consider: \begin{align} \dot S &= A-\beta S(I+\rho_1 T)-(\mu+p)S\\[1ex] \dot V &=pS-\rho_2 \beta V(I+\rho_1 T)-\mu V\\[1ex] \dot L &=l \beta S(I+\rho_1 T)+\rho_2 \beta V(I+\rho_1 T)-(\mu+\delta)L +\rho T\\[1ex] \dot I &=(1-l)\beta S(I+\rho_1 T)+\delta L-(\mu+\alpha+\gamma)I\\[1ex] \dot T &=\gamma I-(\mu+\rho)T \end{align} with $N=S+V+L+I+T$.
to find the equilibriums, we set the above system to $0$: \begin{align} A-\beta S(I+\rho_1 T)-(\mu+p)S&=0\\[1ex] pS-\rho_2 \beta V(I+\rho_1 T)-\mu V&=0\\[1ex] l \beta S(I+\rho_1 T)+\rho_2 \beta V(I+\rho_1 T)-(\mu+\delta)L +\rho T&=0\\[1ex] (1-l)\beta S(I+\rho_1 T)+\delta L-(\mu+\alpha+\gamma)I&=0\\[1ex] \gamma I-(\mu+\rho)T&=0 \end{align}
How we solve this for $S,V,L,T$ in terms of $I$?
For example we see: $$T^*=\frac{\gamma}{\mu+\rho}I^*$$
I tried using solve but this isn't working. This is the code I used:
Solve[A - \[Beta] S (i + \[Rho]1 T) - (\[Mu] + p) S == 0, p S - \[Rho]2 \[Beta] V (i + \[Rho]1 T) - \[Mu] V == 0, l \[Beta] S (i + \[Rho]1 T) + \[Rho]2 \[Beta] V (i + \[Rho]1 T) - (\[Mu] + \[Delta]) L + \[Rho] T == 0, (1 - l) \[Beta] S (i + \[Rho]1 T) + \[Delta] L - (\[Mu] + \[Alpha] + \[Gamma]) i == 0, \[Gamma] i - (\[Mu] + \[Rho]) T == 0, {S, V, L, T}]
Reference(page 4 in pdf, page 839 in journal): https://reader.elsevier.com/reader/sd/pii/S0895717711001932?token=4C8B07AF574B6CAFE11E8CD5FEA143D0E498B5427696EE44C35371A30602EB3CB0825235CB97498301A86775EDC25D17&originRegion=eu-west-1&originCreation=20220124124053
Solve[A - \[Beta] S (i + \[Rho]1 T) - (\[Mu] + p) S == 0, p S - \[Rho]2 \[Beta] V (i + \[Rho]1 T) - \[Mu] V == 0, l \[Beta] S (i + \[Rho]1 T) + \[Rho]2 \[Beta] V (i + \[Rho]1 T) - (\ \[Mu] + \[Delta]) L + \[Rho] T == 0, (1 - l) \[Beta] S (i + \[Rho]1 T) + \[Delta] L - (\[Mu] + \ \[Alpha] + \[Gamma]) i == 0, \[Gamma] i - (\[Mu] + \[Rho]) T == 0, {S, V, L, T}]
$\endgroup$S * I
, both of which are solution variables. $\endgroup$