Math Internal Assessment

Student: Xu,Dejing(Charlotte)

Session number:001762023

Date: March 26, 2010

I. Introduction:

When viral particles of a certain virus enter the human body, they replicate rapidly. In about four hours, the number of viral particles has doubled. The immune system does not respond until there are about 1 million viral particles in the body. The first response of the immune system is fever. The rise in temperature lowers the rate at which the viral particles replicate to 160% every four hours, but the immune system can only eliminate these particular viral particles at the rate of about 50,000 viral particles per hour. Often people do not seek medical attention immediately as they think they have a common cold. If the number of viral particles however, reaches 1012, the person dies. The only thing could affect the growth rate is the immune system. The antiviral medicine only can eliminate the viral particles. 1) Model the initial phase of the illness for a person infected with 10,000 viral particles to determine how long it will take for the body’s immune response to begin. Given: The rate is 200% per 4 hour using r to represent

The initial number of virus is 10,000

Use t to represent time

Use Y to represent the number ofvirus

If regard every 4 hour as a period:

Xn+1=2Xn

∴X1=X0

X2=rX

X3=r X2=r(rX)

X4= rX3=r (rX2)=r(r(rX))

∴the formula for every 1 hour would be:

Xn+1=214Xn

Since X0=10,000

Convert it to function:

Y=10,000×2t4

Solve t:

When immune response begin to work, Y=1,000,000

1,000,000=10,000×2t4

2t4=1,000,00010,000

2t4=100

log100log2=t4

t= 26.58hr

It will take 26.58 hours for the body’s immune response to begin.

hour| viral particles| Fever| hour| viral particles| Fever| 0| 10,000 | | 21| 380,546 | |

1| 11,892 | | 22| 452,548 | |

2| 14,142 | | 23| 538,174 | |

3| 16,818 | | 24| 640,000 | |

4| 20,000 | | 25| 761,093 | |

5| 23,784 | | 26| 905,097 | |

6| 28,284| | 27| 1,076,347 | Yes|

7| 33,636| | 28| 1,280,000 | Yes|

…| …| …| 29| 1,522,185 | Yes|

20| 320,000 | | 30| 1,810,193 | Yes|

From the table above, it determined that the fever start between 26 and27 hours which means immune system started to work at that time.

2) Using a spreadsheet, or otherwise, develop a model for the next phase of the illness, when the immune response has begun but no medications have yet been administered. Use the model to determine how long it will be before the patient dies if the infection is left untreated. Given: The rate is 160% per 4 hour using r to represent

The immune system kills 50000 viral every hour

The initial number of virus is 1,000,000

Use t to represent time

Use Y to represent the number ofvirus

If regard every 4 hour as a period:

Xn+1=1.6Xn-200,000

∴the formula for every 1 hour would be:

Xn+1=1.614Xn-50,000

Since X0=1,000,000

Convert it to function:

Y=1,000,000×1.6t4 - 50,000t

The table above shows that 117 hours after the immune started to work, if no chemical treatment was taken, people will dead. Plus the 26 hours before the immune system start to work which means, when the particles reach to 1012, which is between the 143th and the 144th hour, the person would die. This means 5 days after the virus entered, people will dead. Modeling recovery

An antiviral medication can be administered as soon as a person seeks medical attention. The medication does not affect the growth rate of the viruses but together with the immune response can eliminate 1.2 million viral particles per hour. 3) If the person is to make a full recovery, explain why effective medication must be administered before the number of viral particles reaches 9 to 10 million. Given: The rate is 160% per 4 hour...