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
$N(t)$ 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 $\lambda$.