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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:
Meredith McCormack-Mager Inoculation Strategies for Polio:
Meredith McCormack-Mager Inoculation Strategies for Polio:
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
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
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
Meredith McCormack-Mager Inoculation Strategies for Polio:
Epidemiology
Epidemic
A rapid spread, growth, or development.
EndemicMaintained in a population without external inputs.
Meredith McCormack-Mager Inoculation Strategies for Polio:
Epidemiology
Epidemic
A rapid spread, growth, or development.
EndemicMaintained in a population without external inputs.
Meredith McCormack-Mager Inoculation Strategies for Polio:
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
Meredith McCormack-Mager Inoculation Strategies for Polio:
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
Meredith McCormack-Mager Inoculation Strategies for Polio:
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
Meredith McCormack-Mager Inoculation Strategies for Polio:
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
Meredith McCormack-Mager Inoculation Strategies for Polio:
Epidemiology
EndemicWhat is the end behavior of I(t)?
d(IN
)d(SN
) =
(IN
)′(t)(
SN
)′(t)
> 0
Meredith McCormack-Mager Inoculation Strategies for Polio:
Epidemiology
EndemicWhat is the end behavior of I(t)?
d(IN
)d(SN
) =
(IN
)′(t)(
SN
)′(t)
> 0
Meredith McCormack-Mager Inoculation Strategies for Polio:
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
Meredith McCormack-Mager Inoculation Strategies for Polio:
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
Meredith McCormack-Mager Inoculation Strategies for Polio:
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
Meredith McCormack-Mager Inoculation Strategies for Polio:
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
Meredith McCormack-Mager Inoculation Strategies for Polio:
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
Meredith McCormack-Mager Inoculation Strategies for Polio:
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
Meredith McCormack-Mager Inoculation Strategies for Polio:
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
Meredith McCormack-Mager Inoculation Strategies for Polio:
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
Meredith McCormack-Mager Inoculation Strategies for Polio:
Model: Vaccinating 1-4 Year-Olds
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
Meredith McCormack-Mager Inoculation Strategies for Polio:
Model: Vaccinating 1-4 Year-Olds
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
Meredith McCormack-Mager Inoculation Strategies for Polio:
Model: Vaccinating 1-4 Year-Olds
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
Meredith McCormack-Mager Inoculation Strategies for Polio:
Model: Vaccinating 1-4 Year-Olds
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
Meredith McCormack-Mager Inoculation Strategies for Polio:
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
Meredith McCormack-Mager Inoculation Strategies for Polio:
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
Meredith McCormack-Mager Inoculation Strategies for Polio:
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
ρ = birth rate =0.038
θ = maturation rate = 0.25
δC = child mortality rate = .124
δA = adult mortality rate = .013
0.244 < α
Meredith McCormack-Mager Inoculation Strategies for Polio:
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
ρ = birth rate =0.038
θ = maturation rate = 0.25
δC = child mortality rate = .124
δA = adult mortality rate = .013
0.244 < α
Meredith McCormack-Mager Inoculation Strategies for Polio:
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
ρ = birth rate =0.038
θ = maturation rate = 0.25
δC = child mortality rate = .124
δA = adult mortality rate = .013
0.244 < α
Meredith McCormack-Mager Inoculation Strategies for Polio:
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
ρ = birth rate =0.038
θ = maturation rate = 0.25
δC = child mortality rate = .124
δA = adult mortality rate = .013
0.244 < α
Meredith McCormack-Mager Inoculation Strategies for Polio:
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
Meredith McCormack-Mager Inoculation Strategies for Polio:
Results
Minimum α = 0.78
α = 0.6
α = 0.9
Meredith McCormack-Mager Inoculation Strategies for Polio:
Future Work
I Combination model for vaccination
I Continued analysis of age structures
I Proof of equilibrium state taking into consideration deathfrom disease
Meredith McCormack-Mager Inoculation Strategies for Polio:
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:
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