Vaccination Strategies for Epidemic Models

Douglas B. Meade

Department of Mathematics University of South Carolina Columbia, SC 29208

Phone:  (803) 777-6183

E-mail:  meade@math.sc.edu
WWW URL: http://www.math.sc.edu/˜ meade/

27 May 1999

May 1999  IMA Mathematical Biology Seminar  0

Vaccination Strategies for Epidemic Models

Douglas B. Meade

Department of Mathematics University of South Carolina Columbia, SC 29208

Phone:  (803) 777-6183

E-mail:  meade@math.sc.edu
WWW URL: http://www.math.sc.edu/˜ meade/

27 May 1999

May 1999  IMA Mathematical Biology Seminar  1

S I R Model

• Assumptions – constant population size (N = S + I + R = 1)

– mass-action kinetics for infection

– exponentially distributed mean time to recovery and natural death

– no death from infection

– no vaccination

Ref: Shulgin, Stone, and Agur, Bull Math Bio (1998) 60, 1123–1148.

S I R Model — No Vaccination

Ṡ = μ − μS − βSI
İ =  − μI + βSI − γI
Ṙ =  − μR  + γI
Ṅ = 0

May 1999  IMA Mathematical Biology Seminar  3

S I R Model — No Vaccination

• Equilibria

S  I
infection-free 1 0
endemic  1/R_{0} \frac{μ}{β}(R_{0} − 1)

• Reproduction Rate: R_{0} = \frac{β}{μ+γ} • Stability

R_{0} < 1  R_{0} > 1
infection-free globally stable  unstable
endemic  unstable  globally stable

May 1999  IMA Mathematical Biology Seminar  4

S I R Model — Constant Vaccination

• Assumptions – life-long immunization at birth

Ṡ = (1 − ρ)μ − μS − βSI
İ =  − μI + βSI − γI
Ṙ = ρμ − μR + γI
Ṅ = 0

May 1999  IMA Mathematical Biology Seminar  5

S I R Model — Constant Vaccination

• Equilibria

S  I
infection-free 1 − ρ  0
endemic  1/R_{0} \frac{μ}{β}(R_{0} − 1) − \frac{μ}{μ+γ} ρ

• Reproduction Rate: R_{ρ} = (1 − ρ) R_{0} • Critical Vaccination Rate: ρ_{0} = 1 − \frac{1}{R_{0}} • Stability

R_{ρ} < 1 (ρ > ρ_{0}) R_{ρ} > 1 (ρ < ρ_{0})
infection-free globally stable  unstable
endemic  unstable  globally stable

May 1999  IMA Mathematical Biology Seminar  6

S I R Model — Pulse Vaccination

• Assumptions – fixed portion of susceptibles vaccinated every T years

Ṡ = μ − μS − βSI  − ∑_{n=0}^{∞} ρS(nT^{−})δ(t − nT)
İ =  − μI + βSI − γI
Ṙ = − μR + γI + ∑_{n=0}^{∞} ρS(nT^{−})δ(t − nT)
Ṅ = 0

May 1999  IMA Mathematical Biology Seminar  7

S I R Model — Pulse Vaccination

• Infection-Free Equilibrium – initial value problem:

Ṡ = μ − μS − ∑_{n=0}^{∞} ρS(nT^{−})δ(t − nT),  S(0) = S_{0}

– solution:

S(t) = \{ \begin{matrix}S^{(n)}(t), & t ∈ [nT, (n + 1)T) \\ S_{n+1}, & t = (n + 1)T \end{matrix}
S^{(n)}(t) = 1 − (1 − S_{n})e^{−μ(t−nT)}
S_{n+1} = S((n + 1)T^{+})
= (1 − ρ)S^{(n)}((n + 1)T^{−})

May 1999  IMA Mathematical Biology Seminar  8

S I R Model — Pulse Vaccination

– construction of periodic equilibrium solution:

∗ if S_{∞} = lim_{n→∞} S_{n} exists, then S_{∞} is a fixed point of

F (x) = (1 − ρ)(1 − (1 − x)e^{−μT})

∗ unique fixed point is locally stable

S_{∞} = \frac{(1 − ρ)(e^{μT} − 1)}{e^{μT} − (1 − ρ)}

∗ periodic equilibrium solution:

S^{(∞)}(t) = 1 − (1 − S_{∞})e^{−μ(t mod T)}

May 1999  IMA Mathematical Biology Seminar  9

S I R Model — Pulse Vaccination

– local stability:

\frac{1}{T} ∫_{0}^{T} S dt < S_{c}

where S_{c} = 1/R_{0} is the endemic equilibrium with no vaccination

– maximum vaccination period:

\frac{(μT − ρ)(e^{μT} − 1) + μρT}{μT (e^{μT} − (1 − ρ))} = \frac{μ + γ}{β}

Application: Rubella (Measles)

• Parameters

mean infectious period 3.5 days γ = 100
mean lifespan  50 years μ = 0.02
contact rate  β = 1800

• No Vaccination:

– R_{0} ≈ 18 ⟹ endemic state is stable

– endemic equilibrium: S ≈ 0.05

• Constant Vaccination: – ρ_{0} ≈ 0.95

– reasonable for school age, but not pre-school (35-50%)

May 1999  IMA Mathematical Biology Seminar  11

S E I R Model

• Assumptions – exposed period prior to becoming infectious

– non-constant population size (N = S + E + I + R) ∗ immigration

∗ birth

∗ death from infection

– mass-action kinetics for infection

– exponentially distributed mean time to recovery and natural death

– no vaccination

S E I R Model — No Vaccination

Ṡ = Λ + μ_{0}N − μ_{1}S − βSI
Ė =  − μ_{1}E + βSI − δE
İ =  − μ_{1}I  + δE − γI − ϵI
Ṙ =  − μ_{1}R  + γI
Ṅ = Λ − (μ_{1} − μ_{0})N − ϵI

Note: • all parameters are positive

• to prevent exponential growth in total population, assume death-dominant (μ_{1} > μ_{0})

• S I R model obtained with ϵ = 0, δ = ∞, \frac{Λ}{μ_{1}−μ_{0}} = 1

May 1999  IMA Mathematical Biology Seminar  13

S E I R Model — No Vaccination

• Equilibria

infection-free  endemic
S  \frac{Λ}{μ_{1}−μ_{0}}  \frac{(μ_{1}+δ)(μ_{1}+γ+ϵ)}{βδ}
E  0  \frac{μ_{1}(μ_{1}+γ+ϵ)}{βδ} \frac{A_{0}}{B_{0}}
I  0  \frac{μ_{1}}{β} \frac{A_{0}}{B_{0}}
R  0  \frac{γ}{β} \frac{A_{0}}{B_{0}}
N  \frac{Λ}{μ_{1}−μ_{0}}  \frac{(βΛ+ϵμ_{1})(μ_{1}+δ)(μ_{1}+γ+ϵ)−ϵδβΛ}{βB_{0}}

where

A_{0} = βδΛ − (μ_{1} − μ_{0})(μ_{1} + δ)(μ_{1} + γ + ϵ)
B_{0} = ϵδΛ + (μ_{1} − μ_{0})(μ_{1} + δ)(μ_{1} + γ + ϵ)

May 1999  IMA Mathematical Biology Seminar  14

S E I R Model — No Vaccination

• Reproduction Rate: R_{0} = \frac{Λ}{μ_{1}−μ_{0}} \frac{δ}{μ_{1}+δ} \frac{β}{μ_{1}+γ+ϵ} • Stability

R_{0} < 1  R_{0} > 1
infection-free globally stable  unstable
endemic  unstable  globally stable

May 1999  IMA Mathematical Biology Seminar  15

S E I R Model — Constant Vaccination

• Assumptions – life-long immunization at birth

Ṡ = Λ + (1 − ρ)μ_{0}N − μ_{1}S − βSI
Ė =  − μ_{1}E + βSI − δE
İ =  − μ_{1}I  + δE − γI − ϵI
Ṙ =  ρμ_{0}N − μ_{1}R  + γI
Ṅ = Λ − (μ_{1} − μ_{0})N − ϵI

May 1999  IMA Mathematical Biology Seminar  16

S E I R Model — Constant Vaccination

• Equilibria

infection-free  endemic
S \frac{Λ}{μ_{1}−μ_{0}} (1 − \frac{μ_{0}}{μ_{1}} )  \frac{(μ_{1}+δ)(μ_{1}+γ+ϵ)}{βδ}
E  0  \frac{μ_{1}(μ_{1}+γ+ϵ)}{βδ} \frac{A_{ρ}}{B_{ρ}}
I  0  \frac{μ_{1}}{β} \frac{A_{ρ}}{B_{ρ}}
R  \frac{Λ}{μ_{1}−μ_{0}} \frac{μ_{0}}{μ_{1}}  \frac{γ}{β} \frac{A_{ρ}}{B_{ρ}} + \frac{μ_{0}}{μ_{1}} ρN
N  \frac{Λ}{μ_{1}−μ_{0}}  \frac{(βΛ+ϵμ_{1})(μ_{1}+δ)(μ_{1}+γ+ϵ)−ϵδβΛ}{βB_{ρ}}

where

A_{ρ} = A_{0} − \frac{μ_{0}}{μ_{1}} ρβδΛ,  B_{ρ} = B_{0} + \frac{μ_{0}}{μ_{1}} ρϵδμ_{0}

May 1999  IMA Mathematical Biology Seminar  17

S E I R Model — Constant Vaccination

• Reproduction Rate: R_{ρ} = (1 − \frac{μ_{0}}{μ_{1}} ρ) R_{0}

• Critical Vaccination Rate: ρ_{0} = \frac{μ_{1}}{μ_{0}} (1 − \frac{1}{R_{0}} )

• Stability

R_{ρ} < 1 (ρ > ρ_{0}) R_{ρ} > 1 (ρ < ρ_{0})
infection-free globally stable  unstable
endemic  unstable  globally stable

May 1999  IMA Mathematical Biology Seminar  18

S E I R Model — Pulse Vaccination

• Assumptions – fixed portion of susceptibles vaccinated every T years

Ṡ = Λ + μ_{0}N − μ_{1}S − βSI

− ∑_{n=0}^{∞} ρS(nT^{−})δ(t − nT)

Ė =  − μ_{1}E + βSI − δE
İ =  − μ_{1}I  + δE − γI − ϵI
Ṙ =  − μ_{1}R  + γI

+ ∑_{n=0}^{∞} ρS(nT^{−})δ(t − nT)

Ṅ = Λ − (μ_{1} − μ_{0})N − ϵI

May 1999  IMA Mathematical Biology Seminar  19

S E I R Model — Pulse Vaccination

• Infection-Free Equilibrium

– initial value problem:

Ṡ = Λ + μ_{0}N − μ_{1}S − ∑_{n=0}^{∞} ρS(nT^{−})δ(t − nT)
Ṙ =  − μ_{1}R + ∑_{n=0}^{∞} ρS(nT^{−})δ(t − nT)
Ṅ = Λ − (μ_{1} − μ_{0})N

May 1999  IMA Mathematical Biology Seminar  20

S E I R Model — Pulse Vaccination

– solution:

S(t) = \{ \begin{matrix}S^{(n)}(t), & t ∈ [nT, (n + 1)T) \\ S_{n+1}, & t = (n + 1)T \end{matrix}

where, for all n=0, 1, 2, …,

S^{(n)}(t) = \frac{Λ}{μ_{1} − μ_{0}} + (S_{n} − \frac{Λ}{μ_{1} − μ_{0}} ) e^{−μ_{1}(t−nT)}

+ (N_{0} − \frac{Λ}{μ_{1} − μ_{0}} ) (1 − e^{−μ_{0}(t−nT)}) e^{−(μ_{1}−μ_{0})t}

S_{n+1} = S((n + 1)T^{+})
= (1 − ρ)S^{(n)}((n + 1)T^{−})

May 1999  IMA Mathematical Biology Seminar  21

S E I R Model — Pulse Vaccination

– construction of periodic equilibrium solution:

∗ if S_{∞} = lim_{n→∞} S_{n} exists, then S_{∞} is a fixed point of

F (x) = (1 − ρ) (\frac{Λ}{μ_{1} − μ_{0}} + (x − \frac{Λ}{μ_{1} − μ_{0}} ) e^{−μ_{1}T} )

∗ unique fixed point is locally stable

S_{∞} = \frac{Λ}{μ_{1} − μ_{0}} \frac{(1 − ρ)(e^{μT} − 1)}{e^{μT} − (1 − ρ)}

May 1999  IMA Mathematical Biology Seminar  22

S E I R Model — Pulse Vaccination

∗ asymptotically periodic equilibrium solution:

S^{(∞)}(t) = \frac{Λ}{μ_{1} − μ_{0}} + (S_{∞} − \frac{Λ}{μ_{1} − μ_{0}} ) e^{−μ_{1}(t mod T)}

+ (N_{0} − \frac{Λ}{μ_{1} − μ_{0}} ) (1 − e^{−μ_{0}(t mod T)})

× e^{−(μ_{1}−μ_{0})t}

– …

May 1999  IMA Mathematical Biology Seminar  23

Application: Rubella (Measles)

• Parameters

mean exposed period 9 days δ = 40
mean infectious period 6 days  γ = 60
mean lifespan  80 years μ = 0.0125
contact rate  “high”  β =
immigration rate  ???  Λ =
net birth rate  ???  μ_{0} =
net death rate  ???  μ_{1} =