Determing the differences between two sets of differential eqs

In summary, the conversation discusses two different systems of equations for a virus infection model. The main difference between the two systems is the inclusion of a term for the production of virus particles by actively infected cells in system 1, while system 2 does not have this term. This leads to differences in the amount of infected cells and virus particles in the two systems. The conversation also delves into the process of virus production and infection by actively infected cells.
  • #1
J6204
56
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.

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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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  • #2
J6204 said:
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.
 
  • #3
Mark44 said:
@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?
 
  • #4
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?
 
  • #5
bigfooted said:
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?
 
  • #6
J6204 said:
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.
 
  • #7
bigfooted said:
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?
 
  • #8
J6204 said:
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.
 

FAQ: Determing the differences between two sets of differential eqs

1. What is the purpose of determining the differences between two sets of differential equations?

The purpose of determining the differences between two sets of differential equations is to compare and contrast their solutions and understand how they behave differently under different conditions. This can provide insights into the underlying systems and help in making predictions and decision-making in various fields such as physics, engineering, and economics.

2. How do you determine the differences between two sets of differential equations?

The differences between two sets of differential equations can be determined by first solving each set separately and then comparing the solutions. This can be done by graphing the solutions or by calculating the differences between the values at specific points. Another method is to use mathematical techniques such as substitution and elimination to manipulate the equations and identify their differences.

3. What are the main challenges in determining the differences between two sets of differential equations?

One of the main challenges in determining the differences between two sets of differential equations is the complexity of the equations and the number of variables involved. Another challenge is ensuring that the equations are solved accurately and consistently. Additionally, the interpretation of the differences and their significance can also be challenging, as it requires a deep understanding of the underlying systems and their behavior.

4. In what fields or applications is determining the differences between two sets of differential equations commonly used?

Determining the differences between two sets of differential equations is commonly used in fields such as physics, engineering, economics, and biology. It is also used in various applications such as modeling and predicting the behavior of physical systems, analyzing economic systems, and understanding biological processes.

5. Can determining the differences between two sets of differential equations be automated?

Yes, determining the differences between two sets of differential equations can be automated using computer programs and software. These tools can solve the equations and compare the solutions, making the process faster and more accurate. However, it is still important for a scientist to have a deep understanding of the underlying systems and the equations to interpret the results and identify any potential errors or limitations of the automated process.

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