Could you please explain me the first term of second equation and third equation in the following model?

Question

I have attached the image of the model. Could you please explain me the first term of second equation and third equation in the following model. (I didn't get the meaning of terms in bracket.)

Answer 1

This is a density dependence interaction term between components $a_i$ and $m_i$ which represent parasites and antibodies respectively, with intrinsic growth rate $\rho$ and half saturation constant $\phi$.

Most simpler models have an interaction term that looks something like $\rho a_i m_i$ but there is an underlying physical assumption with this term, called mass action: namely that things are well mixed. In many cases such as an epidemiological model this is not a particularly good assumption, and while many models make such an assumption anyway, many also do not.

In ecological contexts things of the form $$f(p_i) = \frac{p_i}{p_i + \phi} $$ are called Hollings type II interactions, and if the $p_i$ terms were squared it would be Hollings type III: $$g(p_i) =\frac{p_i^2}{p_i^2+\phi^2}. $$

With both of these types of interactions they are density dependent in the sense that $$\lim_{p_i\rightarrow\infty} f(p_i) = \lim_{p_i\rightarrow\infty} g(p_i) = 1.$$ Meaning that no matter how abundant the parasite $$\rho a_i m_i > \rho a_i m_i\frac{p_i}{p_i + \phi} $$ (similar result for $g(p_i)$)

Thus as mentioned above in comments such interaction terms decrease the recrutment rate of component $a_i$.

Below I have included plots of a function of the form of $f(p_i)$, and another of $g(p_i)$, while restricting ourselves to the positive quadrant (assumption of biological models):

where Hollings type II is of form $f(x) = \frac{bx}{a+x}$ and Hollings type III is of form $g(x) = \frac{bx^2}{a^2+x^2}$