I have a system of ODEs with a pathogen population $(P)$ being modelled as a logistic growth as,
${dP\over dt}=rP(1-{P\over k})$, where $r$ is the replication rate and $k$ is the carrying capacity term and both these parameters are constant values.
Now I want to introduce antibiotic effect so that the antibiotic will inhibit the growth of pathogen.For this I am planning on introducing a constant $\alpha \in (0,1)$ so that the replication will be reduced as $\alpha r$. But I want this replication to depend on the antibiotic concentration. So if I model the system as,
${dP\over dt}=\alpha A rP(1-{P\over k})\\
{dA\over dt}=-dA$
where $A$ is the antibiotic concentration, will this be right? ($d$ is the antibiotic decay rate)
Or, should this be,
${dP\over dt}= \alpha {1\over A} rP(1-{P\over k})\\ {dA\over dt}=-dA$
as, when $A$ is high the replication should be reduced more (because the antibiotic will be more effective at high concentrations), and as $A$ decreases the reduction in the replication will be less?
Out of these two systems which one maps this relationship correctly or is there any other way to model this?
HINT :
$${dP\over dt}=\alpha A rP(1-{P\over k})$$ $${dA\over dt}=-DA$$
Solving the separable ODE ${dA\over dt}=-DA$ leads to $$A=c_1e^{-Dt}$$
$$\frac{dP}{dt}=\alpha c_1e^{-Dt} rP(1-\frac{P}{k})$$
Again this is a separable ODE : $$\frac{k\:dP}{P(k-P)}=\alpha c_1re^{-Dt} dt$$ $$k\int \frac{dP}{P(k-P)}=\alpha c_1r\int e^{-Dt} dt$$ $$\ln\left|\frac{P}{k-P}\right|=-\frac{\alpha c_1r}{D}e^{-Dt}+c_2$$ $$\frac{P}{k-P}=c_3\exp\left(-\frac{\alpha c_1r}{D}e^{-Dt}\right)$$ Then you can solve it for $P$.
Compute the $c_1$ and $c_3$ according to the initial conditions.
On the same way, in case of $${dP\over dt}=\alpha \frac{1}{A} rP(1-{P\over k})$$ $${dA\over dt}=-DA$$ the result is : $$\frac{P}{k-P}=c'_3\exp\left(\frac{\alpha r}{c'_1D}e^{Dt}\right)$$ Then solve it for $P$.
Compute the $c'_1$ and $c'_3$ according to the initial conditions.
Now, compare the behaviour of the two functions $P(t)$ and determine which one is convenient for the model.