Modified SIR models to Forecast Epidemics: Modeling Gonorrhea in Erie County

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Modified SIR models to Forecast
EpidemicsModeling Gonorrhea in Erie County
Joel Adornetto, Neil Miller Advisor Dr. Saziye
Bayram, Dr. Joaquin Carbonara Buffalo State
College
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Biological Background for Gonorrhea

• Gonorrhea is a sexually transmitted disease and
  is the second most common disease in the United
  States.
• It is estimated that 700,000 persons in the
  United States get new Gonorrheal infections each
  year only half of these are reported.
• Large amounts of reliable data exists about
  Gonorrhea.
• Any sexually active person can be infected with
  gonorrhea.

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Goals of Modeling a Disease

• Although medical advances have reduced the
  consequences of infectious disease, preventing
  infections in the first place is preferable to
  treating them.
• Once a model has been formulated that captures
  the main features of the progression and
  transmission of a particular disease, it can be
  used to predict the effects of different
  strategies for disease eradication or control.
• Infectious disease modeling, though often
  inexact, has enormous potential to help improve
  human lives.

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Erie County Data

• The Erie County Center for Disease Surveillance,
  has provided the number of cases of Gonorrhea
  reported between 2001 and 2006.
• The data is organized in two ways
• Yearly (6 data points) with Age and Sex
  distribution
• Monthly (72 data points) only given number of
  cases

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Yearly Erie County Data
To simplify our model we decided to use the
following groups Ages 10-19 are considered to be
Teenager Ages 20 are considered to be
Adults Ages 10 and under are not considered in
the model
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Yearly Erie County Data
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Monthly Erie County Data
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Monthly Erie County Data
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SIR and SIS Models

• An SIR model consists of three group
• Susceptible Those who may contract the disease
• Infected Those infected
• Recovered Those with natural immunity or those
  that have died.
• An SIS model consists of two group
• Susceptible Those who may contract the disease
• Infected Those infected

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Important Parameters

• a is the transmission coefficient, which
  determines the rate ate which the disease travels
  from one population to another.
• ? is the recovery rate (I persons)/(days
  required to recover)
• R0 is the basic reproduction number.
• (Number of new cases arising from one infective)
  x (Average duration of infection)
• If R0 1 then ?I 0 and an epidemic occurs

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SIR and SIS Models

• SIR Model
• SIS Model

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Basic SIS model for Gonorrhea
PARAMETERS
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Basic SIS model for Gonorrhea

• Since the population is assumed to be constant
• We can simplify the equations by eliminating the
  susceptible group.
• S N I, where N is the total population

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Basic SIS model for Gonorrhea
The next step in the process is to find an
equilibrium solution. This means that the
population does not change in the next time step.
This is found by setting the change in the
infected population to zero and solving for the
infected female and male population.
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Basic SIS model for Gonorrhea
The existence of only one nonzero equilibrium
point shows that there is at most one endemic
level of the disease. Increasing the removal
rates is the way that these endemic levels can be
decreased. The precise formulas allow prediction
of the endemic level expected from any relative
removal rate a public health program might
achieve.
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Basic SIS model for Gonorrhea
Biological analysis

• This must be positive, since we cannot have a
  negative infected population
• Also, since it has a positive value, it means
  that Gonorrhea will always be present
• Thus, data collection for statistical estimates
  of the infection and removal rates can help
  judge where a disease is likely to remain
  endemic.

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Assumptions for our Erie County Model

• The total population is constant through each
  time step
• i.e. Ntotal Infected Susceptible
• Only heterosexual interaction is accounted for
• The age group 0-12 will not be considered
• We are constructing a modified SIS model
• There are eight groups in our model

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Population in Erie County
Information provided by the 2000 Erie County
Census Teenagers are assumed to be the ages of
10 19 Adults are assumed to be over the age of
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Our SIS Model for Gonorrhea
Susceptible/Infected Female Teenagers (Age
10-19)
Susceptible/Infected Female Adults (Age 20)
Susceptible/Infected Male Teenagers (Age 10-19)
Susceptible/Infected Male Adults (Age 20)
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Closer Look of our SIS Model
Susceptible/Infected Female Teenagers (Age
10-19)
SFT Susceptible Female Teenagers IFT Infected
Female Teenagers SMA Susceptible Male Adults IMT
Infected Male Teenagers Alpha Transmission
Rate Gamma Removal Rate
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Closer Look of our SIS Model
Susceptible/Infected Female Teenagers (Age
10-19)
Since the population is constant we can rewrite
some of our parameters to reduce the number or
equations we have. The number of infected male
teenagers total male teenagers - susceptible
male teenagers IMT NMT SMT IFT NFT SFT
This set of equations becomes
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Findings

• There is a non zero equilibrium solution for our
  model
• We believe that the data we were provided is on
  the tail end of an equilibrium solution, with
  small perturbations
• The transmission rate needs to be extremely small
  to get similar infected populations

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Challenges

• Assumptions must be made, and this of course can
  lead to error
• What is the transmission rate for a given disease
• How long is the recovery period
• Further difficulty is encountered when little
  data is available on a particular subject
• i.e. our model does not consider homosexual
  relations, as we have no data specific to the
  infected persons sexual habits
• Collecting the data yourself could be a solution
• Multiple variables make graphing data very
  difficult or impossible
• We made the assumption that we have a constant
  population to reduce the number of equations.
  This made it possible to write Mathematica code
  to run the model vs. time.
• Increasing the number of groups increases the
  accuracy and usefulness of a model but it
  typically makes the model more difficult.

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Conclusions

• Modeling is an effective way to determine
  dynamics of a population.
• Creating a model to analysis a population lets
  you see how transmission/removal rates affect the
  susceptible class.
• Personally collecting data to formulate a model
  can be beneficial to a specific disease.
• Further analysis on our model should tell us the
  specific transmission rates for all eight groups.
  We can do this by starting off with the known
  removal rates and the equilibrium equations, and
  then work backwards to find Ro and alpha.
• Hopefully we will eventually be able to forecast
  future outbreak, by monitoring the transmission
  rates and comparing it to a critical threshold
  value.

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References

• Erie County Department of Health
• Sexually Transmitted Disease Resource Site
• for Disease Control (CDC)
• Mathematical Models in Biology, An Introduction
• Elizabeth Allman John Rhodes