inoculation strategies for polio: - modeling the effects ...€¦ · inoculation strategies for...
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![Page 1: Inoculation Strategies for Polio: - Modeling the Effects ...€¦ · Inoculation Strategies for Polio: Modeling the E ects of a Growing Population on Public Health Outcomes Meredith](https://reader034.vdocuments.mx/reader034/viewer/2022051808/600a1d94242f6c202b20cfa7/html5/thumbnails/1.jpg)
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:
![Page 2: Inoculation Strategies for Polio: - Modeling the Effects ...€¦ · Inoculation Strategies for Polio: Modeling the E ects of a Growing Population on Public Health Outcomes Meredith](https://reader034.vdocuments.mx/reader034/viewer/2022051808/600a1d94242f6c202b20cfa7/html5/thumbnails/2.jpg)
Meredith McCormack-Mager Inoculation Strategies for Polio:
![Page 3: Inoculation Strategies for Polio: - Modeling the Effects ...€¦ · Inoculation Strategies for Polio: Modeling the E ects of a Growing Population on Public Health Outcomes Meredith](https://reader034.vdocuments.mx/reader034/viewer/2022051808/600a1d94242f6c202b20cfa7/html5/thumbnails/3.jpg)
Meredith McCormack-Mager Inoculation Strategies for Polio:
![Page 4: Inoculation Strategies for Polio: - Modeling the Effects ...€¦ · Inoculation Strategies for Polio: Modeling the E ects of a Growing Population on Public Health Outcomes Meredith](https://reader034.vdocuments.mx/reader034/viewer/2022051808/600a1d94242f6c202b20cfa7/html5/thumbnails/4.jpg)
Meredith McCormack-Mager Inoculation Strategies for Polio:
![Page 5: Inoculation Strategies for Polio: - Modeling the Effects ...€¦ · Inoculation Strategies for Polio: Modeling the E ects of a Growing Population on Public Health Outcomes Meredith](https://reader034.vdocuments.mx/reader034/viewer/2022051808/600a1d94242f6c202b20cfa7/html5/thumbnails/5.jpg)
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:
![Page 6: Inoculation Strategies for Polio: - Modeling the Effects ...€¦ · Inoculation Strategies for Polio: Modeling the E ects of a Growing Population on Public Health Outcomes Meredith](https://reader034.vdocuments.mx/reader034/viewer/2022051808/600a1d94242f6c202b20cfa7/html5/thumbnails/6.jpg)
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:
![Page 7: Inoculation Strategies for Polio: - Modeling the Effects ...€¦ · Inoculation Strategies for Polio: Modeling the E ects of a Growing Population on Public Health Outcomes Meredith](https://reader034.vdocuments.mx/reader034/viewer/2022051808/600a1d94242f6c202b20cfa7/html5/thumbnails/7.jpg)
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:
![Page 8: Inoculation Strategies for Polio: - Modeling the Effects ...€¦ · Inoculation Strategies for Polio: Modeling the E ects of a Growing Population on Public Health Outcomes Meredith](https://reader034.vdocuments.mx/reader034/viewer/2022051808/600a1d94242f6c202b20cfa7/html5/thumbnails/8.jpg)
Epidemiology
Epidemic
A rapid spread, growth, or development.
EndemicMaintained in a population without external inputs.
Meredith McCormack-Mager Inoculation Strategies for Polio:
![Page 9: Inoculation Strategies for Polio: - Modeling the Effects ...€¦ · Inoculation Strategies for Polio: Modeling the E ects of a Growing Population on Public Health Outcomes Meredith](https://reader034.vdocuments.mx/reader034/viewer/2022051808/600a1d94242f6c202b20cfa7/html5/thumbnails/9.jpg)
Epidemiology
Epidemic
A rapid spread, growth, or development.
EndemicMaintained in a population without external inputs.
Meredith McCormack-Mager Inoculation Strategies for Polio:
![Page 10: Inoculation Strategies for Polio: - Modeling the Effects ...€¦ · Inoculation Strategies for Polio: Modeling the E ects of a Growing Population on Public Health Outcomes Meredith](https://reader034.vdocuments.mx/reader034/viewer/2022051808/600a1d94242f6c202b20cfa7/html5/thumbnails/10.jpg)
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:
![Page 11: Inoculation Strategies for Polio: - Modeling the Effects ...€¦ · Inoculation Strategies for Polio: Modeling the E ects of a Growing Population on Public Health Outcomes Meredith](https://reader034.vdocuments.mx/reader034/viewer/2022051808/600a1d94242f6c202b20cfa7/html5/thumbnails/11.jpg)
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:
![Page 12: Inoculation Strategies for Polio: - Modeling the Effects ...€¦ · Inoculation Strategies for Polio: Modeling the E ects of a Growing Population on Public Health Outcomes Meredith](https://reader034.vdocuments.mx/reader034/viewer/2022051808/600a1d94242f6c202b20cfa7/html5/thumbnails/12.jpg)
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:
![Page 13: Inoculation Strategies for Polio: - Modeling the Effects ...€¦ · Inoculation Strategies for Polio: Modeling the E ects of a Growing Population on Public Health Outcomes Meredith](https://reader034.vdocuments.mx/reader034/viewer/2022051808/600a1d94242f6c202b20cfa7/html5/thumbnails/13.jpg)
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:
![Page 14: Inoculation Strategies for Polio: - Modeling the Effects ...€¦ · Inoculation Strategies for Polio: Modeling the E ects of a Growing Population on Public Health Outcomes Meredith](https://reader034.vdocuments.mx/reader034/viewer/2022051808/600a1d94242f6c202b20cfa7/html5/thumbnails/14.jpg)
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:
![Page 15: Inoculation Strategies for Polio: - Modeling the Effects ...€¦ · Inoculation Strategies for Polio: Modeling the E ects of a Growing Population on Public Health Outcomes Meredith](https://reader034.vdocuments.mx/reader034/viewer/2022051808/600a1d94242f6c202b20cfa7/html5/thumbnails/15.jpg)
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:
![Page 16: Inoculation Strategies for Polio: - Modeling the Effects ...€¦ · Inoculation Strategies for Polio: Modeling the E ects of a Growing Population on Public Health Outcomes Meredith](https://reader034.vdocuments.mx/reader034/viewer/2022051808/600a1d94242f6c202b20cfa7/html5/thumbnails/16.jpg)
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:
![Page 17: Inoculation Strategies for Polio: - Modeling the Effects ...€¦ · Inoculation Strategies for Polio: Modeling the E ects of a Growing Population on Public Health Outcomes Meredith](https://reader034.vdocuments.mx/reader034/viewer/2022051808/600a1d94242f6c202b20cfa7/html5/thumbnails/17.jpg)
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:
![Page 18: Inoculation Strategies for Polio: - Modeling the Effects ...€¦ · Inoculation Strategies for Polio: Modeling the E ects of a Growing Population on Public Health Outcomes Meredith](https://reader034.vdocuments.mx/reader034/viewer/2022051808/600a1d94242f6c202b20cfa7/html5/thumbnails/18.jpg)
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:
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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:
![Page 20: Inoculation Strategies for Polio: - Modeling the Effects ...€¦ · Inoculation Strategies for Polio: Modeling the E ects of a Growing Population on Public Health Outcomes Meredith](https://reader034.vdocuments.mx/reader034/viewer/2022051808/600a1d94242f6c202b20cfa7/html5/thumbnails/20.jpg)
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:
![Page 21: Inoculation Strategies for Polio: - Modeling the Effects ...€¦ · Inoculation Strategies for Polio: Modeling the E ects of a Growing Population on Public Health Outcomes Meredith](https://reader034.vdocuments.mx/reader034/viewer/2022051808/600a1d94242f6c202b20cfa7/html5/thumbnails/21.jpg)
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:
![Page 22: Inoculation Strategies for Polio: - Modeling the Effects ...€¦ · Inoculation Strategies for Polio: Modeling the E ects of a Growing Population on Public Health Outcomes Meredith](https://reader034.vdocuments.mx/reader034/viewer/2022051808/600a1d94242f6c202b20cfa7/html5/thumbnails/22.jpg)
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:
![Page 23: Inoculation Strategies for Polio: - Modeling the Effects ...€¦ · Inoculation Strategies for Polio: Modeling the E ects of a Growing Population on Public Health Outcomes Meredith](https://reader034.vdocuments.mx/reader034/viewer/2022051808/600a1d94242f6c202b20cfa7/html5/thumbnails/23.jpg)
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:
![Page 24: Inoculation Strategies for Polio: - Modeling the Effects ...€¦ · Inoculation Strategies for Polio: Modeling the E ects of a Growing Population on Public Health Outcomes Meredith](https://reader034.vdocuments.mx/reader034/viewer/2022051808/600a1d94242f6c202b20cfa7/html5/thumbnails/24.jpg)
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:
![Page 25: Inoculation Strategies for Polio: - Modeling the Effects ...€¦ · Inoculation Strategies for Polio: Modeling the E ects of a Growing Population on Public Health Outcomes Meredith](https://reader034.vdocuments.mx/reader034/viewer/2022051808/600a1d94242f6c202b20cfa7/html5/thumbnails/25.jpg)
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:
![Page 26: Inoculation Strategies for Polio: - Modeling the Effects ...€¦ · Inoculation Strategies for Polio: Modeling the E ects of a Growing Population on Public Health Outcomes Meredith](https://reader034.vdocuments.mx/reader034/viewer/2022051808/600a1d94242f6c202b20cfa7/html5/thumbnails/26.jpg)
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:
![Page 27: Inoculation Strategies for Polio: - Modeling the Effects ...€¦ · Inoculation Strategies for Polio: Modeling the E ects of a Growing Population on Public Health Outcomes Meredith](https://reader034.vdocuments.mx/reader034/viewer/2022051808/600a1d94242f6c202b20cfa7/html5/thumbnails/27.jpg)
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:
![Page 28: Inoculation Strategies for Polio: - Modeling the Effects ...€¦ · Inoculation Strategies for Polio: Modeling the E ects of a Growing Population on Public Health Outcomes Meredith](https://reader034.vdocuments.mx/reader034/viewer/2022051808/600a1d94242f6c202b20cfa7/html5/thumbnails/28.jpg)
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:
![Page 29: Inoculation Strategies for Polio: - Modeling the Effects ...€¦ · Inoculation Strategies for Polio: Modeling the E ects of a Growing Population on Public Health Outcomes Meredith](https://reader034.vdocuments.mx/reader034/viewer/2022051808/600a1d94242f6c202b20cfa7/html5/thumbnails/29.jpg)
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:
![Page 30: Inoculation Strategies for Polio: - Modeling the Effects ...€¦ · Inoculation Strategies for Polio: Modeling the E ects of a Growing Population on Public Health Outcomes Meredith](https://reader034.vdocuments.mx/reader034/viewer/2022051808/600a1d94242f6c202b20cfa7/html5/thumbnails/30.jpg)
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:
![Page 31: Inoculation Strategies for Polio: - Modeling the Effects ...€¦ · Inoculation Strategies for Polio: Modeling the E ects of a Growing Population on Public Health Outcomes Meredith](https://reader034.vdocuments.mx/reader034/viewer/2022051808/600a1d94242f6c202b20cfa7/html5/thumbnails/31.jpg)
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:
![Page 32: Inoculation Strategies for Polio: - Modeling the Effects ...€¦ · Inoculation Strategies for Polio: Modeling the E ects of a Growing Population on Public Health Outcomes Meredith](https://reader034.vdocuments.mx/reader034/viewer/2022051808/600a1d94242f6c202b20cfa7/html5/thumbnails/32.jpg)
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:
![Page 33: Inoculation Strategies for Polio: - Modeling the Effects ...€¦ · Inoculation Strategies for Polio: Modeling the E ects of a Growing Population on Public Health Outcomes Meredith](https://reader034.vdocuments.mx/reader034/viewer/2022051808/600a1d94242f6c202b20cfa7/html5/thumbnails/33.jpg)
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:
![Page 34: Inoculation Strategies for Polio: - Modeling the Effects ...€¦ · Inoculation Strategies for Polio: Modeling the E ects of a Growing Population on Public Health Outcomes Meredith](https://reader034.vdocuments.mx/reader034/viewer/2022051808/600a1d94242f6c202b20cfa7/html5/thumbnails/34.jpg)
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:
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Results
Minimum α = 0.78
α = 0.6
α = 0.9
Meredith McCormack-Mager Inoculation Strategies for Polio:
![Page 36: Inoculation Strategies for Polio: - Modeling the Effects ...€¦ · Inoculation Strategies for Polio: Modeling the E ects of a Growing Population on Public Health Outcomes Meredith](https://reader034.vdocuments.mx/reader034/viewer/2022051808/600a1d94242f6c202b20cfa7/html5/thumbnails/36.jpg)
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:
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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: