# HIV model

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• Jan 19th 2011, 03:56 PM
mathsohard
HIV model
Cells that are susceptible to HIV infection are called T(target) cells. Let T(t) be the population of uninfected T-cells, T*(t) that of the infected T-cells, and V(t) the population of the HIV virus. A model for the rate of change of the infected T-cells is

dT*/dt = kVT - gT*, (eq1)

where g is the rate of clearance of infected cells by the body, and k is the rate constant for the infection of the T-cells by the virus. The equation for the virus is the same as

dV/dt = P-cV, (eq2)

but now the production of the virus can be modeled by
P(t) = NgT*(t).

Here N is the total number of virions produced by an infected T-cell during its lifetime. Since 1/g is the length of its lifetime, NgT*(t) is the total rate of production of V(t).

At least during the initial stages of infection, T can be treated as an approximate constant. Equations (eq1) and (eq2) are the two coupled equations for the two variables T*(t) and V(t).

A drug therapy using RT (reverse transcriptase) inhibitors blocks infection, leading to k~=0. Setting K = 0 in (eq 1), solve for T*(t). Substitute it into (eq 2) and solve for V(t). Show that the solution is

V(t) = [V(0)/(c-g)][ce^(gt) - ge^(-ct)].

• Jan 19th 2011, 04:24 PM
dwsmith
Quote:

Originally Posted by mathsohard
Cells that are susceptible to HIV infection are called T(target) cells. Let T(t) be the population of uninfected T-cells, T*(t) that of the infected T-cells, and V(t) the population of the HIV virus. A model for the rate of change of the infected T-cells is

dT*/dt = kVT - gT*, (eq1)

where g is the rate of clearance of infected cells by the body, and k is the rate constant for the infection of the T-cells by the virus. The equation for the virus is the same as

dV/dt = P-cV, (eq2)

but now the production of the virus can be modeled by
P(t) = NgT*(t).

Here N is the total number of virions produced by an infected T-cell during its lifetime. Since 1/g is the length of its lifetime, NgT*(t) is the total rate of production of V(t).

At least during the initial stages of infection, T can be treated as an approximate constant. Equations (eq1) and (eq2) are the two coupled equations for the two variables T*(t) and V(t).

A drug therapy using RT (reverse transcriptase) inhibitors blocks infection, leading to k~=0. Setting K = 0 in (eq 1), solve for T*(t). Substitute it into (eq 2) and solve for V(t). Show that the solution is

V(t) = [V(0)/(c-g)][ce^(gt) - ge^(-ct)].

Equation 1 separation of variables. First solve that one.
• Jan 19th 2011, 08:29 PM
mathsohard
How do I solve that one ??? do you mean setting k = 0 and dT*/T = -gdt???
How can I solve that and substitute to the eq2???
• Jan 20th 2011, 07:11 AM
mathsohard
Can anyone help me more on this problem???
• Jan 20th 2011, 08:24 AM
islam
hello dear
i will make u the first equation
u have dT*/dt = kVT - gT* when k=0 it becomes
dT*/dt =0 - gT* which is:
dT*/dt = - gT*
now divide the whole equation by T* and multiply by dt u get:

dT*/T*=- gdt now integrate both sides u get
ln(T*)=-gt+c (c is a constant)
T*=e^(-gt+c)
best regards(Nod)
• Jan 20th 2011, 10:40 AM
mathsohard
ok I understood the first one, and then what should I do?
• Jan 20th 2011, 11:11 AM
mathsohard
I thought that first I had to put T*=e^(-gt + c) in to P(t) and plug that into eq 2 but that doesn't give me a proper solution... :(
Help !!!
• Jan 20th 2011, 12:41 PM
mathsohard
Umm..... or do I consider P as 0???? I have no idea how to do this please help
• Jan 20th 2011, 12:47 PM
dwsmith
Quote:

Originally Posted by mathsohard
Umm..... or do I consider P as 0???? I have no idea how to do this please help

Have you taken Differential equations?

Equation two can be separated as well.
• Jan 20th 2011, 12:53 PM
mathsohard
Equation two is not composed of multiplication... Can you give me more explanation please?
• Jan 20th 2011, 12:54 PM
dwsmith
Quote:

Originally Posted by mathsohard
Equation two is not composed of multiplication... Can you give me more explanation please?

After you answer if you have taking Differential Equations which I suspect is no.
• Jan 20th 2011, 12:59 PM
mathsohard
What do you mean by taking differential equations?? I am totally lost, I got T* by setting k = 0 from equation 1, which is Ce^gt if that is what you meant,
but I have no idea where to plug that in
• Jan 20th 2011, 01:08 PM
dwsmith
Quote:

Originally Posted by mathsohard
What do you mean by taking differential equations?? I am totally lost, I got T* by setting k = 0 from equation 1, which is Ce^gt if that is what you meant,
but I have no idea where to plug that in

If you ask a DE question, you should have some idea how to solve it.

Your second equations can be separated.

http://www.mathhelpforum.com/math-he...ial-38182.html

Number 2.
• Jan 20th 2011, 01:13 PM
mathsohard
How can I factor V out.... or just ignore P???
• Jan 20th 2011, 01:15 PM
dwsmith
Quote:

Originally Posted by mathsohard
How can I factor V out.... or just ignore P???

$\displaystyle\frac{dV}{dt}=(P-cV)\Rightarrow dV=(P-cV)dt=\cdots$
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