Graph-theoretical%20Models%20of%20the%20Spread%20and%20Control%20of%20Disease%20Fred%20Roberts,%20DIMACS

1
Graph-theoretical Models of the Spread and
Control of DiseaseFred Roberts, DIMACS
smallpox
2
Models of the Spread and Control of Disease
through Social Networks

• Diseases are spread through social networks.
• Contact tracing is an important part of any
  strategy to combat outbreaks of infectious
  diseases, whether naturally occurring or
  resulting from bioterrorist attacks.

3
The Basic Model
Social Network Graph Vertices People Edges
contact State of a Vertex simplest model 1
if infected, 0 if not infected (SI Model) More
complex models SI, SEI, SEIR, etc. S
susceptible, E exposed, I infected, R
recovered (or removed)
4
More About States
Once you are infected, can you be cured? If you
are cured, do you become immune or can you
re-enter the infected state? We can build a
digraph reflecting the possible ways to move from
state to state in the model.
5
The State Diagram for a Smallpox Model
The following diagram is from a Kaplan-Craft-Wein
(2002) model for comparing alternative responses
to a smallpox attack. This has been considered by
the CDC, Dept. of Homeland Security, Dept. of
Health and Human Services, etc.
6
(No Transcript)
7
The Stages

• Row 1 Untraced and in various stages of
  susceptibility or infectiousness.
• Row 2 Traced and in various stages of the queue
  for vaccination.
• Row 3 Unsuccessfully vaccinated and in various
  stages of infectiousness.
• Row 4 Successfully vaccinated dead

8
Aside Related Work

• The model to be presented arose from the study of
  opinion formulation in groups.
• Similar models are used in distributed computing.
• The models we will study are sometimes called
  threshold networks.

9
The Model Moving From State to State
Let si(t) give the state of vertex i at time
t. Simplified Model Two states 0 and 1. Times
are discrete t 0, 1, 2,
10
First Try Majority Processes
Basic Majority Process You change your state at
time t1 if a majority of your neighbors have
the opposite state at time t. (No change in
case of ties) Useful in models of spread of
opinion. Disease interpretation? Cure if
majority of your neighbors are uninfected. Does
this make sense?
11
Majority Processes II
Irreversible Majority Process You change your
state from 0 to 1 at time t1 if a
majority of your neighbors have state 1 at time
t. You never leave state 1. (No change in case
of ties) Disease interpretation? Infected if
sufficiently many of your neighbors are infected.
12
Basic Majority Process
13
(No Transcript)
14
(No Transcript)
15
Irreversible Majority Process
16
(No Transcript)
17
(No Transcript)
18
Aside Distributed Computing
Majority processes are studied in distributed
computing. Goal Eliminate damage caused by
failed processors (vertices) or at least to
restrict their influence. Do this by maintaining
replicated copies of crucial data and, when a
fault occurs, letting a processor change state
if a majority of its neighbors are in a different
state. Other applications of similar ideas in
distributed computing distributed database
management, quorum systems, fault local mending.
19
Second Try Threshold Processes
Basic k-Threshold Process You change your state
at time t1 if at least k of your neighbors
have the opposite state at time t. Disease
interpretation? Cure if sufficiently many of your
neighbors are uninfected. Does this make sense?
20
Threshold Processes II
Irreversible k-Threshold Process You change
your state from 0 to 1 at time t1 if at
least k of your neighbors have state 1 at
time t. You never leave state 1. Disease
interpretation? Infected if sufficiently many of
your neighbors are infected. Special Case k
1 Infected if any of your neighbors is
infected.
21
Basic 2-Threshold Process
22
(No Transcript)
23
(No Transcript)
24
Irreversible 2-Threshold Process
25
(No Transcript)
26
(No Transcript)
27
Complications to Add to Model

• k 1, but you only get infected with a certain
  probability.
• You are automatically cured after you are in the
  infected state for d time periods.
• You become immune from infection (cant re-enter
  state 1) once you enter and leave state 1.
• A public health authority has the ability to
  vaccinate a certain number of vertices, making
  them immune from infection.

28
Periodicity
State vector s(t) (s1(t), s2(t), ,
sn(t)). First example, s(1) s(3) s(5)
, s(0) s(2) s(4) s(6) Second example
s(1) s(2) s(3) ... In all of these
processes, because there is a finite set of
vertices, for any initial state vector s(0),
the state vector will eventually become periodic,
i.e., for some P and T, s(tP) s(t) for
all t gt T. The smallest such P is called the
period.
29
Periodicity II
First example the period is 2. Second example
the period is 1. Both basic and irreversible
threshold processes are special cases of
symmetric synchronous neural networks. Theorem
(Goles and Olivos, Poljak and Sura) For
symmetric, synchronous neural networks, the
period is either 1 or 2.
30
Periodicity III
When period is 1, we call the ultimate state
vector a fixed point. When the fixed point is
the vector s(t) (1,1,,1) or (0,0,,0), we
talk about a final common state. One problem of
interest Given a graph, what subsets S of the
vertices can force one of our processes to a
final common state with entries equal to the
state shared by all the vertices in S in the
initial state?
31
Periodicity IV
Interpretation Given a graph, what subsets S
of the vertices should we plant a disease with so
that ultimately everyone will get it? (s(t) ?
(1,1,,1)) Economic interpretation What set of
people do we place a new product with to
guarantee saturation of the product in the
population? Interpretation Given a graph, what
subsets S of the vertices should we vaccinate
to guarantee that ultimately everyone will end up
without the disease? (s(t) ? 0,0,,0))
32
Conversion Sets
Conversion set Subset S of the vertices that
can force a k-threshold process to a final common
state with entries equal to the state shared by
all the vertices in S in the initial state. (In
other words, if all vertices of S start in same
state x 1 or 0, then the process goes to a
state where all vertices are in state
x.) Irreversible k-conversion set if
irreversible process.
33
1-Conversion Sets
k 1. What are the conversion sets in a basic
1-threshold process?
34
1-Conversion Sets
k 1. The only conversion set in a basic
1-threshold process is the set of all vertices.
For, if any two adjacent vertices have 0 and 1 in
the initial state, then they keep switching
between 0 and 1 forever. What are the
irreversible 1-conversion sets?
35
Irreversible 1-Conversion Sets
k 1. Every single vertex x is an
irreversible 1-conversion set if the graph is
connected. We make it 1 and eventually all
vertices become 1 by following paths from x.
36
Conversion Sets for Odd Cycles
C2p1 2-threshold process. What is a conversion
set?
37
Conversion Sets for Odd Cycles
C2p1. 2-threshold process. Place p1 1s in
alternating positions.
38
(No Transcript)
39
(No Transcript)
40
Conversion Sets for Odd Cycles
We have to be careful where we put the initial
1s. p1 1s do not suffice if they are next to
each other.
41
(No Transcript)
42
(No Transcript)
43
Irreversible Conversion Sets for Odd Cycles
What if we want an irreversible conversion set
under an irreversible 2-threshold process? Same
set of p1 vertices is an irreversible
conversion set. Moreover, everyone gets infected
in one step.
44
Vaccination Strategies
Mathematical models are very helpful in comparing
alternative vaccination strategies. The problem
is especially interesting if we think of
protecting against deliberate infection by a
bioterrorist.
45
Vaccination Strategies
If you didnt know whom a bioterrorist might
infect, what people would you vaccinate to be
sure that a disease doesnt spread very much?
(Vaccinated vertices stay at state 0 regardless
of the state of their neighbors.) Try odd cycles
again. Consider an irreversible 2-threshold
process. Suppose your adversary has enough supply
to infect two individuals. Strategy 1 Mass
vaccination make everyone 0 and immune in
initial state.
46
Vaccination Strategies
In C5, mass vaccination means vaccinate 5
vertices. This obviously works. In practice,
vaccination is only effective with a certain
probability, so results could be different. Can
we do better than mass vaccination? What does
better mean? If vaccine has no cost and is
unlimited and has no side effects, of course we
use mass vaccination.
47
Vaccination Strategies
What if vaccine is in limited supply? Suppose we
only have enough vaccine to vaccinate 2
vertices. Consider two different vaccination
strategies
Vaccination Strategy I
Vaccination Strategy II
48
Vaccination Strategy I Worst Case (Adversary
Infects Two)Two Strategies for Adversary
Adversary Strategy Ia
Adversary Strategy Ib
49
The alternation between your choice of a
defensive strategy and your adversarys choice of
an offensive strategy suggests we consider the
problem from the point of view of game
theory.The Food and Drug Administration is
studying the use of game-theoretic models in the
defense against bioterrorism.
50
Vaccination Strategy I Adversary Strategy Ia
51
Vaccination Strategy I Adversary Strategy Ib
52
Vaccination Strategy II Worst Case (Adversary
Infects Two)Two Strategies for Adversary
Adversary Strategy IIa
Adversary Strategy IIb
53
Vaccination Strategy II Adversary Strategy IIa
54
Vaccination Strategy II Adversary Strategy IIb
55
Conclusions about Strategies I and II

• If you can only vaccinate two individuals
• Vaccination Strategy II never leads to more than
  two infected individuals, while Vaccination
  Strategy I sometimes leads to three infected
  individuals (depending upon strategy used by
  adversary).
• Thus, Vaccination Strategy II is better.

56
Vaccination Strategies

• Stephen Hartke (2004) worked on a different
  problem
• You can only vaccinate one person per time period
  and you can only infect one person per time
  period.
• The vaccinator and infector alternate turns.
• What is a good strategy for the vaccinator?

57
The Mathematics of k-Conversion Sets
k-conversion sets are complex. Consider the
graph K4 x K2.
58
k-Conversion Sets II
Exercise (a). The vertices a, b, c, d, e form a
2-conversion set. (b). However, the vertices
a,b,c,d,e,f do not. Interpretation Immunizing
one more person can be worse! (Planting a
disease with one more person can be worse if you
want to infect everyone.) Note the same does
not hold true for irreversible k-conversion
sets.
59
k-Conversion Sets in Regular Graphs
G is r-regular if every vertex has degree
r. Set of vertices is independent if there are no
edges. Theorem (Dreyer 2000) Let G (V,E)
be a connected r-regular graph and D be a set
of vertices. (a). D is an irreversible
r-conversion set iff V-D is an independent
set. (b). D is an r-conversion set iff V-D
is an independent set and D is not an
independent set.
60
k-Conversion Sets in Regular Graphs II
Corollary (Dreyer 2000) (a). The size of the
smallest irreversible 2- conversion set in Cn
is ceilingn/2. (b). The size of the smallest
2-conversion set in Cn is ceiling(n1)/2.
This result agrees with our observation.
61
k-Conversion Sets in Regular Graphs III
Proof (a). Cn is 2-regular. The largest
independent set has size floorn/2. Thus, the
smallest D so that V-D is independent has
size ceilingn/2. (b). If n is odd, taking
the first, third, , nth vertices around the
cycle gives a set that is not independent and
whose complement is independent. If n is even,
every vertex set of size n/2 with an
independent complement is itself independent, so
an additional vertex is needed.
62
k-Conversion Sets in Graphs of Maximum Degree r
Theorem (Dreyer 2000) Let G (V,E) be a
connected graph with maximum degree r and S be
the set of all vertices of degree lt r. If D is
a set of vertices, then (a). D is an
irreversible r-conversion set iff S?D and V-D
is an independent set. (b). D is an
r-conversion set iff S?D, V-D is an independent
set, and if S ?, then D is not an independent
set.
63
NP-Completeness
Problem k-CONVERSION SET Given a positive
integer d and a graph G, does G have a
k-conversion set of size at most
d? IRREVERSIBLE k-CONVERSION SET
Similar. Theorem (Dreyer 2000) k-CONVERSION
SET and IRREVERSIBLE k-CONVERSION SET are
NP-complete for fixed k gt 2. (Whether or not
they are NP-complete for k 2 remains open.)
64
NP-Completeness II
Proof Reduction from a case of the independent
set problem. Problem INDEPENDENT SET Given a
positive integer m and a graph G, does G
have an independent set of size ? m? Theorem
(Fricke, Hedetniemi, Jacobs) For fixed k?3,
INDEPENDENT SET is NP-complete for the class of
k-regular, non-bipartite graphs. (The problem is
in P for bipartite graphs.) Given
65
NP-Completeness III
Given an instance of INDEPENDENT SET for a
k-regular, non-bipartite graph G Does G have
an independent set of size ? m? Construct an
instance of IRREVERSIBLE k-CONVERSION SET Does
G have an irreversible k-conversion set of size ?
V(G) - m? D is an irreversible k-conversion
set iff V-D is independent. So There is an
irreversible k-conversion set of size ? V(G) -
m iff there is an independent set of size ? m.
66
NP-Completeness IV
Non-bipartiteness is not used here. It is used
in the similar proof that k-CONVERSION SET is
NP-complete. D is a k-conversion set iff D is
independent and V(G) D is independent. The
construction is the same. We need the fact that
if D is not a k-conversion set in G, then either
V(G) D is not independent or both V(G) D and
D are independent. The latter is impossible in a
non-bipartite graph.
67
k-Conversion Sets in Trees
68
k-Conversion Sets in Trees
The simplest case is when every internal vertex
of the tree has degree gt k. Leaf vertex of
degree 1 internal vertex not a
leaf. What is a 2-conversion set here?
69
All leaves have to be in it. This will suffice.
70
(No Transcript)
71
k-Conversion Sets in Trees
So k 2 is easy. What about k gt 2? Also
easy. Proposition (Dreyer 2000) Let T be a
tree and every internal vertex have degree gt k,
where k gt 1. Then the smallest k-conversion set
and the smallest irreversible k-conversion set
have size equal to the number of leaves of the
tree.
72
k-Conversion Sets in Trees
A vertex cover of graph G is a set S of vertices
so that for every edge u,v, either u or v is in
S. If T is a tree, let LT denote the set of
leaves. Proposition (Dreyer 2000) Let T be a
tree with every internal vertex of degree gt k.
Then S is a k-conversion set (an irreversible
k-conversion set) iff S C?LT where C is a
vertex cover of the edges in V-LT.
73
k-Conversion Sets in Trees
What if not every internal vertex has degree gt
k? If there is an internal vertex of degree lt k,
it will have to be in any k-conversion set and
will never change sign. So, to every neighbor,
this vertex v acts like a leaf, and we can break
T into deg(v) subtrees with v a leaf in each. If
every internal vertex has degree ? k, one can
obtain analogous results to those for the gt k
case by looking at maximal connected subsets of
vertices of degree k.
74
k-Conversion Sets in Trees
Dreyer presents an O(n) algorithm for finding the
size of the smallest irreversible 2-conversion
set in a tree of n vertices. The method builds
on the generalization of the theorem
characterizing irreversible k-conversion sets in
the case where every internal vertex has degree ?
k. The algorithm can be readily extended to find
the size of the smallest irreversible
k-conversion set.
75
k-Conversion Sets in Grids
Let G(m,n) be the rectangular grid graph with
m rows and n columns.
G(3,4)
76
Toroidal Grids
The toroidal grid T(m,n) is obtained from the
rectangular grid G(m,n) by adding edges from
the first vertex in each row to the last and from
the first vertex in each column to the
last. Toroidal grids are easier to deal with
than rectangular grids because they form regular
graphs Every vertex has degree 4. Thus, we can
make use of the results about regular graphs.
77
T(3,4)
78
4-Conversion Sets in Toroidal Grids
Theorem (Dreyer 2000) In a toroidal grid T(m,n)
(a). The size of the smallest 4-conversion set
is maxn(ceilingm/2), m(ceilingn/2) m or n
odd mn/2 1 m, n even (b). The size of
the smallest irreversible 4-conversion set is as
above when m or n is odd, and it is mn/2
when m and n are even.

79
Part of the Proof Recall that D is an
irreversible 4-conversion set in a 4-regular
graph iff V-D is independent. V-D
independent means that every edge u,v in G
has u or v in D. In particular, the ith row
must contain at least ceilingn/2 vertices in D
and the ith column at least ceilingm/2 vertices
in D (alternating starting with the end vertex of
the row or column). We must cover all rows and
all columns, and so need at least
maxn(ceilingm/2), m(ceilingn/2) vertices
in an irreversible 4-conversion set.
80
4-Conversion Sets for Rectangular Grids
More complicated methods give Theorem (Dreyer
2000) The size of the smallest 4-conversion set
and smallest irreversible 4-conversion set in a
grid graph G(m,n) is 2m 2n - 4
floor(m-2)(n-2)/2
81
4-Conversion Sets for Rectangular Grids
Consider G(3,3) 2m 2n - 4
floor(m-2)(n-2)/2 8. What is a smallest
4-conversion set and why 8?
82
4-Conversion Sets for Rectangular Grids
Consider G(3,3) 2m 2n - 4
floor(m-2)(n-2)/2 8. What is a smallest
4-conversion set and why 8? All boundary
vertices have degree lt 4 and so must be included
in any 4-conversion set. They give a conversion
set.
83
2-Conversion Sets for Grids
Let Ci,k(G) be the smallest irreversible k-convers
ion set in graph G. Theorem (Flocchini, Lodi,
Luccio, Pagli, and Santoro) Ci,2G(m,n)
ceiling(mn/2) Ci,2T(m,n)
ceiling(mn/2) 1.
84
3-Conversion Sets for Grids
For 3-conversion sets, the best we have are
bounds Theorem (Flocchini, Lodi, Luccio, Pagli,
and Santoro) (m-1)(n-1)1/3 ?
Ci,3G(m,n) ? (m-1)(n-1)1/3 3m2n-3/4
5 ceiling(mn1/2) ? Ci,3T(m,n) ?
ceiling(mn/2 m/6 2/3)
85
2- and 3-Conversion Sets for Small Grids
For the special case of 2xn and 3xn grids, Dreyer
has more precise results.
86
Bounds on the Size of the Smallest Conversion Sets
In general, it is difficult to get exact values
for the size of the smallest k-conversion and
irreversible k-conversion set in a graph. So,
what about bounds? Sample result Theorem
(Dreyer, 2000) If G is an r-regular graph with
n vertices, then Ci,k(G) ? (1 r/2k)n for k ? r
? 2k.
87
Bounds on the Size of the Smallest Conversion Sets
Conjecture (Dreyer) For a graph G on n vertices,
the size of the smallest k-conversion set is ?
n/k, and this bound is asymptotically sharp.
88
More Realistic Models

• Many oversimplifications in our models. For
  instance
• What if you stay infected only a certain number
  of days?
• What if you are not necessarily infective for the
  first few days you are sick?
• What if your threshold k for changes from 0 to 1
  changes depending upon how long you have been
  uninfected?

89
More Realistic Models
Consider an irreversible process in which you
stay in the infected state (state 1) for d time
periods after entering it and then go back to the
uninfected state (state 0). Consider a
k-threshold process in which we vaccinate a
person in state 0 once k-1 neighbors are infected
(in state 1). Etc. -- let your imagination roam
free ...
90
More Realistic Models

• Our models are deterministic. How do
  probabilities enter?
• What if you only get infected with a certain
  probability if you meet an infected person?
• What if vaccines only work with a certain
  probability?
• What if the amount of time you remain infective
  exhibits a probability distribution?

91
Alternative Model with Probabilities
Consider an irreversible 1-threshold process in
which you stay infected for d time periods and
then enter the uninfected state. Assume that you
get infected with probability p if at least one
of your neighbors is infected. What is the
probability that an epidemic will end with no one
infected?
92
The Case d 2, p 1/2
Consider the following initial state
93
The Case d 2, p 1/2
With probability 1/2, vertex a does not get
infected at time 1. Similarly for vertex
b. Thus, with probability 1/4, we stay in the
same states at time 1.
94
The Case d 2, p 1/2
Suppose vertices are still in same states at time
1 as they were at time 0. With probability 1/2,
vertex a does not get infected at time 2.
Similarly for vertex b. Also after time 1,
vertices c and d have been infected for two
time periods and thus enter the uninfected
state. Thus, with probability 1/4, we get to the
following state at time 2
95
(No Transcript)
96
The Case d 2, p 1/2
Thus, with probability 1/4 x 1/4 1/16, we
enter this state with no one infected at time
2. However, we might enter this state at a later
time. It is not hard to show (using the theory
of finite Markov chains) that we will end in
state (0,0,0,0). (This is the only absorbing
state in an absorbing Markov chain.). Thus with
probability 1 we will eventually kill the disease
off entirely.
97
The Case d 2, p 1/2
Is this realistic? What might we do to modify
the model to make it more realistic?
98
How do we Analyze this or More Complex Models for
Graphs?

• We can use the model to do what if experiments.

99
What-if Experiments

• What if we adopt a particular vaccination
  strategy?
• What happens if we try different plans for
  quarantining infectious individuals?

100

• There is much more analysis of a similar nature
  that can be done with graph-theoretical models.
  Let your imagination run free!