To make it easy, I'll use my [EcoEvo package][1].

First time, you'll need to install it:
```
PacletInstall["EcoEvo", "Site" -> "http://raw.githubusercontent.com/cklausme/EcoEvo/master"]
```

Then, load the package to get started:
```
<<EcoEvo`
```

Both models are a little bit funny, in that total population size is conserved.  Thus, model (i) is effectively one-dimensional and model (ii) is effectively two-dimensional.

Model (i): SI
--------

```
SetModel[{
  Pop[pop] -> {
    Component[s] -> {Equation :> -β s i/n + γ i},
    Component[i] -> {Equation :> β s i/n - γ i}
  },
  Parameters :> {β > 0, γ > 0, n > 0}
}]
```

Let's go straight to the phase planes using [PlotEcoPhasePlane][2], manually adding the total population constraint as a pink straight line.  The no-disease case (disease-free equilibrium `eq[[1]]` is stable, the other equilibrium `eq[[2]]` is biologically meaningless since `s > n` and `i < 0`):
```
β = 0.95; γ = 1; n = 1;
Show[
 PlotEcoPhasePlane[{s, 0.9, 1.1}, {i, -0.1, 0.1}],
 Plot[n - s, {s, 0.9, 1.1}, PlotStyle -> Pink],
 RuleListPlot[{{s -> n, i -> 0}, {s -> (n γ)/β, i -> n - (n γ)/β}}, PlotMarkers -> {True, False}]
 ]
```
[![enter image description here][3]][3]

and the endemic case (endemic equilibrium `eq[[2]]` is stable):
```
β = 4; γ = 1; n = 1;
Show[
 PlotEcoPhasePlane[{s, 0, 1}, {i, 0, 1}],
 Plot[n - s, {s, 0, n}, PlotStyle -> Pink],
 RuleListPlot[{{s -> n, i -> 0}, {s -> (n γ)/β, i -> n - (n γ)/β}}, PlotMarkers -> {False, True}]
]
```
[![enter image description here][4]][4]

Again, because of the total-population constraint, this is effectively a one-dimensional system with either two or one feasible (non-negative) equilibria.  Note that the two different isoclines ($S$ and $I$) overlap completely because of this and just look gold.

Model (ii): SIR
-------------
Here we can get rid of the degeneracy by defining `r := n - s - i`, then work in the SI phase-plane.

```
r := n - s - i;
SetModel[{
  Pop[pop] -> {
    Component[s] -> {Equation :> -β s i/n + ξ r},
    Component[i] -> {Equation :> β s i/n - γ i}
  },
  Parameters :> {β > 0, γ > 0, ξ > 0, n > 0}
}]
```
To verify your analytical results:
```
eq = SolveEcoEq[]
```
[![enter image description here][5]][5]

```
EcoEigenvalues[eq[[1]]]
```
[![enter image description here][6]][6]

```
EcoEigenvalues[eq[[2]]]
```
[![enter image description here][7]][7]

The eigenvalues of the non-trivial equilibrium are ugly, but we can check stability using Routh-Hurwitz criteria in [EcoStableQ][8]:
```
Simplify[EcoStableQ[eq[[2]]]]
```
[![enter image description here][9]][9]

On to the phase planes.  The $S$-isocline is blue, the $I$-isocline is gold.

No-disease case (`eq[[1]]` is stable, `eq[[2]]` is a biologically meaningless saddle point):
```
β = 0.95; γ = 1; ξ = 1; n = 1;
eq = SolveEcoEq[]
N[EcoEigenvalues[eq[[2]]]]
Show[
 PlotEcoPhasePlane[{s, 0.9, 1.1}, {i, -0.1, 0.1}],
 RuleListPlot[eq, PlotMarkers -> EcoStableQ[eq]]
]
(* {{s -> 1., i -> 0}, {s -> 1.05263, i -> -0.0263158}} *)
(* {-1.02384, 0.0488359} *)
```
[![enter image description here][10]][10]

Endemic case 1 (`eq[[2]]` is a stable focus, due to complex eigenvalues):
```
β = 4; γ = 1; ξ = 1; n = 1;
eq = SolveEcoEq[]
N[EcoEigenvalues[eq[[2]]]]
Show[
 PlotEcoPhasePlane[{s, 0, 1.1}, {i, 0, 1.1}],
 RuleListPlot[eq, PlotMarkers -> EcoStableQ[eq]]
]
(* {{s -> 1, i -> 0}, {s -> 1/4, i -> 3/8}} *)
(* {-1.25 + 1.19896 I, -1.25 - 1.19896 I} *)
```
[![enter image description here][11]][11]

Endemic case 2 (`eq[[2]]` is a stable node, due to negative real eigenvalues):
```
β = 4; γ = 1; ξ = 10; n = 1;
eq = SolveEcoEq[]
N[EcoEigenvalues[eq[[2]]]]
Show[
 PlotEcoPhasePlane[{s, 0, 1.1}, {i, 0, 1.1}],
 RuleListPlot[eq, PlotMarkers -> EcoStableQ[eq]]
]
(* {{s -> 1, i -> 0}, {s -> 1/4, i -> 15/22}} *)
(* {-9.60337, -3.1239} *)
```
[![enter image description here][12]][12]


  [1]: https://github.com/cklausme/EcoEvo
  [2]: https://www.wolframcloud.com/obj/EcoEvo/docs/ref/PlotEcoPhasePlane.nb
  [3]: https://i.sstatic.net/kecl7m.png
  [4]: https://i.sstatic.net/kCuVjm.png
  [5]: https://i.sstatic.net/eGvLy.png
  [6]: https://i.sstatic.net/YoFfb.png
  [7]: https://i.sstatic.net/6VcNf.png
  [8]: https://www.wolframcloud.com/obj/EcoEvo/docs/ref/EcoStableQ.nb
  [9]: https://i.sstatic.net/l3S5d.png
  [10]: https://i.sstatic.net/riA6lm.png
  [11]: https://i.sstatic.net/wqDxhm.png
  [12]: https://i.sstatic.net/j3s5bm.png