Modeling the Course of a Viral Illness and Its Treatment
Modeling the Course of a Viral Illness and its Treatment
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 of virus
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 |
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 of virus
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 |