Determing the differences between two sets of differential eqs

[J6204]

Homework Statement

Given the following figure and the following variables and parameters, I have been able to come up with the set of differential equation below the image. My question is how does the system of equations 1 which I produced myself differ from the set of equations 2. Below I have a further explanation of this question. The image below was used to create my system of equations 1.

Homework Equations

$\Gamma$: rate of production of susceptible T-cells
$\tau$: fraction of T-cells susceptible to attack by HIV
$\mu$: removal rate of T-cells
$\beta$: rate of T-cell infection
p: fraction of infected T-cells that are latently infected
$\alpha$: rate that latent T-cells become activated
$\delta$: death rate/removal of actively infected T-cells
$\pi$: rate that virus is produced by actively infected T-cells
$\sigma$: rate of virus removal

System of Equations 1
$\frac{dR}{dt} = \Gamma \tau - \mu R - \beta VR$
$\frac{dL}{dt} = p \beta VR-\mu L - \alpha L$
$\frac{dE}{dt} = (1-p)\beta V R+ \alpha L - \delta E - \pi E$
$\frac{dV}{dt} = \pi E - \sigma V - \beta V R$

System of Equations 2
$\frac{dR}{dt} = \Gamma \tau - \mu R - \beta VR$
$\frac{dL}{dt} = p \beta VR-\mu L - \alpha L$
$\frac{dE}{dt} = (1-p)\beta V R+ \alpha L - \delta E$
$\frac{dV}{dt} = \pi E - \sigma V$

The Attempt at a Solution

So clearly there is a difference between the number of infected T cells in system of equations 1 and 2. System of equations 1 includes the term $\pi E$ while system of equations 2 does not in equation 3. Why is this?

There is a difference between the amount of virus in system of equations 1 and 2. System 1 includes the loss of term $\beta VR$ while the system of equations of 2 in equation 4. Why is this?

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[Mark44]

The image below was used to create my system of equations 1.

@J6204, the image you posted is essentially unreadable, as 2/3 of it is blacked out. Please post another image without the extra blacked-out spaces.

 

[J6204]

@J6204, the image you posted is essentially unreadable, as 2/3 of it is blacked out. Please post another image without the extra blacked-out spaces.

I have edited that in, could you help me?

 

[bigfooted]

Well, for your first question, virus is made by using a fraction $\pi$ of the infected cells E. This fraction of infected cells is directly used to create the virus. What happens to the cells $\pi E$ in scenario 1 and scenario 2?

As for the other term: a fraction of the virus is used to actively infect susceptible cells. So what happens to this fraction after they have infected the susceptible cells?

 

[J6204]

Well, for your first question, virus is made by using a fraction $\pi$ of the infected cells E. This fraction of infected cells is directly used to create the virus. What happens to the cells $\pi E$ in scenario 1 and scenario 2?

As for the other term: a fraction of the virus is used to actively infect susceptible cells. So what happens to this fraction after they have infected the susceptible cells?

well for pi E isn't doesn't appear in the system of equations 2, so is it because the infected cell produces the virus particles but stays intact, or maybe they are they negligible in the second system?

 

[bigfooted]

well for pi E isn't doesn't appear in the system of equations 2, so is it because the infected cell produces the virus particles but stays intact, or maybe they are they negligible in the second system?

A virus can be produced from the actively infected cells without destroying the infected cells (the process is called budding).
In the same way, a virus can infect cells without destroying itself.

 

[J6204]

A virus can be produced from the actively infected cells without destroying the infected cells (the process is called budding).
In the same way, a virus can infect cells without destroying itself.

does this mean that it can stay at actively infected cells or it can move to a virus? meaning it doesn't have to go from actively infected cell to virus it can stay at actively infected?

 

[epenguin]

does this mean that it can stay at actively infected cells or it can move to a virus? meaning it doesn't have to go from actively infected cell to virus it can stay at actively infected?

This is more readable and comprehensible than the version you have posted under Biology

Simply if the π process is not there, or you could say if π = 0, the actively infected cells do not cause production of new virus.

Just by looking at the scheme you should be able to say what the long-term tendencies of the various schemes will be, that is what the situation will be after a long time. This is what you usually try to do, before trying to solve any equations.

If you expect that after a long time there is a stationary state, you then solve the equations for that stationary state which are just algebraic equations, not differential equations. Only after that you solve the differential equations. And in fact probably you don’t even go for solving the full differential equations immediately, but first solve simpler version(s) that you get by making simplifying assumptions.

Anyway your next job is to tell us what you think happens long-term.