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There are data on a pathogen population $N(t)$ measured over time (t). I am using the ODE of, ${dN\over dt}=-\lambda N$ to model its decline over time.

So, $ N(t)=N_0 e^{- \lambda t}$.
However, I want to obtain a maximum likelihood function related to $N(t)$. For that I have to get the probability distribution function to represent $N(t)$.

To convert $N(t)$ to a PDF, I used a normalizing factor $c$,
and $c={\lambda\over N_0}$.

So, can I represent $N(t)$ as a exponential distribution? However, as $N(t)$ is discrete I don't know what the distribution should be.

The final aim is to using maximum likelihood estimation (MLE), to estimate the parameter $\lambda$.
The data is as follows:

$t=[0;1;2;3;4;5;6]; $

$N(t)=[350000,210000,80000,20000,100,100,100]; $

Although I can simply fit the $N(t)$ data using a method such as least squares, I want to know how to do it using MLE and how to find the PDF

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$N(t)$ is a deterministic population model, telling you population at time $t$. It decays with parameter $\lambda$.

You wish to estimate the parameter $\lambda$.

You have the observations $(t, N(t))$ for various $t$.

What is the likelihood of observing $(t, N(t))$ for a specific $t$?

You would need to come up with a sensible model (with parameter $\lambda$) for this. At present, your model is deterministic. If $N(t) \sim Exp( \lambda t)$, that means that you'd expect $N(t)$ to take large values with less probability as $t$ increases, which seems fairly reasonable.

At present, your $N_0 e^{-\lambda t}$ isn't a probability density, not sure where you got that idea.

Define $f_t(x;\lambda)$ to be the density function at population size $x$ with parameter $\lambda$ at time $t$.

Then what you'd do is take the likelihood of your data.

Likelihood $ = \prod_{t=1}^T f_{t}(N(t); \lambda)$.

Take the natural logarithm of your likelihood.

To maximise it, you differentiate the logarithm you just found with respect to $\lambda$, then set it equal to zero.

To be honest what I'd do here is do a least squares fit of $\sum_{t=1}^{T} (N(t)-N_0 e^{-\lambda t})^2$ and minimise this under $\lambda$.

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  • $\begingroup$ Thank you for the answer. Actually, the part that I am struggling with is defining $f_t(x;\lambda)$. Can you please give any help on defining that. I know $N_0 e^{-\lambda t}$ is not a probability density and that's why a normalizing factor of $c={\lambda\over N_0}$ was used as the PDF should integrate to 1. $\endgroup$ – sam_rox Apr 5 at 7:35
  • $\begingroup$ Yeah I can see the problem. I was thinking about $f_t(x;\lambda) = c e^{-\lambda t x}$. Then the chance of getting a high value for $x$ goes down as $t$ increases. $\endgroup$ – George Dewhirst Apr 5 at 7:38
  • $\begingroup$ If it is$ f_t(N(t);\lambda) = c e^{-\lambda t N(t)}$ does it integrate to 1? Shouldn't it be $ f_t(N(t);\lambda) = c e^{-\lambda t \over N(t)}$ $\endgroup$ – sam_rox Apr 5 at 7:57
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    $\begingroup$ Yeah it integrates to $1$ by changing what $c$ is $\endgroup$ – George Dewhirst Apr 5 at 8:04
  • $\begingroup$ @ George Dewhirst Thank you very much for helping me out so far. I also calculated the cumulative distribution function (CDF) and the expected population size of time $t$ of the PDF $f_t(N(t);\lambda) = c e^{-\lambda t N(t)}$. Can you please let me know if these are correct. CDF= $1-{N_0e^{-\lambda N(t)t}\over N(t)}$ and Expected value= $1 \over \lambda N_0$ $\endgroup$ – sam_rox Apr 5 at 11:23

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