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. One recent analysis resolved this conflict by hypothesizing that stigmas had individual and group-evolutionary benefits in the past but are now maladaptive because of intervening societal transitions. Here, we present a quantitative theory of infectious disease stigmatization. Using a four-compartment model of stigmatization against a chronic disease, we show a stigma ratio, being the ratio of net transmissions by stigmatized people to net transmissions by unstigmatized people, predicts the impact of stigmatization on lifetime infection risk. When stigmatized people are segregated from the rest of the population and there are no alternative interventions that reduce transmission, stigmatization can reduce prevalence and infection risk. When stigmas do not lead to segregation but do discourage behavior change and reduce access to medical interventions, stigmatization acts to increases the lifetime risk of infection in the community. We further show that fear of stigmas can create policy resistance to healthcare access. The societal consequences of fear are worse when effective medical treatment is available. We conclude that stigma's can be adaptive, but good healthcare and leaky ostracism can make stigmas against chronic infectious disease maladaptive, and that the deprecation of stigmas is a natural transition in the modern urban societies.

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. Yet our foresight of individual responses to public health investment remains limited. This paper formulates a population game describing the prevention of infectious disease transmission when community health depends on the interactions of individual and public investments. We compare three common relationships between public and individual investments and explain how each relationship alters policy responses and health outcomes. Our methods illustrate how identifying system interactions between nature and society can help us anticipate policy responses.

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. In this paper, we exploit an alternative theory based on Ellner's premise that species invasions are best interpreted not as waves, but as random walks, and that the discreteness of living organisms is fundamentally important. Using a density-dependent invasion model in a stationary environment with indivisible (atomic) individuals where reproduction and dispersal are stochastic and independent, we show 4 key properties of Ellner's invasions previously suggested by simulation analysis: (1) greater spatial dispersal stochasticity quickens invasions, (2) greater demographic stochasticity slows invasions, (3) negative density-dependence slows invasions, and (4) greater temporal dispersal stochasticity quickens invasions. We prove the first three results by using generating functions and stochastic-dominance methods to rank furthest-forward dispersal distributions. The fourth result is proven in the special case of atomless theory, but remains an open conjecture in atomic theory. In addition, we explain why, unlike atomless invasions, an infinitely wide atomic invasion in two-dimensions can travel faster than a finite-width invasion and a one-dimensional invasion. The paper concludes with a classification of invasion dynamics based on dispersal kernel tails.

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. In this paper, we explore how the risks of disease emergence are altered by the aggregation of introduction events within bridge communities. In viscous bridge communities, repeated introductions can elevate the local prevalence of immunity. This local herd immunity can form a barrier reducing the opportunities for disease emergence. In some situations, reducing exposure rates counterintuitively increases the emergence hazards because of off-setting reductions in local immunity. Increases in population mixing can also increase emergence hazards, even when average contact rates are conserved. Our theory of bridge communities may help guide prevention and explain historical emergence events, where disruption of stable economic, political or demographic processes reduced population viscosity at ecological interfaces.

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." Since behavior changes can be costly, we would like to know the optimal social distancing behavior. But the benefits of changes in behavior depend on the course of the epidemic, which itself depends on our behaviors. Differential population game theory provides a method for resolving this circular dependence. Here, I present the analysis of a special case of the differential SIR epidemic population game with social distancing when the relative infection rate is linear, but bounded below by zero. Equilibrium solutions are constructed in closed-form for an open-ended epidemic. Constructions are also provided for epidemics that are stopped by the deployment of a vaccination that becomes available a fixed-time after the start of the epidemic. This can be used to anticipate a window of opportunity during which mass vaccination can significantly reduce the cost of an epidemic.

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. To better understand how age-dependent variations in behavior can help individuals deal with infection risks, we study an epidemiological game in an SI model with two life-history stages where social distancing behaviors that reduce exposure rates are age-dependent. When considering a special case of the general model, we show that there is a unique Nash equilibrium when the infection pressure is a monotone function of aggregate exposure rates, but non-monotone effects can appear even in our special case. The non-monotone effects sometimes result in three Nash equilibria, two of which have local invasion potential simultaneously. Returning to a general case, we also describe a game with continuous age-structure using partial-differential equations, numerically identify some Nash equilibria, and conjecture about uniqueness.

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. In this paper, we provide a rigorous mathematical proof that this method not only conserves the fixed points of a Boolean network, but also conserves the complex attractors of general asynchronous Boolean models wherein at each time step a randomly selected node is updated. This method thus allows one to infer the long-term dynamic properties of a large-scale system from those of the corresponding reduced model.

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. When individuals make decisions based on past epidemics, we find that individuals with many contacts vaccinate, whereas individuals with few contacts do not. However, the threshold number of contacts above which to vaccinate is highly dependent on the overall network structure of the population and has the potential to oscillate more wildly than has been observed empirically. When we increase the number of prior seasons that individuals recall when making vaccination decisions, behavior and thus disease dynamics become less variable. For some networks, we also find that higher flu transmission rates may, counterintuitively, lead to lower (vaccine-mediated) disease prevalence. Our work demonstrates that rich and complex dynamics can result from the interaction between infectious diseases, human contact patterns, and behavior.

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. In addition, we show how path-integral approaches can be applied to the study of models in which strategies are fixed waiting times rather than exponential random variables. These methods can be applied to a wide variety of decision problems with population-dynamic feedbacks.

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. This paper formulates a differential-game to identify how individuals would best use social distancing and related self-protective behaviors during an epidemic. The epidemic is described by a simple, well-mixed ordinary differential equation model. We use the differential game to study potential value of social distancing as a mitigation measure by calculating the equilibrium behaviors under a variety of cost-functions. Numerical methods are used to calculate the total costs of an epidemic under equilibrium behaviors as a function of the time to mass vaccination, following epidemic identification. The key parameters in the analysis are the basic reproduction number and the baseline efficiency of social distancing. The results show that social distancing is most beneficial to individuals for basic reproduction numbers around 2. In the absence of vaccination or other intervention measures, optimal social distancing never recovers more than 30% of the cost of infection. We also show how the window of opportunity for vaccine development lengthens as the efficiency of social distancing and detection improve.

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. A third measure of reproductive success, which combines properties of both the basic reproductive number and the ultimate proliferation rate, is the discounted reproductive number R(d). The discounted reproductive number is a measure of reproductive success that is an individual's expected lifetime offspring production discounted by the background population growth rate. Here, we draw attention to the discounted reproductive number by providing an explicit definition and a systematic application framework. We describe how the discounted reproductive number overcomes the limitations of both the standard reproductive numbers and proliferation rates, and show that R(d) is closely connected to Fisher's reproductive values for different life-history stages.

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. The analyses presented here are important not only for HCV infection but should also be relevant for modeling other infections with hepatotropic viruses, such as hepatitis B virus.

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 transmission dynamics are described by an SIS model and individuals are only allowed to invest in daily prevention measures like hygiene. The analysis shows that disassortative mixing may lead to multiple Nash equilibria when there are two interacting subpopulations affecting disease prevalence. The dynamic stability of these equilibria is analysed under the assumption that strategies change slowly in the direction of self-interest. When mixing is disassortative, interior Nash equilibria are always unstable. When mixing is positively assortative, there is a unique Nash equilibrium that is globally stable.

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. This does not rule out the possibility of multiple stable equilibria, but we provide two sufficient conditions for the uniqueness of the endemic equilibrium, and show that these conditions ensure uniqueness in several common special cases. Our results indicate that the within-host dynamics of immunity can, in principle, have important consequences for population-level dynamics, but also suggest that this would require strong non-monotone effects in the immune response to infection. Neutralizing antibody titer data for measles are used to demonstrate the biological application of our theory.

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. This provides a rigorous justification of the concept of "endemic stability," and demonstrates that folk-practices may have been historically justified.

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. For epidemic influenza, the elderly have the greatest risk of influenza mortality, yet children are responsible for most of the transmission. The long-standing recommendations of the Centers for Disease Control follow the dictates of individual self-interest and prioritize the elderly for vaccination. However, preferentially vaccinating children may dramatically reduce community-wide influenza transmission. A potential obstacle to this is that the personal utility of vaccination is lower for children than it is for the elderly. We parameterize an epidemiological game-theoretic model of influenza vaccination with questionnaire data on actual perceptions of influenza and its vaccine to compare Nash equilibria vaccination strategies driven by self-interest with utilitarian strategies for both epidemic and pandemic influenza. Our results reveal possible strategies to bring Nash and utilitarian vaccination levels into alignment.

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. In contrast to previous work proposing that universal vaccination is impossible in a game theoretic context, we show that optimal individual behavior can vary between universal vaccination and no vaccination, depending on the relative costs and benefits to individuals. By coupling game models and epidemic models, we demonstrate that the pursuit of self-interest often leads to stable dynamics but can lead to oscillations in vaccine uptake over time. The instability is exacerbated in populations that are more homogeneous with respect to their perceptions of vaccine and infection risks. This research illustrates the importance of applying temporal models to an inherently temporal situation, namely, the time evolution of vaccine coverage in an informed population with a voluntary vaccination policy.

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. The second is the distributed-infectives model, often described by the diffusion of infected individuals. However, neither approach is ideal when individuals move within home ranges. This paper presents a unified modeling hypothesis, called the restricted-movement model. We use this model to predict spatial spread in settings where infected individuals move within overlapping home ranges. Using mathematical and computational approaches, we show that our restricted-movement model has three limits: the distributed-contacts model, the distributed-infectives model, and a third, less studied advective distributed-infectives limit. We also calculate approximate upper bounds for the rates of an epidemic's spatial spread. Guidelines are suggested for determining which limit is most appropriate for a specific disease.

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. This paper presents a theory for the optimization of antibiotic cycling based on a density-independent model of transmission and immigration of evolutionarily static strains. In the case of two pathogen strains, I show that the population's asymptotic growth rate is a monotonically increasing function of the oscillation period under certain common assumptions. Monte Carlo simulations show that this result fails in more general settings, but suggest that antibiotic cycling seldom provides a significant improvement over alternative mixing practices. The results support the findings of other researchers that antibiotic cycling does not offer significant advantages over idealized conventional practice. However, cycling strategies may be preferable in some special cases.