Given the following figure and the following variables and parameters, I have been able to come up with the set of differential equation below the image. My question is how does the system of equations $1$ which I produced myself differ from the set of equations $2$. Below I have a further explanation of this question. The image below was used to create my system of equations $1$.
Variables
$R(t)$: number of susceptible T-cells (number, $ R(t) \geq$ 0).
$L(t)$: number of latently infected T-cells (number, $L(t) \geq 0).$
$E(t)$: number of actively infected T-cells (number, $E(t) \geq 0).$
$V(t)$: amount of virus (number, $V(t) \geq 0).$
Parameters
$\Gamma$: rate of production of susceptible T-cells (number/time, $\Gamma \geq 0).$
$\tau$: fraction of T-cells susceptible to attack by HIV (unitless, $0 \leq \tau \leq 1).$
$\mu$: removal rate of T-cells (1/time, $\mu\geq0$).
$\beta$: rate of T-cell infection (1/number/time, $\beta>0$).
$p$: fraction of infected T-cells that are latently infected (unitless, $1 \geq p \geq 0)$.
$\alpha$: rate that latent T-cells become activated (1/time, $\alpha \geq 0).$
$\delta$: death rate/removal of actively infected T-cells (1/time, $\delta \geq \mu).$
$\pi$: rate that virus is produced by actively infected T-cells (1/time, $\pi >0$).
$\sigma$: rate of virus removal (1/time, $\sigma >0$).
System of Equations 1 $$\frac{dR}{dt} = \Gamma \tau - \mu R - \beta VR $$ $$\frac{dL}{dt} = p \beta VR-\mu L - \alpha L$$ $$\frac{dE}{dt} = (1-p)\beta V R+ \alpha L - \delta E - \pi E$$ $$\frac{dV}{dt} = \pi E - \sigma V - \beta V R$$
System of Equations 2 $$\frac{dR}{dt} = \Gamma \tau - \mu R - \beta VR $$ $$\frac{dL}{dt} = p \beta VR-\mu L - \alpha L$$ $$\frac{dE}{dt} = (1-p)\beta V R+ \alpha L - \delta E $$ $$\frac{dV}{dt} = \pi E - \sigma V $$
So clearly there is a difference between the number of infected T cells in system of equations $1$ and $2$. System of equations $1$ includes the term $\pi E$ while system of equations $2$ does not in equation 3. Why is this?
There is a difference between the amount of virus in system of equations $1$ and $2$. System 1 includes the loss of term $\beta VR$ while the system of equations of $2$ in equation 4. Why is this?
