= Solution
There is a consistency issue in the printed model. If $w\equiv w_0$ is imposed while retaining its evolution equation, then $0=\beta v-\gamma w_0$ forces $v\equiv\gamma w_0/\beta$. Thus the full two-equation system has no nonconstant <travelling wave> with exactly constant $w$. Its constant solution exists only when $I=w_0-f(\gamma w_0/\beta)$. The intended wave calculation must treat the <recovery variable> as externally frozen, leaving the scalar <reaction-diffusion equation>
$$
v_t=Dv_{xx}+g(v),\qquad g(v)=f(v)+I-w_0.
$$
For this frozen-recovery reduction, let
$$
a_\pm=\frac{1+\alpha\pm\sqrt{1-\alpha+\alpha^2}}3.
$$
The <cubic polynomial> $f$ has a local minimum at $a_-$ and a local maximum at $a_+$. Hence the condition
$$
\boxed{w_0-f(a_+)<I<w_0-f(a_-)}
$$
gives three distinct real roots $r_1<r_2<r_3$ of $g$, with
$$
g(v)=-(v-r_1)(v-r_2)(v-r_3).
$$
The outer roots are stable for the scalar spatially uniform dynamics and the middle root is unstable. Seek a <travelling wave> $v(x,t)=V(z)$, where $z=x-ct$. The <travelling-wave reduction of a reaction-diffusion system> gives
$$
DV''+cV'-(V-r_1)(V-r_2)(V-r_3)=0.
$$
Let $\delta=r_3-r_1$ and choose
$$
V(z)=r_1+\frac{\delta}{1+\exp[\delta(z-z_0)/\sqrt{2D}]}.
$$
It has $V(-\infty)=r_3$, $V(+\infty)=r_1$ and
$$
V'=-\frac{(V-r_1)(r_3-V)}{\sqrt{2D}},\qquad
V''=\frac{(V-r_1)(r_3-V)(r_1+r_3-2V)}{2D}.
$$
Substitution leaves
$$
(V-r_1)(r_3-V)\left[\frac{r_1+r_3}{2}-r_2-\frac c{\sqrt{2D}}\right]=0.
$$
Consequently the <frozen-recovery cubic front> is
$$
\boxed{v(x,t)=r_1+\frac{r_3-r_1}{1+\exp\left[\frac{r_3-r_1}{\sqrt{2D}}(x-ct-z_0)\right]},\qquad
c=\sqrt{\frac D2}(r_1+r_3-2r_2).}
$$
Reflection $x\mapsto-x$ reverses the <front> and its speed. A nonzero speed requires $r_2\ne(r_1+r_3)/2$; equality gives a stationary <front> and is the <Maxwell balance for a scalar reaction-diffusion front>. Since the roots sum to $1+\alpha$, this stationary value is $I=w_0-f((1+\alpha)/3)$. For the simple admissible choice $I=w_0$, the roots are $0,\alpha,1$ and
$$
\boxed{V(z)=\frac1{1+e^{(z-z_0)/\sqrt{2D}}},\qquad c=\sqrt{\frac D2}(1-2\alpha).}
$$
This particular <front> moves when $\alpha\ne1/2$. At $\alpha=1/2$, other currents inside the displayed open interval give moving <fronts>.
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