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Carrie Manore
Summary of Research
My general research interests are in numerical analysis, dynamical systems, and
applied mathematics, specifically in numerical methods for partial and ordinary
differential equations and the mathematical modeling of infectious diseases and
ecological systems. My current research is the mathematical modeling of multi-host
pathogens in the context of competition between species for resources and spatial
heterogeneity.
Modeling disease and population dynamics is increasingly important as we try to
understand the effects of mechanisms such as pathogen transmission and competition on
an ecological system. Although mathematical advances have been made in this area,
analysis of even basic models that combine the dynamics of disease and two interacting
species is difficult and can often be intractable. Mathematical analysis of such models in
a spatially explicit environment gives more important insight into these systems.
Motivated by participation in the IGERT Ecosystem Informatics program and
collaboration with colleagues from other disciplines, I explore the interactions between
and among disease, competition, and spatial heterogeneity from a mathematical modeling
perspective.
The types of disease models we consider in my thesis are compartmental
Susceptible-Infectious-Recovered ordinary differential equation models. Competition
between species is modeled using Lotka-Volterra competition which assumes that a
species is both self-regulating and regulated by the presence of another species. Space is
modeled using multi-patch, spatially explicit models that are graphs with systems of
differential equations at each vertex. Multi-patch (metapopulation) models can be viewed
as a discrete approximation of diffusion or as a model for discrete, or patchy,
environments. For the case of spatio-temporal dynamics of disease spread, a multi-patch
model would consist of an SIR model on each vertex of a graph with connection between
some or all of the vertices.
I am currently working on four research projects. The first is an
exploration of the dynamics of a model for two competing species that share a directly
transmitted pathogen. We consider a general competition model with density-independent
death rates and a multi-host disease that spreads by either mass action or frequency
incidence. We compute the basic reproduction number, derive analytic forms for
equilibria where possible, and perform local and global stability analysis of the equilibria,
including the disease-free and infected coexistence equilibria. This general analysis gives
us important qualitative information about the factors influencing species coexistence or
exclusion in ecological communities.
The second pursues the important question, “Could exotic species alter disease
transmission dynamics and facilitate invasion?” By modeling and analyzing the
dynamics of multi-host pathogen and vector communities with a coupled system of
ordinary differential equations, we hope to understand how the forces of infection
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Carrie Manore
between multiple species affected by a single pathogen combine with competition and
metapopulations to allow invasion by exotic species that may be competitively excluded
in the absence of the pathogen. In particular, we model the spread of the aphid-vectored
Barley Yellow Dwarf Virus (BYDV) on multiple patches for two host grass species,
including seasonality and age structure for the grasses. Using numerical simulations, the
basic reproduction number, and sensitivity analysis, we are investigating how the virus
modulates competition between different hosts as we change initial conditions and
parameters in the system, including host composition. This work is being done in
collaboration with zoologists at Oregon State University.
The third project extends the two species competition and disease model to a
spatially explicit environment. The question of how disease affects patchy populations,
including whether or not a disease might drive a population to extinction or coexistence,
is an important one to biologists today. We construct multi-patch models that include
disease as well as competition dynamics between multiple species. As with the model in
the first project, we perform local and global stability analysis and derive analytic
expressions for equilibria and basic reproduction numbers. These results help us
investigate how the forces of infection and competition combine and are implicated in
determining community structure on a spatially heterogeneous environment.
The fourth (in collaboration with colleagues at Los Alamos National Laboratory)
investigates the spread of multi-host animal diseases in multiple patches, or counties, in
the United States. We use rinderpest, a virus closely related to human measles and
canine distemper that affects cloven-hoofed animals such as cows, pigs, and sheep, as a
case study. This virus is quite virulent, highly transmissible and causes high mortality in
naive populations. If rinderpest were to emerge in the United States, the loss in livestock
would likely be devastating. In order to suggest effective responses to an introduction of
rinderpest, we adapt a spatially explicit stochastic model for multi-host animal diseases in
the United States specifically to the behavior and primary hosts of rinderpest. The
mathematical model helps us estimate the extent of spread in and the relevance of each of
the considered host groups and different counties or regions of the United States. We use
sensitivity analysis to determine important parameters and effective mitigation strategies.
We also explore the role of space by initiating the epidemic in various locations and
simulating the results.
Future work will include an exploration of the use of partial differential equations
to model the spatial spread of species or pathogens, using optimal control methods to
determine best methods of disease control, and incorporating stochasticity in the models
where appropriate. The broad impact of our work is a partnership between ecologists and
mathematicians to develop significant mathematical advances in the theory and
application of mathematical models for the spread of pathogens and parasite mediated
competition between species on metapopulations. By applying the mathematical models
that will be developed to understand disease and competition dynamics in spatially
explicit environments, we will contribute significant scientific knowledge to the
management of disease and exotic species.