Net sci13

Optimal Resource Allocation to Control EpidemicOutbreaks in Arbitrary Networks

Victor M. Preciado, Michael Zargham, Chinwendu Enyioha,Ali Jadbabaie, and George Pappas

Dept. of Electrical and Systems EngineeringUniversity of Pennsylvania

January 3, 2014

Victor M. Preciado Optimal Resource Allocation to Control Epidemic Outbreaks in Arbitrary Networks 1

Problem Description

Consider the SIS viral spreading process in a network of individuals:

I Spreading rate β and recovery rate δ

Objective: Control an epidemic outbreak by distributing resourcesthroughout the network of contacts:

I Antidote allocation in node i increases its recovery rate, δiI 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?


Related Work

Applications:

I Epidemiology (Bailey, 1975; Anderson and May, 1991),

I Computer viruses (Garetto et al., 2003),

I Viral marketing (Leskovec et al., 2007).

Related work:

I Epidemics in Networks (Newman, 2002; Pastor-Satorras andVespignani, 2002; Wang et al., 2003; Ganesh et al., 2005; VanMieghem et al., 2009; etc.)

I Epidemics Control (Wan et al., 2008; Chen et al., 2008; Chung etal., 2009; Borgs et al., 2011; Gourdin et al., 2011; Darabi andScoglio, 2012; etc.)

We solve exactly –without relaxations or heuristics–the optimal resourceallocation problem in weighted and directed networks of nonidenticalagents in polynomial time.


In this talk...

Outline of the presentation:

I Dynamic Model: Heterogeneous SIS epidemic model

I Epidemic Control: Spectral condition for epidemic stability

I Resource Allocation: We propose a convex framework to allocatetwo types of containment resources:

I Preventive resources to protect nodes against the spreading (such asvaccines in a viral infection)

I Corrective resources to neutralize the spreading after it has reached anode (such as antidotes)

I Simulations: Numerical results in the (weighted and directed) airtransportation network


Some Notation

Consider a (possibly weighted and/or directed) graph G = (V, E) with nnodes and m edges

I A is the (possibly weighted and/or non-symmetric) adjacency matrix

I λ1, λ2, . . . , λn are the eigenvalues of A

I 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

I ρ (A) > 0 is a simple eigenvalue of A

I Au = ρ (A)u, for some u ∈ Rn++

I ρ (M) = inf{λ ∈ R : Mu ≤ λu for u ∈ Rn

++

}


Spreading Processes in Networks

Consider the N-intertwined SIS model [Van Mieghem et al., 09]:

I pi (t) denotes the probability of node i being infected at time t ≥ 0

I βi > 0 denotes the (node-dependent) infection rate

I δi > 0 denotes the (node-dependent) curing rate

Epidemic evolution: After a mean-field approximation of the stochasticSIS dynamics, the infection probabilities evolve as follows:

dpi (t)

dt= (1− pi (t))βi

n∑j=1

aijpj (t)− δipi (t) (1)

Proposition (Stability for low densities of infection)

Define B = diag(βi ) and D = diag(δi ). An initial infection p (0) willconverge to zero exponentially fast if

maxi<{λi (BA− D)} ≤ −ε, for some ε > 0.


Network Protection Problem

Protection resources: Two types are available:I Preventive resources able to defend nodes against the spreading

(such as vaccines in a viral infection process)I Corrective resources able to neutralize the spreading after it has

reached a node (such as antidotes)

We assume that both preventive and corrective resources havenode-dependent associated cost functions, fi (βi ) and gi (δi ):


Network Protection Problem

Budget-constrained allocation problem:Given the following elements:

I A (positively) weighted, directed network with adjacency matrix AGI Protection cost functions fi (βi ) and gi (δi )

I Bounds on the infection and recovery rates 0 < βi≤ βi ≤ βi and

0 < δi ≤ δi ≤ δi , i = 1, . . . , n,

I A total budget C

Find the cost-optimal distribution of vaccines and antidotes to maximizethe exponential decay rate ε.


Optimal Resource Allocation

Mathematical formulation:

maximizeε,{βi ,δi}ni=1

ε (2)

subject to maxi< [λi (diag (βi )AG − diag (δi ))] ≤ −ε, (3)

n∑i=1

fi (βi ) + gi (δi ) ≤ C , (4)

βi≤ βi ≤ βi , (5)

δi ≤ δi ≤ δi , i = 1, . . . , n, (6)

In what follows, propose a convex formulation to solve both thebudget-constrained allocation problem in weighted, directed networksusing geometric programming (GP)


Geometric Programming

Geometric programs are quasiconvex optimization problems that can beeasily transformed into convex programs.

I Let x1, . . . , xn > 0 denote n decision variables

I Define x , (x1, . . . , xn) ∈ Rn++

I In the context of GP, a monomial h(x) is defined ash(x) , dxa11 xa22 . . . xann with d > 0 and ai ∈ R

I A posynomial function q(x) is defined as a sum of monomials, i.e.,

q(x) ,∑K

k=1 ckxa1k1 xa2k2 . . . xankn , 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


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, andequal 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:

minimizeλ,{ui}ni=1,x

λ (8)

subject to

∑nj=1 Mij (x) uj

λui≤ 1, i = 1, . . . , n, (9)

fi (x) ≤ 1, i = 1, . . . ,m. (10)


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 (β∗i )

and gi(

∆ + 1− δ̂∗i)

, where ∆ , max{δi}ni=1

and β∗i ,δ̂∗i are the optimal

solution for βi and δ̂i in the following GP:

minimizeλ,{ui ,βi ,δ̂i ,ti}n

i=1

λ (11)

subject toβi∑n

j=1 Aijuj + δ̂iui

λui≤ 1, (12)

n∑k=1

fk (βk) + gk (tk) ≤ C , (13)(ti + δ̂i

)/(∆ + 1

)≤ 1, (14)

∆ + 1− δi ≤ δ̂i ≤ ∆ + 1− δi , (15)

βi≤ βi ≤ βi , i = 1, . . . , n. (16)


Numerical Results for Vaccine Allocation

Cost-optimal protection strategy against epidemic outbreaks thatpropagate 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 onprotection (in abscissas) is made on that node.


Numerical Results for Vaccine Allocation

Cost-optimal protection strategy against epidemic outbreaks thatpropagate through the air transportation network:

Figure : Results from the budget-constrained allocation problem. From left toright, we have (a) a scatter plot with the investment on correction versusprevention per node, (b) a scatter plot with the investment on protection pernode and the in-degrees, and (c) a scatter plot with the investment onprotection per node versus PageRank centralities.


Extensions not covered in this talk...

I Rate-constrained allocation problem: What is the minimum budgetrequired to achieved a desired exponential decay rate of theinfection? [PZEJP,14]

I Design of edge weights: What is we include the possibility ofreducing the contact rate associated to edges? [PZS13]

I Epidemics in metapopulations: Nodes are cities with internalepidemic dynamics and edges are roads with adjustable traffic [PZ13]

I Decentralized implementation: Ongoing work...


Conclusions

I We have analyzed the problem of allocating protection resources(antidotes and vaccines) in a network to control an epidemicoutbreak

I We have writen this control problem as an eigenvalue designproblem under cost constraints

I We have casted this design problems into a geometric program,under certain conditions on the cost functions

I This GP can be efficiently solved using standard convex optimizationtools

I We have illustrated our results in an air traffic network


References (in ArXiv.org)

I Preciado et al., “Optimal Resource Allocation for Network Protection: AGeometric Programming Approach”

I Preciado et al., “Traffic Control for Network Protection Against SpreadingProcesses”

I Preciado and Zargham, “Traffic Optimization to Control EpidemicOutbreaks in Metapopulation Models”

I Enyioha et al., “Epidemic Control via Geometric Programming,”submitted to ACC 2014

I Preciado et al., “A Convex Framework for Optimal Investment on DiseaseAwareness in Social Networks”


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