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### Mathematical Models of Disease Spread

### Syndromic Surveillance

### Themes of our Meetings

### Themes of our Meetings

### Themes of our Meetings

### Themes of our Meetings

### Themes of our Meetings

### Themes of our Meetings

### Themes of our Meetings

### Themes of our Meetings

### Climate and Health

### Climate and Health

### Climate and Health

### Climate and Health

### Malaria

### Malaria

### Dengue Fever

### Biodiversity

### Biodiversity

### Biodiversity

### Bio-Math Connect Institute

### Bio-Math Connect Institute

### Bio-Math Connect Institute

Why Now?

- Do you want to catch your students’ attention?
- How about talking about:
- Swine flu?

Why Now?

- Do you want to catch your students’ attention?
- How about talking about:
- Climate change?

Why Now?

- Do you want to catch your students’ attention?
- How about talking about:
- Tree-climbing lions?
- These are all things I have gotten involved with.
- How did a mathematician come to do that?

Why Now?

- Do you want to catch your students’ attention?
- How about talking about:
- Smallpox?
- These are all things I have gotten involved with.
- How did a mathematician come to do that?

In 2002, I was invited to join the Secretary of Health

and Human Services’ Smallpox Modeling Group.

How did a mathematician come to do that?

Mathematical models of infectious diseases go back to Daniel Bernoulli’s mathematical analysis of smallpox in 1760.

Understanding infectious systems requires being able to reason about highly complex biological systems, with hundreds of demographic and epidemiological variables.

Intuition alone is insufficient to fully understand the dynamics of such systems.

Experimentation or field trials are often prohibitively expensive or unethical and do not always lead to fundamental understanding.

Therefore, mathematical modeling becomes an important experimental and analytical tool.

Mathematical models have become important tools in analyzing the spread and control of infectious diseases, especially when combined with powerful, modern computer methods for analyzing and/or simulating the models.

Great concern about the deliberate introduction of diseases by bioterrorists has led to new challenges for mathematical modelers.

anthrax

Great concern about possibly devastating new diseases like avian influenza or H1N1 virus (swine flu) has also led to new challenges for mathematical modelers.

The size and overwhelming complexity of modern epidemiological problems -- and in particular the defense against bioterrorism -- calls for new approaches and tools.

As a result, in 2002, DIMACS launched a “special focus” on mathematical and computational epidemiology that has paired mathematicians, computer scientists, and statisticians with epidemiologists, biologists, public health professionals, physicians, etc.

Why Now?

- I have long been interested in applications of
- mathematics.
- I was even interested in mathematical problems
- in biology very early in my career.
- As a graduate student in the 1960s, I worked on
- a problem posed by Nobel prize winning geneticist
- Seymour Benzer.

- The problem was: How can you understand the
- “fine structure” inside the gene without being able
- to see inside?

- The problem involved molecular biology.
- Molecular biology is a prime example of the new biology.
- Increasingly, many biological phenomena are being viewed as involving the processing of information.
- Biology has become an information science.
- Modern computer and information science have played an important role in such major biological accomplishments as sequencing the human genome, and are of fundamental importance in the rapidly-evolving concept of “digital biology”

- The problem was: How can you understand the
- “fine structure” inside the gene without being able
- to see inside?
- Classically, geneticists had treated the chromosome
- as a linear arrangement of genes.
- Benzer asked in 1959: Was the same thing true
- for the “fine structure” inside the gene?

- The problem was: How can you understand the
- “fine structure” inside the gene without being able
- to see inside?
- The Question: was the gene fundamentally linear?

- Or: was the gene fundamentally circular?

- Or: was the gene fundamentally like a figure-8?

- At the time, we could not observe the fine structure
- directly.
- Benzer studied mutations.
- He assumed mutations involved “connected
- substructures” of the gene.
- By gathering mutation data, he was able to surmise
- whether or not two mutations overlapped.

i,j entry is 1 if mutations Si and Sj overlap, 0 otherwise.

- If we represent the tabular (matrix) information
- as a graph, we say that the graph is an interval graph
- if it is consistent with a linear arrangement.
- Interval graphs have been very important in genetics.
- Long after Benzer’s problem was solved using other
- methods, interval graphs played a crucial role in
- physical mapping of DNAand more generally in the
- mapping of the human genome.

- Given a graph, is it an interval graph?

c

a

b

d

e

- We need to find intervals on the line that have the same overlap properties

a

b

e

d

c

Why Now?

- So how did I get from Benzer’s problem to
- Modeling smallpox for the Secretary of Health
- and Human Services?
- Studying swine flu?
- Studying climate change
- Looking for tree-climbing lions
- It has become increasingly clear that biology has
- become an information science.
- The key idea involves DNA.

Deoxyribonucleic acid, DNA, is the basic building block of inheritance and carrier of genetic information.

DNA can be thought of as a chain consisting of bases.

Each base is one of four possible chemicals:

Thymine (T), Cytosine (C), Adenine (A), Guanine (G)

Some DNA chains:

GGATCCTGG, TTCGCAAAAAGAATC

Real DNA chains are long:

Algae (P. salina): 6.6x105 bases long

Slime mold (D. discoideum): 5.4x107 bases long

Insect (D. melanogaster – fruit fly): 1.4x108 bases long

Bird (G. domesticus): 1.2x109 bases long

Human (H. sapiens): 3.3x109 bases long

The sequence of bases in DNA encodes certain genetic information.

In particular, it determines long chains of amino acids known as proteins.

Fundamental methods of combinatorics (the mathematics of counting) are important in mathematical biology.

How many possible DNA chains are there in humans?

How many sequences of 0’s and 1’s are there of length 2?

There are 2 ways to choose the first digit and no matter how we choose the first digit, there are two ways to choose the second digit.

Thus, there are 2x2 = 22 = 4 ways to choose the sequence.

00, 01, 10, 11

How many sequences are there of length 3?

By similar reasoning: 2x2x2 = 23.

Product Rule: If something can happen in n1 ways and no matter how the first thing happens, a second thing can happen in n2 ways, then the two things together can happen in n1 x n2 ways.

More generally, if something can happen in n1 ways and no matter how the first thing happens, a second thing can happen in n2 ways, and no matter how the first two things happen a third thing can happen in n3 ways, … then all the things together can happen in n1 x n2 x n3 ways.

How many possible DNA chains are there in humans?

How many DNA chains are there with two bases?

Answer (Product Rule): 4x4 = 42 = 16.

There are 4 choices for the first base and, for each such choice, 4 choices for the second base.

How many with 3 bases?

How many with n bases?

How many with 3 bases? 43 = 64

How many with n bases? 4n

How many human DNA chains are possible?

4^(3.3x109)

This is greater than 10^(1.98x109)

(1 followed by 198 million zeroes!)

How many human DNA chains are possible?

4^(3.3x109)

This is greater than 10^(1.98x109)

(1 followed by 198 million zeroes!)

A simple counting argument helps us to understand the

remarkable diversity of life.

A simple counting argument helps us to understand the

remarkable diversity of life.

Mathematical modeling will help us protect the remarkable diversity of life on our planet.

Mathematical ecology and population biology have a long history.

Modern mathematical methods allow us to deal with huge ecosystems and understand massive amounts of ecological data.

2003: NSF & NIH asked me to organize a Workshop: Information Processing in the Biological Organism(A Systems Biology Approach)

The potential for dramatic new biological knowledge arises from investigating the complex interactions of many different levels of biological information.

Levels of Biological Information

DNA

mRNA

Protein

Protein interactions and biomodules

Protein and gene networks

Cells

Organs

Individuals

Populations

Ecologies

The workshop investigated information processing in biological organisms from a systems point of view.

The list of parts is a necessary but not sufficient condition for understanding biological function.

Understanding how the parts work is also important.

But it is not enough. We need to know how they work together. This is the systems approach.

The Workshop Was Organized Around Four Themes:

- Genetics to gene-product information flows.
- Signal fusion within the cell.
- Cell-to-cell communication.
- Information flow at the system level, including
- environmental interactions.

Information

processing

between bacteria

helps this squid

in the dark.

Bonnie Bassler

Princeton Univ.

Bacteria process the information about the local density of other bacteria. They use this to produce luminescence.The process involved can be modeled by a mathematical model involving quorum sensing.Similar quorum sensing has been observed in over 70 species

Mdm2-YFP

Example 2: The P53-MDM2 Feedback Loop and DNA Damage RepairKohn, Mol Biol Cell, 1999

Uri Alon, Weizmann Institute

Galit Lahav, Harvard University

Network motifs are conceptual units that are dynamic and larger than single components such as genes or proteins. Such motifs have helped to understand the nonlinear dynamics of the process by which the P53 - MDM2 feedback loop contributes to the regulation of DNA damage repair.

Stress signals

Cell cycle arrest

G1/S G2/M

no

yes

Apoptosis

DNA repair

MDM2

p53

One cell death =

Protection of the whole organism

Is the damage

repairable?

Example 3: Mathematical Modeling of Multiscale phenomena arising in excitation/contraction coupling in the ventricle

Raimond Winslow, Johns Hopkins

Canine Heart

The models study the stochastic behavior of calcium release channels.

- Model components range in size from 10 nanometers to 10 centimeters.
- The work has application to the connection between heart failure and sudden cardiac death.

Ca2+ Release

Channels (RyR)

<-10 nm->

L-Type Ca2+

Channel

The Mathematics of InfectiousDisease is Different from theMathematics of Diseases Like Heart Disease

AIDS

The Mathematics of InfectiousDisease

anthrax

- So how did I get involved with smallpox and swine flu?
- We had made plans to launch a DIMACS special focus on Computational and Mathematical Epidemiology in Fall 2002.
- Then came the Sept. 2001 World Trade Center attacks followed by the anthrax attacks.
- We were all set to go with an epi special focus.
- We started early, with an emphasis on bioterrorism motivated by the anthrax attacks.

Smallpox

- Smallpox is one of the diseases considered a bioterrorism threat.
- It was wiped out in the wild by 1979 by a World Health Organization campaign.
- However, there are still two sources of smallpox in labs and there is danger also of genetically engineered smallpox.
- Consideration of bioterrorism led us into epidemiological modeling.

Models of the Spread and Control of Disease through Social Networks

AIDS

- 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.

A Model: Moving From State to State

Social Network = Graph

Vertices = People

Edges = contact

Let si(t) give the state of vertex i

at time t.

Simplified Model: Two states:

= susceptible, = infected (SI Model)

Times are discrete: t = 0, 1, 2, …

A Model: Moving From State to State

More complex models: SI, SEI,

SEIR, etc.

S = susceptible, E = exposed,

I = infected, R = recovered

(or removed)

measles

SARS

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 directed graph reflecting the possible ways to move from state to state in the model.

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.

Great concern about the deliberate introduction of diseases by bioterrorists has led to new challenges for mathematical scientists.

This is a major reason for our interest in smallpox – which has been eradicated in the natural world.

smallpox

Speaking of Smallpox

- Students can get heavily involved.
- One group of high school students at the Charter School of Wilmington, Delaware developed smallpox models that led to their winning first place at the New Castle Science Fair and an invitation to the International Science and Engineering Fair in Cleveland, Ohio.
- They were later invited to present their work to a group of researchers at a workshop on epidemiological modeling at DIMACS.

Homeland Security What Can Mathematics Do?

My interest in disease got

directed to bioterrorism after the World Trade Center attacks and following anthrax attacks.

anthrax

Homeland Security What Can Mathematics Do?

I gave a talk to Congressmen and their staffers on Capitol Hill in September 2004.

Bioterrorist Event Detection

- Modern data-gathering methods bring with them new challenges for mathematicians.
- They allow us to get early warning of the outbreak of an infectious disease – whether naturally-occurring or caused by a bioterrorist.
- “Biosurveilliance.”
- Method called “syndromic surveillance”

New Data Types for Public Health Surveillance

- Managed care patient encounter data
- Pre-diagnostic/chief complaint (ED data)
- Over-the-counter sales transactions
- Drug store
- Grocery store
- 911-emergency calls
- Ambulance dispatch data
- Absenteeism data
- ED discharge summaries
- Prescription/pharmaceuticals
- Adverse event reports

Many New Mathematical Methods and Approaches under Development

- Spatial-temporal “scan statistics”
- Statistical process control (SPC)
- Bayesian applications
- “Market-basket” association analysis
- Text mining
- Rule-based surveillance
- Change-point techniques

- Has gotten me and DIMACS involved in a partnership with CDC: Centers for Disease Control and Prevention
- CDC has just launched a new program on mathematical modeling of disease.

Early warning is critical

- This is a crucial factor underlying government’s plans to place networks of sensors/detectors to warn of a bioterrorist attack

The BASIS System

Two Fundamental Problems

- Sensor Location Problem (SLP):
- Choose an appropriate mix of sensors
- decide where to locate them for best protection and early warning

Two Fundamental Problems

- Pattern Interpretation Problem (PIP): When sensors set off an alarm, help public health decision makers decide
- Has an attack taken place?
- What additional monitoring is needed?
- What was its extent and location?
- What is an appropriate response?

The work on bioterrorism and epidemiology led to the designation of DIMACS as a U.S. Department of Homeland Security “Center of Excellence” in 2006.

Endemic and emerging diseases of Africa provide new and complex challenges for mathematical modeling.

HIV/AIDS

Malaria

Tuberculosis

Endemic and emerging diseases of Africa provide new and complex challenges for mathematical modeling.

Because of modern transportation systems, no one in the world is safe from diseases originating elsewhere.

Major new health threats such as avian influenza or H1N1 virus (swine flu) present especially complex challenges to modelers in the context of developing countries.

Two DIMACS workshops and a student short course were held in South Africa in 2006-2008, aimed at:

Studying challenges for mathematical models arising from the diseases of Africa

Understanding special challenges from diseases in resource-poor countries.

Bringing together U.S. and African researchers and students to collaborate in solving these problems.

Laying the groundwork for future collaborations to address problems of public health and disease in Africa.

Major New DIMACS African Initiative on BioMath:

Clinic on Meaningful Modeling of Disease – South Africa, May 2009.

Workshop and student short course on “Economic Epidemiology” in Uganda, July-August 2009.

Workshop and student short course on “Conservation Biology” in Kenya, 2010.

Mathematics of ecological reserves

Workshop and student short course on “Genetics and Disease Control” in Madagascar, 2011.

Are genetically altered crops safe? What about malaria control by genetically modifying mosquitoes?

Our African Initiative has already led to themes that should be of interest to high school students.

Tree-climbing lion, Queen Elizabeth National Park, Uganda

Major New DIMACS African Initiative on BioMath:

Workshop and student short course on “Economic Epidemiology” in Uganda, July-August 2009.

In recent years, mathematical modeling has had an increasing influence on the theory and practice of disease management and control.

Modeling has played an important role in shaping public health policy decisions in a number of countries.

Gonorrhea, HIV/AIDS, BSE, FMD, measles, rubella, pertussis (UK, US, Netherlands, Canada)

measles

FMD

Modeling has provided insights leading to “optimal” treatment strategies

Immuno-pathogenesis of HIV/AIDS and use of highly active anti-retroviral therapy

Modeling has played a role in

shaping vaccine design and

determining threshold coverage

levels for vaccine-preventable diseases:

measles, rubella, polio

AIDS

During SARS outbreaks in 2003, modelers and public health officials worked hand-in-hand to devise effective control strategies in a number of countries.

Earlier, similar importance of efforts to control FMD.

- Mathematical Modeling of Diseases that Inflict a Significant Burden on Africa
- HIV/AIDS
- TB
- Malaria
- Diseases of Animals

AIDS orphans, Zambia

Mathematical Modeling of Diseases that Inflict a Significant Burden on Africa

- HIV/AIDS
- Modeling/evaluation of

preventive and therapeutic strategies

- Allocation of anti-retroviral drugs
- Evolution and transmission of drug-resistant strains
- Interaction with other infections: TB, malaria – co-infection a major theme in mathematical epidemiology

Mathematical Modeling of Diseases that Inflict a Significant Burden on Africa

- Malaria
- New methods of control (e.g.,

insecticide-treated cattle)

- Climate and disease (e.g.,

global warming and effect on

mosquito populations)

- Led to new DIMACS initiative

on climate and health

Mathematical Modeling of Diseases that Inflict a Significant Burden on Africa

- Diseases of Animals
- Bovine tuberculosis (in domestic and wild populations)
- Avian influenza
- Trypanosomiasis

Mathematical Modeling of Diseases that Inflict a Significant Burden on Africa

- Diseases of Plants
- Major threat to the food supply.
- In U.S.. DHS has established two
- research centers at universities
- that deal with protection of the
- food supply.

Modeling Issues from Threat of Emerging Diseases in Resource-poor Countries

- Special issues arising from:
- Slow communication
- Short supplies of vaccines

and prophylactics

- Difficulty of imposing

quarantines

- Special emphasis on problems

arising from avian or pandemic influenza

Optimization of Scarce Public Health Resources

- How to handle shortages of drugs and vaccines, physical facilities, and trained personnel.
- Not just an issue in Africa
- Mathematical methods to:
- Allocate medicines to optimize impact
- Assign trained personnel to

most critical jobs

- Design efficient transportation plans.
- Design efficient dispensing plans.

- Explore protocols for vaccination
- for major diseases in Africa
- Discuss potential for vaccines for HIV, malaria
- Use of computer simulations to allow comparison of vaccination strategies when field trials are prohibitively expensive
- Identify major modeling challenges unique to Africa: e.g., age-structured, health-status-related models
- DIMACS Vaccination Modeling Group

DIMACS Workshop on Climate and Disease

- April 7-8, 2008
- DIMACS Initiative on Climate and Health
- 2008-2010

Global warming

Dust storm in Mali

Concerns about global warming.

Resulting impact on health

Of people

Of animals

Of plants

Of ecosystems

Some early warning signs:

1995 extreme heat event in Chicago

514 heat-related deaths

3300 excess emergency admissions

2003 heat wave in Europe

35,000 deaths

Food spoilage on Antarctica

expeditions

Not cold enough to store food

in the ice

Some early warning signs:

Malaria in the African Highlands

Dengue epidemics

Floods, hurricanes

The challenge of climate change: Malaria springs up in areas it wasn’t in before.

Highlands of Kenya

Potential for Malaria in the US – Texas, Florida, Washington, …

A key role for modelers:

Aid in early warning:

surveillance.

For each disease, effect of climate raises its own complex modeling challenges

Malaria a case in point

Climate change impacts:

Transmission rate changes with climate conditions

Affects dynamics of both host and mosquito

Affects time lag for parasite development as a function of temperature

Affects time lag for development of

symptoms

Affects length of time patient remains immune

Rainfall affects the carrying capacity for larvae

Mosquito-borne disease.

Appearing in places it hasn’t appeared before.

Large outbreak in Brazil starting in 2008.

Seems climate-related.

Excessive rainfall leads to excessive cases

Complex interaction among weather, mosquito populations, people’s responses to weather conditions, etc.

A conundrum: 50,000 cases along the Rio Grande between Mexico and Texas.

Almost all in Mexico.

Why?

Extinction and danger for species due to global warming.

Example: challenges for the polar bear.

Some predict a 15 to 37% reduction in biodiversity due to global warming.

But, other factors may be more important than climate – e.g., pollution, changes in land use.

Some factors such as habitat loss are intertwined with climate change.

The effects of climate change on biodiversity need to be modeled on different temporal and spatial scales.

By the kilometer or by the continent?

By the year or by the decade?

Issues present challenges to

biologists and mathematicians alike

and together.

DIMACS workshop on Conservation Biology (Kenya, 2010) to explore:

Mathematical models of biodiversity

Mathematics of setting up an ecological reserve

BMC was born from these kinds of themes.

- Bio-Math has become a major topic at the undergraduate level and at the graduate and postgraduate level.
- Why not in the schools?

Thesis: Exposing biology students to the importance of mathematical methods in biology will help them appreciate biology more.

- Thesis: Exposing mathematics students to their usefulness in modern biological problems will help them appreciate mathematics more.

Thesis: Exposing students to the bio-math interface will open up new horizons for them and expose them to new career opportunities and new opportunities for further education.

- Thesis: Exposing students to the bio-math interface will motivate them as students.

to explore these ideas.

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