The Probability and Duration of Epidemics in Stochastic ...
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The Probability and Duration of Epidemics in Stochastic MultiStage and MultiGroup Models Linda J. S. Allen Texas Tech University Lubbock, Texas IMA Workshop Ecological and Biological Systems University of Minnesota June 6, 2018
Transcript of The Probability and Duration of Epidemics in Stochastic ...
The Probability and Duration of Epidemics in Stochastic MultiStage
and MultiGroup Models
Linda J. S. Allen Texas Tech University
Lubbock, Texas
Acknowledgement
• WAMB Group: Christina Edholm, Blessing Emerenini, Anarina Murillo, Omar Saucedo, Nika Shakiba, Xueying Wang, and Angela Peace
Outline of Talk
I. Stochastic SIR Epidemic
II. Multistage/Multigroup Epidemic Models
IV. Implications for Public Heath
I.The Classic SIR Epidemic Model
R0 = β
dt = γI
S(0) > 0, I(0) > 0, R(0) ≥ 0, and N = S + I +R.
Duration and Final Size of the Classic SIR Epidemic Model
Final size increases with R0
Duration increases near R0 = 1.
Table: Final Size and Duration N = 500, γ = 0.2, I(0) = 2
R0 Final Size Duration D I(D) < 1 I(D) < 0.5
0.9 19 31.1 59.5 1.1 105 166.8 197.03 1.5 296 103.7 112.8
2 401 71.7 77.5
R0 ln
( S(∞)
S(0)
) +N
Stochastic Formulation of the SIR Model as Continuous-Time Markov Chains (CTMC)
S(t), I(t) ∈ {0, 1, . . . , N} with transition probabilities
=
βsit/N + o(t), (k, j) = (s− 1, i+ 1), γit+ o(t), (k, j) = (s, i− 1), 1− [βsi/N + γi]t+ o(t), (k, j) = (s, i), o(t), otherwise.
Probability of a Minor and Major Outbreak The terms ”minor” and ”major” outbreaks come from Whittle (1955) and
are applicable when R0 1 and R0 1 and when N is large.
N = 500, R0 = 2, I(0) = 2
0 50 100
Figure: γ = 0.2, Pminor = 0.25, Pmajor = 0.75
Whittle’s Formula for Probability of a Minor or Major Outbreak
Pminor =
0 50 100
50 100 150
q u e n c y
γ = 0.2, Pminor = 1, Pmajor = 0 ODE Final Size =19; Dur=31-60
Stochastic SIR when R0 = 1.1 N = 500, I(0) = 2
0 50 100
50 100 150 200
q u e n c y
γ = 0.2, Pminor = 0.83, Pmajor = 0.17 ODE Final Size=105; Dur=167-197
Stochastic SIR when R0 = 2 N = 500, I(0) = 2
0 50 100
Final Size
50 100 150
q u e n c y
γ = 0.2, Pminor = 0.25, Pmajor = 0.75 ODE Final Size=401; Dur=72-78
Near the DFE, the Dynamics of I(t) are Approximated by a Birth/Death Process
pi,j(t) = P(I(t) = j|I(0) = i)
pi,j(t) = P(I(t+ t) = j|I(t) = i)
=
1− (β + γ)it+ o(t), j = i,
o(t), otherwise.
The mean of I(t) is
E(I(t)|I(0) = i) = ie(β−γ)t
Whittle’s Result Follows from a Birth/Death Approximation for the Infectious State I(t)
Gi(u, t) = E(uI(t)|I(0) = i) =
∞∑ j=0
pi,j(t)u j,
∂G1
f(u) = βu2
β + γ +
The function f is the infinitesimal offspring probability generating function.
The Solution pi,0(t) is the Cumulative Distribution for Time to Extinction, Given I(0) = i
Ti = time to extinction, conditioned on extinction
Given I(0) = i,
R0 → 1
Ref: Tritch & Allen, 2018
The PDF for Time to Extinction (Conditioned on Extinction) has a “Fat” Tail as R0 Approaches One.
0 5 10 time
0 5 10 time
0.6 R0=0.25 R0=0.7 R0=0.9
Figure: Probability density functions for R0 close to one, γ = 1.
The PDF from Birth/Death is a Good Fit to the Duration of Minor Outbreaks in SIR Models
0 10 20 30 40 time
0
0.05
0.1
0
0.05
0
0.05
0.1
0.15
0
0.1
0.2
R0=0.9
Figure: Pdfs from 106 sample paths. Overlay plot in red is fT3Pext;
γ = 1, β = R0, N = 1000 with I(0) = 3.
II. MultiStage Epidemic Models
R I S E1 E2
R0 = βδ1δ2
(δ1 + µ1)(δ2 + µ2)(γ + µ)
Stochastic MultiStage Epidemic Models
Pminor =
e2 2 q
i 3, R0 > 1
The probabilities q1, q2, q3 follow from a multitype branching process for E1, E2, I.
MultiGroup Epidemic Models
N and Nk are constant.
We Define the Reproduction Number in Terms of Host Susceptibility or Host Infectivity
Incidence rate for group k:
Sk
N
If transmission depends primarily on either Host Susceptibility or Host Infectivity, then
βkj =
R0 =
N , m = S, I
The Stochastic Two-Group Model is Studied in Terms of the Probability of a Minor or a Major Outbreak
Assume I1(0) = i1 and I2(0) = i2.
Pminor =
Pmajor =
1− qi11 q i2 2 , R0 > 1
• The probabilities q1 and q2 follow from a multitype branching process approximation for I1 and I2.
• Smaller values of qi lead to a greater probability of a major outbreak!
With Host Susceptibility, the Probability of a Major Outbreak is Greater if Initiated by the Group with
Smallest Recovery Rate
Suppose βSk depends on Host Susceptibility and Group 1 has a smaller recovery rate than Group 2,
γ1 < γ2
Then the probability a major outbreak is greater if initiated by Group 1 than by Group 2.
q1 < q2.
R0 .
Ref: Nandi & Allen
The Group with the Smallest Recovery Rate Has a Higher Probability of Initiating a Major Outbreak. Host Susceptibility
γ1 = 0.2 < 1 = γ2
Figure: R0 = 2.4, N1 = 200, N = 1000, R01 = 10, R02 = 0.5
Pminor = (q1) 2 = 0.111, Pminor = (q2)
2 = 0.444
Host Susceptibility
2 (0)=0
Duration
0
0.02
0.04
0.06
0.08
I 1 (0)=0, I
2 (0)=2
Duration
0
0.1
0.2
0.3
0.4
2 = 0.444
After a Major Outbreak is Initiated the Final Size Differs in Each Group
Host Susceptibility Group 1
50 100 150 200
Group 2
Group 1
Group 2
q u e n c y
Figure: R0 = 2.4, Top I1(0) = 2, I2(0) = 0; Bottom: I1(0) = 0,
I2(0) = 2, ODE Final Size1= 186, Final Size2= 388 Ref: Magal et al. 2016
Pminor = (q1) 2 = 0.111, Pminor = (q2)
2 = 0.444
With Host Infectivity, the Probability of a Major Outbreak is Greater for the Group with Largest
Reproduction Number.
Suppose βIj depends on Host Infectivity, and Group 1 has a larger reproduction number than Group 2
R01 = βI1 γ1
= R02
Then the probability a major outbreak is greater if initiated by Group 1 than by Group 2
q1 < q2
R0 .
Ref: Nandi & Allen
When Initiated by the Group with the Largest Reproduction Number, Epidemics Occur More
Frequently and Earlier. Host Infectivity
R01 = 10 > 0.5 = R02
Figure: R0 = 2.4, γ1 = 0.2, γ2 = 1, N1 = 200, N = 1000
Pminor = (q1)2 = 0.102 Pminor = (q2)2 = 0.817
The Duration of a Minor and Major Outbreaks
Host Infectivity
2 (0)=2
Duration
0
0.1
0.2
0.3
0.4
0.5
I 1 (0)=2, I
2 (0)=0
Duration
0
0.02
0.04
0.06
0.08
0.1
2 = 0.817
After a Major Outbreak is Initiated, the Final Size Differs for Each Group
Host Infectivity Group 1
F re
q u
e n
c y
Figure: R0 = 2.4, Top I1(0) = 2, I2(0) = 0; Bottom: I1(0) = 0,
I2(0) = 2, ODE Final Size1= 175, Final Size2= 665
Pminor = (q1) 2 = 0.102 Pminor = (q2)
2 = 0.817
III. We Applied a Two-Group, MultiStage Model to MERS, Middle East Respiratory Syndrome
• MERS identified 2012 from outbreak in Saudi Arabia.
• Case fatality rate ≈ 35%
• Group 1=NonSuperspreaders=NS
• Group 2= SuperSpreaders=SS
We Modeled MERS Assuming High Host Infectivity Determines a Superspreader.
R0 = β1
N1 N
Parameter MERS Baseline Range
β1 transmission rate NS 0.06 (0.04, 0.08)) β2 transmission rate SS 0.6931∗∗ (0.4, 0.8)
α−1 i latent period 6.3 (2-8)
δ−1 i duration of asymp. stage 0.4 (0.1, 2) µji disease induced death rate 0.08 (0.02, 0.14) γi recovery rate 0.075 (0.05, 0.1)
For N1 = N2 and N = 2000,
R0 = 2.36
Initiated by NS Initiated by SS
Increase R0
Increase Prop. SS, R0 = 2.36
Time to Reach a Total of 50 Infected Occurs Earlier if Initiated by SS
R0 = 2.36
IV. Implications for Public Health
• For R0 ≈ 1 the duration of outbreaks increases. Public Health: Reduction of R0 ≈ 1 through preventive measures or intervention increases time until eradication which allows pathogens more time to adapt to their host.
• Transmission depends on host susceptibility and infectivity. Public Health: Reduction of both transmission mechanisms, through vaccination, quarantine, drugs, etc, will have the greatest impact on reducing final size.
• Superspreading events have been associated with recent outbreaks (MERS, SARS, Ebola). Public Health: Identification of superspreading events through early tracking contacts, earlier diagnosis and treatment strategies, and community education are important.
Thank You
• National Science Foundation
Linda J. S. Allen Texas Tech University
Lubbock, Texas
Acknowledgement
• WAMB Group: Christina Edholm, Blessing Emerenini, Anarina Murillo, Omar Saucedo, Nika Shakiba, Xueying Wang, and Angela Peace
Outline of Talk
I. Stochastic SIR Epidemic
II. Multistage/Multigroup Epidemic Models
IV. Implications for Public Heath
I.The Classic SIR Epidemic Model
R0 = β
dt = γI
S(0) > 0, I(0) > 0, R(0) ≥ 0, and N = S + I +R.
Duration and Final Size of the Classic SIR Epidemic Model
Final size increases with R0
Duration increases near R0 = 1.
Table: Final Size and Duration N = 500, γ = 0.2, I(0) = 2
R0 Final Size Duration D I(D) < 1 I(D) < 0.5
0.9 19 31.1 59.5 1.1 105 166.8 197.03 1.5 296 103.7 112.8
2 401 71.7 77.5
R0 ln
( S(∞)
S(0)
) +N
Stochastic Formulation of the SIR Model as Continuous-Time Markov Chains (CTMC)
S(t), I(t) ∈ {0, 1, . . . , N} with transition probabilities
=
βsit/N + o(t), (k, j) = (s− 1, i+ 1), γit+ o(t), (k, j) = (s, i− 1), 1− [βsi/N + γi]t+ o(t), (k, j) = (s, i), o(t), otherwise.
Probability of a Minor and Major Outbreak The terms ”minor” and ”major” outbreaks come from Whittle (1955) and
are applicable when R0 1 and R0 1 and when N is large.
N = 500, R0 = 2, I(0) = 2
0 50 100
Figure: γ = 0.2, Pminor = 0.25, Pmajor = 0.75
Whittle’s Formula for Probability of a Minor or Major Outbreak
Pminor =
0 50 100
50 100 150
q u e n c y
γ = 0.2, Pminor = 1, Pmajor = 0 ODE Final Size =19; Dur=31-60
Stochastic SIR when R0 = 1.1 N = 500, I(0) = 2
0 50 100
50 100 150 200
q u e n c y
γ = 0.2, Pminor = 0.83, Pmajor = 0.17 ODE Final Size=105; Dur=167-197
Stochastic SIR when R0 = 2 N = 500, I(0) = 2
0 50 100
Final Size
50 100 150
q u e n c y
γ = 0.2, Pminor = 0.25, Pmajor = 0.75 ODE Final Size=401; Dur=72-78
Near the DFE, the Dynamics of I(t) are Approximated by a Birth/Death Process
pi,j(t) = P(I(t) = j|I(0) = i)
pi,j(t) = P(I(t+ t) = j|I(t) = i)
=
1− (β + γ)it+ o(t), j = i,
o(t), otherwise.
The mean of I(t) is
E(I(t)|I(0) = i) = ie(β−γ)t
Whittle’s Result Follows from a Birth/Death Approximation for the Infectious State I(t)
Gi(u, t) = E(uI(t)|I(0) = i) =
∞∑ j=0
pi,j(t)u j,
∂G1
f(u) = βu2
β + γ +
The function f is the infinitesimal offspring probability generating function.
The Solution pi,0(t) is the Cumulative Distribution for Time to Extinction, Given I(0) = i
Ti = time to extinction, conditioned on extinction
Given I(0) = i,
R0 → 1
Ref: Tritch & Allen, 2018
The PDF for Time to Extinction (Conditioned on Extinction) has a “Fat” Tail as R0 Approaches One.
0 5 10 time
0 5 10 time
0.6 R0=0.25 R0=0.7 R0=0.9
Figure: Probability density functions for R0 close to one, γ = 1.
The PDF from Birth/Death is a Good Fit to the Duration of Minor Outbreaks in SIR Models
0 10 20 30 40 time
0
0.05
0.1
0
0.05
0
0.05
0.1
0.15
0
0.1
0.2
R0=0.9
Figure: Pdfs from 106 sample paths. Overlay plot in red is fT3Pext;
γ = 1, β = R0, N = 1000 with I(0) = 3.
II. MultiStage Epidemic Models
R I S E1 E2
R0 = βδ1δ2
(δ1 + µ1)(δ2 + µ2)(γ + µ)
Stochastic MultiStage Epidemic Models
Pminor =
e2 2 q
i 3, R0 > 1
The probabilities q1, q2, q3 follow from a multitype branching process for E1, E2, I.
MultiGroup Epidemic Models
N and Nk are constant.
We Define the Reproduction Number in Terms of Host Susceptibility or Host Infectivity
Incidence rate for group k:
Sk
N
If transmission depends primarily on either Host Susceptibility or Host Infectivity, then
βkj =
R0 =
N , m = S, I
The Stochastic Two-Group Model is Studied in Terms of the Probability of a Minor or a Major Outbreak
Assume I1(0) = i1 and I2(0) = i2.
Pminor =
Pmajor =
1− qi11 q i2 2 , R0 > 1
• The probabilities q1 and q2 follow from a multitype branching process approximation for I1 and I2.
• Smaller values of qi lead to a greater probability of a major outbreak!
With Host Susceptibility, the Probability of a Major Outbreak is Greater if Initiated by the Group with
Smallest Recovery Rate
Suppose βSk depends on Host Susceptibility and Group 1 has a smaller recovery rate than Group 2,
γ1 < γ2
Then the probability a major outbreak is greater if initiated by Group 1 than by Group 2.
q1 < q2.
R0 .
Ref: Nandi & Allen
The Group with the Smallest Recovery Rate Has a Higher Probability of Initiating a Major Outbreak. Host Susceptibility
γ1 = 0.2 < 1 = γ2
Figure: R0 = 2.4, N1 = 200, N = 1000, R01 = 10, R02 = 0.5
Pminor = (q1) 2 = 0.111, Pminor = (q2)
2 = 0.444
Host Susceptibility
2 (0)=0
Duration
0
0.02
0.04
0.06
0.08
I 1 (0)=0, I
2 (0)=2
Duration
0
0.1
0.2
0.3
0.4
2 = 0.444
After a Major Outbreak is Initiated the Final Size Differs in Each Group
Host Susceptibility Group 1
50 100 150 200
Group 2
Group 1
Group 2
q u e n c y
Figure: R0 = 2.4, Top I1(0) = 2, I2(0) = 0; Bottom: I1(0) = 0,
I2(0) = 2, ODE Final Size1= 186, Final Size2= 388 Ref: Magal et al. 2016
Pminor = (q1) 2 = 0.111, Pminor = (q2)
2 = 0.444
With Host Infectivity, the Probability of a Major Outbreak is Greater for the Group with Largest
Reproduction Number.
Suppose βIj depends on Host Infectivity, and Group 1 has a larger reproduction number than Group 2
R01 = βI1 γ1
= R02
Then the probability a major outbreak is greater if initiated by Group 1 than by Group 2
q1 < q2
R0 .
Ref: Nandi & Allen
When Initiated by the Group with the Largest Reproduction Number, Epidemics Occur More
Frequently and Earlier. Host Infectivity
R01 = 10 > 0.5 = R02
Figure: R0 = 2.4, γ1 = 0.2, γ2 = 1, N1 = 200, N = 1000
Pminor = (q1)2 = 0.102 Pminor = (q2)2 = 0.817
The Duration of a Minor and Major Outbreaks
Host Infectivity
2 (0)=2
Duration
0
0.1
0.2
0.3
0.4
0.5
I 1 (0)=2, I
2 (0)=0
Duration
0
0.02
0.04
0.06
0.08
0.1
2 = 0.817
After a Major Outbreak is Initiated, the Final Size Differs for Each Group
Host Infectivity Group 1
F re
q u
e n
c y
Figure: R0 = 2.4, Top I1(0) = 2, I2(0) = 0; Bottom: I1(0) = 0,
I2(0) = 2, ODE Final Size1= 175, Final Size2= 665
Pminor = (q1) 2 = 0.102 Pminor = (q2)
2 = 0.817
III. We Applied a Two-Group, MultiStage Model to MERS, Middle East Respiratory Syndrome
• MERS identified 2012 from outbreak in Saudi Arabia.
• Case fatality rate ≈ 35%
• Group 1=NonSuperspreaders=NS
• Group 2= SuperSpreaders=SS
We Modeled MERS Assuming High Host Infectivity Determines a Superspreader.
R0 = β1
N1 N
Parameter MERS Baseline Range
β1 transmission rate NS 0.06 (0.04, 0.08)) β2 transmission rate SS 0.6931∗∗ (0.4, 0.8)
α−1 i latent period 6.3 (2-8)
δ−1 i duration of asymp. stage 0.4 (0.1, 2) µji disease induced death rate 0.08 (0.02, 0.14) γi recovery rate 0.075 (0.05, 0.1)
For N1 = N2 and N = 2000,
R0 = 2.36
Initiated by NS Initiated by SS
Increase R0
Increase Prop. SS, R0 = 2.36
Time to Reach a Total of 50 Infected Occurs Earlier if Initiated by SS
R0 = 2.36
IV. Implications for Public Health
• For R0 ≈ 1 the duration of outbreaks increases. Public Health: Reduction of R0 ≈ 1 through preventive measures or intervention increases time until eradication which allows pathogens more time to adapt to their host.
• Transmission depends on host susceptibility and infectivity. Public Health: Reduction of both transmission mechanisms, through vaccination, quarantine, drugs, etc, will have the greatest impact on reducing final size.
• Superspreading events have been associated with recent outbreaks (MERS, SARS, Ebola). Public Health: Identification of superspreading events through early tracking contacts, earlier diagnosis and treatment strategies, and community education are important.
Thank You
• National Science Foundation
