In this presentation, we introduce an epidemiological framework to study the questions associated with the dynamics and control of epidemics arising from the deliberate release of biological agents, like smallpox, on a city where transportation systems are critical. The city is divided into neighborhoods and individuals are stratified by whether or not they use public transportation and by their epidemiological status. The transmission of the agent, among neighborhoods, is driven by contacts on the mass transportation system. This framework is applied to a city, with some of the characteristics of New York City. Simulations suggest that effective vaccination campaigns for smallpox must take into account the impact of temporary residents (tourists); and any delays in response to an attack may result serious consequences.

To devise effective strategies to minimize the impact and spread of infectious diseases, we must use all of the tools available to advance epidemic models from qualitative insight to quantitative predictions. I will describe a flexible, stochastic agent-based decision simulation model for understanding the spread of a disease within a major city and compare it with a class of deterministic differential equation models. Although the agent-based model can include far more detail than the differential equation model, we have fewer analysis tools to understand the underlying dynamics in the detailed simulation. I will describe an approach to define a simple differential equation model that captures the average course of an epidemic as defined by an agent-based model of a million people and over ten thousand locations. The differential equation model uses the same parameters and initial conditions as the agent based model, and then can be analyzed directly to predict the course of the epidemic in the more complex agent based model.

Behavior changes of infected individuals have big impact on the prevention and control of infectious diseases. To gain insight into effects of such behavior changes, we present two susceptible-infective-removed (SIR) epidemic models with infection-age (stage) structure such that infected individuals change their behavior variously based on their infection ages or stages. We derive explicit formulas for the reproductive number and investigate existence of the endemic equilibrium. Effects of the behavior changes on the transmission dynamics of the diseases are also discussed.

Disease is an abstraction to be examined, diagnosed, treated and sometimes researched by the medical community. Illness is the concrete personal experience of disease by the patient. We briefly remember how humans responded to The Black Death of the 14th Century and how human behavior affected the spread of the epidemic. We contrast that time with more recent responses to spreading epidemics. Finally, I will discuss the necessity of understanding the dynamics of both the disease and illness models, as well as the interactions between them, in order to address the new challenges of emerging diseases. A novel approach to the complex interplay between science, people, and systems might be looking at the formation of identity groups of "sick" people and how their networked experiences affect their behavior in combating the spread of the disease.

Recently, the analysis of real world networks has received a lot of attention. In this talk, I will present several real world systems that have been analyzed from a network point of view. The high levels of clustering, short average distance between nodes, and the power law degree distributions observed in such systems seem to be quasi-universal properties. Finally, I will talk about the implications that the structure of such systems have on superimposed dynamics, particularly, in the spread of computer and human viruses on the Internet and social networks respectively.

A model that incorporates behavior changes in response to a smallpox attack is introduced. The model assumes that once a smallpox case appears in the population, people would modify their behavior, in such a way that it would reduce the probability of infection. In this paper we compare two models, one with behavioral changes and another without them. The population in the first model is divided into two classes, non-core (low active individuals) and core (highly active individuals). Low active individuals reduce their number of daily contacts due to the news of a smallpox outbreak in the population, whereas highly active individuals continue having a high average number of contacts per day. We analyze these two models and determine the number of cases and deaths produced during a single outbreak. Vaccination is then added to both models and separate and combined effects of these two control measures are analyzed. In this complex setting, the basic reproductive number Ro is derived and discussed. Our simulation results suggest that by reducing the number of daily contacts, the impact of the disease is reduced. However, if people continue to have a high average number of contacts, eradication of the epidemic does not occur fast and a large number of people die due to smallpox.

I construct and analyze the directed, time-resolved social network that underlies an example of scientific discovery and subsequent knowledge diffusion. This empirical example is based on historical information, from publication records, interviews of the participants and institutional co-occurence. Such empirical considerations strongly inform the construction of social networks from automatic methods. In addition I characterize quantitatively how the social network evolves in time and changes its properties from the initial process of knowledge discovery to a later period of knowledge diffusion. I also highlight specific node relationships as social network motifs and compare them to properties of epidemiological models for the spread of diseases.

Abstract: Despite high pertussis vaccination coverage in US children, reported pertussis incidence has increased in infants, adolescents, and adults. Potential new vaccination policies using the new acellular pertussis vaccines are: 1) universal vaccination of adolescents with the Tdap vaccine, 2) universal vaccination of adolescents and adults using Tdap, and 3) vaccination of family members of newborns to provide a protective cocoon with reduced transmissions to infants from household members. Computer simulations are used to compare the effectiveness per vaccination dose of these three vaccination policies in the US.

Malaria is amongst the reemergent vector-borne diseases with highest public health and economic impact at a global scale. In areas where the Anopheline mosquitoes transmitting malaria tend to blood feed on animals as well as on humans, the presence of livestock can increase the risk of disease transmission to humans. In these areas, the insecticide treatment of cattle has been proposed as a novel approach to control malaria, by killing host-seeking mosquitoes. This approach has recently started to be evaluated and promising results were obtained from trials in Pakistan: the treatment of cattle with pyrethroids decreased the incidence of malaria with a similar efficacy to the traditional indoor insecticide spraying but with much lower costs¹. However, several factors underlying the effectiveness of insecticide treated cattle remain poorly understood. This study aims to investigate some of these key factors; namely, the required coverage of treatment and the degree of vector preference for feeding on animals over humans. These questions are addressed by developing on the Ross-Macdonald model for malaria transmission, further extended to incorporate the vector feeding behaviour on animal and human populations. Understanding the circumstances under which insecticide treated cattle can be effective against human malaria will be a major contribution to the optimization of this control strategy in a given setting. Most importantly, it will enable the impact of the strategy in different settings to be estimated. The quantitative framework developed in this study will be an important step towards this direction.

Invasion of diseases into new territory is a worldwide problem. Examples include West Nile fever in the US, HIV in Africa and Asia, and dengue in Latin America. Traditionally, the spatial spread of disease has been modeled using a local process, diffusion, to model dispersal. However, if dispersal is non-local, diffusion can greatly underestimate speeds of invasion. In this talk, I will discuss integro-differential-equation models that incorporate knowledge about the dispersal of disease propagules and infected hosts to describe disease infection.

Models for infectious diseases are reviewed where the length of the exposed (latent) period can be shortened by re-infections, starting with a model proposed by Ken Cooke (1967) and analyzed by Frank Hoppensteadt and Paul Waltman (1970, 1971). In these models, which can either be formulated as transport equations with variable speed or state-dependent functional differential equations, it is assumed that, after the first infection, an individual is exposed to constant additional infections and becomes infectious as soon as the accumulated infection reaches a certain threshold. ODE models where re-infections only occur at a certain rate, very similarly to the first infection, have been considered for Hepatitis A by Liu (1993) and Liu and van den Driessche (1995) and for tuberculosis by Feng, Castillo-Chavez, and Capurro (2000). Combining aspects from both type of models, a PDE model is proposed and analyzed (joint work with Maia Martcheva) in which the latent stage is divided into a progressive latent stage and a quiescent latent stage. With certain rates, individuals pass from the progressive latent stage to the infectious stage or drop to the quiescent stage from which they are lifted back by re-infection. The model analysis pays particular attention to the existence of multiple endemic equilibria via subcritical bifurcation from the disease-free equilibrium.

The rise of Islamic terrorism around the world is considered to be the result of many complex and interrelated issues associated with globalization and cultural penetration of the West into predominantly Muslim regions. Many of the perceived causes for this social unrest, such as US military bases located in sacred regions or alliances between the West and corrupt secular regimes, have been well-stated by Islamic fundamentalists as obvious determinants of social conflict. Other causal factors, especially long term ones, may not be as obvious and might be better understood by applying methods from Computational Economics and Sociology; agent-based; approaches, to this complex and important situation. We are developing a simulation framework for this purpose. Our end goal is to provide policy makers with decision support based on socio-economic computer experiments -- scenario generation representing known militant and terrorist groups, ethnic and culturally defined groups of agents, Western and Eastern regimes, and their interrelated political economies. In my presentation I will describe the building blocks of this model and demonstrate our progress made to date.

The World Health Organization (WHO) is responsible for making annual vaccine strains recommendation to countries around the globe. The recently proposed antigenic distance hypothesis suggested that vaccine efficacy of repeated annual vaccination can be enhanced by taking into account the immunization history of vaccinees. In this work, we use the antigenic distance hypothesis as a basis to formulate the vaccine selection problem as a discrete-time stochastic dynamic program with a high- dimensional continuous state space. We discuss the techniques that were developed for solving this dynamic program, and present an effective and robust heuristic policy. Finally, we compare the performance of the WHO policy, the heuristic policy, the no- vaccine policy within the context of our model.

Optimal control theory is applied to a system of ordinary differential equations modeling a two-strain tuberculosis model. we consider (time dependent) optimal control strategies associated with case holding and case finding based on a two-strain TB model. The case finding effort is incorporated by adding a control term that identifies and cures a fraction of latent individuals so that the rate at which latent individuals develop the disease will be reduced. The case holding effort is incorporated by adding a control term that may lower the treatment failure rate of individuals with active sensitive TB so that the incidence of acquired drug-resistant TB will be reduced. Our objective functional balances the effect of minimizing the cases of latent and infectious drug-resistant TB and minimizing the cost of implementing the control treatments.

In this talk I will give an overview of the social, cultural and economic reasons for why the poor, the uneducated, and the marginalized do not access health care systems. Often, even when the manifestations of the disease are serious, and had it been us the course of action would have been obvious, they have compelling reasons for not seeking help. Finally I will present some thoughts on the impact of "rogue economies" (alcohol, tobacco, drugs, trafficking in people and in arms) on the rural in the developing world.

The typical pattern when a new disease invades a population is that an epidemic may occur, with an increase in the number of infectives followed by a decrease to zero. By quarantining infectives it may be possible to decrease the size of the epidemic substantially. The problem is more difficult if infectives can not be identified rapidly and with certainty. In addition, in practice the behaviour of a model may be very sensitive to parameter values.

We develop a mathematical model that incorporates partial cross-immunity to next-to- kin strains. The immunity status of the host is the antibody levels corresponding to all the strains that each host has immunity to. Antibody immune response of the host population is captured by an index-set notation where the index specifies the immune- competence level against a particular strain. In contrast to previous modeling frameworks, the population here is structured into non-intersecting subclasses. Since multiple infection with influenza strains is unlikely to occur, we do not imbed superinfection with the same or different strains as part of our model. This framework allows for a biologically interpretable range of cross-immunity levels as observed during the yearly influenza epidemics (antigenic drift). Using the proposed framework of cross-immunity, we provide specific cases to illustrate the impact of the assumptions on the strain state space. Furthermore, we utilize the proposed notation to provide threshold conditions for the invasion of a new strain and show the existence of an endemic multi-strain equilibrium in a particular case.

For rumor spreading, we measure the rate of initial growth for two discrete time models: Daley-Kendall (DK) and Susceptible-Infective-Recovered (SIR). We use a particular type of random networks proposed by Liu-Lai-Ye-Dasgupta. In such topologies our numerical experimentations (various realizations of the rumor epidemics over different networks) suggest that the growth of rumor spreading relates to the connectivity of the network. We also, observe that the rate of initial growth is very sensitive to the network topology, the contact process modeling the dynamics, and the location of the initial spreader.

The complexity of dynamics and behavior of agent-based models (ABM) is challenging for the novice as well as the experts in the field. For example, a ABM system can exhibit a variety of different final states from the identical initial conditions and problem specification. These sensitivities in ABM evolution can capture the chaotic analogs in the deterministic differential equations that represent similar systems, but the sensitivities can also be unique to ABM. This talk will introduce a simple foraging ABM as a model problem. We will then show how processes of self-organization lead to sequential developmental stages in the evolution of the system. A key observation is that the role diversity is central to understanding the progress of the system, both within selective (select from diversity to achieve higher performance) and synergistic (combine diversity without selection to achieve higher performance) processes. The developmental viewpoint is argued to be generally applicable across many decentralized systems (e.g., stock markets, ecosystems, epidemics). The developmental viewpoint is a powerful tool to explain many of the observational inconsistencies of ABM systems. For example, one can predict from knowing the developmental stage of the system when sensitivities affect local or global performance - a critical issue when evaluating the robustness of system to changes. For added intrigue, the developmental viewpoint expressed above is in conflict with the generally held view that evolutionary systems do not have "directions" in their progress.

A model for the transmission dynamics of a single strain of dengue, a mosquito-transmitted disease that models the growth rate of the mosquito egg population via a general Kolmogorov recruitment function is constructed. The model is capable of supporting multiple equilibria. In the special case of a unique positive disease-free equilibrium, its global stability is established. The basins of attraction of disease-free equilibria are identified under some assumptions. Illustrative examples are provided. Possible strategies for dengue control are discussed on this framework.

Two patch discrete-time S-I-S epidemic models with multiple pathogen strains and dispersal are presented. In the single patch model, the basic reproduction number, R0, determines strain dominance and persistence. The strain with the largest R0 dominates the system whenever the magnitude of its R0 is greater than one. The invasion of a resident (dominant) strain on a complex attractor by a more lethal strain is illustrated. In the two-patch epidemic model, both the dispersal probabilities and basic reproduction numbers for each of the strains determine strain persistence. The two-patch model supports coexistence of two or more of the competing strains.

Analysis of epidemiological data early in the course of an epidemic can provide insights into sucessful control strategies and give quantitive predictions for successful control measures. We modelled the recent Severe Acute Respiratory Syndrome epidemics as a system of deterministic ordinary differential equations. We analyzed epidemiological data from Singapore, Hong Kong and Ontario through April to determine basic parameters of the epidemic model including the basic reproductive number (R = 1.2) and to predict the eventual control of the outbreak in Ontario. We conclude that relative to the freely growing epidemic, both diagnosis within 3 days of the onset of symptoms and isolation leading an order of magnitude reduction in the number of new cases are necessary to achieve control.

How do fanatic behaviors in vulnerable populations spread? Can a simple epidemic model (i.e.,a caricature) for the spread of fanaticism give insights into this question? I will briefly outline the work carried out in this direction--for the spread of an ideology in a homogenous environment--by Castillo-Chavez and Song. I will outline extensions that focus on models for competing ideologies and provide some preliminary results.

We use a system of coupled populations experiencing an outbreak to frame a discussion of issues involving the spread and control of infectious diseases in interacting populations such as the inhabitants of paired border cities such as San Diego and Tijuana. Analysis shows that conditions in either city can cause the outbreak to be sustained in both populations regardless of conditions in the other city, and implies that control policies must be global.

Globalization and the possibility of bioterrorist acts have accentuated the need for the rapid development of theoretical and practical mathematical frameworks that may be useful in our efforts to anticipate, prevent and respond to acts of destabilization. The conference's goal is to instigate exchanges between experts and those interested in working (or already working) in problems at the interface of epidemiology, public health, control and mathematics. In this talk, I will attempt to facilitate these exchanges by highlighting issues and challenges at the interface of homeland security, public health policy and mathematics.