arino - Multiscale challenges in modelling the initial spatial spread of an infectious pathogen

In a world where global travel and movement of goods is the norm, understanding the initial phase of spread of an infectious pathogen is paramount. Indeed, determining which locations are the potential "next targets" of a budding epidemic helps public health authorities allocate resources more effectively, which can in turn bring the nascent event under control before it reaches epidemic scale. However, the initial spatial spread is governed by processes occurring at a variety of scales which act in the origin location, during transport and in the destination location. In this talk, I will discuss some of the issues arising in this context and present some (very partial) methodology that has been developed to address these issues.

12:15 PM
02:30 PM

Lunch Break

02:30 PM
03:15 PM
Eben Kenah - Pairwise survival analysis: Contact intervals, regression, and phylogenetics

When integrating epidemiologic data with pathogen phylogenetics, the likelihood for the transmission model is often a branching-process likelihood based on a generation interval distribution and offspring distribution. We show that a misspecified likelihood can lead to severely biased estimates with or without a pathogen phylogeny. Writing the likelihood as a survival likelihood with failure times in pairs---a process we call pairwise survival analysis---accounts for time spent at risk of infection. In a simple example with three infections, we show that a pairwise survival likelihood produces more accurate source attribution and parameter estimation. In a mass-action model with negligible depletion of susceptibles, the pairwise survival likelihood depends only on information about infected individuals in the limit of a large population. However, this asymptotic likelihood has cumulative hazard terms that have no counterpart in a branching process likelihood. As an example of the flexibility of pairwise survival analysis, we describe a pairwise accelerated failure time model that can be used to estimate covariate effects on infectiousness and susceptibility. This model---modified to account for the buildup of immunity---will be used to estimate the efficacy of the Ebola vaccine based on the WHO ring vaccination trial in Guinea. This trial collected data on individuals exposed to infection who escaped as well as Ebola virus genetic sequences. Finally, we describe a pruning algorithm for calculating an approximate likelihood using both epidemiologic data and a pathogen phylogeny. Pathogen genetics can improve statistical efficiency and reduce bias, but this depends on good epidemiologic study design and a good likelihood for transmission.

03:15 PM
04:00 PM
Sean Moore - Local and regional dynamics of arbovirus transmission: the role of mismatched spatial heterogeneity

Background: Mathematical models of transmission dynamics are routinely fitted to epidemiological time series, which must inevitably be aggregated at some spatial scale. Weekly case reports of chikungunya have been made available nationally for numerous countries in the western hemisphere since late 2013, and numerous models have made use of this data set for forecasting and inferential purposes. Motivated by an abundance of literature suggesting that the transmission of this mosquito-borne pathogen is localized at scales much finer than nationally, we fitted models at three different spatial scales to weekly case reports from Colombia to explore limitations of analyses of nationally aggregated time series data.

Methods: We adapted the recently developed DTK-Dengue model for modeling chikungunya virus (CHIKV) transmission, given the numerous similarities of these viruses vectored by a common mosquito vector. We fitted versions of this model specified at different spatial scales to weekly case reports aggregated at different spatial scales: (1) single-patch national model fitted to national data; (2) single-patch departmental models fitted to departmental data; and (3) multi-patch departmental models fitted to departmental data, where the multiple patches refer to municipalities within a department. We compared the consistency of simulations from fitted models with empirical data.

Results: We found that model consistency with epidemic dynamics improved with increasing spatial granularity of the model. Specifically, the sum of single-patch departmental model fits better captured national-level temporal patterns than did a single-patch national model. Likewise, multi-patch departmental model fits better captured departmental-level temporal patterns than did single-patch departmental model fits. Furthermore, inferences about municipal-level incidence based on multi-patch departmental models fitted to departmental-level data were positively correlated with municipal-level data that were withheld from model fitting.

Conclusions: Our model performed better when posed at finer spatial scales, due to better matching between human populations with locally relevant risk. Confronting spatially aggregated models with spatially aggregated data imposes a serious structural constraint on model behavior by averaging over epidemiologically meaningful spatial variation in drivers of transmission, impairing the ability of models to reproduce empirical patterns.

04:00 PM
05:00 PM

Panel Discussion

05:00 PM

Shuttle pick-up from MBI

05:00 PM
07:00 PM

FMD Modeling Group Working Dinner

Wednesday, April 25, 2018
Time Session
08:00 AM

Shuttle to MBI

08:15 AM
09:00 AM

Breakfast

09:00 AM
09:45 AM
James Hyman - Multiscale Modeling the Transmission of Infectitious Diseases

TBD

09:45 AM
10:30 AM
Anna Jolles - A pandemonium of parasites: predictability in the face of complex parasite interaction networks?

Human activities are altering host-parasite interactions and their ecological context at unprecedented rates, reducing our ability to anticipate disease impacts and predict transmission dynamics in host populations of interest. These changes lend urgency to understanding the mechanisms underlying variation in disease dynamics, by developing data-driven models that scale up from within-host processes affecting parasite invasion and persistence, to population-level disease dynamics.

In this talk, I focus on interactions among co-infecting parasites as drivers of disease outcomes and dynamics. Using African buffalo and their parasite community as a model system, I demonstrate that co-infecting parasites contribute strongly to incidence and duration of concurrent infections, perhaps forming a modular interaction network. I propose trait-based community analysis as a possible approach for increasing the predictability of outcomes of parasite invasions, using Mycobacterium bovis, the causative agent of bovine TB, as a case study. M. bovis infection leads to changes in parasite infracommunity structure in buffalo, but few other parasites seem to affect the incidence of M. bovis itself. Instead, genetically driven variation in host defense responses against M. bovis underlies variation in infection risk. Linking ecological processes with evolutionary responses, I discuss how co-infection might be shifting both host and pathogen population genetic structure – an area in need of further investigation.

10:30 AM
10:45 AM

Break

10:45 AM
11:30 AM
Cristina Lanzas - Transmission and infection dynamics of Clostridium difficile

Clostridium difficile is an anaerobic human pathogen that forms spores, produce toxins and resides in the gut. Clostridium difficile infection (CDI) is an important hospital acquired infection that causes diarrhea, pseudomembranous colitis, and possibly death. In the last decade, the incidence and severity of C. difficile infection has increased at alarming rates throughout North America, especially in the elderly. Antimicrobial therapy is often a strong and independent risk factor for CDI because it disrupts the indigenous microbiota, which provides protection against C. difficile colonization. Epidemiological and within host models for Clostridium difficile and integration of both scales will be discussed.

11:30 AM
12:15 PM
Renata Ivanek - Epidemiology of environmentally transmitted infections: models and empirical data

For environmentally-transmitted infections the contaminated environments, such as food, water, soil, objects, and contact surfaces, represent vehicles for transmission of the infection to susceptible hosts. The causative agents of these diseases exhibit rich dynamics in the host and in the environment and require modeling and empirical studies at both the host and pathogen scales. This presentation will describe what we have learned about the spread of these pathogens between the host populations and their environments, and about the pathogens’ dynamics in the heterogeneous environments. The focus will be on findings of compartmental and agent based models and selected empirical studies. Several human/animal pathogens, including Salmonella, pathogenic Escherichia coli and Listeria, will be used as examples. Finally, the existing knowledge gaps in terms of modeling and empirical data to support or validate models will be discussed.

12:15 PM
02:30 PM

Lunch Break

02:30 PM
03:15 PM
Michael Kirby - Modeling the temporal evolution of the host immune response to infection

We will present two explorations into to the temporal dynamics of the host immune response. The first concerns the real-time modeling telemetry data generated by collaborative cross (CC) mice. The CC mice are monitored in a healthy state for seven days and are subsequently challenged with a salmonella infection. Temperature and activity data is collected at one minute intervals and modeled in real time using a method for time-series prediction that also serves as a novelty detector to quantify time to symptoms and full blown infection. The analysis seeks to identify and characterize aspects of tolerance to high pathogen loads exhibited by certain CC lines. We are also exploring dynamical systems models to explain this phenomenon. The second study relates to the early prognosis of subject outcome in human challenges to influenza viruses. We demonstrate algorithms that can predict if a subject will become symptomatic, or not, as early as 5 hours after exposure. (The mice study is being conducted in collaboration with David Threadgill and Helene Andrews-Polymenis of Texas A&M. Aspects of the dynamical modeling are recent collaborations with Judy Day (UTK) and Helene Andrews-Polymenis.)

03:15 PM
04:00 PM
Grzegorz Rempala - Micro and macro SIR models

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.

04:00 PM
05:00 PM

Panel Discussion

05:00 PM

Shuttle pick-up from MBI

Thursday, April 26, 2018
Time Session
08:00 AM

Shuttle to MBI

08:15 AM
09:00 AM

Breakfast

09:00 AM
09:45 AM
Juan B. Gutierrez - Multiscale Systems Biology: A Case Study Linking Molecular Dynamics to Epidemiological Processes of Malaria

The advent of high-throughput molecular technologies in particular, and the broad availability of data, in general, have forced the quantitative biology community to rethink how to conceive, build, and validate mathematical models. In this talk I will demonstrate how molecular and cellular processes are related to the epidemiology of malaria. We will explore (i) asymptomaticity at the epidemiological level, (ii) telemetry analysis identifying a systemic response to the disease before traditional symptoms show, (iii) the cellular models that explain this phenomenon as an interaction between the immune system and infected red blood cells, (iv) mathematical models that link cellular and transcriptional time series, (v) transcriptomic analysis, and finally (vi) high-throughput in silico drug discovery to solve an epidemiological problem. All these linked analyses provide a comprehensive picture that no single scale can produce alone. The usefulness of models under this light takes on new meanings, and this broad scope requires the cooperation of scientists coming from very different intellectual traditions. In this talk we will also explore how an information system that delivers Adaptive Learning for Interdisciplinary Collaborative Environments (ALICE) is used to train scientists in this new normal.

09:45 AM
10:30 AM
Maia Martcheva - Identifiability Issues of an Immuno-Epidemiological Model: The case of Rift Valley Fever Virus

We discuss the structural and practical identifiability of a nested immuno-epidemiological model of arbovirus diseases, where host–vector transmission rate, host recovery, and disease-induced death rates are governed by the within-host immune system. We incorporate the newest ideas and the most up-to-date features of numerical methods to fit multi-scale models to multi-scale data. For an immunological model, we use Rift Valley Fever Virus (RVFV) time-series data obtained from livestock under laboratory experiments, and for an epidemiological model we incorporate a human compartment to the nested model and use the number of human RVFV cases reported by the CDC during the 2006–2007 Kenya outbreak. We show that the immunological model is not structurally identifiable for the measurements of time-series viremia concentrations in the host. Thus, we study the scaled version of the immunological model and prove that it is structurally globally identifiable. After fixing estimated parameter values for the immunological model derived from the scaled model, we develop a numerical method to fit observable RVFV epidemiological data to the nested model for the remaining parameter values of the multi-scale system. For the given (CDC) data set, Monte Carlo simulations indicate that only three parameters of the epidemiological model are practically identifiable when the immune model parameters are fixed. Alternatively, we fit the multi-scale data to the multi-scale model simultaneously. Monte Carlo simulations for the simultaneous fitting suggest that the parameters of the immunological model and the parameters of the immuno-epidemiological model are practically identifiable.

Research performed in collaboration with Necibe Tuncer, Hayriye Gulbudak, and Vincent Cannataro.

10:30 AM
10:45 AM

Break

10:45 AM
11:30 AM
Hayriye Gulbudak - Vector-Borne Pathogen and Host Evolution in a Structured Immuno-Epidemiological System

Vector-borne disease transmission is a common dissemination mode used by many pathogens to spread in a host population. Similar to directly transmitted diseases, the within-host interaction of a vector-borne pathogen and a host's immune system influence the pathogen's transmission potential between hosts via vectors. Yet there are few theoretical studies on virulence-transmission tradeoffs and evolution in vector-borne pathogen-host systems. Here we consider an immuno-epidemiological model that links the within-host dynamics to between-host circulation of a vector-borne disease. On the immunological scale, the model mimics antibody-pathogen dynamics for arbovirus diseases, such as Rift Valley Fever and West Nile Virus. The within-host dynamics govern transmission and host mortality and recovery in an age-since-infection structured host-vector-borne pathogen epidemic model. By considering multiple pathogen strains and multiple competing host populations differing in their within-host replication rate and immune response parameters, respectively, we derive evolutionary optimization principles for both pathogen and host. Invasion analysis shows that the $mathcal R_0$ maximization principle holds for the vector-borne pathogen. For the host, we prove that evolution favors minimizing case fatality ratio (CFR). These results are utilized to compute host and pathogen evolutionary trajectories, and to determine how model parameters affect evolution outcomes. We find that increasing the vector inoculum size increases the pathogen $mathcal R_0$, but can either increase or decrease the pathogen virulence (the host CFR), suggesting that vector inoculum size can contribute to virulence of vector-borne diseases in distinct ways.

11:30 AM
12:15 PM
Winston Garira - A Categorization Framework for Multiscale Models of the Dynamics of Infections

The development of multiscale models of the dynamics of infections is a scientific endevour whose progress has so far resulted in the development of a wide variety of multiscale models of the dynamics of infections with different structure and mathematical representations which are associated with the different levels of organization of an infectious disease system (cell level, tissue level, host level, etc.). In this talk I will present a framework for categorization of multiscale models of the dynamics of infections. Such a categorizarion framework is useful in categorizing and classifying the different types of multiscale models of infection dynamics and in turn in bring some order to the discussion on the structure of multiscale models of infectious disease systems. This categorization of the different types of multiscale models of infectious disease systems may be found useful as a basis for further refinement, in the discourse of multiscale modelling of the dynamics of infections. It will enable infectious disease modelers to discuss the general principles that should apply to each category of multiscale models. Instead of discussing these principles repeatedly whenever a multiscale model of an infectious disease system is being developed, infectious disease modelers will be able to refer to the generic discussion for the category of the multiscale model concerned, and focus instead on the application of that discussion to the particular multiscale model of an infectious disease system being developed, and on issues peculiar to that multiscale model. While this categorization cannot be claimed to be unique, I think that it constitutes a good starting point, which may be found useful as a basis for further refinement in the discourse of multiscale modelling of the dynamics of infections.

12:15 PM
02:30 PM

Lunch Break

02:30 PM
03:15 PM
Lauren Childs - Incorporating within-host heterogeneity into transmission models

Mathematical models are a key tool in the study of the spread of infectious diseases such as influenza, malaria and dengue. In particular, transmission models have been successful at determining the most promising intervention strategies, despite the fact that many of these models assume all individuals experience identical infections. Here, we discuss the importance of incorporating heterogeneity into transmission models and consider when particular types of heterogeneities can reasonably be ignored.

03:15 PM
04:00 PM
Daniel Linder - Analysis of the Plague at Eyam

The rodent-flea-human transmission route of Y. pestis is widely known and has been medically confirmed to cause the bubonic form of plague. However, the observed rapidity of plague outbreaks during the second pandemic appear to have happened on timescales that are not readily explained by this transmission mechanism alone. We propose a statistical method to analyze such data that is capable of distinguishing the important mechanisms of outbreak dynamics. We implement the methodology on data from the latter half of the outbreak at Eyam in the summer of 1666.

04:00 PM
05:00 PM

Panel Discussion

05:00 PM

Shuttle pick-up from MBI

06:30 PM
07:00 PM

Cash Bar

07:00 PM
09:00 PM

Banquet in the Fusion Room @ Crowne Plaza Hotel

Friday, April 27, 2018
Time Session
08:00 AM

Shuttle to MBI

08:15 AM
09:00 AM

Breakfast

09:00 AM
10:00 AM

Summaries of the week and what scales are well developed/ well linked

10:00 AM
10:30 AM

Break

10:30 AM
11:30 AM

Thoughts on future directions and areas in need of development

11:30 AM
12:00 PM

Conclusions - Thank You

12:00 PM

Shuttle pick up (one to hotel, one to airport)

Name Email Affiliation
Aguilar, Jacob jacobaguilar@uga.edu Mathematics, University of Georgia
Al-Mamun, Mohammad mohammad.al-mamun@yale.edu Department of Epidemiology of Microbial Diseases, Yale School of Public Health, Yale University
Aminian, Manuchehr manuchehr.aminian@colostate.edu Mathematics, Colorado State University
Arino, Julien arinoj@cc.umanitoba.ca Mathematics, University of Manitoba
Bagheri, Neda n-bagheri@northwestern.edu Chemical & Biological Engineering, Northwestern University
Bastian, Caleb caleb.bastian@gmail.com Princeton University
Beck-Johnson, Lindsay L.Beck-Johnson@colostate.edu Department of Biology, Colorado State University
Bekele, Bewketu bewketu.bekele@univen.ac.za Mathematics and Applied Mathematics, University of Venda
Cazer, Casey clcazer@gmail.com Population Medicine and Diagnostic Sciences, Cornell University
Childs, Lauren lchilds@vt.edu Mathematics, Virginia Polytechnic Institute and State University
Faraimunashe, Chirove chirovef@gmail.com Mathematics, Statistics and Computer Science, University of KwaZulu-Natal
Fonseca, Luis llfonseca@gatech.edu Biomedical Engineering, Georgia Institute of Technology
Garabed, Rebecca garabed.1@osu.edu Veterinary Preventive Medicine, The Ohio State University
Garira, Winston Winston.Garira@univen.ac.za Department of Math. & Applied Math. C/o Modelling Health and Environmental Linkages Research Group (MHELRG), University of Venda - South Africa
Gonzalez, Karen kg67392@uga.edu Bioinformatics, University of Georgia
Gulbudak, Hayriye Mathematics, University of Louisiana at Lafayette
Gupta, Anuj anujg@gatech.edu BMED and Bioinformatics, Georgia Institute of Technology
Gutierrez, Juan B. jgutierr@uga.edu Mathematics, University of Georgia
Hyman, James mhyman@tulane.edu Mathematics, Tulane University
Islam, Md rafiul.islam@ttu.edu Mathematics and Statistics, Texas Tech University
Ivanek, Renata ri25@cornell.edu Population Medicine and Diagnostic Sciences, Cornell University
Jolles, Anna jollesa@science.oregonstate.edu College of Veterinary Medicine / Department of Integrative Biology, Oregon State University
Just, Winfried mathjust@gmail.com Department of Mathematics, Ohio University
Kenah, Eben College of Public Health, The Ohio State University
Kirby, Michael Kirby@math.colostate.edu Mathematics, Colorado State University
Lanzas, Cristina clanzas@ncsu.edu Population Health and Pathobiology, North Carolina State University
Linder, Daniel Biostatistics, Georgia Health Sciences University
Lindstrm, Tom tomli@ifm.liu.se IFM,
Martcheva, Maia maia@ufl.edu Mathematics, University of Florida
Moncada Giraldo, Diego Mauricio dmm21096@uga.edu Bioinformatics, University of Georgia
Moore, Sean smoore15@nd.edu Biological Sciences, University of Notre Dame
Osborne, Matthew Mathematics, The Ohio State University
Parsons, Todd todd.parsons@upmc.fr Laboratoire de Probabilités, Statistique et Modélisation, Sorbonne Université
Pomeroy, Laura Veterinary Preventive Medicine, Ohio State University
Qu, Zhuolin zqu1@tulane.edu Mathematics, Tulane University
Rempala, Grzegorz Biostatistics, OSU
Trippe, Elizabeth edt37727@uga.edu Institute of Bioinformatics, University of Georgia
Voit, Eberhard eberhard.voit@bme.gatech.edu Dept. of Biomedical Engineering, Georgia Tech and Emory University
Multiscale challenges in modelling the initial spatial spread of an infectious pathogen

In a world where global travel and movement of goods is the norm, understanding the initial phase of spread of an infectious pathogen is paramount. Indeed, determining which locations are the potential "next targets" of a budding epidemic helps public health authorities allocate resources more effectively, which can in turn bring the nascent event under control before it reaches epidemic scale. However, the initial spatial spread is governed by processes occurring at a variety of scales which act in the origin location, during transport and in the destination location. In this talk, I will discuss some of the issues arising in this context and present some (very partial) methodology that has been developed to address these issues.

Hybrid models predict emergent dynamics of multiscale cell populations

Computational models are essential tools that can be used to simultaneously explain and guide biological intuition. With increasingly high-resolution, high-throughput, and dynamic experimental data, computational biologists are better equipped to develop informed models that to characterize complex cellular responses and direct experimental design. I introduce an agent-based model as an intuitive, modular, and flexible framework to study emergence of heterogeneous cell populations. We use this framework to interrogate the inherent multiscale nature of cells—reinforcing how “the whole is greater than the sum of its parts”—and to predict cell population dynamics from the composition of simpler biological modules. Elucidating the compositionality problem is fundamental to advancing our understanding of basic science; to promoting the impact of synthetic biology; and to designing precise dynamic therapeutic strategies.

Transition Dynamics Convey Key Programs in Cellular Populations

The epithelial-mesenchymal transition (EMT) is a key cellular program of growth and development and of mature cells, such as wound healing. Evidence for its role in pathology, such as cancer invasion, has been accumulating following from intensive study, yet little is known about the dynamics of the transition per se. Using a multidisciplinary approach, this talk explores the dynamics of the transition and its role in conveying distinct gene programs. The global spatiotemporal distribution of metastatic (dysregulatory) burden is found to be critically sensitive to key non-linear dynamics.

Exploring the role of geographic scale on foot and mouth disease outbreaks

Foot and mouth disease (FMD) is a viral infection of cloven-hoofed animals, including livestock species such as cattle, pigs, and sheep. FMD is highly transmissible and outbreaks in non-endemic countries are very expensive. Therefore, there is interest from policy makers in understanding potential outbreaks and identifying ways to reduce losses. The United States Disease Outbreak Simulation (USDOS) is a simulation model that allows for exploration of national-scale infection patterns and control of potential U.S. FMD outbreaks in cattle. USDOS is a premises-level model, incorporating information on both local spread and long-distance transmission events, which are informed by cattle shipment networks. In this presentation, I will discuss how the national scale outcomes of potential FMD outbreaks in the U.S. are influenced by local, regional, and national factors and how these may be altered by control actions.

Different FMD outbreak models incorporate geographic scale in different ways and in systems, like FMD, where there are multiple quality models, it can be difficult for policy makers to select a single model on which to base high-stakes decisions. Here I will present an international collaborative project on the development of a multi-model ensemble that includes four different FMD models that are commonly used in policy. Ensemble modeling methods provide a standardized and transparent way of producing a single, interpretable projection from multiple model outputs. Here we use the multi-model ensemble and data from the initial weeks of the 2001 U.K. FMD outbreak to explore whether the method could improve the accuracy of the model predictions early in an epidemic before the outcome of the outbreak is known.

Incorporating within-host heterogeneity into transmission models

Mathematical models are a key tool in the study of the spread of infectious diseases such as influenza, malaria and dengue. In particular, transmission models have been successful at determining the most promising intervention strategies, despite the fact that many of these models assume all individuals experience identical infections. Here, we discuss the importance of incorporating heterogeneity into transmission models and consider when particular types of heterogeneities can reasonably be ignored.

Modelling coupled within host and population dynamics of R5 and X4 HIV infection

Most existing models have considered the immunological processes occurring within the host and the epidemiological processes occurring at population level as decoupled systems. We present a new model using continuous systems of non linear ordinary differential equations by directly linking the within host dynamics capturing the interactions between Langerhans cells, CD4[Formula: see text] T-cells, R5 HIV and X4 HIV and the without host dynamics of a basic compartmental HIV/AIDS model. The model captures the biological theories of the cells that take part in HIV transmission. The study incorporates in its analysis the differences in time scales of the fast within host dynamics and the slow without host dynamics. In the mathematical analysis, important thresholds, the reproduction numbers, were computed which are useful in predicting the progression of the infection both within the host and without the host. The study results showed that the model exhibits four within host equilibrium points inclusive of three endemic equilibria whose effects translate into different scenarios at the population level. All the endemic equilibria were shown to be globally stable using Lyapunov functions and this is an important result in linking the within host dynamics to the population dynamics, because the disease free equilibrium point ceases to exist. The effects of linking were observed on the endemic equilibrium points of both the within host and population dynamics. Linking the two dynamics was shown to increase in the viral load within the host and increase in the epidemic levels in the population dynami

A Categorization Framework for Multiscale Models of the Dynamics of Infections

The development of multiscale models of the dynamics of infections is a scientific endevour whose progress has so far resulted in the development of a wide variety of multiscale models of the dynamics of infections with different structure and mathematical representations which are associated with the different levels of organization of an infectious disease system (cell level, tissue level, host level, etc.). In this talk I will present a framework for categorization of multiscale models of the dynamics of infections. Such a categorizarion framework is useful in categorizing and classifying the different types of multiscale models of infection dynamics and in turn in bring some order to the discussion on the structure of multiscale models of infectious disease systems. This categorization of the different types of multiscale models of infectious disease systems may be found useful as a basis for further refinement, in the discourse of multiscale modelling of the dynamics of infections. It will enable infectious disease modelers to discuss the general principles that should apply to each category of multiscale models. Instead of discussing these principles repeatedly whenever a multiscale model of an infectious disease system is being developed, infectious disease modelers will be able to refer to the generic discussion for the category of the multiscale model concerned, and focus instead on the application of that discussion to the particular multiscale model of an infectious disease system being developed, and on issues peculiar to that multiscale model. While this categorization cannot be claimed to be unique, I think that it constitutes a good starting point, which may be found useful as a basis for further refinement in the discourse of multiscale modelling of the dynamics of infections.

Vector-Borne Pathogen and Host Evolution in a Structured Immuno-Epidemiological System

Vector-borne disease transmission is a common dissemination mode used by many pathogens to spread in a host population. Similar to directly transmitted diseases, the within-host interaction of a vector-borne pathogen and a host's immune system influence the pathogen's transmission potential between hosts via vectors. Yet there are few theoretical studies on virulence-transmission tradeoffs and evolution in vector-borne pathogen-host systems. Here we consider an immuno-epidemiological model that links the within-host dynamics to between-host circulation of a vector-borne disease. On the immunological scale, the model mimics antibody-pathogen dynamics for arbovirus diseases, such as Rift Valley Fever and West Nile Virus. The within-host dynamics govern transmission and host mortality and recovery in an age-since-infection structured host-vector-borne pathogen epidemic model. By considering multiple pathogen strains and multiple competing host populations differing in their within-host replication rate and immune response parameters, respectively, we derive evolutionary optimization principles for both pathogen and host. Invasion analysis shows that the $mathcal R_0$ maximization principle holds for the vector-borne pathogen. For the host, we prove that evolution favors minimizing case fatality ratio (CFR). These results are utilized to compute host and pathogen evolutionary trajectories, and to determine how model parameters affect evolution outcomes. We find that increasing the vector inoculum size increases the pathogen $mathcal R_0$, but can either increase or decrease the pathogen virulence (the host CFR), suggesting that vector inoculum size can contribute to virulence of vector-borne diseases in distinct ways.

Multiscale Systems Biology: A Case Study Linking Molecular Dynamics to Epidemiological Processes of Malaria

The advent of high-throughput molecular technologies in particular, and the broad availability of data, in general, have forced the quantitative biology community to rethink how to conceive, build, and validate mathematical models. In this talk I will demonstrate how molecular and cellular processes are related to the epidemiology of malaria. We will explore (i) asymptomaticity at the epidemiological level, (ii) telemetry analysis identifying a systemic response to the disease before traditional symptoms show, (iii) the cellular models that explain this phenomenon as an interaction between the immune system and infected red blood cells, (iv) mathematical models that link cellular and transcriptional time series, (v) transcriptomic analysis, and finally (vi) high-throughput in silico drug discovery to solve an epidemiological problem. All these linked analyses provide a comprehensive picture that no single scale can produce alone. The usefulness of models under this light takes on new meanings, and this broad scope requires the cooperation of scientists coming from very different intellectual traditions. In this talk we will also explore how an information system that delivers Adaptive Learning for Interdisciplinary Collaborative Environments (ALICE) is used to train scientists in this new normal.

Multiscale Modeling the Transmission of Infectitious Diseases

TBD

Epidemiology of environmentally transmitted infections: models and empirical data

For environmentally-transmitted infections the contaminated environments, such as food, water, soil, objects, and contact surfaces, represent vehicles for transmission of the infection to susceptible hosts. The causative agents of these diseases exhibit rich dynamics in the host and in the environment and require modeling and empirical studies at both the host and pathogen scales. This presentation will describe what we have learned about the spread of these pathogens between the host populations and their environments, and about the pathogens’ dynamics in the heterogeneous environments. The focus will be on findings of compartmental and agent based models and selected empirical studies. Several human/animal pathogens, including Salmonella, pathogenic Escherichia coli and Listeria, will be used as examples. Finally, the existing knowledge gaps in terms of modeling and empirical data to support or validate models will be discussed.

A pandemonium of parasites: predictability in the face of complex parasite interaction networks?

Human activities are altering host-parasite interactions and their ecological context at unprecedented rates, reducing our ability to anticipate disease impacts and predict transmission dynamics in host populations of interest. These changes lend urgency to understanding the mechanisms underlying variation in disease dynamics, by developing data-driven models that scale up from within-host processes affecting parasite invasion and persistence, to population-level disease dynamics.

In this talk, I focus on interactions among co-infecting parasites as drivers of disease outcomes and dynamics. Using African buffalo and their parasite community as a model system, I demonstrate that co-infecting parasites contribute strongly to incidence and duration of concurrent infections, perhaps forming a modular interaction network. I propose trait-based community analysis as a possible approach for increasing the predictability of outcomes of parasite invasions, using Mycobacterium bovis, the causative agent of bovine TB, as a case study. M. bovis infection leads to changes in parasite infracommunity structure in buffalo, but few other parasites seem to affect the incidence of M. bovis itself. Instead, genetically driven variation in host defense responses against M. bovis underlies variation in infection risk. Linking ecological processes with evolutionary responses, I discuss how co-infection might be shifting both host and pathogen population genetic structure – an area in need of further investigation.

Pairwise survival analysis: Contact intervals, regression, and phylogenetics

When integrating epidemiologic data with pathogen phylogenetics, the likelihood for the transmission model is often a branching-process likelihood based on a generation interval distribution and offspring distribution. We show that a misspecified likelihood can lead to severely biased estimates with or without a pathogen phylogeny. Writing the likelihood as a survival likelihood with failure times in pairs---a process we call pairwise survival analysis---accounts for time spent at risk of infection. In a simple example with three infections, we show that a pairwise survival likelihood produces more accurate source attribution and parameter estimation. In a mass-action model with negligible depletion of susceptibles, the pairwise survival likelihood depends only on information about infected individuals in the limit of a large population. However, this asymptotic likelihood has cumulative hazard terms that have no counterpart in a branching process likelihood. As an example of the flexibility of pairwise survival analysis, we describe a pairwise accelerated failure time model that can be used to estimate covariate effects on infectiousness and susceptibility. This model---modified to account for the buildup of immunity---will be used to estimate the efficacy of the Ebola vaccine based on the WHO ring vaccination trial in Guinea. This trial collected data on individuals exposed to infection who escaped as well as Ebola virus genetic sequences. Finally, we describe a pruning algorithm for calculating an approximate likelihood using both epidemiologic data and a pathogen phylogeny. Pathogen genetics can improve statistical efficiency and reduce bias, but this depends on good epidemiologic study design and a good likelihood for transmission.

Modeling the temporal evolution of the host immune response to infection

We will present two explorations into to the temporal dynamics of the host immune response. The first concerns the real-time modeling telemetry data generated by collaborative cross (CC) mice. The CC mice are monitored in a healthy state for seven days and are subsequently challenged with a salmonella infection. Temperature and activity data is collected at one minute intervals and modeled in real time using a method for time-series prediction that also serves as a novelty detector to quantify time to symptoms and full blown infection. The analysis seeks to identify and characterize aspects of tolerance to high pathogen loads exhibited by certain CC lines. We are also exploring dynamical systems models to explain this phenomenon. The second study relates to the early prognosis of subject outcome in human challenges to influenza viruses. We demonstrate algorithms that can predict if a subject will become symptomatic, or not, as early as 5 hours after exposure. (The mice study is being conducted in collaboration with David Threadgill and Helene Andrews-Polymenis of Texas A&M. Aspects of the dynamical modeling are recent collaborations with Judy Day (UTK) and Helene Andrews-Polymenis.)

Transmission and infection dynamics of Clostridium difficile

Clostridium difficile is an anaerobic human pathogen that forms spores, produce toxins and resides in the gut. Clostridium difficile infection (CDI) is an important hospital acquired infection that causes diarrhea, pseudomembranous colitis, and possibly death. In the last decade, the incidence and severity of C. difficile infection has increased at alarming rates throughout North America, especially in the elderly. Antimicrobial therapy is often a strong and independent risk factor for CDI because it disrupts the indigenous microbiota, which provides protection against C. difficile colonization. Epidemiological and within host models for Clostridium difficile and integration of both scales will be discussed.

Analysis of the Plague at Eyam

The rodent-flea-human transmission route of Y. pestis is widely known and has been medically confirmed to cause the bubonic form of plague. However, the observed rapidity of plague outbreaks during the second pandemic appear to have happened on timescales that are not readily explained by this transmission mechanism alone. We propose a statistical method to analyze such data that is capable of distinguishing the important mechanisms of outbreak dynamics. We implement the methodology on data from the latter half of the outbreak at Eyam in the summer of 1666.

Challenges when building a continental scale livestock disease spread model: Animal movements, farm locations, and computation

Applied epidemiological modeling offers powerful tools to inform policy decisions regarding control actions, identify spatial hotspots, or predict the course of an outbreak. However, modeling is often challenged by limited information about the system. For instance, in the USA, there are no federal databases with animal movements that can be used to model disease spread via these contacts. I will present some of our ongoing work to circumvent this issue by developing the United States Animal Movement Model (USAMM), which is parameterized based on interstate certificates of veterinary inspection and can be used to predict movement across the USA, including within state shipments.

In an international collaboration with researchers from Linköping University (Sweden), Colorado State University (USA), United States Department of Agriculture, and University of Warwick (UK), we connect USAMM predictions to a disease spread model (United States Disease Outbreak Simulator, USDOS), which also accounts for local spread, to simulate outbreak of livestock diseases. This faces another challenge: where are the farms? As with animal movements, there are no federal databases with farm locations, and the most detailed information available is census data from the National Agricultural Statistics Service (NASS) at the county level. Using predictions based on the Farm Location and Animal Population Simulator (FLAPS), I will present some of our work focused on determining whether fine-scale farm distribution matters when modeling disease outbreaks on the US farm population, using Foot and Mouth Diseases as an example. For this purpose, we use both analytical methods and stochastic simulations. The latter involves simulations among nearly 850000 farms, and efficient computation is crucial. I will also present an approach to speed up simulations for spatially explicit disease simulations.

Identifiability Issues of an Immuno-Epidemiological Model: The case of Rift Valley Fever Virus

We discuss the structural and practical identifiability of a nested immuno-epidemiological model of arbovirus diseases, where host–vector transmission rate, host recovery, and disease-induced death rates are governed by the within-host immune system. We incorporate the newest ideas and the most up-to-date features of numerical methods to fit multi-scale models to multi-scale data. For an immunological model, we use Rift Valley Fever Virus (RVFV) time-series data obtained from livestock under laboratory experiments, and for an epidemiological model we incorporate a human compartment to the nested model and use the number of human RVFV cases reported by the CDC during the 2006–2007 Kenya outbreak. We show that the immunological model is not structurally identifiable for the measurements of time-series viremia concentrations in the host. Thus, we study the scaled version of the immunological model and prove that it is structurally globally identifiable. After fixing estimated parameter values for the immunological model derived from the scaled model, we develop a numerical method to fit observable RVFV epidemiological data to the nested model for the remaining parameter values of the multi-scale system. For the given (CDC) data set, Monte Carlo simulations indicate that only three parameters of the epidemiological model are practically identifiable when the immune model parameters are fixed. Alternatively, we fit the multi-scale data to the multi-scale model simultaneously. Monte Carlo simulations for the simultaneous fitting suggest that the parameters of the immunological model and the parameters of the immuno-epidemiological model are practically identifiable.

Research performed in collaboration with Necibe Tuncer, Hayriye Gulbudak, and Vincent Cannataro.

Local and regional dynamics of arbovirus transmission: the role of mismatched spatial heterogeneity

Background: Mathematical models of transmission dynamics are routinely fitted to epidemiological time series, which must inevitably be aggregated at some spatial scale. Weekly case reports of chikungunya have been made available nationally for numerous countries in the western hemisphere since late 2013, and numerous models have made use of this data set for forecasting and inferential purposes. Motivated by an abundance of literature suggesting that the transmission of this mosquito-borne pathogen is localized at scales much finer than nationally, we fitted models at three different spatial scales to weekly case reports from Colombia to explore limitations of analyses of nationally aggregated time series data.

Methods: We adapted the recently developed DTK-Dengue model for modeling chikungunya virus (CHIKV) transmission, given the numerous similarities of these viruses vectored by a common mosquito vector. We fitted versions of this model specified at different spatial scales to weekly case reports aggregated at different spatial scales: (1) single-patch national model fitted to national data; (2) single-patch departmental models fitted to departmental data; and (3) multi-patch departmental models fitted to departmental data, where the multiple patches refer to municipalities within a department. We compared the consistency of simulations from fitted models with empirical data.

Results: We found that model consistency with epidemic dynamics improved with increasing spatial granularity of the model. Specifically, the sum of single-patch departmental model fits better captured national-level temporal patterns than did a single-patch national model. Likewise, multi-patch departmental model fits better captured departmental-level temporal patterns than did single-patch departmental model fits. Furthermore, inferences about municipal-level incidence based on multi-patch departmental models fitted to departmental-level data were positively correlated with municipal-level data that were withheld from model fitting.

Conclusions: Our model performed better when posed at finer spatial scales, due to better matching between human populations with locally relevant risk. Confronting spatially aggregated models with spatially aggregated data imposes a serious structural constraint on model behavior by averaging over epidemiologically meaningful spatial variation in drivers of transmission, impairing the ability of models to reproduce empirical patterns.

The population genentics of pathogen virulence

Life history theory provides a powerful framework to understand the evolution of pathogens in both epidemic and endemic situations. This framework, however, relies on the assumption that pathogen populations are very large and that one can neglect the effects of demographic stochasticity. In my talk, I will present an alternative approach, based in population genetics, which will explore the effects of finite population size on the evolution of pathogen virulence and transmission. I will show that demographic stochasticity introduces additional evolutionary forces that can affect qualitatively the dynamics and the evolutionary outcome. In particular, I will discuss scenarios where finite population size can either select for lower or higher virulence.

Modeling endemic foot-and-mouth disease in Cameroon

Foot-and-mouth disease virus (FMDV) threatens animal health and leads to considerable economic losses worldwide. Endemic disease dynamics occur throughout Asia and Africa; however, most models of infectious disease dynamics represent locations throughout North America, Europe, and Australia that experience epidemic dynamics or have been FMD-free for decades. This dichotomy has resulted in a void in our understanding of natural FMD dynamics. Therefore, we investigated disease dynamics in the Far North Region, Cameroon, where FMDV is endemic and five serotypes have been detected in cattle. First, we characterized and quantified serotype-specific transmission and immunodynamics in cattle. Second, we investigated if pastoral mobility plays a role in endemic maintenance. This work underscores the importance of using empirical data in models to understand processes that influence the transmission of infectious diseases at multiple scales.

Micro and macro SIR models

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.

Dynamic Models of Malaria

Malaria is a complex disease that afflicts 200 million individuals worldwide and causes about 500,000 deaths per year. It is transmitted by Anopheles mosquitoes and caused by one of several Plasmodium species, which have a complicated life cycle that is distributed between the mosquito and the human host. The molecular events, host-pathogen interactions, physiological host responses, and the global reach of the disease create a truly multi-scale system. I will present five vignettes describing mathematical models of molecular and cellular events during malaria infections in non-human primates, which offer excellent models of the human disease processes. Three vignettes will focus on different aspects related to the dynamics of red blood cells and reticulocytes, whose disappearance is the dominant driver of malarial anemia, one of the most severe hallmarks of the disease. The other two vignettes will demonstrate how insights into the meaning of transcriptomic changes during the disease can be gained through models of concomitant processes at the metabolic and physiological levels.

This work is joint work with Luis L. Fonseca and Anuj Gupta.

### Posters

A Hierarchy of Reduced Mathematical Models for Wolbachia Transmission in Mosquitoes to Control Mosquito-borne Diseases

We create and analyze a hierarchy of reduced models for the spread of a Wolbachia bacteria infection in mosquitoes that can help predict the effectiveness of efforts to control the spread of Zika, chikungunya, dengue fever and other mosquito-borne diseases. Mosquitoes that are infected with some strains of the Wolbachia bacteria are much less effective at transmitting zoonotic diseases. The infection will persist in a wild mosquito population only if the fraction of infected mosquitoes exceeds a minimum threshold. This threshold can be characterized as a backward bifurcation for a system of nine ordinary differential equations modeling the complex vertical transmission of the bacteria infection in a heterosexual mosquito population. Although the large system of differential equations capture the detailed transmission dynamics, they are difficult to analyze. We derive a seven-equation, a four-equation and a two-equation system of differential equations that are formulated in terms of the more accurate nine-equation model and capture the important properties of the original system. The reduced models preserve key dimensionless numbers, the ratios of infected and uninfected male and female mosquitoes, and accurately capture the backward bifurcation threshold.


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# Delta connected

Jump to: navigation, search

We say that a topological space $(X,\tau)$ is $\delta$-connected if for every $p,q \in X$ there exists a continuum $A$ such that $p,q \in A$ and $A$ is hereditarily decomposable.

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# What is the best way to use a 2-class classifier for a multi-category case?

I have a 3 class sample labeled data set, which I have divided into 2 parts. I am using the first part to train the two-class perceptron classifier.

One approach is to train $\binom{3}{2}$ two-class classifiers on training data for two classes at a time, and then use voting to classify the test samples for a multi-category case. The obvious problem with the above approach is the presence of ambiguous regions like the ones shown in the image below:

Pattern classification by Duda, Hart, Stork suggests training $c$ different linear discriminant functions, where $c$ is the number of unique classes, such that

$$g_{i}(x) = W_{i}^tX + w_{i0} \quad \quad \quad i = 1,..., c$$

and assigning $X$ to $\omega_{i}$ if $g_{i}(X) > g_{j}(X)$. The resulting classifier is called in the text book as a linear machine.

Following is an illustration from the book showing decision boundary produced by a linear machine for 3-class problem.

My doubt is how is the training process of the $c$ linear discriminant functions $g_{i}(X)$ different from training of $\binom{c}{2}$ 2-class classifiers?

UPDATE: The one vs rest approach mentioned in DavidDLewis's answer is not the one I am referring to. With the one vs rest approach, there are infact even more ambiguous regions. I am referring to linear machine classifier using which there are no ambiguous regions. See illustration of 1 vs rest approach below:

• I believe the 'linear machine' is a 'one vs. rest' approach, but the resultant classifiers are used differently than in your diagram. – shabbychef Sep 2 '11 at 20:11
• It would be really helpful, if you could please elaborate on the usage differences you mentioned in your comment. – stressed_geek Sep 4 '11 at 9:54
• imagine $k$ 1-v-rest classifiers as each describing regions where the $i$th class dominates the rest, _i.e._ $\left\{x|g_i(x) \le c_i\right\}$. This can result in the ambiguous region in your bottom figure. The linear machine approach trains the classifiers in the same way, but then defines the region where the $i$th class dominates as $\left\{x|g_i(x) \le g_j(x),\,\forall j\right\}$. – shabbychef Sep 4 '11 at 23:29


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# Partial Differentiation with Indicial Notation (Ritz Method for FEM)

1. Aug 22, 2012

### bugatti79

Folks,

I am stuck on an example which is partial differenting a functional with indicial notation

The functional $\displaystyle I(c_1,c_2,...c_N)=\frac{1}{2} \int_0^1 \left [ \left (\sum\limits_{j=1}^N c_j \frac{d \phi_j}{dx}\right )^2-\left(\sum\limits_{j=1}^N c_j \phi_j\right)^2+2x^2 \left(\sum\limits_{j=1}^N c_j \phi_j \right)\right ]dx$

Differentiating this wrt to $c_i$ ie

$\displaystyle \frac{\partial I}{\partial c_i}= \int_0^1 \left[ \frac{d \phi_i}{dx} \left(\sum\limits_{j=1}^N c_j \frac{ d\phi_j}{dx} \right )-\phi_i \left(\sum\limits_{j=1}^N c_j \phi_j \right) + \phi_i x^2 \right] dx$

I dont understand how this last line is obtained. If we focus on the first term. I realize that there is a chain rule procedure. My attempt on the first term in first eqn was

$\displaystyle \frac{\partial I}{\partial c_i}= \frac{1}{2}\int_0^1 \left[ 2\left(\sum\limits_{j=1}^N c_j \frac{ d\phi_j}{dx} \right )\frac{d}{dc_i}\left(\sum\limits_{j=1}^N c_j\frac{ d\phi_j}{dx}\right) \right]dx$

Not sure how to handle the indicial notation or how proceed any further.
Any help will be greatly appreciated....thanks

2. Aug 22, 2012

### voko

$$\frac{d}{dc_i}\left(\sum\limits_{j=1}^N c_j\frac{ d\phi_j}{dx}\right) = \frac{d}{dc_i}\left(c_1\frac{ d\phi_1}{dx} + ... + c_i\frac{ d\phi_i}{dx} + ... + c_N\frac{ d\phi_N}{dx}\right) = 0 + ... + \frac{ d\phi_i}{dx} + ... 0$$

3. Aug 22, 2012

### bugatti79

Thank you so much. So the i is some integer between j=1 and j=N.

Cheers


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# [OS X TeX] LastPage Reference

Herbert Schulz herbs at wideopenwest.com
Tue Mar 6 20:58:17 EST 2007

```On Mar 6, 2007, at 7:50 PM, Roussanka Loukanova wrote:

>
>> Howdy,
>>
>> Not sure, but do you need to explicitly have
>>
>> \usepackage{lastpage}
>>
>> to use LastPage? Maybe one of the other packages loads it?
>
> It seems that lastpage.sty doesn't get loaded after commenting it,
> i.e.,
> with
> % \usepackage{lastpage}
> I do not find
> (/usr/local/gwTeX/texmf.texlive/tex/latex/lastpage/lastpage.sty)
> resp.
> (/usr/local/texlive/2007/texmf-dist/tex/latex/lastpage/lastpage.sty)
> in the log file.
>
> However, the setting LastPage wold be available for an additional
> pass of latex, then in a 2nd round, the number of LastPage is not
> available. (This is my interpretation of what I see in the log
> file: I do not know if I'm right.)
>

The LastPage cross reference is defined in the lastpage package.
There are some comments about that in the hyperref package but it
wasn't clear to me that it was defined there.

>> Also \ifthenelse has three arguments:
>>
>> \ifthenelse{test}{true commands}{else commands}
>>
>> so
>>
>> \lfoot{\ifthenelse{\value{page}=\pageref{LastPage}}{\tiny\today}}
>
>> should have an extra {} pair for the else (assuming it should be
>> blank if it's not the last page).
>
> For strictness, you are right. But since  Maurino's tex code
> documentation: Without being absolutely sure, ifthenelse.sty is
> extending (they say compatible with) ifthen.sty and adds lasy
> evaluation of the logical operators:  in cases when P is true, "if
> P then Q else R" wouldn't look for the else proposition, i.e. it
> the same as "if P then Q".
>
>>
>> Finally, in the command
>>
>> \cfoot{\ifthenelse{\not\value{page}=\pageref{LastPage}}
>> {\scriptsize (over)}}
>>
>> the \not is only acting on \value{page} and I believe you mean it
>> to be
>
> Again, because Maurino's tex code typeset ok (I've ran it several
> times), it looked to me that \not (number1 = number 2) has also
> abbreviated form
> \not number1 = number 2. The documentation is more clear to me in
> this aspect (than for the lasy evaluation). But they treat "number1
> = number 2" as a syntactic unit: atomic proposition, which hints to
> parsing of
> \not number1 = number 2 as equivalent to \not (number1 = number 2).
>
>>
>> \not{\value{page}=\pageref{LastPage}} as well as missing the extra
>> {}; i.e., it should be
>>
>> \cfoot{\ifthenelse{\not{\value{page}=\pageref{LastPage}}}
>> {\scriptsize (over)}{}}
>>
>> which is the same as the simpler
>>
>> \cfoot{\ifthenelse{\value{page}=\pageref{LastPage}}{}{\scriptsize
>> (over)}}.
>
> I think, it is; and I think that one would be on the safe side by
> following the strict (and clear) syntax as you suggested. Esp. with
> the drawbacks of the lazy evaluation (or may I just do not like it).
>
> Roussanka

All I know is that I get different results if I define it one way or
the other and I get the expected one if I include the braces: \not{...}.

Good Luck,

Herb Schulz
(herbs at wideopenwest.com)


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# Conducting pairwise ranking with XGBoost

I am trying to build a ranking model using xgboost, which seems to work, but am not sure however of how to interpret the predictions. I haven't been able to find relevant documentation or examples on this particular task, so I am unsure if I'm either failing to correctly build a ranking model, which gives nonsensical output, or if I'm just not able to make sense of it.

This is what I've tried so far:

library(xgboost)

#generate some random data and target
train <- matrix(sample(c(runif(100), rep(0, 100)), 2000, replace = T), ncol = 20)
target <- sample(c(0, 1, 2), 100, replace = T)
test <- matrix(sample(c(runif(100), rep(0, 100)), 1000, replace = T), ncol = 20)

xgbTrain <- xgb.DMatrix(train, label = target)
xgbTest <- xgb.DMatrix(test)

params <- list(booster = 'gbtree',
objective = 'rank:pairwise')

rankModel <- xgb.train(params, xgbTrain, 10, watchlist = list(tr = xgbTrain), eval_metric = 'ndcg')

print(predict(rankModel, xgbTest))

Which prints:

 [1] -0.94882607 -0.58704734  1.64421082  1.75749838 -0.82614720  0.30855450 -0.23618424  2.03426790  0.18326324  1.54318023  1.60956442 -0.26394469
[13]  2.18902779 -0.82873511  2.01967430 -0.55936623 -0.77433181 -0.95465362  0.30088556  2.07671452 -0.98471618 -0.55923200 -0.96335471 -0.64822674
[25]  1.62699771 -0.78486574 -0.80902183 -1.09310806 -0.75859952  0.76334584  0.64056098  0.48438376  0.48405507  1.67765403  0.63187867  0.66793621
[37]  1.00044215 -0.87645602  1.60261607  0.56360698 -1.14934421  1.68098438  2.03522825 -0.95139503 -1.02189994 -0.94723392 -0.69983196 -1.20132697
[49]  1.26217628  0.79300237  0.79510546  2.24825048  1.84355474  1.10526133 -1.37557507 -1.00994503  1.77173507  1.13348961  0.79858458 -0.67265379
[61]  1.01461124  0.48507297 -0.83393693  1.67525303  1.24216270  0.02764088  0.91251671  2.04108095  2.17401123 -0.62811875  0.73657835  2.39004827
[73] -0.18949705  0.60755765  2.26420307 -1.23398590  0.59091914  1.65885520 -0.30570436  2.00285411  0.67023963  2.06346416  1.69319844  2.27233768
[85]  0.32537588  2.29775119  0.08256459 -1.16774368  0.78462589 -1.01925206 -0.81494403  0.50675780 -0.65493345 -0.17356467 -0.76901770 -0.86279511
[97] -1.08725667  1.94247663 -0.96650219  0.88204288

• It looks like you're missing target values for your test set so it will be difficult for you to assess the predictions there. However, you can look at predictions on the training set against the true target values to get a sense of what the model is doing. plot(target ~ predict(rankModel, xgbTrain)) should give you some good insight. – assumednormal Nov 28 '15 at 16:01
• What I don't understand is that I would be supposed to get a ranking through "predict". Is that not the case? Here I get predictions, that might be accurate if rounded, but that only contains information on the "first"of the ranking. – Gerome Pistre Nov 29 '15 at 0:32


text:
Circlet of Rhaelyx

Circlet of Rhaelyx
Combine this with the Jewel of Rhaelyx
Item ID: melvorD:Circlet_of_Rhaelyx
Category: Patreon
Type: Misc
Sells For: 750,000
Item Sources:
• Any action in: , , , , , ,
Item Uses:
Part of 100% Completion: Yes

The Circlet of Rhaelyx is one of the two component pieces of the (the other being the ) which provides a 10% chance to preserve resources when training non-combat skills.

Item Sources

The Circlet of Rhaelyx is a very rare drop while training , , , , , , and . The chance of getting a Circlet depends on your total level, which can be seen on the Statistics page.

The base drop rate of the Circlet is $\displaystyle{ \frac{1}{10,000,000} }$ per action taken. The drop rate increases by $\displaystyle{ \frac{1}{2,500,000,000} }$ for each Mastery level you have in any skill, to a max of $\displaystyle{ \frac{1}{100,000} }$ which is reached at 24,750 total mastery level. After that point, additional Mastery levels will not increase the chance further.


text:
# User talk:99.241.166.168

## Welcome to Wikipedia!

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## Reference Errors on 27 January

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## February 2014

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• group]] {{math|''Sn''}}. Here, {{math|''n''}} is a [[natural number]]; if {{math|''n'' > 1}}, then {{math|''Bn''}} is an [[infinite group]]. Braid groups find applications in [[
• ≤ ''i'' ≤ ''n''−2}} and in the second group of relations, {math|{!}''i'' − ''j''{!} ≥ 2}}. This presentation leads to generalisations of braid groups called [[Artin group]]s. The cubic

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## Reproducing kernel Hilbert space

Edits were rolled back as something corrupted the page and it was not obvious how to fix it. The error showed as Failed to parse(PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \begin{cases} T_K: L_2(X) \to L_2(X) \\ [T_K f](\cdot) = \int_X K(\cdot,t) f(t) d\mu(t) \end{cases} in the third line of Integral Operators and Mercer Theorem section  Ronhjones  (Talk) 20:18, 14 February 2014 (UTC)

## Blockquotes

Please stop putting theorems in
 tags. These tags generate a box with quotation marks, and quotation marks are inappropriate for theorems. Ozob (talk) 15:06, 19 February 2014 (UTC)

Let me expand on that: Don't use
 or
 tags to set off theorems, definitions, or proofs from the main body of the text. In fact, don't use
 tags except for quotations, and don't use
 tags at all. Furthermore, TeX's default formatting for subscripts and superscripts is just fine, and you should not put \nolimits tags on them. Ozob (talk) 15:34, 19 February 2014 (UTC)

Couple of things, on theorems, I did that because sometimes a theorem is several paragraphs long and it is not obvious where it ends. Also it does not display any quotation marks for me so I don't know what you are referring to there.

As for the subscripts, the \nolimit option is appropriate when you don't have a superscript, other wise the formula's vertical placement will be non-symmetric and closer to the text that appears before it.

So I disagree on your second point, but the first point I was trying to address a problem, maybe you have a better solution.

Long theorems are often indented using a colon at the start of each line (in the same way that I am indenting my reply to you right now; by the way, indenting replies is standard practice). This is sufficent to set off the theorem and does not distract the reader.
It seems that most users, and in particular all logged out users, will not currently see quotation marks around blockquote tags. Logged in users who have the "Typography Refresh" option enabled (like me) will. Eventually this option will become the default, and when that happens everyone will see quotation marks around blockquote tags. Even if this were not the case, it would be poor semantics to use an HTML tag intended for quotations to set off theorems.
I don't understand your statement about \nolimits. With \limits, the subscript is centered underneath the symbol, as in:
$\prod_p (1 - p^{-s})^{-1}.$
With \nolimits, the subscript is placed asymmetrically on the right of the symbol:
$\prod\nolimits_p (1 - p^{-s})^{-1}.$
Is this what you're referring to? If not, then I am misunderstanding you.
(By the way, you should sign your comments. Use four tildes (the ~ character) and the software will fill in a signature for you.) Ozob (talk) 03:05, 20 February 2014 (UTC)
It seems from your reply on my talk page that you're concerned about vertical asymmetry, not horizontal. I agree extreme vertical asymmetry should be avoided; too much whitespace at the top or bottom of a formula does not look good. But I find a small amount of whitespace, even asymmetric whitespace, to be acceptable. Thus, I do not mind the look of the first displayed equation I gave above, nor do I mind:
$\bigcup_n S^n, \qquad \sum_n \frac{x^n}{n!}, \qquad \varprojlim_n k[x] / (x^n).$
I find formulas with horizontal asymmetry much more jarring, and so I dislike:
$\bigcup\nolimits_n S^n, \qquad \sum\nolimits_n \frac{x^n}{n!}, \qquad \varprojlim\nolimits_n k[x] / (x^n).$
I understand you to say that you prefer the latter. This is fine. However, you have been changing articles to reflect your preference, and this is not such a good thing to do. This kind of situation, where there is more than one conventional way to do things, is usually handled by a recommendation called WP:RETAIN. The text of WP:RETAIN talks about national varieties of English (most commonly, this is British versus American English), but the same principal has been applied in many other situations. The general principal is: Use the style of the first major contributor to the article. Thus, if the earliest major contributor used \limits (even implicitly), then you should use \limits; if they used \nolimits, then you should use \nolimits. Changing an article to be internally consistent is acceptable. Articles should always be consistent with normal usage (an article that uses $\lim^{n\to\infty}$ should be changed, even if the first major contributor disagrees), and in particular if a convention is widespread within a certain specialty, then it's acceptable to conform to that field's convention. But changes between optional styles are specifically discouraged because they tend to lead to WP:Edit warring. This is why, for example, I don't search through physics articles changing every occurrence of $S = \int d^4x\,L$ to $S = \int L\,d^4x$; I prefer the latter, but the former is an acceptable alternative used by some authors, and I am supposed to accept it (at least on Wikipedia).
One other thing, since I've brought up WP:RETAIN. There are some editors who feel strongly about the use of [itex], {{math}}, and plain italics. The mathematics WikiProject (see Wikipedia:WikiProject Mathematics and especially Wikipedia talk:WikiProject Mathematics) has had discussions on this issue several times. There is no consensus as to what style is the best, and there are editors who strongly believe in all three. WP:RETAIN is sometimes applied to the proper way to put mathematics into Wikipedia, so you may encounter editors who oppose your changes. That's more a reflection on history than it is on you and your edits. Ozob (talk) 15:22, 20 February 2014 (UTC)

## Errors

Please make sure that your formatting changes do not otherwise change articles. This edit introduced an error (an ungrammatical sentence). Ozob (talk) 06:32, 2 March 2014 (UTC)

## Format changes

Hi there,

I don't understand why you're changing the math formatting even when there is no need to do so. Please stop. -- Taku (talk) 01:21, 4 March 2014 (UTC)

## May 2014

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## June 2014

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## July 2014

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## Formatting change

Please do not make a formatting change. It's not an improvement. -- Taku (talk) 11:20, 14 July 2014 (UTC)


text:
Chapter 1 - Foundations for Algebra - 1-5 Adding and Subtracting Real Numbers - Lesson Check: 6

$-7$ $1.5-8.5=-7$

Work Step by Step

If we have 1.5 and take away 8.5, we are left with -7. In other words, if we start at 1.5 on the number line and move 8.5 units to the left, we are at -7. Thus, we conclude: $1.5-8.5=-7$

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.


text:
## Z test of mean

### From OPOSSEM

$Z = \frac {\bar{y}-\mu_0}{\sigma_{\mu_y}}$, where $\mu_0$ is the test value.


text:
# How do you graph y=-1-3cos4x?

Jun 15, 2018

Know how the cosine curve behaves and some standard values of it.

#### Explanation:

First off, rearanging terms gives, $y = - 3 \cos 4 x - 1$.

This means you will have a negative cosine curve, and because of the $- 1$ the curves center will be moved down by 1.

Also notice how for all $x$, $\cos \left(x\right)$ is between $- 1 , 1$, and because $\cos x$ is multiplied by $3$ in $y$ it will now oscillate between $- 3 , 3$.

We now know the center and its max/min value. Now for the shape.

Start with plugging in values into $y$ that will give us known values for cosine which is:
$\cos \left(0\right) = 1$,
$\cos \left(\setminus \frac{\pi}{2}\right) = 0$,
$\cos \left(\setminus \pi\right) = - 1$,
$\cos \left(\frac{3 \setminus \pi}{2}\right) = 0$,
$\cos \left(2 \setminus \pi\right) = 1$.
(These values can easily be seen in the unitcircle where cosine is the x value.)

We do only need these values to know how the curve will look for all $x$ as cosine is periodic and will repeat forever. This will also apply to the negative side as cosine is an even function ($\cos \left(- x\right) = \cos \left(x\right)$).

So we want to insert som $x$ into $y$ such that we get these known values. Lets start off with getting $\cos \left(0\right) = 1$ in $y$.

$4 x = 0 \implies y \left(0\right) = - 4$
$4 x = \setminus \frac{\pi}{2} \implies y \left(\setminus \frac{\pi}{8}\right) = - 1$
$4 x = \setminus \pi \implies y \left(\setminus \frac{\pi}{4}\right) = 2$
$4 x = \frac{3 \setminus \pi}{2} \implies y \left(\frac{3 \setminus \pi}{8}\right) = - 1$
$4 x = 2 \setminus \pi \implies y \left(\setminus \frac{\pi}{2}\right) = - 4$

Now insert these $\left(x , y\right)$ pairs into a cordinate system and connect the dots! Note, as its the negative cosine we want to plot, the function will start by increasing from $x = 0$ instead of decreasing.


text:
The traditional “evaporation∕melt and blow” mechanism of $CO2$ laser cutting of aluminum nitride (AlN) chip carriers and heat sinks suffers from energy losses due to its high thermal conductivity, formation of dross, decomposition to aluminum, and uncontrolled thermal cracking. In order to overcome these limitations, a thermochemical method that uses a defocused laser beam to melt a thin layer of AlN surface in oxygen environment was utilized. Subsequent solidification of the melt layer generated shrinkage and thermal gradient stresses that, in turn, created a crack along the middle path of laser beam and caused material separation through unstable crack propagation. The benefits associated with thermal stress fracture method over the traditional method are improved cut quality, higher cutting speed, and lower energy losses.

1.
1999,
Machining of Ceramics and Composites
,
S.
Jahanmir
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M.
Ramulu
, and
P.
Koshy
, eds.
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,
New York
.
2.
Migliore
,
L.
, and
Ozkan
,
A.
, 2003, “
Laser Cutting of Aluminum Nitride
,”
Proc. SPIE
0277-786X,
5063
, pp.
505
508
.
3.
Quintero
,
F.
,
Varas
,
F.
,
Pou
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J.
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Lusquiños
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R.
, and
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, 2005, “
Theoretical Analysis of Material Removal Mechanisms in Pulsed Laser Fusion Cutting of Ceramics
,”
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666
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4.
Zhang
,
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, and
Modest
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M. F.
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Energy Requirements for Ablation or Decomposition of Ceramics During CO2 and Nd:YAG Laser Machining
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text:
# On the proof: $\exp(A)\exp(B)=\exp(A+B)$ , where uses the hypothesis $AB=BA$?

I was seeing the proof that $\exp(A)\exp(B)=\exp(A+B)$ on link

Show that $e^{A+B}=e^A e^B$

where uses the hypothesis $AB=BA$?

Thanks!

• The use of the Binomial Theorem. Apr 15 '14 at 20:54
• Thanks Hayden, but I can't understand. To use the Binomial Theorem I don't need change the position A and B, so I could not use AB=BA. Apr 15 '14 at 20:57
• Now I see, Thanks Hayden and all of you!! Apr 15 '14 at 21:00

The step that claims $$\sum_{m=0}^l{\frac{l!}{m!(l-m)!}A^{m}B^{l-m}}=(A+B)^l$$ For example, we would usually say that $$(A+B)^2=A^2+AB+BA+B^2$$ Only because $$AB=BA$$ can this be simplified.
The commutativity of $A$ and $B$ is used where the double summation becomes a single summation. Consider as an example the term in that sum with $\ell =2$: $(A+B)^2 = A^2+AB+BA+B^2$.
The binomial formula $$(A+B)^l=\sum_{k=0}^l{l\choose k}A^kB^{l-k}$$ is valid if $AB=BA$.
• Not quite: $(A+B)^{1}=A+B$ is valid even if $AB\ne BA$. I think you meant "if" instead of "only if." (sorry to be picky!)
• In the proof $l$ take the values from $0$ to $\infty$.