Modeling of Disease Spreading on Trees
Daniel Guo
Mentor: Professor Partha Dey
PRIMES Conference
May 16, 2015
Daniel Guo (PRIMES Conference)
Modeling of Disease Spreading on Trees
May 16, 2015
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Introduction and Definitions
1
Introduction and Definitions
2
The Problem
3
Results and Continuation
4
References
Daniel Guo (PRIMES Conference)
Modeling of Disease Spreading on Trees
May 16, 2015
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Introduction and Definitions
Motivation
Motivation:
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Modeling of Disease Spreading on Trees
May 16, 2015
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Introduction and Definitions
Motivation
Motivation:
Diseases
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Modeling of Disease Spreading on Trees
May 16, 2015
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Introduction and Definitions
Motivation
Motivation:
Diseases
Spread of “information” (rumors, etc.)
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Modeling of Disease Spreading on Trees
May 16, 2015
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Introduction and Definitions
Motivation
Motivation:
Diseases
Spread of “information” (rumors, etc.)
→ the motivation for the question
Daniel Guo (PRIMES Conference)
Modeling of Disease Spreading on Trees
May 16, 2015
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Introduction and Definitions
Graphs and Trees
Definition
Graph: A simple graph (V , E ) consists of a set representing vertices, V,
and a set of unordered pairs of elements of V representing edges, E .
Daniel Guo (PRIMES Conference)
Modeling of Disease Spreading on Trees
May 16, 2015
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Introduction and Definitions
Graphs and Trees
Definition
Graph: A simple graph (V , E ) consists of a set representing vertices, V,
and a set of unordered pairs of elements of V representing edges, E .
Daniel Guo (PRIMES Conference)
Modeling of Disease Spreading on Trees
May 16, 2015
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Introduction and Definitions
Graphs and Trees
Definition
Tree: A tree is a graph that is connected and has no loops.
Daniel Guo (PRIMES Conference)
Modeling of Disease Spreading on Trees
May 16, 2015
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Introduction and Definitions
Graphs and Trees
Definition
Tree: A tree is a graph that is connected and has no loops.
This implies that
Daniel Guo (PRIMES Conference)
Modeling of Disease Spreading on Trees
May 16, 2015
5 / 22
Introduction and Definitions
Graphs and Trees
Definition
Tree: A tree is a graph that is connected and has no loops.
This implies that
Lemma
Tree: A tree is a graph with |V | = |E | + 1 and no loops.
Daniel Guo (PRIMES Conference)
Modeling of Disease Spreading on Trees
May 16, 2015
5 / 22
Introduction and Definitions
Graphs and Trees
Definition
Tree: A tree is a graph that is connected and has no loops.
This implies that
Lemma
Tree: A tree is a graph with |V | = |E | + 1 and no loops.
Daniel Guo (PRIMES Conference)
Modeling of Disease Spreading on Trees
May 16, 2015
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Introduction and Definitions
More Definitions
Definition
Binary Tree: A binary tree is a tree such that for each vertex V , there are
at most 2 children.
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Modeling of Disease Spreading on Trees
May 16, 2015
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Introduction and Definitions
More Definitions
Definition
Binary Tree: A binary tree is a tree such that for each vertex V , there are
at most 2 children.
Definition
Perfect Binary Tree: A perfect binary tree is a binary tree with 2N − 1
vertices such that the last level is completely full.
Daniel Guo (PRIMES Conference)
Modeling of Disease Spreading on Trees
May 16, 2015
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Introduction and Definitions
More Definitions
Definition
Binary Tree: A binary tree is a tree such that for each vertex V , there are
at most 2 children.
Definition
Perfect Binary Tree: A perfect binary tree is a binary tree with 2N − 1
vertices such that the last level is completely full.
Note that such a tree is unique, not including labeling or directed edges.
Daniel Guo (PRIMES Conference)
Modeling of Disease Spreading on Trees
May 16, 2015
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Introduction and Definitions
Perfect Binary Tree
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Modeling of Disease Spreading on Trees
May 16, 2015
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The Problem
1
Introduction and Definitions
2
The Problem
3
Results and Continuation
4
References
Daniel Guo (PRIMES Conference)
Modeling of Disease Spreading on Trees
May 16, 2015
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The Problem
Depiction of the Problem
Given any tree, nodes = people
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Modeling of Disease Spreading on Trees
May 16, 2015
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The Problem
Depiction of the Problem
Given any tree, nodes = people
Nodes are represented with a Boolean variable for infection
Daniel Guo (PRIMES Conference)
Modeling of Disease Spreading on Trees
May 16, 2015
9 / 22
The Problem
Depiction of the Problem
Given any tree, nodes = people
Nodes are represented with a Boolean variable for infection
Infection can jump from any initial node to any other node in the
sub-tree that it is the root of
Daniel Guo (PRIMES Conference)
Modeling of Disease Spreading on Trees
May 16, 2015
9 / 22
The Problem
Depiction of the Problem
Given any tree, nodes = people
Nodes are represented with a Boolean variable for infection
Infection can jump from any initial node to any other node in the
sub-tree that it is the root of
Each node has a chance to be infected at any given point
Daniel Guo (PRIMES Conference)
Modeling of Disease Spreading on Trees
May 16, 2015
9 / 22
The Problem
Depiction of the Problem
Given any tree, nodes = people
Nodes are represented with a Boolean variable for infection
Infection can jump from any initial node to any other node in the
sub-tree that it is the root of
Each node has a chance to be infected at any given point
Infection rate decreases with distance from node
Daniel Guo (PRIMES Conference)
Modeling of Disease Spreading on Trees
May 16, 2015
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The Problem
An Example
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Modeling of Disease Spreading on Trees
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The Problem
An Example
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Modeling of Disease Spreading on Trees
May 16, 2015
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The Problem
An Example
Daniel Guo (PRIMES Conference)
Modeling of Disease Spreading on Trees
May 16, 2015
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The Problem
Questions
1 How long will it take to reach layer n or below?
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Modeling of Disease Spreading on Trees
May 16, 2015
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The Problem
Questions
1 How long will it take to reach layer n or below?
2 How many infected nodes are there when this occurs?
Daniel Guo (PRIMES Conference)
Modeling of Disease Spreading on Trees
May 16, 2015
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The Problem
Solving the Problem(s)
Program was devised to simulate the binary tree and infection process
Daniel Guo (PRIMES Conference)
Modeling of Disease Spreading on Trees
May 16, 2015
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The Problem
Solving the Problem(s)
Program was devised to simulate the binary tree and infection process
Used MATLAB
Daniel Guo (PRIMES Conference)
Modeling of Disease Spreading on Trees
May 16, 2015
14 / 22
The Problem
Solving the Problem(s)
Program was devised to simulate the binary tree and infection process
Used MATLAB
Modeled the infection time between two points with exponential
distributions
Daniel Guo (PRIMES Conference)
Modeling of Disease Spreading on Trees
May 16, 2015
14 / 22
The Problem
Solving the Problem(s)
Program was devised to simulate the binary tree and infection process
Used MATLAB
Modeled the infection time between two points with exponential
distributions
Exponential distributions had rates of 2−k (k 1−α − (k + 1)1−α )
Daniel Guo (PRIMES Conference)
Modeling of Disease Spreading on Trees
May 16, 2015
14 / 22
The Problem
Solving the Problem(s)
Program was devised to simulate the binary tree and infection process
Used MATLAB
Modeled the infection time between two points with exponential
distributions
Exponential distributions had rates of 2−k (k 1−α − (k + 1)1−α )
α is a predetermined constant such that α > 1
Daniel Guo (PRIMES Conference)
Modeling of Disease Spreading on Trees
May 16, 2015
14 / 22
The Problem
Solving the Problem(s)
Program was devised to simulate the binary tree and infection process
Used MATLAB
Modeled the infection time between two points with exponential
distributions
Exponential distributions had rates of 2−k (k 1−α − (k + 1)1−α )
α is a predetermined constant such that α > 1
k is the difference in layers between the infecting/infected nodes
Daniel Guo (PRIMES Conference)
Modeling of Disease Spreading on Trees
May 16, 2015
14 / 22
The Problem
Solving the Problem(s)
There are 3 important properties of an exponential distribution that we
can use
Daniel Guo (PRIMES Conference)
Modeling of Disease Spreading on Trees
May 16, 2015
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The Problem
Solving the Problem(s)
There are 3 important properties of an exponential distribution that we
can use
Property
Memoryless: If X = exp(r ), then P(X > x + y |X > y ) = P(X > x).
Daniel Guo (PRIMES Conference)
Modeling of Disease Spreading on Trees
May 16, 2015
15 / 22
The Problem
Solving the Problem(s)
There are 3 important properties of an exponential distribution that we
can use
Property
Memoryless: If X = exp(r ), then P(X > x + y |X > y ) = P(X > x).
Property
Minimum: If X1 , X2 , . . . , Xn , are all exponential with rates r1 , r2 , . . . , rn ,
then min(X1 , X2 , . . . , Xn ) = exp(r1 + r2 + · · · + rn ).
Daniel Guo (PRIMES Conference)
Modeling of Disease Spreading on Trees
May 16, 2015
15 / 22
The Problem
Solving the Problem(s)
There are 3 important properties of an exponential distribution that we
can use
Property
Memoryless: If X = exp(r ), then P(X > x + y |X > y ) = P(X > x).
Property
Minimum: If X1 , X2 , . . . , Xn , are all exponential with rates r1 , r2 , . . . , rn ,
then min(X1 , X2 , . . . , Xn ) = exp(r1 + r2 + · · · + rn ).
Property
Probability: If X1 , X2 , . . . , Xn , are all exponential with rates r1 , r2 , . . . , rn ,
then the probability than Xi is the minimum of X1 , X2 , . . . , Xn is
ri
r1 +r2 +···+rn .
Daniel Guo (PRIMES Conference)
Modeling of Disease Spreading on Trees
May 16, 2015
15 / 22
The Problem
Solving the Problem(s)
Properties can shorten simulation time
1 Pick a point that will infect, probability property
Daniel Guo (PRIMES Conference)
Modeling of Disease Spreading on Trees
May 16, 2015
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The Problem
Solving the Problem(s)
Properties can shorten simulation time
1 Pick a point that will infect, probability property
2 Pick point to be infected
Daniel Guo (PRIMES Conference)
Modeling of Disease Spreading on Trees
May 16, 2015
16 / 22
The Problem
Solving the Problem(s)
Properties can shorten simulation time
1 Pick a point that will infect, probability property
2 Pick point to be infected
3 Time generated from exp(sum of all infected node rates), minimum
property
Daniel Guo (PRIMES Conference)
Modeling of Disease Spreading on Trees
May 16, 2015
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Results and Continuation
1
Introduction and Definitions
2
The Problem
3
Results and Continuation
4
References
Daniel Guo (PRIMES Conference)
Modeling of Disease Spreading on Trees
May 16, 2015
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Results and Continuation
Results
The results seem to model a curve that is slightly skewed right
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Modeling of Disease Spreading on Trees
May 16, 2015
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Results and Continuation
Results
Daniel Guo (PRIMES Conference)
Modeling of Disease Spreading on Trees
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Results and Continuation
Predictions
Conjecture
If 1 < α < 2, prediction is polynomial in n with degree α − 1
Daniel Guo (PRIMES Conference)
Modeling of Disease Spreading on Trees
May 16, 2015
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Results and Continuation
Predictions
Conjecture
If 1 < α < 2, prediction is polynomial in n with degree α − 1
Conjecture
If α ≥ 2, prediction is linear in n
Daniel Guo (PRIMES Conference)
Modeling of Disease Spreading on Trees
May 16, 2015
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Results and Continuation
Future Goals
Expand to different types of trees
Daniel Guo (PRIMES Conference)
Modeling of Disease Spreading on Trees
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References
Acknowledgments
Professor Partha Dey for mentoring me on this project
PRIMES and Tanya Khovanova for offering this opportunity
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