Solving an ODE Problem: Calculating Deaths & Synthesis

[Apostolos]

Hi everyone,

I have a problem understanding an ODE and using it to find something particular. Consider the following :

ODE : dC/dt= S-r*C
where

S: synthesis rate
r : death rate
C: population
Co: initial population

the analystical solution is simply C(t) =S/r -(S/r-Co)*exp(-r*t)

My questions are : i) How can we calculate the population that has died at time t
ii) How can we calculate the population that has synthesized at time t
My problem in understanding is relevant the rates S and r and how these can be applied to evaluate the exact quantity of population that has followed death and generation.
So If I want to find how many have died (or synthesized) at time t (given r or S) is it simply the multiplication with C(t) or with C(t-1)?

I would appreciate your help with this problem. Thanks a lot.

 

[Marioeden]

Firstly, I'd have expected the equation to be dC/dt = (S-r)*C
because otherwise if you have r=0, i.e. no people dying,
then your equation says dC/dt = S which means C = Co + St
which is counter-intuitive as you would expect exponential growth
so you might want to just double check that!

Anyways, with regards to the population that has died...
Let D be the population that have died at time t.

Then we can say that dD/dt = r*C
and as we know C explicity as a function of time, we can just plug in and integrate.

With regards to your Synthesis rate, let X be the total population synthesised at time t.
Now, according to your current model, dX/dt = S, so X = Xo + St (which as I mentioned before is counter-intuitive).

If your initial ODE was dC/dt = (S-r)*C,
then you have that dX/dt = S*C
and as you have C explicity, just plug in and integrate.

Hope that all makes sense and answers your questions :)

 

[Apostolos]

Thank you very much for your response! That finally makes sense to me... :)

However, the model takes into acount protein population and therefore the synthesis rate is independent of the current population...

Apostolos

 

[Apostolos]

Dear Marioeden,

something is wrong with the solution you have provided. I don't know why but the died population is greater than the current population when I use the method you suggested!

I have integrated as you said using the analytic solution for C and I get :

D(t)=s*t+(Co-s/r)*(1-exp(-r*t))+Do

I have used as initial conditions Do=0 and Co=0. s=2 ,r =0.0693

Do you have any idea why I get this imbalance? Is that the cumulative result?

I would appreciate any help on that! Thanks a lot!

Apostolos

 

[Apostolos]

I am wondering...considering the previous model.

Is it true that ??:

C(current time)=

C(prev.time) + S(current time) - r*[ C(prev.time) + S(current time)]

where C : population
S : newly synthesised polulation (assume that is independt of population size(const.))
r : death rate of population (assume constant)
and if also the population that dies at any time t is labeled with D we must have :

D(current time)=r*[ C(prev.time) + S(current time)] ??


Please anyone who can take a look at it and let me know an opinion about it.

Thanks!

 

[Marioeden]

Dear Marioeden,

something is wrong with the solution you have provided. I don't know why but the died population is greater than the current population when I use the method you suggested!

I have integrated as you said using the analytic solution for C and I get :

D(t)=s*t+(Co-s/r)*(1-exp(-r*t))+Do

I have used as initial conditions Do=0 and Co=0. s=2 ,r =0.0693

Do you have any idea why I get this imbalance? Is that the cumulative result?

I would appreciate any help on that! Thanks a lot!

Apostolos

This is because the D that we've found is the total population that have died.

For example, we'd expect that over the history of the Earth, more people have died in total than are currently living now.

As such if you want to find the population that died in the interval [T, T+K],
then just evaluate D(T+K) - D(T)

 

[Apostolos]

So that's the cumulative...as I thought!
By the way, I think an easier way to evaluate that and in the same time avoid the error of the exponential terms that comes after integration would be to say simply that :
D(t)=r*C(t-1)

...which was and my initial thought.

Thanks a lot for your responses! I appreciate it!

Apostolos

 

[Marioeden]

So that's the cumulative...as I thought!
By the way, I think an easier way to evaluate that and in the same time avoid the error of the exponential terms that comes after integration would be to say simply that :
D(t)=r*C(t-1)

...which was and my initial thought.

Thanks a lot for your responses! I appreciate it!

Apostolos

You'll have to explain your reasoning behind that, I don't quite see why D(t) = r*C(t-1)
Unless you're dealing with finite time intervals, but if that were the case you'd be solving difference equations, not differential equations

 

[Apostolos]

Indeed I am solving the problem over a finite time interval.I just use the ODE to find the numerical values at the time points of interest after discretisation. Is that wrong you think?
If the model considers two parallel processes, death and synthesis with rates r and s, how do these rates apply ?
Lets say at t=0 there is an initial population and no death or generation. At t=1 a proportion r of the initial population dies while population S is generated. This forms the population at time t=1. At t=2 the same aplies over the population C(1) and so on...
Is not that the same with the assumption of the ODE model?

I have checked it and numerically very small difference occurs when I aim to find the population of died population using the ODE or by using the recursive relationship I prementioned.

If it is clear to you and there is something totally wrong please let me know your opinion to make corrections in my methods.

Thank you

Apostolos

 

[Marioeden]

Ah, right, so you actually want to be using difference equations, not differential equations!

So, ODE's are when you have a continuous model i.e. at every infinitesimal point in time there is synthesis and death, but here you clearly have a discrete problem.

In other words, you want to solve the equation C(t+1) = S - r*C(t)
To solve this, first solve the homogeneous equation:
C(t+1) + r*C(t) = 0
which has the solution C(t) = A*(-r)^t
where A is some arbitrary constant.

Then finding a particular solution to the forced equation, you have C(t) = S/(1+r)

Thus the solution is given by C(t) = s/(1+r) + [Co - S/(1+r)]*(-r)^t
where Co = C(0)

The interesting thing here, is that the population tends to a stable limit of C = S/(1+r)Lastly, if synthesis actually did depend on the population at that time, then you have
C(t+1) = (S-r)*C(t)
which yields that solution:
C = Co(S - r)^t

 

[Chestermiller]

Hi everyone,

I have a problem understanding an ODE and using it to find something particular. Consider the following :

ODE : dC/dt= S-r*C
where

S: synthesis rate
r : death rate
C: population
Co: initial population

the analystical solution is simply C(t) =S/r -(S/r-Co)*exp(-r*t)

My questions are : i) How can we calculate the population that has died at time t
ii) How can we calculate the population that has synthesized at time t
My problem in understanding is relevant the rates S and r and how these can be applied to evaluate the exact quantity of population that has followed death and generation.
So If I want to find how many have died (or synthesized) at time t (given r or S) is it simply the multiplication with C(t) or with C(t-1)?

I would appreciate your help with this problem. Thanks a lot.

This is an interesting question. The cumulative amount that has been synthesized up to time t is just St. This plus C_o would be the amount you would have if the rate of destruction were zero.
The amount you actually have left at time t is C(t). So that amount that died must be St + C_o minus that amount that is left at time t. You can also obtain this same result by using the original differential equation to integrate rC (the destruction rate) from t = 0 to t = t.