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 <h1>Epidemiological Modeling</h1>

 <img src="images/sis.gif" align=RIGHT HSPACE=10 VSPACE=10 width=25%>
 <p> Compartmental disease models are some of the most widely used
 models of infectious disease. In these models individuals (or portions of
 an individual) are considered to be in a compartments based on their
 disease status. By convention, commonly used compartmental models are
 designated by the first letters of the compartments and the order
 people move between them. So, the model for a disease where people
 start out Susceptible, become Infectious after catching the disease,
 and then after some time clear the disease and become Susceptible again
 would be an SIS model. The four most commonly used compartments
 are <u>S</u>usceptible, <u>E</u>xposed, <u>I</u>nfectious, and
 <u>R</u>emoved. <u>S</u>usceptible contains those individuals
 who can still be infected with disease. <u>E</u>xposed contains individuals
 who have been infected with the disease, but have not yet become
 infectious themselves. <u>I</u>nfectious contains those individuals who
 are infected and infectious. <u>R</u>emoved (sometimes called <u>R</u>ecovered)
 contains those individuals who have been removed from the system
 either through death or immunity. The disease state of a population
 is speccified by knowing the nuber of individuals (or proportion of
 the population) that is in each compartment.

  <p> There are two main types of compartmental models: deterministic
  and stochastic. In deterministic model, rates of movements between
  compartments are used to represent the processes by which individuals
  are infected, become ill and eventually recover from disease. These
  rates specify a set of differential or difference equations, which
  can be evaluated to give that state of the model at a future time
  point. In deterministic models compartments need not contain an
  integer number of individuals.

  <p>Deterministic models only give the <i>expected</i> behavior of
  the disease in a population under a set of assumptions about
  the natural history and epidemiology of the disease. Often
  we would also like to know how the disease behaves in more
  extreme cases, and what the distribution of this behavior
  is. Stochastic models provide a method for addressing this
  issue. In a stochastic model, whole individuals are moved
  between compartments based on a random draw from some specified
  distribution. To get informative results from a stochastic
  compartmental model it is necessary to do hundreds, or even
  thousands of experiments.

 <p>Compartamental models also may have "closed" or "open" populations.
 In closed populations the set of individuals represented by the model
 is static, no one comes in and no one comes out. It is of note that
 in such populations any disease that confers permanent immunity
 after infection will eventually become extinct. In an open
 population there is continuous immigration and emigration from
 the population, usually via birth and death.

 <p>STEM comes with deterministic implementations for three commonly used
 compartmental models: SI(S), SIR(S), and SIER(S). These models are
 implemented as having open populations, but closed populations can
 be easily simulated by setting the immigration/emigration rate to
 zero. Which model is appropriate for a modeling project depends
 on both the disease of interest and the timescale that we are
 interested in modeling our results.

 <h2>The SI(S) Model</h2>
 <a href="simath.html">
 <img src="images/sis.gif" align=CENTER HSPACE=10 VSPACE=10 width=24% border=0>
 </a>

 <p>SI(S) are useful for modeling diseases with a short latent period (the time
 between being infected and becoming infectious) that either confer no long term
 immunity (in the case of SIS models) or result in permanent infection (in the case
 of SI models). Diseases of this type include the common cold and many
 macro-parasite infections. Susceptible individuals are infected at the
 rate of <em>&beta;I</em>, at which time they lose infectiousness
 at the rate of <em>&gamma;</em>. Mathematical details of this model
 can be viewed <a href="simath.html">here</a>.


 <h2>The SIR(S) Model</h2>
 <a href="sirmath.html">
 <img src="images/sirs.gif" align=CENTER HSPACE=10 VSPACE=10 width=36% border=0>
 </a>

 <p> SIR(S) models are useful for modeling diseases with a short latent
 period that confer immunity after infection, whether permanently (an SIR
 model), or temporarily (an SIRS model). Such a model might be useful for
 modeling diseases such as influenza or chicken-pox. Susceptible individuals are infected
 at the rate of <em>&beta;I</em>. They then loose infectiousness at the rate of
 <em>&gamma;</em> and then become immune to further infection. Immunity is lost
 at the rate of <em>&sigma;</em>.  Mathematical details of this model can be
 viewed <a href="sirmath.html">here</a>.

 <h2>The SEIR(S) Model</h2>
 <a href="seirmath.html">
 <img src="images/seirs.gif" align=CENTER HSPACE=10 VSPACE=10 width=48% border=0>
 </a>

 <p> SEIR(S) models are useful for modeling diseases that have a long latent
 period in which infected individuals are not themselves infectious.  Measles,
 influenza and smallpox are all diseases that might be appropriately characterized
 by this model. Susceptible individuals are infected  at the rate of <em>&beta;I</em>.
 These individuals become infectious at the rate of <em>&epsilon;</em>, and
 then loose infectiousness at the rate of
 <em>&gamma;</em>, becoming immune to further infection. Immunity is lost
 at the rate of <em>&sigma;</em>. Mathematical details of this model can be
 viewed <a href="swirmath.html">here</a>.


 <h2>Further Reading</h2>
 <ol>
 	<li> Hethcote, H. W. 2000. The Mathematics of Infectious Diseases.
 	SIAM Rev. 42, 4 (Dec. 2000), 599-653. DOI= http://dx.doi.org/10.1137/S0036144500371907
 	<li>  Ford, D. A., Kaufman, J. H., Eiron, I.,2006. An extensible spatial
 		and temporal epidemiological modeling system.
 		Int J Health Geogr. 5, 4 (Jan. 2006). DOI=http://dx.doi.org/10.1186/1476-072X-5-4

 </ol>

 </body>
 </html>
	<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN">

	<html>
	<head>
	<meta http-equiv="Content-Type" content="text/html; charset=iso-8859-1">
	<title>Epidemiological Modeling Overview</title>
	</head>

	<body>
	<h1>Epidemiological Modeling</h1>

	<img src="images/sis.gif" align=RIGHT HSPACE=10 VSPACE=10 width=25%>
	<p> Compartmental disease models are some of the most widely used
	models of infectious disease. In these models individuals (or portions of
	an individual) are considered to be in a compartments based on their
	disease status. By convention, commonly used compartmental models are
	designated by the first letters of the compartments and the order
	people move between them. So, the model for a disease where people
	start out Susceptible, become Infectious after catching the disease,
	and then after some time clear the disease and become Susceptible again
	would be an SIS model. The four most commonly used compartments
	are <u>S</u>usceptible, <u>E</u>xposed, <u>I</u>nfectious, and
	<u>R</u>emoved. <u>S</u>usceptible contains those individuals
	who can still be infected with disease. <u>E</u>xposed contains individuals
	who have been infected with the disease, but have not yet become
	infectious themselves. <u>I</u>nfectious contains those individuals who
	are infected and infectious. <u>R</u>emoved (sometimes called <u>R</u>ecovered)
	contains those individuals who have been removed from the system
	either through death or immunity. The disease state of a population
	is speccified by knowing the nuber of individuals (or proportion of
	the population) that is in each compartment.

	<p> There are two main types of compartmental models: deterministic
	and stochastic. In deterministic model, rates of movements between
	compartments are used to represent the processes by which individuals
	are infected, become ill and eventually recover from disease. These
	rates specify a set of differential or difference equations, which
	can be evaluated to give that state of the model at a future time
	point. In deterministic models compartments need not contain an
	integer number of individuals.

	<p>Deterministic models only give the <i>expected</i> behavior of
	the disease in a population under a set of assumptions about
	the natural history and epidemiology of the disease. Often
	we would also like to know how the disease behaves in more
	extreme cases, and what the distribution of this behavior
	is. Stochastic models provide a method for addressing this
	issue. In a stochastic model, whole individuals are moved
	between compartments based on a random draw from some specified
	distribution. To get informative results from a stochastic
	compartmental model it is necessary to do hundreds, or even
	thousands of experiments.

	<p>Compartamental models also may have "closed" or "open" populations.
	In closed populations the set of individuals represented by the model
	is static, no one comes in and no one comes out. It is of note that
	in such populations any disease that confers permanent immunity
	after infection will eventually become extinct. In an open
	population there is continuous immigration and emigration from
	the population, usually via birth and death.

	<p>STEM comes with deterministic implementations for three commonly used
	compartmental models: SI(S), SIR(S), and SIER(S). These models are
	implemented as having open populations, but closed populations can
	be easily simulated by setting the immigration/emigration rate to
	zero. Which model is appropriate for a modeling project depends
	on both the disease of interest and the timescale that we are
	interested in modeling our results.

	<h2>The SI(S) Model</h2>
	<a href="simath.html">
	<img src="images/sis.gif" align=CENTER HSPACE=10 VSPACE=10 width=24% border=0>
	</a>

	<p>SI(S) are useful for modeling diseases with a short latent period (the time
	between being infected and becoming infectious) that either confer no long term
	immunity (in the case of SIS models) or result in permanent infection (in the case
	of SI models). Diseases of this type include the common cold and many
	macro-parasite infections. Susceptible individuals are infected at the
	rate of <em>βI</em>, at which time they lose infectiousness
	at the rate of <em>γ</em>. Mathematical details of this model
	can be viewed <a href="simath.html">here</a>.


	<h2>The SIR(S) Model</h2>
	<a href="sirmath.html">
	<img src="images/sirs.gif" align=CENTER HSPACE=10 VSPACE=10 width=36% border=0>
	</a>

	<p> SIR(S) models are useful for modeling diseases with a short latent
	period that confer immunity after infection, whether permanently (an SIR
	model), or temporarily (an SIRS model). Such a model might be useful for
	modeling diseases such as influenza or chicken-pox. Susceptible individuals are infected
	at the rate of <em>βI</em>. They then loose infectiousness at the rate of
	<em>γ</em> and then become immune to further infection. Immunity is lost
	at the rate of <em>σ</em>. Mathematical details of this model can be
	viewed <a href="sirmath.html">here</a>.

	<h2>The SEIR(S) Model</h2>
	<a href="seirmath.html">
	<img src="images/seirs.gif" align=CENTER HSPACE=10 VSPACE=10 width=48% border=0>
	</a>

	<p> SEIR(S) models are useful for modeling diseases that have a long latent
	period in which infected individuals are not themselves infectious. Measles,
	influenza and smallpox are all diseases that might be appropriately characterized
	by this model. Susceptible individuals are infected at the rate of <em>βI</em>.
	These individuals become infectious at the rate of <em>ε</em>, and
	then loose infectiousness at the rate of
	<em>γ</em>, becoming immune to further infection. Immunity is lost
	at the rate of <em>σ</em>. Mathematical details of this model can be
	viewed <a href="swirmath.html">here</a>.


	<h2>Further Reading</h2>
	<ol>
	<li> Hethcote, H. W. 2000. The Mathematics of Infectious Diseases.
	SIAM Rev. 42, 4 (Dec. 2000), 599-653. DOI= http://dx.doi.org/10.1137/S0036144500371907
	<li> Ford, D. A., Kaufman, J. H., Eiron, I.,2006. An extensible spatial
	and temporal epidemiological modeling system.
	Int J Health Geogr. 5, 4 (Jan. 2006). DOI=http://dx.doi.org/10.1186/1476-072X-5-4

	</ol>

	</body>
	</html>