Inoculation Strategies for Polio: - Modeling the Effects ...€¦ · Inoculation Strategies for Polio: Modeling the E ects of a Growing Population on Public Health Outcomes Meredith

Inoculation Strategies for Polio:Modeling the Effects of a Growing Population on Public Health

Outcomes

Meredith McCormack-Mager

July 23, 2014

Meredith McCormack-Mager Inoculation Strategies for Polio:




Model

S I R- - -

6 6 6

?

ρN βS IN γI

δS δI δR

dI

Figure : Basic SIR Model with Non-Constant Population and Death fromDisease


Model

S I R- - -

6 6 6

?

ρN βS IN γI

δS δI δR

dI



Model

S I R- - -

6 6 6

?

ρN βS IN γI

δS δI δR

dI



Epidemiology

Epidemic

A rapid spread, growth, or development.

EndemicMaintained in a population without external inputs.


Epidemiology

Epidemic

A rapid spread, growth, or development.

EndemicMaintained in a population without external inputs.


Epidemiology

Epidemic

When is I ′(t) > 0?

I ′(t) = βSI

N− δI − dI − γI > 0

R0 :=β

δ + d + γ> 1

R0constant :=β

γ> 1


Epidemiology

Epidemic


I ′(t) = βSI

N− δI − dI − γI > 0

R0 :=β

δ + d + γ> 1

R0constant :=β

γ> 1


Epidemiology

Epidemic


I ′(t) = βSI

N− δI − dI − γI > 0

R0 :=β

δ + d + γ> 1

R0constant :=β

γ> 1


Epidemiology

Epidemic


I ′(t) = βSI

N− δI − dI − γI > 0

R0 :=β

δ + d + γ> 1

R0constant :=β

γ> 1


Epidemiology

EndemicWhat is the end behavior of I(t)?

d(IN

)d(SN

) =

(IN

)′(t)(

SN

)′(t)

> 0


Epidemiology

EndemicWhat is the end behavior of I(t)?

d(IN

)d(SN

) =

(IN

)′(t)(

SN

)′(t)

> 0


Model: Vaccination

S I R- - -

6 6 6

?

(1− α)ρN βS IN γI

δS δI δR

dI

αρN�

Figure : Basic SIR Model with Non-Constant Population, Death fromDisease, and Vaccination of Newborns


Model: Vaccination

S I R- - -

6 6 6

?

(1− α)ρN βS IN γI

δS δI δR

dI

αρN�

Figure : Basic SIR Model with Non-Constant Population, Death fromDisease, and Vaccination of Newborns


Evaluating the Disease Free State

S R-

6 6

(1− α)ρN

δS δR

αρN�

S

N(t) = (1− α)

I

N(t) = 0

R

N(t) = α


Evaluating the Disease Free State

S R-

6 6

(1− α)ρN

δS δR

αρN�

S

N(t) = (1− α)

I

N(t) = 0

R

N(t) = α


Epidemiology

Preventing an Epidemic

When is I ′(t) < 0?

R0 :=β SN

δ + d + γ< 1

β(1− α)

δ + d + γ< 1

1− δ + d + γ

β< α


Model: Split Age Classes

SC IC RC

SA IA RA

- -

-

-

-

6 6 6

? ? ?

? ?

ρNA βSCIC+IAN

βSAIC+IAN

γC IC

γA(1− µ)IA

δCSC δC IC δCRC

δASA dµIA δARA

θSC θRC

Figure : Split Age Class SIR Model


Model: Vaccinating 1-4 Year-Olds

SC IC RC

SA IA RA

- -

-

-

-

6 6 6

? ? ?

? ?

ρNA βSCIC+IAN

βSAIC+IAN

γC IC

γA(1− µ)IA

δCSC δC IC δCRC

δASA dµIA δARA

θSC θRC6

αSC

Figure : Split Age Class SIR Model For 1-5 Year-Old Vaccination Strategy



SC IC RC

SA IA RA

- -

-

-

-

6 6 6

? ? ?

? ?

ρNA βSCIC+IAN

βSAIC+IAN

γC IC

γA(1− µ)IA

δCSC δC IC δCRC

δASA dµIA δARA

θSC θRC6

αSC

Figure : Split Age Class SIR Model For 1-5 Year-Old Vaccination Strategy



R0 :=β SC (t)

N(t)

δC + γC+

β SA(t)N(t)

dµ+ γA(1− µ)< 1

δA(δC + θ) > ρθ

ρ = birth rate =0.038

θ = maturation rate = 0.25

δC = child mortality rate = .124

δA = adult mortality rate = .013

0.0049 6> 0.0095



R0 :=β SC (t)

N(t)

δC + γC+

β SA(t)N(t)

dµ+ γA(1− µ)< 1






0.0049 6> 0.0095



R0 :=β SC (t)

N(t)

δC + γC+

β SA(t)N(t)

dµ+ γA(1− µ)< 1






0.0049 6> 0.0095



R0 :=β SC (t)

N(t)

δC + γC+

β SA(t)N(t)

dµ+ γA(1− µ)< 1






0.0049 6> 0.0095


Model: Vaccinating Newborns (Age 0)

SC IC RC

SA IA RA

-

-

-

-

6 6 6

? ? ?

? ?

βSCIC+IAN

βSAIC+IAN

γC IC

γA(1− µ)IA

δCSC δC IC δCRC

δASA dµIA δARA

θSC θRC

�αρNA(1− α)ρNA

-

Figure : SIR Model For Newborn Vaccination Strategy


Model: Vaccinating Newborns (Age 0)

SC IC RC

SA IA RA

-

-

-

-

6 6 6

? ? ?

? ?

βSCIC+IAN

βSAIC+IAN

γC IC

γA(1− µ)IA

δCSC δC IC δCRC

δASA dµIA δARA

θSC θRC

�αρNA(1− α)ρNA

-

Figure : SIR Model For Newborn Vaccination Strategy


Model: Vaccinating Newborns(Age 0)

R0 :=β SC (t)

N(t)

δC + γC+

β SA(t)N(t)

dµ+ γA(1− µ)< 1

1− δA(δC + θ)

ρθ< 2α





0.244 < α



R0 :=β SC (t)

N(t)

δC + γC+

β SA(t)N(t)

dµ+ γA(1− µ)< 1

1− δA(δC + θ)

ρθ< 2α





0.244 < α



R0 :=β SC (t)

N(t)

δC + γC+

β SA(t)N(t)

dµ+ γA(1− µ)< 1

1− δA(δC + θ)

ρθ< 2α





0.244 < α



R0 :=β SC (t)

N(t)

δC + γC+

β SA(t)N(t)

dµ+ γA(1− µ)< 1

1− δA(δC + θ)

ρθ< 2α





0.244 < α


Results

Minimum α = 0.78

α = 0.78 10 20 30 40 50Weeks

0.002

0.004

0.006

0.008

0.010

Population HNon-dimensionalizedL


Results

Minimum α = 0.78

α = 0.6

α = 0.9


Future Work

I Combination model for vaccination

I Continued analysis of age structures

I Proof of equilibrium state taking into consideration deathfrom disease


Inoculation Strategies for Polio:Modeling the Effects of a Growing Population on Public Health

Outcomes

Meredith McCormack-Mager

July 23, 2014


Documents

Inoculation Strategies for Polio: - Modeling the Effects ...€¦ · Inoculation Strategies for Polio: Modeling the E ects of a Growing Population on Public Health Outcomes Meredith