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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 $1$. enter image description here

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?

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  • $\begingroup$ @David maybe your browser is being extra picky. My Chrome shows a valid cert for the image, which is on imgur.com, a widely used site. $\endgroup$ – mgkrebbs Jan 30 '18 at 22:32
  • $\begingroup$ @mgkrebbs — That was Safari, Mac. Firefox Mac shows only the text "enter image description here" and a link which returns "Your connection is not secure". Chrome Mac shows a broken icon and the same text. Opera Mac sends up a warning. If SE doesn't have a policy about linking to external sites, it probably should have. Better to download the image (if text will not replace it) and mount it onesself. $\endgroup$ – David Jan 30 '18 at 23:20
  • $\begingroup$ @David This is image is located on imgur which is used by stackexchange. There is no problem. $\endgroup$ – Chris Jan 31 '18 at 5:58
  • $\begingroup$ @Chris — Ok. One can't be too careful these days. $\endgroup$ – David Jan 31 '18 at 7:50
  • $\begingroup$ @David Sure, no complain. Do you still have this problem? The most likely reason is that the SSL certificate from Imgur expired - and they didn't renew it on time. If so, I will file a ticket. $\endgroup$ – Chris Jan 31 '18 at 8:39

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