Writing Differential equations to describe a system

Question

I have a system that models the interaction between a pathogen an the immune response. If $P$ is the pathogen and $I$ represents immune response, the differential equations of the system are:

$$ \begin{alignat*}{1} {\mathrm{d}P\over \mathrm{d}t} &= r_1P \left(1-{P\over k}\right)-d_1P \left({I\over I+\sigma}\right)\\ {\mathrm{d}I\over \mathrm{d}t} &= r_2I \left({P \over P+\sigma_2}\right)-d_2I \end{alignat*} $$

This is somewhat similar to this article.
$r_i$, $k$, and $\sigma$ are constants. $\sigma$ represents the pathogen density when the immune response is at half its maximal capacity. $d_i$ is the killing rate through immune response.

I want to change these equations so that some of the pathogens that interact with the immune response do not get killed. So I want it to be modelled such that immune-system cells will engulf pathogens, but a portion of pathogens can survive within the immune-system cells and will not get killed.

If $P_S$ represents the population that survives and if $\alpha$ the proportion of the pathogen that interacts with the immune system and can survive killing, if I change the equations in the following way will it be correct?

$$ \begin{alignat*}{1} {\mathrm{d}P\over \mathrm{d}t} &= r_1P\left(1-{P\over k}\right)-(1-\alpha)d_1P\left({I\over I+\sigma}\right)\\ {\mathrm{d}I\over \mathrm{d}t} &= r_2I\left({P \over P+\sigma_2}\right)-d_2I \\ {\mathrm{d}P_s\over \mathrm{d}t} &= \alpha d_1 P\left({I\over I+\sigma}\right) \end{alignat*} $$

But I am not sure if I should use the death rate $d_1$ for the surviving population $P_s$?

Or is there any other way to show that a proportion of the pathogens that interact with the immune response move into a different compartment that includes surviving pathogen?

Answer 1

1. $\sigma$ is the density of immune cells, not pathogens, at which the immune response is at half of its max capacity.
2. You can consider $d_1$ as the rate at which pathogen cells are "absorbed" by immune cells. All these cells disappear from the original population and hence, the first equation should remain unchanged: $$\dot{P}=r_1P(1−\frac{P}{k})−d_1P(\frac{I}{I+σ})$$
3. Then you split the absorbed pathogens into two groups: survived and not survived. The survived are described by $$\dot{P}_s=\alpha d_1 P\frac{I}{I+\sigma}$$ and the not survived disappear. The death rate of pathogens is $(1-\alpha)d_1$
4. Now you can modify the DEs to take into account the further evolution of those that survived within immune cells. You can add a growth term to account for their propagation within the cell, a term that describe the pathogens that come out from the cell. Finally, there could be a term describing the death rate of the pathogens within the immune cell, i.e., describing those pathogens that did not die during the absorption process, but die as time goes on. So, a possible model could look as follows: $$\begin{aligned}\dot{P}&=r_1P(1−\frac{P}{k})−d_1P(\frac{I}{I+σ})+\gamma P_s\\ \dot{P}_s&=r_2P_s(1−\frac{P_s}{k_2})+\alpha d_1 P\frac{I}{I+\sigma}-\gamma P_s-\delta P_s,\end{aligned}$$ where $r_2$ and $k_2$ are the growth rate of pathogens within immune cells and the respective capacity; $\gamma$ is the inverse of the mean residence time within the cell (i.e., the rate at which pathogens come out of the cell) and $\delta$ is the inverse life duration of a pathogen (that is the rate at which they die).