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Modeling Bird/Human Interaction with Avian Influenza. Mathematical Modeling. Nicole Rogerson and Theresa Holtz North Edgecombe High School Craven Early College. 2008. What’s the Problem?. Tracking the Avian Influenza virus Trends between people and bird populations. - PowerPoint PPT Presentation
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Modeling Bird/Human Interaction with Avian Influenza
Mathematical Modeling
Nicole Rogerson and Theresa HoltzNorth Edgecombe High School
Craven Early College
2008
What’s the Problem?
Tracking the Avian Influenza virus
Trends between people and bird populations
http://tk.files.storage.msn.com/x1pUr2osLO3XWjESyfPMmOH5SY3GtDQ1HGa1nrdHHfAaarZzRWEYkF5x_ueI40CZC329LSFFV2x0Q7pISR9bmXRBJsok8mdQQSLlcWtVyiZEPgB8hzb5YLgZEII6pPG0kAFQKnwPgXBGNQ
Outline
What is avian influenza? Describing our models Results using trial and error Variations in our model A quick review In the future…
http://www.topnews.in/health/files/Bird_Exam.jpg
What is Avian Influenza (H5N1)? Influenza A
Respiratory• Coughing, congestion, sore throat,
muscle aches, fatigue, fever
Spreads through birds the same way the flu spreads through people, except for coughing
It is a potential pandemic
http://ec.europa.eu/health/ph_threats/com/Influenza/images/influenza.jpg
How did we get our models?
SIR MODEL
SusceptiblePopulation
InfectedPopulation
RecoveredPopulation
How did we get our models?
idtdr
isidtdi
sidtds
susceptibles
recovered
infectives
How did we get our models?
infected birds
infected humans
recovered humans
recovered birds
bpdtdq
bpaidtdp
What do these symbols mean?
: transmittivity constant; determines rate of infection for birds
:rate of recovery for birds a=transmittivity constant; determines rate of
infection for humans b= rate of recovery for humans
Assumptions and possible constants in our models
Assume that in every country that had reported a bird flu case, all people are susceptible
Total Human Susceptibles
0
200,000,000
400,000,000
600,000,000
800,000,000
1,000,000,000
1,200,000,000
1,400,000,000
1,600,000,000
1,800,000,000
2,000,000,000
2004 2005 2006 2007
Year
Num
ber o
f Sus
ce[ti
bles
147,555,041
1,470,187,0161,571,237,542
1,720,083,064
Assumptions and possible constants in our models
Constants are susceptible birds
Numbers are high
They don’t vary much over time
Total Birds Susceptible
0
2,000,000
4,000,000
6,000,000
8,000,000
10,000,000
12,000,000
2004 2005 2006 2007
Year
Num
ber S
usce
ptib
le
677,3621,064,694
11,270,536
9,319,991
http://upload.wikimedia.org/wikipedia/en/7/7f/H5n1_spread_%28with_regression%29.png
Infected People from 2004-2007
People Infected – Data
Birds Infected 2004-2007Total Birds Infected
0
200,000
400,000
600,000
800,000
1,000,000
1,200,000
1,400,000
2004 2005 2006 2007
Year
Num
ber I
nfec
ted
42,866115,893
1,107,3071,188,752
Birds Infected-Data
Fit Data to Model
idtdr
kiisdtdi
)( 0
Assume s(t) = constant = s0
Find and to fit infected bird population.
Determining Parameters Decreasing the
rate at which birds get infected: The graph
goes up and turns right
• birds infected over time is smaller
Increasing Transmission Rate - Bird Population -
Increasing the rate at which birds get infected: line went right
then straight up• the birds
infected over time shot up into unrealistic numbers
Recovery Rate
Changing the rate of recovery of the infected bird population: line goes out
into a half parabola
Makes a wide curve
Results-Bird Population
43,000
3a
4b
000,677
0
0
i
s
Fit Human Population Data
Find a and b to fit infected bird population.
bpdtdq
bpaidtdp
Changing Transmission Rate
Changing the rate of infection of the human population: doesn’t really change much even if
totally opposite values are put in Only moves the line farther away from
the y-axis
Increasing Recovery Rate
Increasing the rate of recovery of the humans makes the line go straighter
Decreasing Recovery Rate
Decreasing the recovery rate of humans curves the line
Results – Human Population
43,000i
0.00011a
0.001b
000,677
0
0
6
s
Further Improvements
Birds Humans
43,000
3a
4b
000,677
0
0
i
s
43,000i
0.00011a
0.001b
000,677
0
0
6
s
New Model – Logistic Term
bpdtdq
bpaidtdp
idtdr
ciisdtdi
20 )(
Birds Humans
ci2 – limited growth term
Quadratic Fit p(t) to DataInfected People Over Time
y = 0.0954x2 + 4.1233x
0
50
100
150
200
250
300
350
0 5 10 15 20 25 30 35 40 45
Time(In Months)
Peop
le In
fect
ed
Exponential Fit i(t)=i0ektBirds Infected Over Time
y = 13369e0.1012x
0
200000
400000
600000
800000
1000000
1200000
1400000
1600000
1800000
2000000
0 10 20 30 40 50 60
Time(In Months)
Bird
s In
fect
ed
Modified Models
13,369i
10c
0.1012k
0.06b
0.0003a
(
000,677
0
0
3-
k)/sO
s
Our Findings
Birds aren’t tracked as carefully as humans when it comes to Avian Influenza.
Birds should be tracked as well as humans because they are the ones carrying the virus, not humans.
There wasn’t enough information about the birds, so we didn’t have a lot of data to work with
To the future!
Look at the Spanish flu. Compare results to Avian influenza
H5N1 Look at other strains of Avian
Influenza (H7N2, H7N3,H7N7,H9N2)
A Quick Review
What Avian Influenza H5N1 is. How we got and tested our models. What we found out.
Bibliography Siegel, Marc. Bird Flu Everything you need to know about the
next pandemic . New Jersey: John Wiley & Sons, Inc, 2006. "Bird Flu Everything you need to know about the next
pandemic ." Oie. 09/07/2008. World Organization for Animal Health. 8 Jul 2008 <http://www.oie.int/downld/AVIAN%20INFLUENZA/A2004_AI.php>.
"Cumulative Number of Confirmed Human Cases of Avian Influenza A/(H5N1) Reported to WHO." Epidemic and Pandemic Alert and Response (EPR). 19 June 2008. World Health Organization . 6 Jul 2008 <http://www.who.int/csr/disease/avian_influenza/country/cases_table_2008_06_19/en/index.html>.
"Global: Cumulative number of human cases of avian influenza A/(H5N1)." EINet. 6Jul 2008 <http://depts.washington.edu/einet/?a=printArticle&print=5054>.
"Index of /worldfacts/countries." AirNinja . H Brothers Inc. 1 Jul 2008 <http://www.airninja.com/worldfacts/countries/>.
"Central Intelligence Agency ." The World Factbook. CIA. 6 Jul 2008 <https://www.cia.gov/library/publications/the-world-factbook/index.html>.
Acknowledgments
SVSM Staff Dr. Herman Mr. Glasier Parents (both families) Mrs. Moser Mrs. Land
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