Solving system of nonlinear differential equation in MAPLE

Question

I'am working out on nonlinear differential equation and I need to find the equilibrium point which means all the system is equal to zero. Here is the System of Diferential Equation:

\begin{align*} \frac{dS}{dt} &= \alpha - \beta SV - \delta S \\ \frac{dI}{dt} &= \beta SV - \sigma I \\ \frac{dV}{dt} &= \mu nI - \gamma_1 V - \gamma_2 V - \gamma_3 V - \beta SV \end{align*}

then, to find the equilibrium point set $\frac{dS}{dt} = \frac{dI}{dI} = \frac{dV}{dt} = 0$. It means that I have to solve the system of equation \begin{align*} \alpha - \beta SV - \delta S &=0 \\ \beta SV - \sigma I &= 0\\ \mu nI - \gamma_1 V - \gamma_2 V - \gamma_3 V - \beta SV &=0 \end{align*}

I've got the result by solving it manually but Can I use MAPLE software to solve this problem?

What I've done manually: $V = \frac{\alpha - \delta S}{\beta S}, I = \frac{\alpha - \delta S}{\sigma}, S = \frac{\alpha}{\delta} \text{ or } S = \frac{(\gamma_1 +\gamma_2+\gamma_3)\sigma}{\beta(\mu n - \sigma)} $

Answer 1

Here is how to do it in Maple

eqs:={alpha-beta*S*V-delta*S,beta*S*V-sigma*II,
mu*n*II-gamma[1]*V-gamma[2]*V-gamma[3]*V-beta*S*V};
vars:={S,II,V};
solve(eqs,vars);

Note that I is a reserved symbol in Maple, so I just used another symbol II for it.

Maple yields two solutions $$ \left\{ {\it II}=0,S={\frac {\alpha}{\delta}},V=0 \right\} $$ and $$ \left\{ {\it II}={\frac {\alpha\,\beta\,\mu\,n-\alpha\,\beta\,\sigma- \delta\,\sigma\,\gamma_{{1}}-\delta\,\sigma\,\gamma_{{2}}-\delta\, \sigma\,\gamma_{{3}}}{\beta\,\sigma\, \left( \mu\,n-\sigma \right) }}, S={\frac {\sigma\, \left( \gamma_{{1}}+\gamma_{{2}}+\gamma_{{3}} \right) }{\beta\, \left( \mu\,n-\sigma \right) }}, \\ V={\frac {\alpha\, \beta\,\mu\,n-\alpha\,\beta\,\sigma-\delta\,\sigma\,\gamma_{{1}}- \delta\,\sigma\,\gamma_{{2}}-\delta\,\sigma\,\gamma_{{3}}}{\beta\, \sigma\, \left( \gamma_{{1}}+\gamma_{{2}}+\gamma_{{3}} \right) }} \right\} $$