Title: The Influence of Cross-Immunity on the Coexistence, Invasion and Evolution of Pathogen Strains

Several epidemic models with many co-circulating strains have shown that partial cross-immunity between otherwise identical strains of a pathogen can lead to three solutions: all strains stably coexisting, a subset of strains stably coexisting, all strains coexisting in complex cycles. Here we step back to a three strain model to examine the mechanisms behind some of these solutions. Using a one-dimensional antigenic space, we consider a host population in which two strains are endemic and ask when it can be invaded by a third strain. If the function relating antigenic distance to crossimmunity is a square-root or linear this is always possible. If the function is parabolic it depends on the degree of antigenic similarity between strains and the basic reproductive number. We show that the differences between functional forms occur because their shape determines the importance of secondary infection. The basic reproductive number is important because it affects the size of the tertiary susceptible host population. Since the key property of the cross-immunity function is the shape, these results should apply to a more general class of functions. This agrees with, and explains, the results of previous multistrain models based on a one-dimensional antigenic space: pathogens for which antigenic distance is related to cross-immunity by a function with square-root type curvature will exist as a cloud of strains with no antigenic structuring. Conversely, pathogens for which the relationship has parabolic type curvature will exist in populations with strong antigenic structuring and the number of strains limited by the basic reproductive number.

Title: Annual epidemic in drifting influenza

Immunity to a new drift strain of influenza is comprised of cross-immunity induced by previous infections with related strains. Due to cross-correlations in susceptibility to flu-strains, the immunity is difficult to describe in detail. I will present a new modelling approach, annual epidemic models where cross-immunity is updated independently of the epidemic processes, and show how this approach can be used to understand the speed of flu drift and the selection forces that keep the flu-quasispecies from diverging.

Title: Instabilities in Multi-Strain Disease Models with Antibody-Dependent Enhancement

As we become more sophisticated in our resources to fight disease, pathogens become more resilient in their means to survive. Antibody-dependent enhancement (ADE), a phenomenon in which viral replication is increased rather than decreased by immune sera, has been observed in vitro for a large number of viruses of public health importance, including flaviviruses, coronaviruses, and retroviruses. This increased viral growth rate is thought to increase the infectivity of the secondary infectious class. We study the complex dynamics induced by ADE in multi-strain disease models. In the models, ADE induces the onset of oscillations without external forcing. We derive approximations of the ADE parameter needed to induce oscillations and analyze the associated bifurcations that separate the types of oscillations. We investigate the stability of these dynamics by adding stochastic perturbations to the model. We also present a preliminary analysis of the effect of vaccination strategies. Though the models presented are specifically designed for dengue hemorrhagic fever, our results are applicable to any epidemiological system in which partial immunity increases pathogen replication rates.

Title: Transmission and Control of Seasonal and Pandemic Influenza

Recurrent epidemics of influenza are observed seasonally around the world with considerable health and economic consequences. Major changes in the influenza virus composition through antigenic shifts can give rise to pandemics. The reproduction number provides a measure of the transmissibility of influenza. We estimated the reproduction number across influenza seasons in the United States, France, and Australia for the last 3 decades. In regards to pandemic influenza, we estimated the reproduction number for the first two epidemic waves during the 1918 influenza pandemic in Geneva, Switzerland. I will discuss the public health implications of our findings in terms of controlling regular influenza epidemics and an influenza pandemic of comparable magnitude to that of 1918.

Title: Rapid Evolution and Predator-Prey Dynamics with Variable Costs of Defense

Under certain conditions, predation acts as a selective pressure that drives prey adaptation. It follows then that a necessary piece of understanding population dynamics is to understand the underlying genetic forces that drive those dynamics. Here, we investigate the effects of genetic variability in predator search efficiency and in prey defense against predation on the stability and dynamics of a predator-prey system. In particular, we examine the impact of varied vs. fixed cost of anti-predator defense on population dynamics. Varied cost implies that the cost of anti-predator defense varies with population size, so that, at low prey density, defense is without cost. We assume that the tradeoff for investment in defense is a decrease in fecundity.

Title: Coupling Ecology and Evolution: Malaria and the S-Gene Across Time Scales

Malaria has long been a scourge to humans. The exceptionally high mortality of malaria in some regions has led to strong selection for resistance, even at the cost of increased risk of potentially fatal red blood cell deformities in some offspring. In particular, genes that confers resistance to malaria when they appear in heterozygous individuals are known to lead to sickle-cell anemia. Thus, there is balancing selection against the evolution of resistance, with the strength of that selection dependent upon malaria prevalence. Over longer time scales, the increased frequency of resistance in a population might be expected to decrease the frequency of malaria and reduce selection for resistance. However, possession of the sickle-cell gene leads to longer-lasting parasitaemia in heterozygote individuals, and therefore the presence of resistance may actually increase infection prevalence. We developed a model that explores the interplay among these processes. By coupling the dynamics of the epidemiology of malaria and the genetics of sickle-cell gene, our model allows for joint investigation of impact of malaria on the selection of S-gene and influence of genetic composition of a population on the maintenance of malaria. Our results are based on threshold conditions derived from the model by separating malaria disease dynamics on the fast time scale and the dynamics of S-gene on the slow time scale and by conducting stability analysis.

Title: Evolution of Virulence: Interdependence, Constraints, and Selection using Nested Models

Natural selection acts on virus populations at two distinct but interrelated levels: within individual hosts and between them. Studies of the evolution of virulence typically focus on selection acting at the epidemiological or between-host level and demonstrate the importance of trade-offs between disease transmission and virulence rates. Within-host studies reach similar conclusions regarding trade-offs between transmission and virulence at the level of individual cells. Studies which examine selection at both scales assume that between- and within-host selection are necessarily in conflict. We explicitly examine these ideas and assumptions using a model of within-host viral dynamics nested within a model of between-host disease dynamics. Our approach allows us to evaluate the direction of selection at the within- and between-host levels and identify situations leading to conflict and accord between the two levels of selection. In addition, we are now expanding this framework to examine how, in the instance of two competing strains, these conflicts between selection at different levels is resolved.

Title: Epochal Evolution Shapes the Phylodynamics of Interpandemic Influenza

Influenza A H3N2 viruses are characterized genetically by their limited standing diversity and antigenically by clusters that emerge and replace each other within 2-8 years. By introducing an epidemiological model that allows for differences between the genetic and antigenic properties of influenza's hemagglutinin, we show that these patterns can arise from cluster-specific immunity alone. Central to the formulation is a genotype-to-phenotype mapping, based on neutral networks, with antigenic phenotypes, not genotypes, determining the degree of strain cross-immunity. The model parsimoniously explains well known, as well as previously unremarked, features of interpandemic influenza dynamics and evolution. It captures the observed boom-and-bust pattern of HA1 evolution, with periods of antigenic stasis during which genetic diversity grows, and episodic contraction of this diversity during cluster transitions.

This work is in collaboration with Sarah Cobey (UM-Ann Arbor), Bryan Grenfell (Penn State), and Mercedes Pascual (UM-Ann Arbor).

Title: Pathogen Type Replacement: Theoretical Mechanism

The talk is based on articles with authors: Maia Martcheva, Mimmo Iannelli, Xue-Zhi Li, Xinyang Normal University, China, Benjamin Bolker and Robert D. Holt, University of Florida

Many pathogens are represented by multiple antigenically distinct types. Vaccination provides vaccinated individuals with immunity specific to the particular type that created it. What is observed on population level when a vaccinated campaign is initiated is that those pathogen types that circulated in the population before the vaccination diminish in prevalence while others (typically not included in the vaccine) increase in prevalence.

This phenomenon has been termed the replacement effect. The mechanism of the replacement effect is thought to be the differential effectiveness of the vaccine. In a recent article we reproted that we observed population level strain replacement in an epidemic model with super-infection and "perfect" vaccination, that is vaccination that protects 100 percent all vaccinated individuals against both strains involved in the model. Would strain replacement have occured if the trade-off mechanism in the model was coinfection and not super-infection? If differential effectiveness of the vaccine is not necessary for the replacement effect to occur, what is the causal mechanism of strain replacement? I will address these and other questions in this talk.

Title: Using Semi-Deterministic Models to Describe Dynamics of Highly Infectious Multi-Strain Diseases

Deterministic epidemic models have two key weaknesses when used to de- scibe the long-term dynamics of an evolving antingenically diverse pathogen - they fail to include strain extinction (except as a result of a long-term fitness deficit), and strain creation (e.g. mutation) is modelled as a continuous rather than a discrete (and random) process.

To study the influence of the two above factors on epidemic dynamics we use a semi-deterministic population model with extinction and a stochastic rep- resentation of mutation. Considering the probability of a single mutation as small, we describe mutation on population level as a stochastic Poisson process. Extinction is modelled by eliminating a strain if the number of infected indi- viduals in a popualtion falls below 1. All other aspects of a model are fully deterministic.

We extend the model first outlined by Gupta et al (Science, 1998. 280: p. 912-915) by allowing the degree of cross-immunity against a new strain to vary as a function of the number of novel antigenic alleles in the new strain - instead of there being a single fixed level of cross-immunity experiences by all new strains which share one or more alleles with any previously experienced strain. This allows selection for antigenic diversity (as occurs in influenza) to be modelled more easily, via a cross-immunity function which decays with genetic distance.

As well as introducing the model, this talk will compare the results from our model with those from the original Gupta et al model, and examine the impact of extinction and stochasticity in mutation on model dynamics. We show that both factors tend to lead to deterministic models predicting unrealistically high strain diversity and to also over-estimate strain lifetimes. The dependence of strain diversity and life times on population size, R0 and recovery rates will also be discussed.

Title: Simplified Models of Host Immune Response, and the Evolution of Virulence

Recently, several papers have introduced explicit modelling of hosts' immune response to study the evolution of pathogens (or the coevolution of hosts and parasites), without the need of assuming a priori trade-offs between pathogens' virulence and transmission rates (or other traits).

Following this idea, I extend the submodel for within-host dynamics by Gilchrist and Sasaki (2002) in which functional response in pathogen-immune system interaction, and decay of immune response are considered: different qualitative behaviours (pathogen clearance, pathogen uncontrolled growth, equilibrium or periodic coexistence of pathogen and immune response) are possible, according to parameter values and initial conditions. Passing to the population level, it is seen that, in all cases examined, an intermediate level of pathogen replication (or of hosts' rate of immune response) is selected for. The qualitative properties of the resulting within-host system may vary: I study numerically how this depends on key parameters, such as the decay of hosts' immune response.

In this setting, one can allow very naturally, without ad-hoc assumptions, for multiple infections from the same pathogen strain, or from a competing one. The price is, of course, the mathematical complexity of the resulting system, that makes it very difficult to compute even the invasion coefficient of a second pathogen strain. I will present some preliminary results in that direction.

Title: Networks as Evolutionary Landscapes

Studies of networks and their properties have proliferated in recent years. Traditional models of disease transmission, however, are non-spatial and do not generally restrict the transmission of a pathogen to the limited number of contacts typically made by a host. Such non-spatial models generally predict that diseases should evolve to be highly transmissible, to be benign to the host, and to possess a long infectious period. However, when transmission opportunities are defined by a host contact network, different pathogen attributes may be selected. In this talk, I shall present the results from models of pathogen spread and mutation within structurally defined transmission networks. I will also discuss the implications for more complicated host models which include an interaction between a pathogen and its host's immune system. Networks essentially act as an evolutionary landscape upon which competing pathogen strains must persist or face extinction. Complicated dynamics are driven via a feedback between the infection process and the network structure: a pathogen modifies the network in the process of adapting to it.

Title: The network structure of a college student population influences the dynamics of influenza epidemics.

The student populations of academic institutions are often arranged in insular, densely interconnected social networks.  This structure potentially allows disease, such as influenza, to spread rapidly through the population.  As the threat of avian influenza has received increasing amounts of attention from the media and scientific community, social network models will play an increasingly important role in investigating the dynamics of epidemics. We analyze the structure of the student body at the State University of New York College at Geneseo and use a simulation model to study the spread of influenza through this network.  Network connections between individuals (edges) are based on course registration and student housing data collected from the Fall 2005 semester.  The degree distribution of the network changed daily, according to class schedule. Individuals in this network were infected with influenza.  In general, the dense structure of this network results in large epidemics. These results are being incorporated into the college pandemic influenza preparedness plan. 

Title: Models for Biofilm Dynamics and Phenotypic Resistance

In this talk we describe a model for the interaction of biofilms (complex microbial communities) with the surrounding medium and explore in a rough way the effects of antimicrobial on the selection of resistance strains. The biofilm-environment interaction is explored through differential equations incorporating the dynamics of the fluid medium where biofilms live. The interaction with antimicrobials deals with the phenomenon of phenotypic switches to avoid deleterious effects and is based on a ODE model.