Why Does This Differential Equation Epidemic Model Stump Me?

In summary, the problem is that the CAS gave me a term that was switched and I cannot isolate the variable Y.
  • #1
Kudaros
18
0
I have a logistic issue with differential equations. I have spent four hours working on this problem and it is way past the point of ridiculous.

If anyone can help out that would be great.

It begins like this.

dy/dt = k(P-y)*y. It is an epidemic model, where k is a positive constant relating the rate of infection. P is total population involved. An initial condition of y(0)=t is given. I am to find this particular solution.

The y is multiplied. Strange that it isn't shown as ky(p-y) but perhaps its a hint that I am not getting.

Anyway, I am treating this as a separable equation as that is the tool we are given (the method).

I have tried numerous approaches and have gotten the farthest with this one.

dy/(y(p-y)= kdt

Partial fractions on the left side where A and B are both = to 1/p.

1/py + 1/(p(p-y)) dy = kdt

Then integrate both sides.

ln(y)/p - ln(p-y)/p = kt + c.

Here is where the problem begins (unless I screwed up earlier). A CAS gives me -ln(y-p) (second term left side) . Can anyone explain this? I integrated by U substitution. Why would the variable and constant be switched?

I continued on assuming the CAS was correct and now I cannot isolate Y variable. I have never had this trouble before with math but hey I guess that's the way it goes.

Any help would be greatly appreciated.
 
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  • #2
trust your math skills, not mathematica! (or whatever program you're using). the difference arises if you factor a negative one out before integrating. thus, you are then integrating - 1/p int (1/y-p) using u substitution now requires no extra negative for du. make sense?
 
  • #3
Remember that:
[tex]\int\frac{dx}{x}=\ln(|x|)+C[/itex]
 
  • #4
p-y also makes logical sense, as y is the number of infected persons and cannot be greater than p, ensuring a nonnegative number.

Continuing along the problem however, it seems I cannot isolate the variable Y due to the term p-y. I have attempted numerous approaches finding that same problem. Is there another way to solve differential equations, in general, using the initial condition but without isolating Y?

More specific to this problem, are there any assumptions that can be made about p to eliminate it all together? (a longshot I realize.)

Thanks again!
 

Related to Why Does This Differential Equation Epidemic Model Stump Me?

1. What is a differential equation epidemic model?

A differential equation epidemic model is a mathematical model that is used to describe the spread of an infectious disease within a population. It is based on the principles of differential equations, which are equations that describe how a quantity changes over time.

2. How does a differential equation epidemic model work?

A differential equation epidemic model works by using a set of differential equations to describe the interactions between individuals in a population and the spread of an infectious disease. These equations take into account factors such as the rate of infection, recovery rate, and contact rate between individuals.

3. What are the assumptions made in a differential equation epidemic model?

The assumptions made in a differential equation epidemic model include the assumption of a closed population, where there is no migration in or out of the population, and the assumption of homogeneity, where all individuals in the population are considered to have the same characteristics and level of susceptibility to the disease.

4. How accurate are differential equation epidemic models?

The accuracy of a differential equation epidemic model depends on the quality of the data and the assumptions made. While these models can provide valuable insights and predictions, they are not always accurate due to the complex nature of disease spread and the potential for unforeseen events or variables.

5. What are the limitations of a differential equation epidemic model?

The limitations of a differential equation epidemic model include the assumptions made, the potential for inaccuracies due to incomplete or faulty data, and the inability to account for individual behaviors and variations in disease transmission patterns. These models also do not take into account factors such as interventions or changes in behavior that can affect the spread of the disease.

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