This document presents a method for optimally allocating resources to control epidemic outbreaks in networks. It formulates the problem as maximizing the exponential decay rate of infections by distributing preventive and corrective resources subject to a budget constraint. The problem is cast as a geometric program that can be solved efficiently. Numerical results applying the approach to the air transportation network are presented.

1. Optimal Resource Allocation to Control Epidemic
Outbreaks in Arbitrary Networks
Victor M. Preciado, Michael Zargham, Chinwendu Enyioha,
Ali Jadbabaie, and George Pappas
Dept. of Electrical and Systems Engineering
University of Pennsylvania
January 3, 2014
Victor M. Preciado
Optimal Resource Allocation to Control Epidemic Outbreaks in Arbitrary Networks
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2. Problem Description
Consider the SIS viral spreading process in a network of individuals:
Spreading rate β and recovery rate δ
Objective: Control an epidemic outbreak by distributing resources
throughout the network of contacts:
Antidote allocation in node i increases its recovery rate, δi
Vaccinating a node reduces the rate of spreading through its
incoming links, βi
How should we distribute our resources in the most cost-efficient manner?
Victor M. Preciado
Optimal Resource Allocation to Control Epidemic Outbreaks in Arbitrary Networks
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3. Related Work
Applications:
Epidemiology (Bailey, 1975; Anderson and May, 1991),
Computer viruses (Garetto et al., 2003),
Viral marketing (Leskovec et al., 2007).
Related work:
Epidemics in Networks (Newman, 2002; Pastor-Satorras and
Vespignani, 2002; Wang et al., 2003; Ganesh et al., 2005; Van
Mieghem et al., 2009; etc.)
Epidemics Control (Wan et al., 2008; Chen et al., 2008; Chung et
al., 2009; Borgs et al., 2011; Gourdin et al., 2011; Darabi and
Scoglio, 2012; etc.)
We solve exactly –without relaxations or heuristics–the optimal resource
allocation problem in weighted and directed networks of nonidentical
agents in polynomial time.
Victor M. Preciado
Optimal Resource Allocation to Control Epidemic Outbreaks in Arbitrary Networks
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4. In this talk...
Outline of the presentation:
Dynamic Model: Heterogeneous SIS epidemic model
Epidemic Control: Spectral condition for epidemic stability
Resource Allocation: We propose a convex framework to allocate
two types of containment resources:
Preventive resources to protect nodes against the spreading (such as
vaccines in a viral infection)
Corrective resources to neutralize the spreading after it has reached a
node (such as antidotes)
Simulations: Numerical results in the (weighted and directed) air
transportation network
Victor M. Preciado
Optimal Resource Allocation to Control Epidemic Outbreaks in Arbitrary Networks
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5. Some Notation
Consider a (possibly weighted and/or directed) graph G = (V, E) with n
nodes and m edges
A is the (possibly weighted and/or non-symmetric) adjacency matrix
λ1 , λ2 , . . . , λn are the eigenvalues of A
The spectral radius of A is defined as ρ(A) = maxi (|λi |)
Lemma (Perron-Frobenius Lemma)
If G is strongly connected and positively weighted, then
ρ (A) > 0 is a simple eigenvalue of A
Au = ρ (A) u, for some u ∈ Rn
++
ρ (M) = inf λ ∈ R : Mu ≤ λu for u ∈ Rn
++
Victor M. Preciado
Optimal Resource Allocation to Control Epidemic Outbreaks in Arbitrary Networks
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6. Spreading Processes in Networks
Consider the N-intertwined SIS model [Van Mieghem et al., 09]:
pi (t) denotes the probability of node i being infected at time t ≥ 0
βi > 0 denotes the (node-dependent) infection rate
δi > 0 denotes the (node-dependent) curing rate
Epidemic evolution: After a mean-field approximation of the stochastic
SIS dynamics, the infection probabilities evolve as follows:
dpi (t)
= (1 − pi (t)) βi
dt
n
aij pj (t) − δi pi (t)
(1)
j=1
Proposition (Stability for low densities of infection)
Define B = diag (βi ) and D = diag (δi ). An initial infection p (0) will
converge to zero exponentially fast if
max {λi (BA − D)} ≤ −ε, for some ε > 0.
i
Victor M. Preciado
Optimal Resource Allocation to Control Epidemic Outbreaks in Arbitrary Networks
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7. Network Protection Problem
Protection resources: Two types are available:
Preventive resources able to defend nodes against the spreading
(such as vaccines in a viral infection process)
Corrective resources able to neutralize the spreading after it has
reached a node (such as antidotes)
We assume that both preventive and corrective resources have
node-dependent associated cost functions, fi (βi ) and gi (δi ):
Victor M. Preciado
Optimal Resource Allocation to Control Epidemic Outbreaks in Arbitrary Networks
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8. Network Protection Problem
Budget-constrained allocation problem:
Given the following elements:
A (positively) weighted, directed network with adjacency matrix AG
Protection cost functions fi (βi ) and gi (δi )
Bounds on the infection and recovery rates 0 < β i ≤ βi ≤ β i and
0 < δ i ≤ δi ≤ δ i , i = 1, . . . , n,
A total budget C
Find the cost-optimal distribution of vaccines and antidotes to maximize
the exponential decay rate ε.
Victor M. Preciado
Optimal Resource Allocation to Control Epidemic Outbreaks in Arbitrary Networks
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9. Optimal Resource Allocation
Mathematical formulation:
maximize ε
n
ε,{βi ,δi }
(2)
i=1
subject to max
i
n
[λi (diag (βi ) AG − diag (δi ))] ≤ −ε,
fi (βi ) + gi (δi ) ≤ C ,
(3)
(4)
i=1
β i ≤ βi ≤ β i ,
(5)
δ i ≤ δi ≤ δ i , i = 1, . . . , n,
(6)
In what follows, propose a convex formulation to solve both the
budget-constrained allocation problem in weighted, directed networks
using geometric programming (GP)
Victor M. Preciado
Optimal Resource Allocation to Control Epidemic Outbreaks in Arbitrary Networks
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10. Geometric Programming
Geometric programs are quasiconvex optimization problems that can be
easily transformed into convex programs.
Let x1 , . . . , xn > 0 denote n decision variables
Define x
(x1 , . . . , xn ) ∈ Rn
++
In the context of GP, a monomial h(x) is defined as
a a
a
h(x) dx1 1 x2 2 . . . xn n with d > 0 and ai ∈ R
A posynomial function q(x) is defined as a sum of monomials, i.e.,
K
a1k a2k
ank
q(x)
k=1 ck x1 x2 . . . xn , where ck > 0.
A Geometric Program is an optimization problem of the form:
minimize f (x)
(7)
subject to qi (x) ≤ 1, i = 1, ..., m,
hi (x) = 1, i = 1, ..., p,
where qi and f are posynomial functions, hi are monomials
Victor M. Preciado
Optimal Resource Allocation to Control Epidemic Outbreaks in Arbitrary Networks
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11. A Convex Framework for Optimal Allocation
From Perron-Frobenius, we have the following:
Corollary
Let G be a strongly connected digraph with nonnegative weights. Then,
its eigenvalue with the largest real part, λ1 (AG ), is real, simple, and
equal to the spectral radius ρ (AG ) > 0.
Proposition
Consider the adjacency matrix of a strongly connected digraph M (x)
with entries being either 0 or posynomials with domain x ∈ S ⊆ Rk ,
++
where S =
m
i=1
x ∈ Rk : fi (x) ≤ 1 , fi being posynomials. Then, we
++
can minimize λ1 (M (x)) for x ∈ S solving the following GP:
minimize λ
n
(8)
λ,{ui }i=1 ,x
subject to
Victor M. Preciado
n
j=1
Mij (x) uj
≤ 1, i = 1, . . . , n,
λui
fi (x) ≤ 1, i = 1, . . . , m.
Optimal Resource Allocation to Control Epidemic Outbreaks in Arbitrary Networks
(9)
(10)
11

12. Solution to the Budget-Constrained Allocation
Main Result: Assuming that the cost functions fi and gi are posynomials,
the optimal investment on vaccines and antidotes for node vi are fi (βi∗ )
and gi ∆ + 1 − δi∗ , where ∆
max δ i
n
i=1
and βi∗ ,δi∗ are the optimal
solution for βi and δi in the following GP:
minimize n λ
(11)
λ,{ui ,βi ,δi ,ti }
i=1
βi
n
j=1
Aij uj + δi ui
≤ 1,
(12)
fk (βk ) + gk (tk ) ≤ C ,
subject to
(13)
λui
n
k=1
ti + δi
∆ + 1 ≤ 1,
(14)
∆ + 1 − δ i ≤ δi ≤ ∆ + 1 − δ i ,
β i ≤ βi ≤ β i , i = 1, . . . , n.
Victor M. Preciado
(15)
(16)
Optimal Resource Allocation to Control Epidemic Outbreaks in Arbitrary Networks
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13. Numerical Results for Vaccine Allocation
Cost-optimal protection strategy against epidemic outbreaks that
propagate through the air transportation network:
Figure : Infection rate (in red, and multiplied by 20, to improve visualization)
and recovery rate (in blue) achieved at node vi after an investment on
protection (in abscissas) is made on that node.
Victor M. Preciado
Optimal Resource Allocation to Control Epidemic Outbreaks in Arbitrary Networks
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14. Numerical Results for Vaccine Allocation
Cost-optimal protection strategy against epidemic outbreaks that
propagate through the air transportation network:
Figure : Results from the budget-constrained allocation problem. From left to
right, we have (a) a scatter plot with the investment on correction versus
prevention per node, (b) a scatter plot with the investment on protection per
node and the in-degrees, and (c) a scatter plot with the investment on
protection per node versus PageRank centralities.
Victor M. Preciado
Optimal Resource Allocation to Control Epidemic Outbreaks in Arbitrary Networks
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15. Extensions not covered in this talk...
Rate-constrained allocation problem: What is the minimum budget
required to achieved a desired exponential decay rate of the
infection? [PZEJP,14]
Design of edge weights: What is we include the possibility of
reducing the contact rate associated to edges? [PZS13]
Epidemics in metapopulations: Nodes are cities with internal
epidemic dynamics and edges are roads with adjustable traffic [PZ13]
Decentralized implementation: Ongoing work...
Victor M. Preciado
Optimal Resource Allocation to Control Epidemic Outbreaks in Arbitrary Networks
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16. Conclusions
We have analyzed the problem of allocating protection resources
(antidotes and vaccines) in a network to control an epidemic
outbreak
We have writen this control problem as an eigenvalue design
problem under cost constraints
We have casted this design problems into a geometric program,
under certain conditions on the cost functions
This GP can be efficiently solved using standard convex optimization
tools
We have illustrated our results in an air traffic network
Victor M. Preciado
Optimal Resource Allocation to Control Epidemic Outbreaks in Arbitrary Networks
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17. References (in ArXiv.org)
Preciado et al., “Optimal Resource Allocation for Network Protection: A
Geometric Programming Approach”
Preciado et al., “Traffic Control for Network Protection Against Spreading
Processes”
Preciado and Zargham, “Traffic Optimization to Control Epidemic
Outbreaks in Metapopulation Models”
Enyioha et al., “Epidemic Control via Geometric Programming,”
submitted to ACC 2014
Preciado et al., “A Convex Framework for Optimal Investment on Disease
Awareness in Social Networks”
Victor M. Preciado
Optimal Resource Allocation to Control Epidemic Outbreaks in Arbitrary Networks
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