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S.I.R. Modeling Epidemics with Differential Equations
Ross Beckley, Cametria Weatherspoon, Michael Alexander, Marissa Chandler, Anthony Johnson, Ghan Bhatt
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TOPICS The Model
Var iab les & Parameters , Ana lys is , Assumpt ions
Solut ion Techniques Vaccinat ion Birth/Death Constant Vaccinat ion wi th Bir th/Death Saturat ion of the Suscept ib le Populat ion Infect ion Delay Future of SIR
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VARIABLES & PARAMETERS
[S] is the susceptible population[I] is the infected population[R] is the recovered population1 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
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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:
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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.
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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
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SOLUTION TECHNIQUES Determine equilibrium solutions for [I’] and [S’].
Equilibrium occurs when [S’] and [I’] are 0:
Equilibrium solutions in the form ( and :
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SOLUTION TECHNIQUES Compute the Jacobian Transformation:
General Form:
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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 (+,-)
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SOLUTION TECHNIQUES
Evaluate the Data:Phase portraits are generated via Mathematica.
0.0 0.5 1.0 1.5 2.0
0.0
0.5
1.0
1.5
2.0
Susceptible
Infe
cted
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
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SOLUTION TECHNIQUES
Evaluate the Data:Another example of an S vs. I graph with different
values of [].
𝑹𝟎
12952
Typical Values Flu: 2 Mumps: 5 Pertussis: 9 Measles: 12-18
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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
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CRITICAL VACCINATION Herd Immunity implies that an epidemic can be
prevented if 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.
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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:
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SIR WITH BIRTH AND DEATH
Disease free equilibrium(, )
Epidemic equilibrium, ),
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SIR WITH BIRTH AND DEATH
Jacobian matrix
(,)
(
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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]
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CONSTANT VACCINATION AT BIRTH
Parameters
Susceptible
Infected
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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
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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.
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CONSTANT VACCINATION GRAPHS
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
Susceptible
Infe
cted
• Non epidemic
< 1 p > 95 %
Susceptible Vs. Infected
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0.0 0.5 1.0 1.5 2.0
0.0
0.5
1.0
1.5
2.0
Susceptible
Infe
cted
CONSTANT VACCINATION GRAPHS
• Epidemic > 1 < 95 %
Susceptible Vs. Infected
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CONSTANT VACCINATION GRAPHS
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CONSTANT VACCINATION GRAPHS
Constant Vaccination Moving Towards Disease Free
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SATURATION
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.
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SATURATION
The Equations
Susceptible
)
= growth rate of birth
= capacity of susceptible population Infected
= death rate
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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.
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THE DELAY MODEL
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THE DELAY MODEL
U.S. Center for Disease Control
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FUTURE S.I.R. WORK
Eliminate AssumptionsPopulation DensityAgeGenderEmigration and ImmigrationEconomicsRace