SIR With Pulse Vaccination.pptx

THEORETICAL EXAMINATION OF THE

PULSE VACCINATION POLICY IN THE

SIR EPIDEMIC MODEL

MODEL

SIR model with Pulse Vaccination Strategy

Need to look into the relationship of the

susceptible, infected and recovered.

MEASLES

Viral infection of the respiratory system

Causes rash all over the body

Can also cause death

Contagious

PULSE VACCINATION

Repeated vaccination of susceptible individuals

pulsed in time

Efficient in controlling Measles

Expected to eradicate measles from the entire

population

CONSTANT VACCINATION

Vaccinate a proportion of newborns in a

population.

Vaccinate at least 95% of the newborns

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

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

EQUILIBRIUM

Trivial or Infection Free Complete Eradication

Non Trivial Epidemic Equilibrium

TRIVIAL EQUILIBRIUM

TO BE EVALUATED AT EQUILIBRIUM

NONTRIVIAL EQUILIBRIUM

RO IS THE REPRODUCTIVE RATE OF INFECTION

IF R0> 1, THE NONTRIVIAL EQUILIBRIUM IS LOCALLY

STABLE.

IF R0

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.

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

DERIVATION OF INFECTION-FREE SOLUTION

CONTINUED..

F DETERMINES THE S(T) AFTER EACH PULSE

VACCINATION

STABILITY OF INFECTION-FREE SOLUTION

Perturb the susceptibles and infectives

where s and i are the perturbations

STABILITY CONTINUED

After linearizing the SIR equations using the

perturbations we get,

Infectives population will decrease if

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

PULSE VACCINATION GRAPH

CONCLUSION

Pulse Vaccination is more efficient for the

treatment of Measles than Constant Vaccination.

Measles can result in death.

REFERENCE

Agur, Shulgin and Stone. Theoretical

Examination of the Pulse Vaccination Policy in

the SIR Epidemic Model. Mathematical and

Computer Modelling. 2000. Print.