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Accounting for Fluctuations in Stochastic SIRS Model on Networks
1
1,2
Daniel T. Citron , Christopher R. Myers
1
2
Laboratory of Atomic and Solid State Physics, Cornell University
Institute of Biotechnology, Cornell University
Introduction
Moment Closure
Results
Here we present our analysis of the stochastic Susceptible-InfectedRecovered-Susceptible (SIRS) model of infectious disease dynamics on heterogeneous networks. We perform a moment closure analysis
to obtain approximate analytical predictions for the magnitude
of fluctuations in the endemic state. We use the heterogeneous
mean field (HMF) to approximate dynamics on a network with degree
heterogeneity. This can be used to show how stochastic fluctuations
and spontaneous extinction depend on the size and heterogeneity of the full network. We present results from our simulations
to demonstrate the accuracy and utility of our predictions.
We obtain fluctuation magnitudes by solving for the quasi-stationary
distribution (QSD) of the number of Infecteds in the population. We
use Gaussian moment-closure to approximately solve for the QSD.
1. Derive Kolmogorov forward equation (KFE) for the model.
2. Condition on no extinction (> 0 infected individuals, not reaching
the absorbing state) to obtain KFE for QSD [3].
3. Obtain PDE for the Probability Generating Function for the QSD.
4. Assume rate of extinction is small, ignore nonlinear sink term [4].
5. Obtain PDE for the Cumulant Generating Function K.
6. Assume Gaussian form of QSD, quadratic K:
1 2 2 1 2 2
K (θ, φ, t) ≈ µx θ + µy φ + σxy θφ + σx θ + σy φ . . .
2
2
7. Collect terms in series expansion to derive coupled ODEs for all first
and second moments of the QSD (means {µ} and variances {σ 2 }).
8. Numerically integrate ODEs for {µ} and {σ 2 } as functions of time.
9. Can look for stationary behavior of QSD, {µ}, and {σ 2 } .
10. Knowing the variances {σ 2 } characterizes the magnitude of
stochastic fluctuations. With this technique, we can account for both
finite population size effects and node degree heterogeneity.
The network can support endemic disease (µy > 0) when R0 ≡ βhki/γ
and ρ/γ are sufficiently large. We plot the signal to noise ratio
(SNR) of the number of infected individuals (µy /σy ) to show persistence of endemic disease. Network properties also affect endemic
disease:
Θk ≡
X
0
Waning
P (k |k) I ,
k0
Recovery
Force of infection between degree classes k & k
k0
Contact: [email protected]
20
30
time (t)
40
500
0
20
18
16
14
12
10
8
6
4
2
6
5
4
3
2
SNR µ/σ
7
ρ/γ
R0 =1.1, ρ/γ = .25
2.0 Simulation, N = 1000
1.8
1.6
1.4
1.2
1.0
0.8
0.6
0.4
10-3 10-2 10-1 100 101 102 103
SNR µ/σ
R0
SNR µ/σ
8
R0
R0
550
9
SNR µ/σ
(Y(t) = n)
50
450
6.0
5.6
5.2
4.8
4.4
4.0
3.6
3.2
2.8
2.4
2.0
ρ/γ
Degree Heterogeneity Predicted and simulated SNR µy /σy for a
graph with a skewed degree distribution (hki2 /hk 2 i = .44):
20
40
60
Number infected n
80
100
This material is based upon work supported by the National Science Foundation
Graduate Research Fellowship under Grant No. DGE-1144153. Any opinion, findings, and conclusions or recommendations expressed in this material are those of the
authors and do not necessarily reflect the views of the National Science Foundation.
This work is also supported by the Science & Technology Directorate, Department
of Homeland Security via interagency agreement no. HSHQDC-10-X-00138.
2.0 Moment Closure, N = 1000
1.8
1.6
1.4
1.2
1.0
0.8
0.6
0.4
10-3 10-2 10-1 100 101 102 103
ρ/γ
22
20
18
16
14
12
10
8
6
4
2
2.0 Simulation, N = 1000
1.8
1.6
1.4
1.2
1.0
0.8
0.6
0.4
10-3 10-2 10-1 100 101 102 103
22
20
18
16
14
12
10
8
6
4
2
SNR µ/σ
10
400
2.0 Moment Closure, N = 1000
1.8
1.6
1.4
1.2
1.0
0.8
0.6
0.4
10-3 10-2 10-1 100 101 102 103
Simulation, N = 100
2.0
1.8
1.6
1.4
1.2
1.0
0.8
0.6
0.4
10-3 10-2 10-1 100 101 102 103
ρ/γ
R0
0.00
350
[1] Modeling Infectious Diseases in Humans and Animals (Princeton University
Press, 2008).
[2] Dynamical Processes on Complex Networks (Cambridge University Press, 2008).
[3] I. Nåsell, J. of the Royal Stat. Soc.: Series B61, 309 (1999).
[4] D. Clancy and S. T. Mendy, J. of Math. Bio. 61, 527 (2009).
dIk
= βSk kΘk − γIk
| {z }
|{z}
dt
Infection
R0 =1.2, ρ/γ = .25
0.1
300
References and Acknowledgements
• Assume no degree correlations in network
• Partition graph into degree classes, classes couple together
• Account for degree heterogeneity in the mean field
Infection
0.2
R0 =1.75, ρ/γ =10.
Examples of stochastic trajectories Accurate for small rate of extincin grey, µy and µy ± σ in black, tion (y-intercept); QSD peaked
moment closure prediction in red. away from 0 [3, 4].
Heterogeneous Mean Field Theory
dSk
= −βSk kΘk + ρ (1 − Sk − Ik ),
| {z } |
{z
}
dt
R0 =1.75, ρ/γ =10
0.3
0.10
0.08
0.06
0.04
0.02
0.00250
0.20
0.15
0.10
0.05
0.000
R0
On Networks
• Each node represents a single individual.
• Each node is in one of three states (S, I, R)
• Network defines contacts between individuals
• Network structure accounts for contact heterogeneity [2]
0.4
(Y(t) = n)
Fraction of infected nodes
Similarity to other models
• SIRS −→ SIR as ρ −→ 0 — Disease dies out after initial outbreak
• SIRS −→ SIS as ρ −→ ∞ — Persistent endemic disease
• When does endemic disease persist in the SIRS model?
0.5
QSD Examples
4.0
3.8
3.6
3.4
3.2
3.0
2.8
2.6
2.4
2.2
2.0
ρ/γ
Quantifying Stochastic Fluctuations
Example Trajectories
2.0 Moment Closure, N = 100
1.8
1.6
1.4
1.2
1.0
0.8
0.6
0.4
10-3 10-2 10-1 100 101 102 103
SNR µ/σ
Waning Immunity
Infection spreads through contact; infected hosts develop immunity;
recovered individuals lose immunity over time [1]
R0
SIRS model overview
Population Size Predicted and simulated SNR µy /σy for G(n, p)-type
graphs of two different sizes:
ρ/γ
Future Work
• Measure how extinction rates are affected by both finite size and
degree heterogeneity in graphs with simulations.
• Examine role of low-degree nodes: moment closure predicts they decrease fluctuations in graphs with skewed degree distributions.
• Find a perturbative correction to the initial approximation that accommodates cases where rate of extinction is not small.
• Are two degree classes sufficient to capture network heterogeneity?