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The application of mathematics to epidemiological theory is a rapidly growing field which shows a great
deal of promise, both in terms of pure research and in terms of benefit to policy-makers in determining
the best possible actions for maximizing public health. The researchers at MathEcology are highly
skilled at the development of mathematical models in epidemiology at a variety of levels; below is just
a small sample of the type work we have done in this area.
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| Stochastic Epidemics |
| Smallpox |
| Epidemiological Software |
| Applied Epidemiology |
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Mathematical models of epidemic progression are essential components of the analysis of disease
patterns and the projection of these patterns into the future. These models must provide the means
to parameterize the characteristics of a disease, including infectivity and latency, birth and death
rates, transmission and contact rates, and population density and mobility, all of which can be
difficult to quantify, even in the presence of large volumes of data. The reality of the gaps in the
existing knowledge base and unknown quantities pulls uncertainty and chance into major roles in the
development of any predictive model.
One of the more widely applicable epidemiological models is the discrete SEIR model, in which the
affected population can be divided into four distinct classes based on the disease state of individuals:
susceptible, exposed, infective, and removed or recovered(where the subscript j refers to the specific
spatial population under consideration). The inclusion of such demography can be justified in the
comparison with diseases such as smallpox, in which distinct disease states can be recognized and
measured.
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The inclusion of immigration and emigration for each of the four disease states, and thus the
incorporation of spatial structure into the disease model, doubles the number of transitions and
radically increases the size of the mathematical system, and also the computing time to solve or
simulate it. We combined these aspects to create a stochastic model for the propagation of a
theoretical disease through a spatially structured population.
The fully stochastic model described above is too complicated to allow explicit solutions. The
behavior of the model can, however, be simulated by making use of the flow chart and transitions shown
above and stepping through the epidemic chronologically.
Deterministic and stochastic systems do not, however, form disjoint sets in the space of
epidemiological models. It is conventional wisdom that, for a given stochastic model, there exists an
approximating deterministic model which is an acceptable approximation if the population size is
sufficiently large.
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Such a deterministic model can be considered an appropriate limit of the
stochastic process, with the acknowledgement of the fact that different threshold theorems based on the
reproductive number R0 (defined as the number of cases generated by one infective over the period of
infectivity) exist for the two systems.
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Based on an in-depth study of historical outbreaks of smallpox prior to its global eradication, we
developed a set of parameter values to be incorporated into a deterministic compartmental model for
the disease. These values and other data were combined to create a smallpox timeline delineating the
various stages for discrete ordinary-type smallpox in the average host.
We also investigated epidemiological literature to devise mathematical best- and worst-case scenarios
for smallpox vaccination. The best-case vaccination scenario was developed based on the 1947 smallpox
vaccination campaign in New York City during which, through the voluntary free vaccination program,
over six million people (of the total population of seven million) were immunized within a four week
period. A worst-case vaccination scenario was developed based on early vaccination efforts in India in
the early 1960s, when vaccinators followed a contact-tracing strategy and travelled door-to-door
through villages and cities immunizing individuals through the rotary lancet method.
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Additionally, the effects of vaccination on immuno-compromised populations were researched and
summarized, culminating in a listing of contraindications to vaccination with live vaccinia virus,
potential side-effects, and possible alternative treatment strategies.
This work was incorporated into other research, and presented at the 2004 ASM Biodefense Research
Meeting. The abstract can be provided upon request.
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