Finding differences amongst a system of differential equations

AI Thread Summary
The discussion focuses on the differences between two systems of differential equations modeling T-cell dynamics and virus interactions. System 1 includes the term πE in the equation for actively infected T-cells, while System 2 omits it, leading to a discrepancy in the representation of infected cell dynamics. Additionally, System 1 accounts for the loss term -βVR in the virus equation, which is absent in System 2, raising questions about the accuracy of both models. The contributors note that the omission of these terms in the referenced literature may be considered negligible, but they emphasize the importance of these differences in understanding the system's behavior. Overall, the conversation highlights the need for careful examination of mathematical models in biological contexts.
J6204
Messages
56
Reaction score
2

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


Variables
R(t): number of susceptible T-cells
L(t): number of latently infected T-cells
E(t): number of actively infected T-cells
V(t): amount of virus

Parameters
$$\Gamma$$: 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
assigment2.png
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?
 

Attachments

  • assigment2.png
    assigment2.png
    18.9 KB · Views: 617
Physics news on Phys.org
upload_2018-1-30_17-51-54.png


So set 1 comes from the above picture in Otto (A Biologist's Guide to Mathematical Modeling in Ecology and Evolution) and set 2 comes from Science.

Apparently, both the article in Science and Otto missed the ##-\beta RV## in ##V(t)## but not in ##R(t)##, but you did not. [edit] However, at the bottom in Box 2.4 (continued) they claim it is negligible.

The ##-\pi E## term does not appear in ##dE\over dt## because the infected cell produces the virus particles but stays intact. This is explained in Box 2.4 (continued). So there I side with the Science/Otto sets.

You can check your set of equations by adding them all up. That should yield the in- and outgoing solid arrows plus the ##\pi E## as mentioned.

[edit] my advice: take your time to read the whole thing :smile:
 

Attachments

  • upload_2018-1-30_17-51-54.png
    upload_2018-1-30_17-51-54.png
    15.1 KB · Views: 1,097
BvU said:
View attachment 219387

So set 1 comes from the above picture in Otto (A Biologist's Guide to Mathematical Modeling in Ecology and Evolution) and set 2 comes from Science.

Apparently, both the article in Science and Otto missed the ##-\beta RV## in ##V(t)## but not in ##R(t)##, but you did not. [edit] However, at the bottom in Box 2.4 (continued) they claim it is negligible.

The ##-\pi E## term does not appear in ##dE\over dt## because the infected cell produces the virus particles but stays intact. This is explained in Box 2.4 (continued). So there I side with the Science/Otto sets.

You can check your set of equations by adding them all up. That should yield the in- and outgoing solid arrows plus the ##\pi E## as mentioned.

[edit] my advice: take your time to read the whole thing :smile:

so is my equation for ##dE\over dt## correct for the figure? and what about the last equation ##dV\over dt## why is this different?
 
BvU said:
View attachment 219387

So set 1 comes from the above picture in Otto (A Biologist's Guide to Mathematical Modeling in Ecology and Evolution) and set 2 comes from Science.

Apparently, both the article in Science and Otto missed the ##-\beta RV## in ##V(t)## but not in ##R(t)##, but you did not. [edit] However, at the bottom in Box 2.4 (continued) they claim it is negligible.

The ##-\pi E## term does not appear in ##dE\over dt## because the infected cell produces the virus particles but stays intact. This is explained in Box 2.4 (continued). So there I side with the Science/Otto sets.

You can check your set of equations by adding them all up. That should yield the in- and outgoing solid arrows plus the ##\pi E## as mentioned.

[edit] my advice: take your time to read the whole thing :smile:

I was asked to compare the equations I have made to the equations given to find difference so I guess I was right? there should be some difference?
 
J6204 said:
so is my equation for dEdtdEdtdE\over dt correct for the figure?
No. E does not change when a virus is released.
J6204 said:
I was asked to compare the equations I have made to the equations given to find difference so I guess I was right? there should be some difference?
Was answered in an edit:
BvU said:
Apparently, both the article in Science and Otto missed the ##-\beta RV## in ##V(t)## but not in ##R(t)##, but you did not. [edit] However, at the bottom in Box 2.4 (continued) they claim it is negligible.

BvU said:
[edit] my advice: take your time to read the whole thing :smile:
 
I don't get how to argue it. i can prove: evolution is the ability to adapt, whether it's progression or regression from some point of view, so if evolution is not constant then animal generations couldn`t stay alive for a big amount of time because when climate is changing this generations die. but they dont. so evolution is constant. but its not an argument, right? how to fing arguments when i only prove it.. analytically, i guess it called that (this is indirectly related to biology, im...
Back
Top