joel c. miller - harvard universityscholar.harvard.edu/files/joelmiller/files/2012_09_07...intro to...
TRANSCRIPT
![Page 1: Joel C. Miller - Harvard Universityscholar.harvard.edu/files/joelmiller/files/2012_09_07...Intro to dynamic modeling of epidemics Static Networks Dynamic Populations Edge-based compartmental](https://reader034.vdocuments.us/reader034/viewer/2022050301/5f6a67ac0858bd490c75d0c4/html5/thumbnails/1.jpg)
Intro to dynamic modeling of epidemics Static Networks Dynamic Populations
Edge-based compartmental modeling
Joel C. Miller& Erik Volz
RAPIDD; Harvard School of Public Health; Penn State University
7 Sept 2012
![Page 2: Joel C. Miller - Harvard Universityscholar.harvard.edu/files/joelmiller/files/2012_09_07...Intro to dynamic modeling of epidemics Static Networks Dynamic Populations Edge-based compartmental](https://reader034.vdocuments.us/reader034/viewer/2022050301/5f6a67ac0858bd490c75d0c4/html5/thumbnails/2.jpg)
Intro to dynamic modeling of epidemics Static Networks Dynamic Populations
Intro to dynamic modeling of epidemics
Static NetworksNetwork AssumptionsFinal SizeDynamics
Dynamic PopulationsUnchanging DegreesDormant contacts (binding sites)
![Page 3: Joel C. Miller - Harvard Universityscholar.harvard.edu/files/joelmiller/files/2012_09_07...Intro to dynamic modeling of epidemics Static Networks Dynamic Populations Edge-based compartmental](https://reader034.vdocuments.us/reader034/viewer/2022050301/5f6a67ac0858bd490c75d0c4/html5/thumbnails/3.jpg)
Intro to dynamic modeling of epidemics Static Networks Dynamic Populations
The basic SIR model
• Individuals begin susceptible,
• become infected from contacting infected individuals,
• and eventually recover with immunity.
![Page 4: Joel C. Miller - Harvard Universityscholar.harvard.edu/files/joelmiller/files/2012_09_07...Intro to dynamic modeling of epidemics Static Networks Dynamic Populations Edge-based compartmental](https://reader034.vdocuments.us/reader034/viewer/2022050301/5f6a67ac0858bd490c75d0c4/html5/thumbnails/4.jpg)
Intro to dynamic modeling of epidemics Static Networks Dynamic Populations
The basic SIR model
• Individuals begin susceptible,
• become infected from contacting infected individuals,
• and eventually recover with immunity.
![Page 5: Joel C. Miller - Harvard Universityscholar.harvard.edu/files/joelmiller/files/2012_09_07...Intro to dynamic modeling of epidemics Static Networks Dynamic Populations Edge-based compartmental](https://reader034.vdocuments.us/reader034/viewer/2022050301/5f6a67ac0858bd490c75d0c4/html5/thumbnails/5.jpg)
Intro to dynamic modeling of epidemics Static Networks Dynamic Populations
The basic SIR model
• Individuals begin susceptible,
• become infected from contacting infected individuals,
• and eventually recover with immunity.
![Page 6: Joel C. Miller - Harvard Universityscholar.harvard.edu/files/joelmiller/files/2012_09_07...Intro to dynamic modeling of epidemics Static Networks Dynamic Populations Edge-based compartmental](https://reader034.vdocuments.us/reader034/viewer/2022050301/5f6a67ac0858bd490c75d0c4/html5/thumbnails/6.jpg)
Intro to dynamic modeling of epidemics Static Networks Dynamic Populations
The basic SIR model
• Individuals begin susceptible,
• become infected from contacting infected individuals,
• and eventually recover with immunity.
![Page 7: Joel C. Miller - Harvard Universityscholar.harvard.edu/files/joelmiller/files/2012_09_07...Intro to dynamic modeling of epidemics Static Networks Dynamic Populations Edge-based compartmental](https://reader034.vdocuments.us/reader034/viewer/2022050301/5f6a67ac0858bd490c75d0c4/html5/thumbnails/7.jpg)
Intro to dynamic modeling of epidemics Static Networks Dynamic Populations
The basic SIR model
• Individuals begin susceptible,
• become infected from contacting infected individuals,
• and eventually recover with immunity.
![Page 8: Joel C. Miller - Harvard Universityscholar.harvard.edu/files/joelmiller/files/2012_09_07...Intro to dynamic modeling of epidemics Static Networks Dynamic Populations Edge-based compartmental](https://reader034.vdocuments.us/reader034/viewer/2022050301/5f6a67ac0858bd490c75d0c4/html5/thumbnails/8.jpg)
Intro to dynamic modeling of epidemics Static Networks Dynamic Populations
The basic SIR model
• Individuals begin susceptible,
• become infected from contacting infected individuals,
• and eventually recover with immunity.
![Page 9: Joel C. Miller - Harvard Universityscholar.harvard.edu/files/joelmiller/files/2012_09_07...Intro to dynamic modeling of epidemics Static Networks Dynamic Populations Edge-based compartmental](https://reader034.vdocuments.us/reader034/viewer/2022050301/5f6a67ac0858bd490c75d0c4/html5/thumbnails/9.jpg)
Intro to dynamic modeling of epidemics Static Networks Dynamic Populations
The mass-action model
Let S be the proportion susceptible, I be the proportion infected,and R be the proportion recovered.
βIS γI
S I R
Resulting equations:
S = −βIS , I = βIS − γI , R = γI
![Page 10: Joel C. Miller - Harvard Universityscholar.harvard.edu/files/joelmiller/files/2012_09_07...Intro to dynamic modeling of epidemics Static Networks Dynamic Populations Edge-based compartmental](https://reader034.vdocuments.us/reader034/viewer/2022050301/5f6a67ac0858bd490c75d0c4/html5/thumbnails/10.jpg)
Intro to dynamic modeling of epidemics Static Networks Dynamic Populations
The mass-action model
Let S be the proportion susceptible, I be the proportion infected,and R be the proportion recovered.
βIS γI
S I R
Resulting equations:
S = −βIS , I = βIS − γI , R = γI
![Page 11: Joel C. Miller - Harvard Universityscholar.harvard.edu/files/joelmiller/files/2012_09_07...Intro to dynamic modeling of epidemics Static Networks Dynamic Populations Edge-based compartmental](https://reader034.vdocuments.us/reader034/viewer/2022050301/5f6a67ac0858bd490c75d0c4/html5/thumbnails/11.jpg)
Intro to dynamic modeling of epidemics Static Networks Dynamic Populations
The mass-action model
Let S be the proportion susceptible, I be the proportion infected,and R be the proportion recovered.
βIS γI
S I R
Resulting equations:
S = −βIS , I = βIS − γI , R = γI
![Page 12: Joel C. Miller - Harvard Universityscholar.harvard.edu/files/joelmiller/files/2012_09_07...Intro to dynamic modeling of epidemics Static Networks Dynamic Populations Edge-based compartmental](https://reader034.vdocuments.us/reader034/viewer/2022050301/5f6a67ac0858bd490c75d0c4/html5/thumbnails/12.jpg)
Intro to dynamic modeling of epidemics Static Networks Dynamic Populations
Solutions
S = −βIS , I = βIS − γI , R = γI
5 0 5 10 150.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
t
Infe
ctio
ns
5 0 5 10 150.0
0.2
0.4
0.6
0.8
1.0
t
Cum
ula
tive infe
ctio
ns
Solutions with β = 3, γ = 1
![Page 13: Joel C. Miller - Harvard Universityscholar.harvard.edu/files/joelmiller/files/2012_09_07...Intro to dynamic modeling of epidemics Static Networks Dynamic Populations Edge-based compartmental](https://reader034.vdocuments.us/reader034/viewer/2022050301/5f6a67ac0858bd490c75d0c4/html5/thumbnails/13.jpg)
Intro to dynamic modeling of epidemics Static Networks Dynamic Populations
−5 0 5 10 150.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35Mass ActionHomogeneousPoissonBimodalTruncated Powerlaw
t
Infe
ctio
ns
![Page 14: Joel C. Miller - Harvard Universityscholar.harvard.edu/files/joelmiller/files/2012_09_07...Intro to dynamic modeling of epidemics Static Networks Dynamic Populations Edge-based compartmental](https://reader034.vdocuments.us/reader034/viewer/2022050301/5f6a67ac0858bd490c75d0c4/html5/thumbnails/14.jpg)
Intro to dynamic modeling of epidemics Static Networks Dynamic Populations
Intro to dynamic modeling of epidemics
Static NetworksNetwork AssumptionsFinal SizeDynamics
Dynamic PopulationsUnchanging DegreesDormant contacts (binding sites)
![Page 15: Joel C. Miller - Harvard Universityscholar.harvard.edu/files/joelmiller/files/2012_09_07...Intro to dynamic modeling of epidemics Static Networks Dynamic Populations Edge-based compartmental](https://reader034.vdocuments.us/reader034/viewer/2022050301/5f6a67ac0858bd490c75d0c4/html5/thumbnails/15.jpg)
Intro to dynamic modeling of epidemics Static Networks Dynamic Populations
Inserting partnership structure into the SIR model
• Different people have different numbers of partners.
• Partners are chosen randomly (pop is unclustered).
P(3) = P(1) = 0.5
• Assume static partnerships.
• Assume few initial infections.• Assume disease process occurs at simple rates:
• Transmit at rate β per partnership.• Recover at rate γ.
• Assume closed populations.
![Page 16: Joel C. Miller - Harvard Universityscholar.harvard.edu/files/joelmiller/files/2012_09_07...Intro to dynamic modeling of epidemics Static Networks Dynamic Populations Edge-based compartmental](https://reader034.vdocuments.us/reader034/viewer/2022050301/5f6a67ac0858bd490c75d0c4/html5/thumbnails/16.jpg)
Intro to dynamic modeling of epidemics Static Networks Dynamic Populations
Inserting partnership structure into the SIR model
• Different people have different numbers of partners.
• Partners are chosen randomly (pop is unclustered).
P(3) = P(1) = 0.5
• Assume static partnerships.
• Assume few initial infections.• Assume disease process occurs at simple rates:
• Transmit at rate β per partnership.• Recover at rate γ.
• Assume closed populations.
![Page 17: Joel C. Miller - Harvard Universityscholar.harvard.edu/files/joelmiller/files/2012_09_07...Intro to dynamic modeling of epidemics Static Networks Dynamic Populations Edge-based compartmental](https://reader034.vdocuments.us/reader034/viewer/2022050301/5f6a67ac0858bd490c75d0c4/html5/thumbnails/17.jpg)
Intro to dynamic modeling of epidemics Static Networks Dynamic Populations
Inserting partnership structure into the SIR model
• Different people have different numbers of partners.
• Partners are chosen randomly (pop is unclustered).
P(3) = P(1) = 0.5
• Assume static partnerships.
• Assume few initial infections.• Assume disease process occurs at simple rates:
• Transmit at rate β per partnership.• Recover at rate γ.
• Assume closed populations.
![Page 18: Joel C. Miller - Harvard Universityscholar.harvard.edu/files/joelmiller/files/2012_09_07...Intro to dynamic modeling of epidemics Static Networks Dynamic Populations Edge-based compartmental](https://reader034.vdocuments.us/reader034/viewer/2022050301/5f6a67ac0858bd490c75d0c4/html5/thumbnails/18.jpg)
Intro to dynamic modeling of epidemics Static Networks Dynamic Populations
Inserting partnership structure into the SIR model
• Different people have different numbers of partners.
• Partners are chosen randomly (pop is unclustered).
P(3) = P(1) = 0.5
• Assume static partnerships.
• Assume few initial infections.
• Assume disease process occurs at simple rates:• Transmit at rate β per partnership.• Recover at rate γ.
• Assume closed populations.
![Page 19: Joel C. Miller - Harvard Universityscholar.harvard.edu/files/joelmiller/files/2012_09_07...Intro to dynamic modeling of epidemics Static Networks Dynamic Populations Edge-based compartmental](https://reader034.vdocuments.us/reader034/viewer/2022050301/5f6a67ac0858bd490c75d0c4/html5/thumbnails/19.jpg)
Intro to dynamic modeling of epidemics Static Networks Dynamic Populations
Inserting partnership structure into the SIR model
• Different people have different numbers of partners.
• Partners are chosen randomly (pop is unclustered).
P(3) = P(1) = 0.5
• Assume static partnerships.
• Assume few initial infections.• Assume disease process occurs at simple rates:
• Transmit at rate β per partnership.• Recover at rate γ.
• Assume closed populations.
![Page 20: Joel C. Miller - Harvard Universityscholar.harvard.edu/files/joelmiller/files/2012_09_07...Intro to dynamic modeling of epidemics Static Networks Dynamic Populations Edge-based compartmental](https://reader034.vdocuments.us/reader034/viewer/2022050301/5f6a67ac0858bd490c75d0c4/html5/thumbnails/20.jpg)
Intro to dynamic modeling of epidemics Static Networks Dynamic Populations
Inserting partnership structure into the SIR model
• Different people have different numbers of partners.
• Partners are chosen randomly (pop is unclustered).
P(3) = P(1) = 0.5
• Assume static partnerships.
• Assume few initial infections.• Assume disease process occurs at simple rates:
• Transmit at rate β per partnership.• Recover at rate γ.
• Assume closed populations.
![Page 21: Joel C. Miller - Harvard Universityscholar.harvard.edu/files/joelmiller/files/2012_09_07...Intro to dynamic modeling of epidemics Static Networks Dynamic Populations Edge-based compartmental](https://reader034.vdocuments.us/reader034/viewer/2022050301/5f6a67ac0858bd490c75d0c4/html5/thumbnails/21.jpg)
Intro to dynamic modeling of epidemics Static Networks Dynamic Populations
Intro to dynamic modeling of epidemics
Static NetworksNetwork AssumptionsFinal SizeDynamics
Dynamic PopulationsUnchanging DegreesDormant contacts (binding sites)
![Page 22: Joel C. Miller - Harvard Universityscholar.harvard.edu/files/joelmiller/files/2012_09_07...Intro to dynamic modeling of epidemics Static Networks Dynamic Populations Edge-based compartmental](https://reader034.vdocuments.us/reader034/viewer/2022050301/5f6a67ac0858bd490c75d0c4/html5/thumbnails/22.jpg)
Intro to dynamic modeling of epidemics Static Networks Dynamic Populations
The final size of an epidemic in a network
• Consider a randomly chosen test individual u in thepopulation.
• Disallow infection from u to its partners (allows independenceassumption for partners).
• The probability u is Susceptible or Recovered at the end ofthe epidemic is affected by the status of its partners.
• The fraction of the population that is susceptible S equals theprobability u is susceptible.
S = P(u is susceptible)
• Let v be a random partner of u.
• Defineθ = P(v did not transmit to u)
![Page 23: Joel C. Miller - Harvard Universityscholar.harvard.edu/files/joelmiller/files/2012_09_07...Intro to dynamic modeling of epidemics Static Networks Dynamic Populations Edge-based compartmental](https://reader034.vdocuments.us/reader034/viewer/2022050301/5f6a67ac0858bd490c75d0c4/html5/thumbnails/23.jpg)
Intro to dynamic modeling of epidemics Static Networks Dynamic Populations
The final size of an epidemic in a network
• Consider a randomly chosen test individual u in thepopulation.
• Disallow infection from u to its partners (allows independenceassumption for partners).
• The probability u is Susceptible or Recovered at the end ofthe epidemic is affected by the status of its partners.
• The fraction of the population that is susceptible S equals theprobability u is susceptible.
S = P(u is susceptible)
• Let v be a random partner of u.
• Defineθ = P(v did not transmit to u)
![Page 24: Joel C. Miller - Harvard Universityscholar.harvard.edu/files/joelmiller/files/2012_09_07...Intro to dynamic modeling of epidemics Static Networks Dynamic Populations Edge-based compartmental](https://reader034.vdocuments.us/reader034/viewer/2022050301/5f6a67ac0858bd490c75d0c4/html5/thumbnails/24.jpg)
Intro to dynamic modeling of epidemics Static Networks Dynamic Populations
The final size of an epidemic in a network
• Consider a randomly chosen test individual u in thepopulation.
• Disallow infection from u to its partners (allows independenceassumption for partners).
• The probability u is Susceptible or Recovered at the end ofthe epidemic is affected by the status of its partners.
• The fraction of the population that is susceptible S equals theprobability u is susceptible.
S = P(u is susceptible)
• Let v be a random partner of u.
• Defineθ = P(v did not transmit to u)
![Page 25: Joel C. Miller - Harvard Universityscholar.harvard.edu/files/joelmiller/files/2012_09_07...Intro to dynamic modeling of epidemics Static Networks Dynamic Populations Edge-based compartmental](https://reader034.vdocuments.us/reader034/viewer/2022050301/5f6a67ac0858bd490c75d0c4/html5/thumbnails/25.jpg)
Intro to dynamic modeling of epidemics Static Networks Dynamic Populations
The final size of an epidemic in a network
• Consider a randomly chosen test individual u in thepopulation.
• Disallow infection from u to its partners (allows independenceassumption for partners).
• The probability u is Susceptible or Recovered at the end ofthe epidemic is affected by the status of its partners.
• The fraction of the population that is susceptible S equals theprobability u is susceptible.
S = P(u is susceptible)
• Let v be a random partner of u.
• Defineθ = P(v did not transmit to u)
![Page 26: Joel C. Miller - Harvard Universityscholar.harvard.edu/files/joelmiller/files/2012_09_07...Intro to dynamic modeling of epidemics Static Networks Dynamic Populations Edge-based compartmental](https://reader034.vdocuments.us/reader034/viewer/2022050301/5f6a67ac0858bd490c75d0c4/html5/thumbnails/26.jpg)
Intro to dynamic modeling of epidemics Static Networks Dynamic Populations
The final size of an epidemic in a network
• Consider a randomly chosen test individual u in thepopulation.
• Disallow infection from u to its partners (allows independenceassumption for partners).
• The probability u is Susceptible or Recovered at the end ofthe epidemic is affected by the status of its partners.
• The fraction of the population that is susceptible S equals theprobability u is susceptible.
S = P(u is susceptible)
• Let v be a random partner of u.
• Defineθ = P(v did not transmit to u)
![Page 27: Joel C. Miller - Harvard Universityscholar.harvard.edu/files/joelmiller/files/2012_09_07...Intro to dynamic modeling of epidemics Static Networks Dynamic Populations Edge-based compartmental](https://reader034.vdocuments.us/reader034/viewer/2022050301/5f6a67ac0858bd490c75d0c4/html5/thumbnails/27.jpg)
Intro to dynamic modeling of epidemics Static Networks Dynamic Populations
The final size of an epidemic in a network
• Consider a randomly chosen test individual u in thepopulation.
• Disallow infection from u to its partners (allows independenceassumption for partners).
• The probability u is Susceptible or Recovered at the end ofthe epidemic is affected by the status of its partners.
• The fraction of the population that is susceptible S equals theprobability u is susceptible.
S = P(u is susceptible)
• Let v be a random partner of u.
• Defineθ = P(v did not transmit to u)
![Page 28: Joel C. Miller - Harvard Universityscholar.harvard.edu/files/joelmiller/files/2012_09_07...Intro to dynamic modeling of epidemics Static Networks Dynamic Populations Edge-based compartmental](https://reader034.vdocuments.us/reader034/viewer/2022050301/5f6a67ac0858bd490c75d0c4/html5/thumbnails/28.jpg)
Intro to dynamic modeling of epidemics Static Networks Dynamic Populations
Finding S
u
θθ θ
θθ
θ = P(v did not transmit to u)
Probability a random test individual remains susceptible is
S =∑k
P(k)
θk
= ψ(θ)
whereψ(x) =
∑k
P(k)xk
![Page 29: Joel C. Miller - Harvard Universityscholar.harvard.edu/files/joelmiller/files/2012_09_07...Intro to dynamic modeling of epidemics Static Networks Dynamic Populations Edge-based compartmental](https://reader034.vdocuments.us/reader034/viewer/2022050301/5f6a67ac0858bd490c75d0c4/html5/thumbnails/29.jpg)
Intro to dynamic modeling of epidemics Static Networks Dynamic Populations
Finding S
u
θθ θ
θθ
θ = P(v did not transmit to u)
Probability a random degree k test individual remains susceptible is
S =∑k
P(k)
θk
= ψ(θ)
whereψ(x) =
∑k
P(k)xk
![Page 30: Joel C. Miller - Harvard Universityscholar.harvard.edu/files/joelmiller/files/2012_09_07...Intro to dynamic modeling of epidemics Static Networks Dynamic Populations Edge-based compartmental](https://reader034.vdocuments.us/reader034/viewer/2022050301/5f6a67ac0858bd490c75d0c4/html5/thumbnails/30.jpg)
Intro to dynamic modeling of epidemics Static Networks Dynamic Populations
Finding S
u
θθ θ
θθ
θ = P(v did not transmit to u)
Probability a random degree k test individual remains susceptible is
S =∑k
P(k)θk
= ψ(θ)
whereψ(x) =
∑k
P(k)xk
![Page 31: Joel C. Miller - Harvard Universityscholar.harvard.edu/files/joelmiller/files/2012_09_07...Intro to dynamic modeling of epidemics Static Networks Dynamic Populations Edge-based compartmental](https://reader034.vdocuments.us/reader034/viewer/2022050301/5f6a67ac0858bd490c75d0c4/html5/thumbnails/31.jpg)
Intro to dynamic modeling of epidemics Static Networks Dynamic Populations
Finding S
u
θθ θ
θθ
θ = P(v did not transmit to u)
Probability a random degree k test individual remains susceptible is
S =∑k
P(k)θk = ψ(θ)
whereψ(x) =
∑k
P(k)xk
![Page 32: Joel C. Miller - Harvard Universityscholar.harvard.edu/files/joelmiller/files/2012_09_07...Intro to dynamic modeling of epidemics Static Networks Dynamic Populations Edge-based compartmental](https://reader034.vdocuments.us/reader034/viewer/2022050301/5f6a67ac0858bd490c75d0c4/html5/thumbnails/32.jpg)
Intro to dynamic modeling of epidemics Static Networks Dynamic Populations
Finding θ
v
u
θθ θ
θθ
θθ
Probability a random degree k partner still susceptible is
φS =∑k
θk−1
=ψ′(θ)
ψ′(1)
If T = β/(β+γ), then probability partner does not transmit to u is
θ = φS + (1− T )(1− φS) = 1− T + Tψ′(θ)
ψ′(1)
![Page 33: Joel C. Miller - Harvard Universityscholar.harvard.edu/files/joelmiller/files/2012_09_07...Intro to dynamic modeling of epidemics Static Networks Dynamic Populations Edge-based compartmental](https://reader034.vdocuments.us/reader034/viewer/2022050301/5f6a67ac0858bd490c75d0c4/html5/thumbnails/33.jpg)
Intro to dynamic modeling of epidemics Static Networks Dynamic Populations
Finding θ
v
u
θθ θ
θθ
θθ
Probability a random degree k partner still susceptible is
φS =∑k
Pn(k)θk−1
=ψ′(θ)
ψ′(1)
If T = β/(β+γ), then probability partner does not transmit to u is
θ = φS + (1− T )(1− φS) = 1− T + Tψ′(θ)
ψ′(1)
![Page 34: Joel C. Miller - Harvard Universityscholar.harvard.edu/files/joelmiller/files/2012_09_07...Intro to dynamic modeling of epidemics Static Networks Dynamic Populations Edge-based compartmental](https://reader034.vdocuments.us/reader034/viewer/2022050301/5f6a67ac0858bd490c75d0c4/html5/thumbnails/34.jpg)
Intro to dynamic modeling of epidemics Static Networks Dynamic Populations
Finding θ
v
u
θθ θ
θθ
θθ
Probability a random degree k partner still susceptible is
φS =∑k
kP(k)
〈K 〉θk−1
=ψ′(θ)
ψ′(1)
If T = β/(β+γ), then probability partner does not transmit to u is
θ = φS + (1− T )(1− φS) = 1− T + Tψ′(θ)
ψ′(1)
![Page 35: Joel C. Miller - Harvard Universityscholar.harvard.edu/files/joelmiller/files/2012_09_07...Intro to dynamic modeling of epidemics Static Networks Dynamic Populations Edge-based compartmental](https://reader034.vdocuments.us/reader034/viewer/2022050301/5f6a67ac0858bd490c75d0c4/html5/thumbnails/35.jpg)
Intro to dynamic modeling of epidemics Static Networks Dynamic Populations
Finding θ
v
u
θθ θ
θθ
θθ
Probability a random degree k partner still susceptible is
φS =∑k
kP(k)
〈K 〉θk−1 =
ψ′(θ)
ψ′(1)
If T = β/(β+γ), then probability partner does not transmit to u is
θ = φS + (1− T )(1− φS) = 1− T + Tψ′(θ)
ψ′(1)
![Page 36: Joel C. Miller - Harvard Universityscholar.harvard.edu/files/joelmiller/files/2012_09_07...Intro to dynamic modeling of epidemics Static Networks Dynamic Populations Edge-based compartmental](https://reader034.vdocuments.us/reader034/viewer/2022050301/5f6a67ac0858bd490c75d0c4/html5/thumbnails/36.jpg)
Intro to dynamic modeling of epidemics Static Networks Dynamic Populations
Finding θ
v
u
θθ θ
θθ
θθ
Probability a random degree k partner still susceptible is
φS =∑k
kP(k)
〈K 〉θk−1 =
ψ′(θ)
ψ′(1)
If T = β/(β+γ), then probability partner does not transmit to u is
θ = φS + (1− T )(1− φS) = 1− T + Tψ′(θ)
ψ′(1)
![Page 37: Joel C. Miller - Harvard Universityscholar.harvard.edu/files/joelmiller/files/2012_09_07...Intro to dynamic modeling of epidemics Static Networks Dynamic Populations Edge-based compartmental](https://reader034.vdocuments.us/reader034/viewer/2022050301/5f6a67ac0858bd490c75d0c4/html5/thumbnails/37.jpg)
Intro to dynamic modeling of epidemics Static Networks Dynamic Populations
Final Size
SoR = 1− ψ(θ)
where
θ = 1− T + Tψ′(θ)
ψ′(1)
![Page 38: Joel C. Miller - Harvard Universityscholar.harvard.edu/files/joelmiller/files/2012_09_07...Intro to dynamic modeling of epidemics Static Networks Dynamic Populations Edge-based compartmental](https://reader034.vdocuments.us/reader034/viewer/2022050301/5f6a67ac0858bd490c75d0c4/html5/thumbnails/38.jpg)
Intro to dynamic modeling of epidemics Static Networks Dynamic Populations
Intro to dynamic modeling of epidemics
Static NetworksNetwork AssumptionsFinal SizeDynamics
Dynamic PopulationsUnchanging DegreesDormant contacts (binding sites)
![Page 39: Joel C. Miller - Harvard Universityscholar.harvard.edu/files/joelmiller/files/2012_09_07...Intro to dynamic modeling of epidemics Static Networks Dynamic Populations Edge-based compartmental](https://reader034.vdocuments.us/reader034/viewer/2022050301/5f6a67ac0858bd490c75d0c4/html5/thumbnails/39.jpg)
Intro to dynamic modeling of epidemics Static Networks Dynamic Populations
Calculating Dynamics
The network structure alters the infection process (but not therecoveries)
? γI
S I R
I = 1− S − R , R = γI
We will switch to a partnership-based perspective to find S(t).
![Page 40: Joel C. Miller - Harvard Universityscholar.harvard.edu/files/joelmiller/files/2012_09_07...Intro to dynamic modeling of epidemics Static Networks Dynamic Populations Edge-based compartmental](https://reader034.vdocuments.us/reader034/viewer/2022050301/5f6a67ac0858bd490c75d0c4/html5/thumbnails/40.jpg)
Intro to dynamic modeling of epidemics Static Networks Dynamic Populations
Calculating Dynamics
The network structure alters the infection process (but not therecoveries)
? γI
S I R
I = 1− S − R , R = γI
We will switch to a partnership-based perspective to find S(t).
![Page 41: Joel C. Miller - Harvard Universityscholar.harvard.edu/files/joelmiller/files/2012_09_07...Intro to dynamic modeling of epidemics Static Networks Dynamic Populations Edge-based compartmental](https://reader034.vdocuments.us/reader034/viewer/2022050301/5f6a67ac0858bd490c75d0c4/html5/thumbnails/41.jpg)
Intro to dynamic modeling of epidemics Static Networks Dynamic Populations
Revisiting the test individual
• Consider a randomly chosen test individual u in thepopulation.
• Disallow infection from u to its partners (allows independenceassumption for partners).
• The probability u is Susceptible, Infected, or Recovered attime t is affected by the status of its partners.
• The fraction of the population that is susceptible S(t) equalsthe probability u is susceptible.
S(t) = P(u is susceptible)
• Let v be a random partner of u.
• Defineθ(t) = P(v not yet transmitted to u)
![Page 42: Joel C. Miller - Harvard Universityscholar.harvard.edu/files/joelmiller/files/2012_09_07...Intro to dynamic modeling of epidemics Static Networks Dynamic Populations Edge-based compartmental](https://reader034.vdocuments.us/reader034/viewer/2022050301/5f6a67ac0858bd490c75d0c4/html5/thumbnails/42.jpg)
Intro to dynamic modeling of epidemics Static Networks Dynamic Populations
Revisiting the test individual
• Consider a randomly chosen test individual u in thepopulation.
• Disallow infection from u to its partners (allows independenceassumption for partners).
• The probability u is Susceptible, Infected, or Recovered attime t is affected by the status of its partners.
• The fraction of the population that is susceptible S(t) equalsthe probability u is susceptible.
S(t) = P(u is susceptible)
• Let v be a random partner of u.
• Defineθ(t) = P(v not yet transmitted to u)
![Page 43: Joel C. Miller - Harvard Universityscholar.harvard.edu/files/joelmiller/files/2012_09_07...Intro to dynamic modeling of epidemics Static Networks Dynamic Populations Edge-based compartmental](https://reader034.vdocuments.us/reader034/viewer/2022050301/5f6a67ac0858bd490c75d0c4/html5/thumbnails/43.jpg)
Intro to dynamic modeling of epidemics Static Networks Dynamic Populations
Revisiting the test individual
• Consider a randomly chosen test individual u in thepopulation.
• Disallow infection from u to its partners (allows independenceassumption for partners).
• The probability u is Susceptible, Infected, or Recovered attime t is affected by the status of its partners.
• The fraction of the population that is susceptible S(t) equalsthe probability u is susceptible.
S(t) = P(u is susceptible)
• Let v be a random partner of u.
• Defineθ(t) = P(v not yet transmitted to u)
![Page 44: Joel C. Miller - Harvard Universityscholar.harvard.edu/files/joelmiller/files/2012_09_07...Intro to dynamic modeling of epidemics Static Networks Dynamic Populations Edge-based compartmental](https://reader034.vdocuments.us/reader034/viewer/2022050301/5f6a67ac0858bd490c75d0c4/html5/thumbnails/44.jpg)
Intro to dynamic modeling of epidemics Static Networks Dynamic Populations
Revisiting the test individual
• Consider a randomly chosen test individual u in thepopulation.
• Disallow infection from u to its partners (allows independenceassumption for partners).
• The probability u is Susceptible, Infected, or Recovered attime t is affected by the status of its partners.
• The fraction of the population that is susceptible S(t) equalsthe probability u is susceptible.
S(t) = P(u is susceptible)
• Let v be a random partner of u.
• Defineθ(t) = P(v not yet transmitted to u)
![Page 45: Joel C. Miller - Harvard Universityscholar.harvard.edu/files/joelmiller/files/2012_09_07...Intro to dynamic modeling of epidemics Static Networks Dynamic Populations Edge-based compartmental](https://reader034.vdocuments.us/reader034/viewer/2022050301/5f6a67ac0858bd490c75d0c4/html5/thumbnails/45.jpg)
Intro to dynamic modeling of epidemics Static Networks Dynamic Populations
Revisiting the test individual
• Consider a randomly chosen test individual u in thepopulation.
• Disallow infection from u to its partners (allows independenceassumption for partners).
• The probability u is Susceptible, Infected, or Recovered attime t is affected by the status of its partners.
• The fraction of the population that is susceptible S(t) equalsthe probability u is susceptible.
S(t) = P(u is susceptible)
• Let v be a random partner of u.
• Defineθ(t) = P(v not yet transmitted to u)
![Page 46: Joel C. Miller - Harvard Universityscholar.harvard.edu/files/joelmiller/files/2012_09_07...Intro to dynamic modeling of epidemics Static Networks Dynamic Populations Edge-based compartmental](https://reader034.vdocuments.us/reader034/viewer/2022050301/5f6a67ac0858bd490c75d0c4/html5/thumbnails/46.jpg)
Intro to dynamic modeling of epidemics Static Networks Dynamic Populations
Revisiting the test individual
• Consider a randomly chosen test individual u in thepopulation.
• Disallow infection from u to its partners (allows independenceassumption for partners).
• The probability u is Susceptible, Infected, or Recovered attime t is affected by the status of its partners.
• The fraction of the population that is susceptible S(t) equalsthe probability u is susceptible.
S(t) = P(u is susceptible)
• Let v be a random partner of u.
• Defineθ(t) = P(v not yet transmitted to u)
![Page 47: Joel C. Miller - Harvard Universityscholar.harvard.edu/files/joelmiller/files/2012_09_07...Intro to dynamic modeling of epidemics Static Networks Dynamic Populations Edge-based compartmental](https://reader034.vdocuments.us/reader034/viewer/2022050301/5f6a67ac0858bd490c75d0c4/html5/thumbnails/47.jpg)
Intro to dynamic modeling of epidemics Static Networks Dynamic Populations
Finding S(t)
u
θθ θ
θθ
θ(t) = P(v not yet transmitted to u)
Probability a random test individual still susceptible is
S(t) =∑k
P(k)
θ(t)k
= ψ(θ(t))
whereψ(x) =
∑k
P(k)xk
![Page 48: Joel C. Miller - Harvard Universityscholar.harvard.edu/files/joelmiller/files/2012_09_07...Intro to dynamic modeling of epidemics Static Networks Dynamic Populations Edge-based compartmental](https://reader034.vdocuments.us/reader034/viewer/2022050301/5f6a67ac0858bd490c75d0c4/html5/thumbnails/48.jpg)
Intro to dynamic modeling of epidemics Static Networks Dynamic Populations
Finding S(t)
u
θθ θ
θθ
θ(t) = P(v not yet transmitted to u)
Probability a random degree k test individual still susceptible is
S(t) =∑k
P(k)
θ(t)k
= ψ(θ(t))
whereψ(x) =
∑k
P(k)xk
![Page 49: Joel C. Miller - Harvard Universityscholar.harvard.edu/files/joelmiller/files/2012_09_07...Intro to dynamic modeling of epidemics Static Networks Dynamic Populations Edge-based compartmental](https://reader034.vdocuments.us/reader034/viewer/2022050301/5f6a67ac0858bd490c75d0c4/html5/thumbnails/49.jpg)
Intro to dynamic modeling of epidemics Static Networks Dynamic Populations
Finding S(t)
u
θθ θ
θθ
θ(t) = P(v not yet transmitted to u)
Probability a random degree k test individual still susceptible is
S(t) =∑k
P(k)θ(t)k
= ψ(θ(t))
whereψ(x) =
∑k
P(k)xk
![Page 50: Joel C. Miller - Harvard Universityscholar.harvard.edu/files/joelmiller/files/2012_09_07...Intro to dynamic modeling of epidemics Static Networks Dynamic Populations Edge-based compartmental](https://reader034.vdocuments.us/reader034/viewer/2022050301/5f6a67ac0858bd490c75d0c4/html5/thumbnails/50.jpg)
Intro to dynamic modeling of epidemics Static Networks Dynamic Populations
Finding S(t)
u
θθ θ
θθ
θ(t) = P(v not yet transmitted to u)
Probability a random degree k test individual still susceptible is
S(t) =∑k
P(k)θ(t)k = ψ(θ(t))
whereψ(x) =
∑k
P(k)xk
![Page 51: Joel C. Miller - Harvard Universityscholar.harvard.edu/files/joelmiller/files/2012_09_07...Intro to dynamic modeling of epidemics Static Networks Dynamic Populations Edge-based compartmental](https://reader034.vdocuments.us/reader034/viewer/2022050301/5f6a67ac0858bd490c75d0c4/html5/thumbnails/51.jpg)
Intro to dynamic modeling of epidemics Static Networks Dynamic Populations
How does θ evolve?
v
u
θ
θ
v
u
φS v
u
φI v
u
φR
v
u
1 − θ
• θ = φS + φI + φR .
• θ = −βφI .• Our goal is to find φI in terms of θ.
![Page 52: Joel C. Miller - Harvard Universityscholar.harvard.edu/files/joelmiller/files/2012_09_07...Intro to dynamic modeling of epidemics Static Networks Dynamic Populations Edge-based compartmental](https://reader034.vdocuments.us/reader034/viewer/2022050301/5f6a67ac0858bd490c75d0c4/html5/thumbnails/52.jpg)
Intro to dynamic modeling of epidemics Static Networks Dynamic Populations
How does θ evolve?
θ
v
u
φS v
u
φI v
u
φR
{ v
u
1 − θ
• θ = φS + φI + φR .
• θ = −βφI .• Our goal is to find φI in terms of θ.
![Page 53: Joel C. Miller - Harvard Universityscholar.harvard.edu/files/joelmiller/files/2012_09_07...Intro to dynamic modeling of epidemics Static Networks Dynamic Populations Edge-based compartmental](https://reader034.vdocuments.us/reader034/viewer/2022050301/5f6a67ac0858bd490c75d0c4/html5/thumbnails/53.jpg)
Intro to dynamic modeling of epidemics Static Networks Dynamic Populations
How does θ evolve?
θ
v
u
φS v
u
φI v
u
φR
{ v
u
1 − θ
βφI
γφI
• θ = φS + φI + φR .
• θ = −βφI .
• Our goal is to find φI in terms of θ.
![Page 54: Joel C. Miller - Harvard Universityscholar.harvard.edu/files/joelmiller/files/2012_09_07...Intro to dynamic modeling of epidemics Static Networks Dynamic Populations Edge-based compartmental](https://reader034.vdocuments.us/reader034/viewer/2022050301/5f6a67ac0858bd490c75d0c4/html5/thumbnails/54.jpg)
Intro to dynamic modeling of epidemics Static Networks Dynamic Populations
How does θ evolve?
θ
v
u
φS v
u
φI v
u
φR
{ v
u
1 − θ
βφI
γφI
• θ = φS + φI + φR .
• θ = −βφI .• Our goal is to find φI in terms of θ.
![Page 55: Joel C. Miller - Harvard Universityscholar.harvard.edu/files/joelmiller/files/2012_09_07...Intro to dynamic modeling of epidemics Static Networks Dynamic Populations Edge-based compartmental](https://reader034.vdocuments.us/reader034/viewer/2022050301/5f6a67ac0858bd490c75d0c4/html5/thumbnails/55.jpg)
Intro to dynamic modeling of epidemics Static Networks Dynamic Populations
Finding φR(t)
θ
v
u
φS v
u
φI v
u
φR
{ v
u
1 − θ
βφI
γφI
Because derivatives are proportional, φR = γβ (1− θ)
![Page 56: Joel C. Miller - Harvard Universityscholar.harvard.edu/files/joelmiller/files/2012_09_07...Intro to dynamic modeling of epidemics Static Networks Dynamic Populations Edge-based compartmental](https://reader034.vdocuments.us/reader034/viewer/2022050301/5f6a67ac0858bd490c75d0c4/html5/thumbnails/56.jpg)
Intro to dynamic modeling of epidemics Static Networks Dynamic Populations
Finding φS(t)
v
u
θθ θ
θθ
θθ
Probability a random degree k partner still susceptible is
φS(t) =∑k
θ(t)k−1
=ψ′(θ)
ψ′(1)
![Page 57: Joel C. Miller - Harvard Universityscholar.harvard.edu/files/joelmiller/files/2012_09_07...Intro to dynamic modeling of epidemics Static Networks Dynamic Populations Edge-based compartmental](https://reader034.vdocuments.us/reader034/viewer/2022050301/5f6a67ac0858bd490c75d0c4/html5/thumbnails/57.jpg)
Intro to dynamic modeling of epidemics Static Networks Dynamic Populations
Finding φS(t)
v
u
θθ θ
θθ
θθ
Probability a random degree k partner still susceptible is
φS(t) =∑k
Pn(k)θ(t)k−1
=ψ′(θ)
ψ′(1)
![Page 58: Joel C. Miller - Harvard Universityscholar.harvard.edu/files/joelmiller/files/2012_09_07...Intro to dynamic modeling of epidemics Static Networks Dynamic Populations Edge-based compartmental](https://reader034.vdocuments.us/reader034/viewer/2022050301/5f6a67ac0858bd490c75d0c4/html5/thumbnails/58.jpg)
Intro to dynamic modeling of epidemics Static Networks Dynamic Populations
Finding φS(t)
v
u
θθ θ
θθ
θθ
Probability a random degree k partner still susceptible is
φS(t) =∑k
kP(k)
〈K 〉θ(t)k−1
=ψ′(θ)
ψ′(1)
![Page 59: Joel C. Miller - Harvard Universityscholar.harvard.edu/files/joelmiller/files/2012_09_07...Intro to dynamic modeling of epidemics Static Networks Dynamic Populations Edge-based compartmental](https://reader034.vdocuments.us/reader034/viewer/2022050301/5f6a67ac0858bd490c75d0c4/html5/thumbnails/59.jpg)
Intro to dynamic modeling of epidemics Static Networks Dynamic Populations
Finding φS(t)
v
u
θθ θ
θθ
θθ
Probability a random degree k partner still susceptible is
φS(t) =∑k
kP(k)
〈K 〉θ(t)k−1 =
ψ′(θ)
ψ′(1)
![Page 60: Joel C. Miller - Harvard Universityscholar.harvard.edu/files/joelmiller/files/2012_09_07...Intro to dynamic modeling of epidemics Static Networks Dynamic Populations Edge-based compartmental](https://reader034.vdocuments.us/reader034/viewer/2022050301/5f6a67ac0858bd490c75d0c4/html5/thumbnails/60.jpg)
Intro to dynamic modeling of epidemics Static Networks Dynamic Populations
θ
v
u
ψ′(θ)ψ′(1) v
u
φI v
u
γβ(1 − θ)
{ v
u
1 − θ
βφI
γφI
Since φI = θ − φS − φR
= θ − ψ′(θ)ψ′(1) −
γβ (1− θ), we have
θ = −βφI = −βθ + βψ′(θ)
ψ′(1)+ γ(1− θ)
![Page 61: Joel C. Miller - Harvard Universityscholar.harvard.edu/files/joelmiller/files/2012_09_07...Intro to dynamic modeling of epidemics Static Networks Dynamic Populations Edge-based compartmental](https://reader034.vdocuments.us/reader034/viewer/2022050301/5f6a67ac0858bd490c75d0c4/html5/thumbnails/61.jpg)
Intro to dynamic modeling of epidemics Static Networks Dynamic Populations
θ
v
u
ψ′(θ)ψ′(1) v
u
φI v
u
γβ(1 − θ)
{ v
u
1 − θ
βφI
γφI
Since φI = θ − φS − φR = θ − ψ′(θ)ψ′(1) −
γβ (1− θ)
, we have
θ = −βφI = −βθ + βψ′(θ)
ψ′(1)+ γ(1− θ)
![Page 62: Joel C. Miller - Harvard Universityscholar.harvard.edu/files/joelmiller/files/2012_09_07...Intro to dynamic modeling of epidemics Static Networks Dynamic Populations Edge-based compartmental](https://reader034.vdocuments.us/reader034/viewer/2022050301/5f6a67ac0858bd490c75d0c4/html5/thumbnails/62.jpg)
Intro to dynamic modeling of epidemics Static Networks Dynamic Populations
θ
v
u
ψ′(θ)ψ′(1) v
u
φI v
u
γβ(1 − θ)
{ v
u
1 − θ
βφI
γφI
Since φI = θ − φS − φR = θ − ψ′(θ)ψ′(1) −
γβ (1− θ), we have
θ = −βφI = −βθ + βψ′(θ)
ψ′(1)+ γ(1− θ)
![Page 63: Joel C. Miller - Harvard Universityscholar.harvard.edu/files/joelmiller/files/2012_09_07...Intro to dynamic modeling of epidemics Static Networks Dynamic Populations Edge-based compartmental](https://reader034.vdocuments.us/reader034/viewer/2022050301/5f6a67ac0858bd490c75d0c4/html5/thumbnails/63.jpg)
Intro to dynamic modeling of epidemics Static Networks Dynamic Populations
Final System
We finally have
θ = −βθ + βψ′(θ)
ψ′(1)+ γ(1− θ)
R = γI S = ψ(θ) I = 1− S − R
![Page 64: Joel C. Miller - Harvard Universityscholar.harvard.edu/files/joelmiller/files/2012_09_07...Intro to dynamic modeling of epidemics Static Networks Dynamic Populations Edge-based compartmental](https://reader034.vdocuments.us/reader034/viewer/2022050301/5f6a67ac0858bd490c75d0c4/html5/thumbnails/64.jpg)
Intro to dynamic modeling of epidemics Static Networks Dynamic Populations
−5 0 5 10 150.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35Mass ActionHomogeneousPoissonBimodalTruncated Powerlaw
t
Infe
ctio
ns
![Page 65: Joel C. Miller - Harvard Universityscholar.harvard.edu/files/joelmiller/files/2012_09_07...Intro to dynamic modeling of epidemics Static Networks Dynamic Populations Edge-based compartmental](https://reader034.vdocuments.us/reader034/viewer/2022050301/5f6a67ac0858bd490c75d0c4/html5/thumbnails/65.jpg)
Intro to dynamic modeling of epidemics Static Networks Dynamic Populations
Intro to dynamic modeling of epidemics
Static NetworksNetwork AssumptionsFinal SizeDynamics
Dynamic PopulationsUnchanging DegreesDormant contacts (binding sites)
![Page 66: Joel C. Miller - Harvard Universityscholar.harvard.edu/files/joelmiller/files/2012_09_07...Intro to dynamic modeling of epidemics Static Networks Dynamic Populations Edge-based compartmental](https://reader034.vdocuments.us/reader034/viewer/2022050301/5f6a67ac0858bd490c75d0c4/html5/thumbnails/66.jpg)
Intro to dynamic modeling of epidemics Static Networks Dynamic Populations
Dynamic Populations
We now modify the population to have dynamic partnerships.Existing partnerships break at rate η and are immediately replacedby new connections.
7→
![Page 67: Joel C. Miller - Harvard Universityscholar.harvard.edu/files/joelmiller/files/2012_09_07...Intro to dynamic modeling of epidemics Static Networks Dynamic Populations Edge-based compartmental](https://reader034.vdocuments.us/reader034/viewer/2022050301/5f6a67ac0858bd490c75d0c4/html5/thumbnails/67.jpg)
Intro to dynamic modeling of epidemics Static Networks Dynamic Populations
θ
v
u
φSv
u
φIv
u
φR
v
u
1 − θ
βφI
γφI
ηφRηφS ηφI
πS
πI
πR
γπI
We begin with the previous diagram, but φS , φI , and φR are theprobabilities a stub has not received infection from a partner andcurrently connects to a susceptible, infected, or recovered partner.
![Page 68: Joel C. Miller - Harvard Universityscholar.harvard.edu/files/joelmiller/files/2012_09_07...Intro to dynamic modeling of epidemics Static Networks Dynamic Populations Edge-based compartmental](https://reader034.vdocuments.us/reader034/viewer/2022050301/5f6a67ac0858bd490c75d0c4/html5/thumbnails/68.jpg)
Intro to dynamic modeling of epidemics Static Networks Dynamic Populations
θ
v
u
φSv
u
φIv
u
φR
v
u
1 − θ
βφI
γφI
ηφR
ηφS ηφI
πS
πI
πR
γπI
Consider an edge with a recovered partner breaking. Rate = ηφR .
![Page 69: Joel C. Miller - Harvard Universityscholar.harvard.edu/files/joelmiller/files/2012_09_07...Intro to dynamic modeling of epidemics Static Networks Dynamic Populations Edge-based compartmental](https://reader034.vdocuments.us/reader034/viewer/2022050301/5f6a67ac0858bd490c75d0c4/html5/thumbnails/69.jpg)
Intro to dynamic modeling of epidemics Static Networks Dynamic Populations
θ
v
u
φSv
u
φIv
u
φR
v
u
1 − θ
βφI
γφI
ηφR
ηφS ηφI
πS
πI
πR
γπI
The replacement partner may have any status.
![Page 70: Joel C. Miller - Harvard Universityscholar.harvard.edu/files/joelmiller/files/2012_09_07...Intro to dynamic modeling of epidemics Static Networks Dynamic Populations Edge-based compartmental](https://reader034.vdocuments.us/reader034/viewer/2022050301/5f6a67ac0858bd490c75d0c4/html5/thumbnails/70.jpg)
Intro to dynamic modeling of epidemics Static Networks Dynamic Populations
θ
v
u
φSv
u
φIv
u
φR
v
u
1 − θ
βφI
γφI
ηφR
ηθ
ηφS ηφI
πS
πI
πR
γπI
The same is true for any edge that breaks. The total rate stubsthat have not received infections disconnect is ηθ.
![Page 71: Joel C. Miller - Harvard Universityscholar.harvard.edu/files/joelmiller/files/2012_09_07...Intro to dynamic modeling of epidemics Static Networks Dynamic Populations Edge-based compartmental](https://reader034.vdocuments.us/reader034/viewer/2022050301/5f6a67ac0858bd490c75d0c4/html5/thumbnails/71.jpg)
Intro to dynamic modeling of epidemics Static Networks Dynamic Populations
θ
v
u
φSv
u
φIv
u
φR
v
u
1 − θ
βφI
γφI
ηφR
ηθ
ηθπS ηθπIηθπRηφS ηφI
πS
πI
πR
γπI
The probability the new partner is susceptible, infected, orrecovered is equal to the proportion of stubs belonging toindividuals of that type: πS , πI , and πR .
![Page 72: Joel C. Miller - Harvard Universityscholar.harvard.edu/files/joelmiller/files/2012_09_07...Intro to dynamic modeling of epidemics Static Networks Dynamic Populations Edge-based compartmental](https://reader034.vdocuments.us/reader034/viewer/2022050301/5f6a67ac0858bd490c75d0c4/html5/thumbnails/72.jpg)
Intro to dynamic modeling of epidemics Static Networks Dynamic Populations
θ
v
u
φSv
u
φIv
u
φR
v
u
1 − θ
βφI
βφI φ
Sψ
′′(θ)
ψ′(θ
)
γφI
ηφR
ηθ
ηθπS ηθπIηθπRηφS ηφI
πS
πI
πR
γπI
The simple expressions for φR and πS break.
![Page 73: Joel C. Miller - Harvard Universityscholar.harvard.edu/files/joelmiller/files/2012_09_07...Intro to dynamic modeling of epidemics Static Networks Dynamic Populations Edge-based compartmental](https://reader034.vdocuments.us/reader034/viewer/2022050301/5f6a67ac0858bd490c75d0c4/html5/thumbnails/73.jpg)
Intro to dynamic modeling of epidemics Static Networks Dynamic Populations
Finding πS
θθ θ
θθ
θθ
θ
Probability a random stub belongs to a degree k susceptibleindividual is
πS(t) =∑k
Pn(k)θ(t)k
=θψ′(θ)
ψ′(1)
![Page 74: Joel C. Miller - Harvard Universityscholar.harvard.edu/files/joelmiller/files/2012_09_07...Intro to dynamic modeling of epidemics Static Networks Dynamic Populations Edge-based compartmental](https://reader034.vdocuments.us/reader034/viewer/2022050301/5f6a67ac0858bd490c75d0c4/html5/thumbnails/74.jpg)
Intro to dynamic modeling of epidemics Static Networks Dynamic Populations
Finding πS
θθ θ
θθ
θθ
θ
Probability a random stub belongs to a degree k susceptibleindividual is
πS(t) =∑k
Pn(k)θ(t)k
=θψ′(θ)
ψ′(1)
![Page 75: Joel C. Miller - Harvard Universityscholar.harvard.edu/files/joelmiller/files/2012_09_07...Intro to dynamic modeling of epidemics Static Networks Dynamic Populations Edge-based compartmental](https://reader034.vdocuments.us/reader034/viewer/2022050301/5f6a67ac0858bd490c75d0c4/html5/thumbnails/75.jpg)
Intro to dynamic modeling of epidemics Static Networks Dynamic Populations
Finding πS
θθ θ
θθ
θθ
θ
Probability a random stub belongs to a degree k susceptibleindividual is
πS(t) =∑k
kP(k)
〈K 〉θ(t)k
=θψ′(θ)
ψ′(1)
![Page 76: Joel C. Miller - Harvard Universityscholar.harvard.edu/files/joelmiller/files/2012_09_07...Intro to dynamic modeling of epidemics Static Networks Dynamic Populations Edge-based compartmental](https://reader034.vdocuments.us/reader034/viewer/2022050301/5f6a67ac0858bd490c75d0c4/html5/thumbnails/76.jpg)
Intro to dynamic modeling of epidemics Static Networks Dynamic Populations
Finding πS
θθ θ
θθ
θθ
θ
Probability a random stub belongs to a degree k susceptibleindividual is
πS(t) =∑k
kP(k)
〈K 〉θ(t)k =
θψ′(θ)
ψ′(1)
![Page 77: Joel C. Miller - Harvard Universityscholar.harvard.edu/files/joelmiller/files/2012_09_07...Intro to dynamic modeling of epidemics Static Networks Dynamic Populations Edge-based compartmental](https://reader034.vdocuments.us/reader034/viewer/2022050301/5f6a67ac0858bd490c75d0c4/html5/thumbnails/77.jpg)
Intro to dynamic modeling of epidemics Static Networks Dynamic Populations
πS
πI
πR
γπI
This leads to
πR = γπI , πI = 1− πS − πR , πS =θψ′(θ)
ψ′(1)
![Page 78: Joel C. Miller - Harvard Universityscholar.harvard.edu/files/joelmiller/files/2012_09_07...Intro to dynamic modeling of epidemics Static Networks Dynamic Populations Edge-based compartmental](https://reader034.vdocuments.us/reader034/viewer/2022050301/5f6a67ac0858bd490c75d0c4/html5/thumbnails/78.jpg)
Intro to dynamic modeling of epidemics Static Networks Dynamic Populations
Completing the equations
θ
v
u
φSv
u
φIv
u
φR
v
u
1 − θ
βφI
βφI φ
Sψ
′′(θ)
ψ′(θ
)
γφI
ηφR
ηθ
ηθπS ηθπIηθπRηφS ηφI
φS = ηθπS − ηφS − βφIφSψ′′(θ)/ψ′(θ)
φI = βφIφSψ′′(θ)/ψ′(θ) + ηθπI − (η + γ + β)φI
θ = −βφI
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Intro to dynamic modeling of epidemics Static Networks Dynamic Populations
Final Equations
φS = ηθπS − ηφS − βφIφSψ′′(θ)
ψ′(θ)
φI = βφIφSψ′′(θ)
ψ′(θ)+ ηθπI − (η + γ + β)φI
θ = −βφI
πR = γπI , πI = 1− πS − πR , πS =θψ′(θ)
ψ′(1)
R = γI , I = 1− S − R , S = ψ(θ)
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Intro to dynamic modeling of epidemics Static Networks Dynamic Populations
−5 0 5 10 15 200.00
0.02
0.04
0.06
0.08
0.10
0.12
t
Infe
ctio
ns
Comparison of theory (dashed) with average of 102 simulations ina population of 104 individuals (solid).Negative binomial degree distribution: P(k) =
(k+r−1k
)(1− p)rpk
for r = 4, p = 1/3. ψ(x) = [2/(3− x)]4.β = 5/4, γ = 1, η = 1/2.
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Intro to dynamic modeling of epidemics Static Networks Dynamic Populations
Intro to dynamic modeling of epidemics
Static NetworksNetwork AssumptionsFinal SizeDynamics
Dynamic PopulationsUnchanging DegreesDormant contacts (binding sites)
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Intro to dynamic modeling of epidemics Static Networks Dynamic Populations
Introducing “dormant” contacts
• Consider a population in which individuals have k stubs (or“binding sites”), not all of which are active at any time.
• A dormant stub becomes active at rate η1.
• An active stub becomes dormant at rate η2.
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Intro to dynamic modeling of epidemics Static Networks Dynamic Populations
θ
v
u
φSv
u
φIv
u
φR
v
u
1 − θ
βφI
βφI φ
Sψ
′′(θ)
ψ′(θ
)
γφI
u
φD η2(θ − φD)η1φD
ηθ
ηθπS ηθπI
ηθπR
ηφR
ηφS
ηφI
πS
πI
πR
γπI
ξS
ξI
ξR
γξI
η2ξS
η1πS
η2ξI
η1πI
η2ξR
η1πR
Introduce φD : probability stub is dormant, and let π variablesrepresent dormant stubs, ξ variables represent active stubs.
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Intro to dynamic modeling of epidemics Static Networks Dynamic Populations
θ
v
u
φSv
u
φIv
u
φR
v
u
1 − θ
βφI
βφI φ
Sψ
′′(θ)
ψ′(θ
)
γφI
u
φD η2(θ − φD)η1φD
η1φDπSπ η
1φD π
Iπ
η1φ
D πR
π
η2φR
η2φS
η 2φ I
πS
πI
πR
γπI
ξS
ξI
ξR
γξI
η2ξS
η1πS
η2ξI
η1πI
η2ξR
η1πR
Introduce φD : probability stub is dormant, and let π variablesrepresent dormant stubs, ξ variables represent active stubs.
![Page 85: Joel C. Miller - Harvard Universityscholar.harvard.edu/files/joelmiller/files/2012_09_07...Intro to dynamic modeling of epidemics Static Networks Dynamic Populations Edge-based compartmental](https://reader034.vdocuments.us/reader034/viewer/2022050301/5f6a67ac0858bd490c75d0c4/html5/thumbnails/85.jpg)
Intro to dynamic modeling of epidemics Static Networks Dynamic Populations
Equations
θ = −βφI
φS = −βφIφSψ′′(θ)
ψ′(θ)+ η1
πS
πφD − η2φS
φI = βφIφSψ′′(θ)
ψ′(θ)+ η1
πI
πφD − (η2 + β + γ)φI
φD = η2(θ − φD)− η1φD
ξR = −η2ξR + η1πR + γξI , ξS = (θ − φD)ψ′(θ)
ψ′(1), ξI = ξ − ξS − ξR
πR = η2ξR − η1πR + γπI , πS = φDψ′(θ)
ψ′(1), πI = π − πS − πR
ξ =η1
η1 + η2, π =
η2η1 + η2
R = γI , S = ψ(θ) , I = 1− S − R
![Page 86: Joel C. Miller - Harvard Universityscholar.harvard.edu/files/joelmiller/files/2012_09_07...Intro to dynamic modeling of epidemics Static Networks Dynamic Populations Edge-based compartmental](https://reader034.vdocuments.us/reader034/viewer/2022050301/5f6a67ac0858bd490c75d0c4/html5/thumbnails/86.jpg)
Intro to dynamic modeling of epidemics Static Networks Dynamic Populations
−5 0 5 10 150.00
0.05
0.10
0.15
0.20
t
Infe
ctio
ns
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Intro to dynamic modeling of epidemics Static Networks Dynamic Populations
Hierarchy
DormantContacts
DynamicFixed-Degree
DynamicVariable-Degree
ConfigurationModel
Mixed Pois-son
Mean FieldSocial Het-erogeneity
Mass Action
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Intro to dynamic modeling of epidemics Static Networks Dynamic Populations
Conclusions
Fairly complex population structure can be captured by a smallnumber of equations.
PS: mass action can be written as
ξ = βI
S = S(0)e−ξ , R = R(0) +γξ
β, I = 1− S − R
where ξ(0) = 0.
![Page 89: Joel C. Miller - Harvard Universityscholar.harvard.edu/files/joelmiller/files/2012_09_07...Intro to dynamic modeling of epidemics Static Networks Dynamic Populations Edge-based compartmental](https://reader034.vdocuments.us/reader034/viewer/2022050301/5f6a67ac0858bd490c75d0c4/html5/thumbnails/89.jpg)
Intro to dynamic modeling of epidemics Static Networks Dynamic Populations
Conclusions
Fairly complex population structure can be captured by a smallnumber of equations.
PS: mass action can be written as
ξ = βI
S = S(0)e−ξ , R = R(0) +γξ
β, I = 1− S − R
where ξ(0) = 0.