I am working with a system that includes pharmacokinetic and pharmacodynamic elements. To model the antibiotic effect on bacteria I am using a commonly used Hill type function as,${(\phi_{max} - \phi _{min}){({A \over MIC})}^k}\over {{({A \over MIC})}^k-{\phi_{min} \over \phi_{max}}}$,
where,
$\phi_{max}$ – maximum bacteria growth rate in the absence of antibiotic which is only limited by resources, and so it i limited by the carrying capacity term as, $\phi_{max}=r(1-{P \over k})$, where $k$ is the carrying capacity term and $r$ is the replication rate, and $P$ is the bacterial density that change over time.
$\phi_{min}$ – minimum bacterial growth rate at high concentrations of the antibiotic.
$MIC$-minimum inhibitory concentration of the antibiotic
$k$ - Hill coefficient.
$\phi_{min},k,MIC$ are all constant values.
$A$ is the antibiotic concentration which decays exponentially so that ${dA\over dt}=-dA$
So, my question is, in almost all the articles that I found, the value of $\phi_{min}$ was selected as a negative value. But it is defined as a bacterial growth rate. So, how can a growth rate be negative?
When this function was used in my model and the $\phi_{min}$ value was changed from, negative values to positive values, for negative values I get the desired behaviour (the pathogen get cleared). But, for some positive values, in Matlab I get a warning as,
Unable to meet integration tolerances without reducing the step size below the smallest
value allowed (1.818989e-12) at time t
Could this be because, that it is possible to come up with a situation where, $\phi_{min}=\phi_{max}$ and ${({A \over MIC})}^k=1$, so that the above function reaches $0 \over 0$ case.
I believe this does not happen when $\phi_{min}$ is negative, does it?
In articles, they just chose low negative value for antibiotics that has high killing ability (e.g $\phi_{min}=-3$)and for lesser effective antibiotics $\phi_{min}=-0.01$.
I would like to know if there is mathematical reason for choosing $\phi_{min}$ to be negative? Is my reasoning on $0 \over 0$ case reasonable?