John Cross, epidemic theory, and mathematically modeling the Norwich smallpox epidemic of 1819.

In this paper, we reintroduce Dr. John Cross' neglected and unusually complete historical data set describing a smallpox epidemic occurring in Norwich, England in 1819. We analyze this epidemic data in the context of early models of epidemic spread including the Farr-Evans-Brownlee Normal law, the Kermack-McKendrick square Hyperbolic Secant and SIR laws, along with the modern Volz-Miller random-network law.

ATTRACTOR-BASED MODELS FOR SEQUENCES AND PATTERN GENERATION IN NEURAL CIRCUITS.

Neural circuits in the brain perform a variety of essential functions, including input classification, pattern completion, and the generation of rhythms and oscillations that support processes such as breathing and locomotion [51]. There is also substantial evidence that the brain encodes memories and processes information via of neural activity. In this dissertation, we are focused on the general problem of how neural circuits encode rhythmic activity, as in central pattern generators (CPGs), as well as the encoding of sequences.

Dynamic and game theory of infectious disease stigmas.

Stigmas are a primal phenomena, ubiquitous in human societies past and present. Some evolutionary anthropologists have argued that stigmatization in response to disease is an adaptive behavior because stigmatization may help people and communities reduce the risks they face from infectious diseases and increase reproductive success. On the other hand, some cultural anthropologists and social critics argue that stigmatization has strong negative impacts on community health.

Unintended consequences and the paradox of control: Management of emerging pathogens with age-specific virulence.

We project forward total Zika virus disease (ZVD) under varying hazards of infection and consider how the age distribution of disease burden varies between these scenarios. Pathogens with age structured disease outcomes, such as rubella and Zika virus, require that management decisions consider their impact not only on total disease incidence but also on distribution of disease burden within a population. Some situations exhibit a "paradox of control" in which reductions of overall transmission decrease the total incidence but increase the incidence of severe disease.

Provisioning of Public Health Can Be Designed to Anticipate Public Policy Responses.

Public health policies can elicit strong responses from individuals. These responses can promote, reduce, and even reverse the expected benefits of the policies. Therefore, projections of individual responses to policy can be important ingredients in policy design.

The importance of being atomic: Ecological invasions as random walks instead of waves.

Invasions are one of the most easily identified spatial phenomena in ecology, and have inspired a rich variety of theories for ecologists' and naturalists' consideration. However, a number of arguments over the sensitivities of invasion rates to stochasticity, density-dependence, dimension, and discreteness persist in the literature. The standard mathematical approach to invasions is based on Fisher's analysis of traveling waves solutions for the spread of an advantageous allele.

Population viscosity suppresses disease emergence by preserving local herd immunity.

Animal reservoirs for infectious diseases pose ongoing risks to human populations. In this theory of zoonoses, the introduction event that starts an epidemic is assumed to be independent of all preceding events. However, introductions are often concentrated in communities that bridge the ecological interfaces between reservoirs and the general population.

Equilibria of an epidemic game with piecewise linear social distancing cost.

Around the world, infectious disease epidemics continue to threaten people's health. When epidemics strike, we often respond by changing our behaviors to reduce our risk of infection. This response is sometimes called "social distancing.

A REDUCTION METHOD FOR BOOLEAN NETWORK MODELS PROVEN TO CONSERVE ATTRACTORS.

Boolean models, wherein each component is characterized with a binary (ON or OFF) variable, have been widely employed for dynamic modeling of biological regulatory networks. However, the exponential dependencse of the size of the state space of these models on the number of nodes in the network can be a daunting prospect for attractor analysis of large-scale systems. We have previously proposed a network reduction technique for Boolean models and demonstrated its applicability on two biological systems, namely, the abscisic acid signal transduction network as well as the T-LGL leukemia survival signaling network.

Games of age-dependent prevention of chronic infections by social distancing.

Epidemiological games combine epidemic modelling with game theory to assess strategic choices in response to risks from infectious diseases. In most epidemiological games studied thus-far, the strategies of an individual are represented with a single choice parameter. There are many natural situations where strategies can not be represented by a single dimension, including situations where individuals can change their behavior as they age.

Erratic flu vaccination emerges from short-sighted behavior in contact networks.

The effectiveness of seasonal influenza vaccination programs depends on individual-level compliance. Perceptions about risks associated with infection and vaccination can strongly influence vaccination decisions and thus the ultimate course of an epidemic. Here we investigate the interplay between contact patterns, influenza-related behavior, and disease dynamics by incorporating game theory into network models.

A general approach for population games with application to vaccination.

Reconciling the interests of individuals with the interests of communities is a major challenge in designing and implementing health policies. In this paper, we present a technique based on a combination of mechanistic population-scale models, Markov decision process theory and game theory that facilitates the evaluation of game theoretic decisions at both individual and community scales. To illustrate our technique, we provide solutions to several variants of the simple vaccination game including imperfect vaccine efficacy and differential waning of natural and vaccine immunity.

Game theory of social distancing in response to an epidemic.

Social distancing practices are changes in behavior that prevent disease transmission by reducing contact rates between susceptible individuals and infected individuals who may transmit the disease. Social distancing practices can reduce the severity of an epidemic, but the benefits of social distancing depend on the extent to which it is used by individuals. Individuals are sometimes reluctant to pay the costs inherent in social distancing, and this can limit its effectiveness as a control measure.

An SIS epidemiology game with two subpopulations.

There is significant current interest in the application of game theory to problems in epidemiology. Most mathematical analyses of epidemiology games have studied populations where all individuals have the same risks and interests. This paper analyses the rational-expectation equilibria in an epidemiology game with two interacting subpopulations of equal size where decisions change the prevalence and transmission patterns of an infectious disease.

The discounted reproductive number for epidemiology.

The basic reproductive number, R(0), and the effective reproductive number, R, are commonly used in mathematical epidemiology as summary statistics for the size and controllability of epidemics. However, these commonly used reproductive numbers can be misleading when applied to predict pathogen evolution because they do not incorporate the impact of the timing of events in the life-history cycle of the pathogen. To study evolution problems where the host population size is changing, measures like the ultimate proliferation rate must be used.

ANALYSIS OF HEPATITIS C VIRUS INFECTION MODELS WITH HEPATOCYTE HOMEOSTASIS.

Recently, we developed a model for hepatitis C virus (HCV) infection that explicitly includes proliferation of infected and uninfected hepatocytes. The model predictions agree with a large body of experimental observations on the kinetics of HCV RNA change during acute infection, under antiviral therapy, and after the cessation of therapy. Here we mathematically analyze and characterize both the steady state and dynamical behavior of this model.

Backward bifurcations and multiple equilibria in epidemic models with structured immunity.

Many disease pathogens stimulate immunity in their hosts, which then wanes over time. To better understand the impact of this immunity on epidemiological dynamics, we propose an epidemic model structured according to immunity level that can be applied in many different settings. Under biologically realistic hypotheses, we find that immunity alone never creates a backward bifurcation of the disease-free steady state.

Optimal timing of disease transmission in an age-structured population.

It is a common medical folk-practice for parents to encourage their children to contract certain infectious diseases while they are young. This folk-practice is controversial, in part, because it contradicts the long-term public health goal of minimizing disease incidence. We study an epidemiological model of infectious disease in an age-structured population where virulence is age-dependent and show that, in some cases, the optimal behavior will increase disease transmission.

Resistance mechanisms matter in SIR models.

We compare four SIR-style models describing behavioral or immunological disease resistance that may be both partial and temporary in parameter regions feasible for interpandemic influenza. For the models studied, backward bifurcations and bistability may occur in contexts where resistance is due to behavior change, but they do not occur when resistance originates from an immune response. Care must be exercised to ensure that modeling assumptions about resistance are consistent with the biological mechanisms under study.

Long-standing influenza vaccination policy is in accord with individual self-interest but not with the utilitarian optimum.

Influenza vaccination is vital for reducing infection-mediated morbidity and mortality. To maximize effectiveness, vaccination programs must anticipate the effects of public perceptions and attitudes on voluntary adherence. A vaccine allocation strategy that is optimal for the population is not necessarily optimal for an individual.

Evolving public perceptions and stability in vaccine uptake.

Recent vaccine scares and sudden spikes in vaccine demand remind us that the effectiveness of mass vaccination programs is governed by the public perception of vaccination. Previous work has shown that the tendency of individuals to optimize self-interest can lead to vaccination levels that are suboptimal for a community. We use game theory to relate population-level demand for vaccines to decision-making by individuals with varied beliefs about the costs of infection and vaccination.

A model of spatial epidemic spread when individuals move within overlapping home ranges.

One of the central goals of mathematical epidemiology is to predict disease transmission patterns in populations. Two models are commonly used to predict spatial spread of a disease. The first is the distributed-contacts model, often described by a contact distribution among stationary individuals.

Simple models of antibiotic cycling.

The use of environmental heterogeneity is an old but potentially powerful method for managing biological systems. Determining the optimal form of environmental heterogeneity is a difficult problem. One family of heterogeneous management strategies that has received attention in the medical community is the periodic cycling of antibiotic usage to control antibiotic resistance.

Simulated evolution of selfish herd behavior.

Single species aggregations are a commonly observed phenomenon. One potential explanation for these aggregations is provided by the selfish herd hypothesis, which states that aggregations result from individual efforts to reduce personnel predation risk at the expense of group-mates. Not all movement rules based on the selfish herd hypothesis are consistent with observed animal behavior.

Nonequilibrium thermodynamics and nonlinear kinetics in a cellular signaling switch.

We develop a rigorous nonequilibrium thermodynamics for an open system of nonlinear biochemical reactions responsible for cell signal processing. We show that the quality of the biological switch consisting of a phosphorylation-dephosphorylation cycle, such as those in protein kinase cascade, is controlled by the available intracellular free energy from the adenosine triphosphate (ATP) hydrolysis in vivo: DeltaG=k(B)Tln(([ATP]/K(eq)[ADP]), where K(eq) is the equilibrium constant. The model reveals the correlation between the performance of the switch and the level of DeltaG.