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