sir with pulse vaccination.pptx
DESCRIPTION
This is my presentation on a mathematical based model, the SIR model which involves the effects of pulse vaccination on a population over a time period.TRANSCRIPT
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THEORETICAL EXAMINATION OF THE
PULSE VACCINATION POLICY IN THE
SIR EPIDEMIC MODEL
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MODEL
SIR model with Pulse Vaccination Strategy
Need to look into the relationship of the
susceptible, infected and recovered.
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MEASLES
Viral infection of the respiratory system
Causes rash all over the body
Can also cause death
Contagious
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PULSE VACCINATION
Repeated vaccination of susceptible individuals
pulsed in time
Efficient in controlling Measles
Expected to eradicate measles from the entire
population
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CONSTANT VACCINATION
Vaccinate a proportion of newborns in a
population.
Vaccinate at least 95% of the newborns
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SIR MODEL
Systems of Equations
General Form
S(t)+I(t)+R(t)=1
S=# of individuals susceptible to disease
I=# of infected individuals
R=# of recovered individuals
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MODEL
m is the birth and death rate,
making the life expectancy to be 1/m
B is the contact rate
g is the rate of recovery, making the
mean infectious period 1/g
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EQUILIBRIUM
Trivial or Infection Free Complete Eradication
Non Trivial Epidemic Equilibrium
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TRIVIAL EQUILIBRIUM
TO BE EVALUATED AT EQUILIBRIUM
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NONTRIVIAL EQUILIBRIUM
RO IS THE REPRODUCTIVE RATE OF INFECTION
IF R0> 1, THE NONTRIVIAL EQUILIBRIUM IS LOCALLY
STABLE.
IF R0
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PULSE VACCINATION STRATEGY
T represents the period of pulse vaccination.
represents the time the nth pulse is applied
represents the time just before the nth pulse
is applied.
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INFECTION FREE SOLUTION
No infectious Individuals and Eradication of
disease
The susceptible population S cycles with period T
while the infective population is at equilibrium
I(t)=I*= 0
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DERIVATION OF INFECTION-FREE SOLUTION
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CONTINUED..
F DETERMINES THE S(T) AFTER EACH PULSE
VACCINATION
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STABILITY OF INFECTION-FREE SOLUTION
Perturb the susceptibles and infectives
where s and i are the perturbations
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STABILITY CONTINUED
After linearizing the SIR equations using the
perturbations we get,
Infectives population will decrease if
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STABILITY OF INFECTION-FREE SOLUTION
Solution will be locally stable if
Mean value of S(t) must be less than the
threshold level Sc
Where is the epidemic threshold
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PULSE VACCINATION GRAPH
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CONCLUSION
Pulse Vaccination is more efficient for the
treatment of Measles than Constant Vaccination.
Measles can result in death.
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REFERENCE
Agur, Shulgin and Stone. Theoretical
Examination of the Pulse Vaccination Policy in
the SIR Epidemic Model. Mathematical and
Computer Modelling. 2000. Print.