Anne NiesM34O -CN

8 Nov 2011Paper #2

ln the study of epidemics geometry is used to model the actual and potential

transmission of disease. There are many variables to account for in the study of epidemics, and

geometry allows them to be included in the analytical models to give more accurate predictions

so that disease control experts can more effectively react to threats.

"Epidemic dynamics study is an important theoretic approach toinvestigate the transmission dynamics of infectious disease. lt formulates

mathematical models to describe the mechanisms of disease

transmissions and dynamics of infectious agents. The mathematical

models are based on population dynamics, behavior of disease

transmissions, features of the infectious agents, and the connections with

other social and physiologic factors. Through quantitative and qualitative

analysis, sensitivity analysis, and numeric simulations, mathematical

models can give us good understanding of how infectious diseases

spread, discover general principles governing the transmission dynamics

of the diseases, and identify more important and sensitive parameters, to

make reliable predictions and provide useful prevention and control

strategies and guidance [6]."

BSI

Early in the study of epidemics Kermack and McKendrick created a basic model for

studying epidemics, that continues to be the basis for many models today. ln their model the

population was divided into three compartments: those susceptible (S), those infected (l), and

those removed (R) (those who have recovered or are immune). The rate of flow from one

comparlment to the next is defined by differential functions 1. The goal of those trying to control

the spread of an epidemic is to have Ro < l, where Ro: (SoB)/y, so that the epidemic will die out.

lf Ro > 1, then the epidemic spreads [6].

1 (dsxdt)= - gsl, (drxdt)recovery rate.

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fl,

= BSI - yl, & (dR)/(dt) = yl;where B is the transmission coefficient, and 7 is the

Ultimately the Kermack-McKendrick model proved to be too simple, and since their work

in the late 1920's many other mathematicians and biologists have continued to search for and

discover models that are more accurate. One such discovery is related in The Geometry of

Ecological lnteractions: Simplifying Spacial Complexity where Ulf, Law, and Metz discuss how

apparent wave like dispersals of disease led to 2D modeling and studying of epidemics, and

how the compartmentalization of these 2D models can give realistic results for susceptibility

Ievels of a population. Also, "Cauchemez and Ferguson introduced a new approach to the

analysis of epidemic time series data to take account of partial observation of latency and the

temporal aggregation of observed data. They showed that homogeneous standard models can

miss key features of epidemics in large populations [4]." Ultimately, as more factors with various

impacts on the rate of spread were analyzed, information geometry became the primary method

for analyzing epidemic data.

"The field of information geometry first emerged as a method to

combine geometry and statistical probability theory to tackle problems In

other (than biology) fields including physics and economics. 'The new

field of information geometry draws a lot [of] inspiration from information

processing in biological systems, from the cell to the brain,' [5]."

lnformation geometry is the study of a collection of probability distributions over a vector,

which is used to create a topological2 space that is locally Euclidean (a manifold)with

coordinate system, = = {(3 | | e !/ll For a space to be locally Euclidean, means that you can

break it into small sections that behave like a Euclidean geometry (consider breaking the

surface of a sphere into flat pentagons). Doing this allows researchers to account for many

variables simultaneously, and then to view their model more simplistically while retaining

accountability for all variables.

Using information geometry Dodson extended the results of Britton and Lindenstrand to

produce a more accurate model. Britton and Lindenstrand had found that Ro: l.pLr, where )" is

the rate of infectious contacts and pLris the mean length of infectious period [4]. The goal in their

model remains for Ro < 1, otherwise if Ro > 1 then the "epidemic becomes a major outbreak with

2 A topological space is a set X together with a collection of open subsets T that satisfies the fourconditions.

1. The empty set is in T.

2. XisinT.3. The intersection of a finite number of sets in T is also in T (or if T is the collection of closed

subsets: The intersection of an arbitrary number of sets in T is also in T).

4. The union of an arbitrary number of sets in T is also in T (or if T is the collection of closedsubsets. The union of a finite number of sets in T is also in T) t8l

s The xi-function: ((s) = E (O) fl(1 - (sip)) t8l

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the number infected increasing exponentially [4]," and although they used a single number to

analyze the seriousness of the epidemic, this number is much more reliable than the original Ro

provided by Kermack & McKendrick. Dodson added the consideration of "the McKay bivariate

gamma distributiona for the joint distribution of latency and infectivity, recovering the effects of

variability but allowing possible correlation [4]." This is important because by doing so, Dodson

has moved the model into information geometry without loosing any of its effectiveness.

"ln modeling epidemics information geometry is able to account for

more complex and subtle differences in quantities, such as rates of

infection among populations, than previous approaches. lndeed, the

reason for the growing use of information geometry across biological and

medical research is that it is capable of allowing for non-uniformity in the

systems under study, whether at the scale of a protein, cell, pathway or

ecosystem. ln essence, information geometry combines geometry with

probability theory to model changing, complex and nonlinear systems [5]."

lnformation geometry is an extremely valuable tool because when we are able to model

a complex system we are then able to better analyze it to find potential solutions and to better

understand how variables impact the spread of an epidemic and epidemics in general.

a This is a statistical distribution combined, with a probability distribution. l've left out the equationbecause it's huge & messy, it can be found at the bottom of page 3 of [4].

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Bibliooraohv

[1]Amari, Shun-lchi and Nagaoka, Hiroshi. Methods of lnformation Geometry.

l21Ay, Nihat; Bertschinger, Nils; Jost, Jurgen; and Olbrich, Eckehard. "Ageometric approach to

complexity." Chaos. Volume. 21, lssue. 3, (Sep 2011).037'103-1 - 037103-10. American

I nstitute of Physics. 3 1 Oct 20 1 1 <http://dx.doi. org I 1 0.1 0631 1. 3638446>.

[3] Dieckmann, Ulf; Law, Richard; and Metz, J.A.J. The Geometry of Ecological lnteractions:

Simplifying Spatial Complexity. Cambridge: Cambridge University Press, 2001.

[4] Dodson, C.T.J. "lnformation Geometry and Entropy in a Stochastic Epidemic Rate Process."

<http://www. maths.manchester.ac.uk/-kd/PREPRINTS/lnfGeomlnStochRateProc.pdf>

[5] Hunter, Philip. "Biology isthe new physics." EMBO reportsVolume: 11, lssue: 5, (2010):

350-352. European Molecular Biology Organization. 23 Oct 2011 <http.//dx.doi.org/'10.1038/

embor.2010.55>.

[6] Ma, Zhien and Li, Jia. Dynamical Modeling andAnalysis of Epidemics. Singapore: World

Scientific Publishing Co. Ptc. Ltd., 2009.

[7] Shalizi, Cosma Rohilla. "lnformation Geometry." Notebooks. (20 Feb 2011).31 Oct2011

<http://cscs. umich.ed u I -crshalizilnotebooks/info-geo. htm l>.

[8] Wolfram Math World. 3 1 Oct 201'1 <http://mathworld.wolfram.com/Manifold.html>

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