Infectious Disease Spreads in Cattle Herd: Find # Infected After 6 Days

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In summary, the conversation discusses a problem involving an infectious disease spreading in a herd of cattle. The rate of spread is proportional to the product of the number of infected and uninfected cattle. With an initial condition of 500 uninfected cattle and one infected cow joining the herd, the problem is solved using a differential equation and integration by partial fraction. The final answer is 500 divided by the exponential of the difference of the natural log of 499 and time, plus 1.
  • #1
AAO
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Homework Statement


[/B]
A carrier of an infectious disease joins a herd of 500 initially uninfected cattle. At any
instant in time, the rate at which the disease spreads dx/dt is known to be proportional to
the product of:
(i) the number of infected cattle x(t); and
(ii) the number of uninfected cattle.
If the number of cattle infected after 4 days is 50, how many will have been infected after
6 days?

Homework Equations



The Attempt at a Solution


I interpreted the problem as the following D.E:
dx/dt=k*x*(500-x); where k is a constant to be determined

And the initial condition: x(0)=0
And the given information: x(4)=50

The equation looks to be separable:
dx/(500*k*x-k*x^(2)) = dt

but I cannot integrate the left hand side.

Is my interpreted D.E correct? If yes how to integrate the left hand side?

Appreciate your help.
 
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  • #2
AAO said:
And the initial condition: x(0)=500
And the given information: x(4)=50

Your initial condition here is saying that 500 kettle are infected at t=0 ...
 
  • #3
Orodruin said:
Your initial condition here is saying that 500 kettle are infected at t=0 ...

Thanks I corrected it. Any hints about the solution to this DE?

I tried to use Matlab:
>> syms x(t) k
>> x(t) = dsolve(diff(x,t) == k*x(t)*(500-x(t)) , x(0)==0)

But I got
x(t) = 0

!
 
  • #4
Try splitting the integrand into two by partial fraction decomposition.

Edit: Also, your initial condition is not x(0) = 0. Obviously, if there are no infected cattle, there will be no spread of the disease.
 
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  • #5
How many infected cattle are there at t=0?
If it is zero, transmission rate will surely be zero.
 
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  • #6
AAO said:
Thanks I corrected it. Any hints about the solution to this DE?

I tried to use Matlab:
>> syms x(t) k
>> x(t) = dsolve(diff(x,t) == k*x(t)*(500-x(t)) , x(0)==0)

But I got
x(t) = 0

!
x(0) is not zero. The problem said that one infected cow joined the others, so (presumably), x(0) = 1.
 
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  • #7
Many Thanks All for your great help.

I managed to get the answer assuming that x(0)=1, and performing the integration using partial fraction.

The final answer is:
500
---------------------------
exp(log(499) - t) + 1

I also confirmed this using Matlab.

Once again, thank you very much.
 
  • #8
Shouldn't your original DE have been$$\frac {dx}{dt}= kx(501-x),~x(0) = 1$$
 
  • #9
I believe you are right LCKurtz, I should have done this. Thanks a lot for the hint.
 

FAQ: Infectious Disease Spreads in Cattle Herd: Find # Infected After 6 Days

1. How does infectious disease spread in a cattle herd?

Infectious diseases in cattle herds can spread through direct contact with infected animals, contaminated objects or surfaces, or through the air. Some diseases can also be transmitted through insects or other vectors.

2. What are some common symptoms of infectious diseases in cattle?

Some common symptoms of infectious diseases in cattle include fever, loss of appetite, weight loss, decreased milk production, respiratory issues, and diarrhea. However, symptoms can vary depending on the specific disease.

3. How long does it take for an infectious disease to spread through a cattle herd?

The time it takes for an infectious disease to spread through a cattle herd can vary depending on the specific disease, management practices, and environmental factors. In some cases, it can take just a few days, while in others it may take weeks or even months.

4. How can the spread of infectious diseases in cattle herds be prevented?

Preventing the spread of infectious diseases in cattle herds involves implementing biosecurity measures such as quarantining sick animals, practicing proper hygiene, and limiting contact with other herds. Vaccinations can also help protect against certain diseases.

5. How can we determine the number of infected cattle in a herd after 6 days?

The number of infected cattle in a herd after 6 days can be determined by regularly monitoring and testing the herd for symptoms of disease. This can include physical exams, blood tests, and other diagnostic tests. It is important to consult a veterinarian for accurate and timely detection of infectious diseases in cattle herds.

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