Is it possible to determine the initial population given the final one?

It is possible to determine the initial population of, say, bacteria given a measure at a later time?

Let's say I use the logistic growth model

$$\dot N (t) = rN(t)\left(1 - \frac{N(t)}{K}\right)$$

and let's assume I know $$K$$. Let's imagine I have a mouse and I inject it with a dose of Shiga-toxin producing Escherichia coli (STEC). After 24 hours, the mouse dies, and I measure STEC with qPCR, obtaining a value of 20 million cells.

Can I calculate how many bacteria were in the initial shot, $$N_0$$?

And how many parameters do I need to know? For instance, let's say the mouse simply died some times after the injection. Can I know $$N_0$$ even if the absence of the time parameter?

How do I calculate $$K$$?

Is the logistic equation valid for live models? I understand it is good for chemostats, but is it valid for a mouse. Of course, a mouse can also reach saturation, since if $$K$$ is reached, the mouse dies, as in the example. Or is it a better description for bacterial growth in animals?

Thank you

• can you link any study which shows quantification of bacterial numbers using qPCR. I just want to see how it is done. Commented Nov 29, 2020 at 13:20
• The second here pubmed.ncbi.nlm.nih.gov/… explains also how to make the standard curve for the quantification... Commented Nov 29, 2020 at 20:14

For the logistic equation, $$\frac{\mathrm{d}N(t)}{\mathrm{d}t} = rN(t)\left(\frac{K - N(t)}{K}\right)$$ the solution that allows you to calculate population population size at any time is: $$N(t) = \frac{N(0)Ke^{rt}}{K - N(0) + N(0)e^{rt}}.$$ Looking at this equation algebraically with a set of knowns—$$K$$, $$t$$, and $$N(t)$$—and a set of unknowns—$$r$$ and $$N(0)$$—we see with this one equation that we have two unknowns, which are the intrinsic rate of growth and initial population size. So, the short answer is there is insufficient information to know the initial population size without additionally knowing the intrinsic rate of growth.
Calculating $$K$$ is problematic since $$K$$ is a maximum value. Better would be to use the original Verhuslt logistic equation, where the parameter $$\alpha$$ represents the decrease in population growth rate as a function of individuals added to the population: $$\frac{\mathrm{d}N(t)}{\mathrm{d}t} = rN(t) - \alpha N(t)^2.$$ Note here that is the same if equation if $$K = r/\alpha$$. In the $$K$$ equation, $$K$$ is conflated as both a maximum and an equilibrium, which is never the case in real populations or any extension of the logistic model (e.g., including harvest or species interactions).