# Math Help - HIV model

1. ## 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)].

2. 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.

3. 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???

4. Can anyone help me more on this problem???

5. 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

6. ok I understood the first one, and then what should I do?

7. 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 !!!

8. Umm..... or do I consider P as 0???? I have no idea how to do this please help

9. 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.

10. Equation two is not composed of multiplication... Can you give me more explanation please?

11. 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.

12. 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

13. 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.

14. How can I factor V out.... or just ignore P???

15. 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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