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Modeling Epidemics with Differential Equations. S.i.r . . Ross Beckley, Cametria Weatherspoon , Michael Alexander, Marissa Chandler, Anthony Johnson, Ghan Bhatt. Topics. The Model Variables & Parameters, Analysis, Assumptions Solution Techniques Vaccination Birth/Death

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modeling epidemics with differential equations
Modeling Epidemics with Differential Equations


Ross Beckley, CametriaWeatherspoon, Michael Alexander, Marissa Chandler, Anthony Johnson, Ghan Bhatt

  • The Model
    • Variables & Parameters, Analysis, Assumptions
  • Solution Techniques
  • Vaccination
  • Birth/Death
  • Constant Vaccination with Birth/Death
  • Saturation of the Susceptible Population
  • Infection Delay
  • Future of SIR
variables parameters
Variables & Parameters
    • [S] is the susceptible population
    • [I] is the infected population
    • [R] is the recovered population
    • 1 is the normalized total population in the system
  • The population remains the same size
  • No one is immune to infection
  • Recovered individuals may not be infected again
  • Demographics do not affect probability of infection
variables parameters1
Variables & Parameters
    • [α] is the transmission rate of the disease
    • [β] is the recovery rate
  • The population may only move from being susceptible to infected, infected to recovered:
variables parameters2
Variables & Parameters
  • is the Basic Reproductive Number- the average number of people infected by one person.
  • Initially,
  • The representation for will change as the model is improved and becomes more developed.
  • [] is the metric that most easily represents how infectious a disease is, with respect to that disease’s recovery rate.
conditions for epidemic
Conditions for Epidemic
  • An epidemic occurs if the rate of infection is > 0
    • If , and
      • It follows that an epidemic occurs if
    • Moreover, an epidemic occurs if
solution techniques
Solution Techniques
  • Determine equilibrium solutions for [I’] and [S’]. Equilibrium occurs when [S’] and [I’] are 0:
    • Equilibrium solutions in the form ( and :
solution techniques1
Solution Techniques
  • Compute the Jacobian Transformation:

General Form:

solution techniques2
Solution Techniques
  • Evaluate the Eigenvalues.
    • Our Jacobian Transformation reveals what the signs of the Eigenvalues will be.
    • A stable solution yields Eigenvalues of signs (-, -)
    • An unstable solution yields Eigenvalues of signs (+,+)
    • An unstable “saddle” yields Eigenvalues of (+,-)
solution techniques3
Solution Techniques
  • Evaluate the Data:
    • Phase portraits are generated via Mathematica.
  • Susceptible Vs. Infected Graph
  • Unstable Solutions deplete the susceptible population
  • There are 2 equilibrium solutions
  • One equilibrium solution is stable, while the other is unstable
  • The Phase Portrait converges to the stable solution, and diverges from the unstable solution
solution techniques4
Solution Techniques
  • Evaluate the Data:
    • Another example of an S vs. I graph with different values of [].
  • Typical Values
    • Flu: 2
    • Mumps: 5
    • Pertussis: 9
    • Measles: 12-18





herd immunity
Herd Immunity
  • Herd Immunity assumes that a portion [p] of the population is vaccinated prior to the outbreak of an epidemic.
  • New Equations Accommodating Vaccination:
  • An outbreak occurs if
    • , or
critical vaccination
Critical Vaccination
  • Herd Immunity implies that an epidemic can be preventedif a portion [p] of the population is vaccinated.
    • Epidemic:
    • No Epidemic:
    • Therefore the critical vaccination occurs at , or
      • In this context, [] is also known as the bifurcation point.
sir with birth and death
Sir with birth and death
  • Birth and death is introduced to our model as:

The birth and death rate is a constant rate [m]

The basic reproduction number is now given by:

sir with birth and death1
Sir with birth and death

Epidemic equilibrium

, ),

Disease free equilibrium

(, )

sir with birth and death2
Sir with birth and death
  • Jacobian matrix


  • (
constant vaccination at birth
Constant Vaccination At Birth
  • New Assumptions
    • A portion [p] of the new born population has the vaccination, while others will enter the population susceptible to infection.
    • The birth and death rate is a constant rate [m]
constant vaccination at birth1
Constant Vaccination At Birth
  • Parameters
  • Susceptible
  • Infected
parameters of the model
Parameters of the Model
  • The initial rate at which a disease is spread when one infected enters into the population.
  • p = number of newborn with vaccination

< 1 Unlikely Epidemic

> 1 Probable Epidemic

parameters of the model1
Parameters of the Model
  • = critical vaccination value
    • For measles, the accepted value for , therefore to stymy the epidemic, we must vaccinate 94.5% of the population.
constant vaccination graphs
Constant Vaccination Graphs

Susceptible Vs. Infected

  • Non epidemic
  • < 1
  • p > 95 %

Constant Vaccination Graphs

Susceptible Vs. Infected

  • Epidemic
  • > 1
  • < 95 %
constant vaccination graphs2
Constant Vaccination Graphs

Constant Vaccination Moving Towards Disease Free


New Assumption

  • We introduce a population that is not constant.

S + I + R ≠ 1

  • is a growth rate of the susceptible
  • K is represented as the capacity of the susceptible population.
  • Susceptible


= growth rate of birth

= capacity of susceptible population

  • Infected

= death rate

The Equations

the delay model
The Delay Model
  • People in the susceptible group carry the disease, but become infectious at a later time.
  • [r] is the rate of susceptible population growth.
  • [k] is the maximum saturation that S(t) may achieve.
  • [T] is the length of time to become infectious.
  • [σ] is the constant of Mass-Action Kinetic Law.
    • The constant rate at which humans interact with one another
    • “Saturation factor that measures inhibitory effect”
  • Saturation remains in the Delay model.
  • The population is not constant; birth and death occur.
the delay model2
The Delay Model

U.S. Center for Disease Control

future s i r work
Future S.I.R. Work
  • Eliminate Assumptions
    • Population Density
    • Age
    • Gender
    • Emigration and Immigration
    • Economics
    • Race