Homotopy equivalence onto special fiber The following proposition appears in Peters and Steenbrink's book on Mixed Hodge Structures.

Proposition([Peters--Steenbrink, Proposition C.11]) If $f\colon X\to\Delta$ is proper and smooth over $\Delta^\ast:=\Delta-\{0\}$, then the inclusion $X_0\hookrightarrow X$ is a homotopy equivalence.

Does anyone know a reference for the proof of this result?
 A: As mentioned in the comment, Clemens proved the case for normal crossing divisors. The general case will follow from this result, which I'll provide proof later.

Theorem. (Clemens, 1977) If $X$ is a smooth complex manifold and $f:X\to \Delta$ is a proper family. Suppose $f$ is smooth
over $\Delta^*$, and $X_0=f^{-1}(0)$ is normal crossing divisor, then there is a strong
deformation retract from the total space $X$ to the central fiber
$X_0$.

Remark 1. Here $X_0$ being a normal crossing divisor means $X_0=\cup_im_iD_i$ and the $D_i$ are smooth proper varieties meeting transversely among each other and $m_i\ge 1$. The local equation of a point $x_0\in X_0$ in a neighborhood of $X$ is analytically equivalent to $t=x_1^{m_1}\cdots x_k^{m_k}$.
Note a normal crossing divisor is different from simple normal crossing divisor, in which the central divisor is reduced ($m_i=1$) and the family is called semi-stable. To get a semi-stable family from $X\to \Delta$, one need to take a base change with respect to a finite cover $p:\tilde{\Delta}\to \Delta$ where $\deg(p)$ is the l.c.m. of $m_i$. (People studying asymptotic Hodge theory prefer to work with a semi-stable family, that's why various literature, e.g., by David Morrison, on Clemens-Schmid sequence work with a semi-stable family, even though the existence of the limiting mixed Hodge structure only rely on normal crossing condition.)
Remark 2. A strong deformation retract $F:X\times I\to X$ means $F$ is a deformation retract ($F_0=F(\cdot,0)=Id_X$ and $F_1=F(\cdot,1)$ sends $X\to X_0$) and satisfies $F(x,t)=x$ whenever $x\in X_0$.
Now let's prove the general case based on Clemens' theorem.

Claim: Let $Y\to \Delta$ be a proper family and smooth over
$\Delta^*$. Then there is a strong deformation retract of $Y$ onto
$Y_0$.

Proof. By Hironaka, there is a log resolution
$$\sigma: X\to Y,$$
where $\sigma$ is isomorphism over $Y\setminus Y_0^{sing}$ and $X_0=\sigma^{-1}(Y_0)$ is a normal crossing divisor.
We define $G:Y\times I\to Y$ by
$$
G(y,t)=\begin{cases}\sigma F(\sigma^{-1}(y),t),\: y\notin Y_0;\\
y,\:\:\:\:\:\:\:\:\:\:\:\: y\in Y_0.
\end{cases}
$$
I claim that $G$ is a strong deformation retract of $Y$ onto $Y_0$. In fact, it suffices to check the continuity. Since the points know where to flow outside $Y_0$, it suffices to check continuity on $Y_0$. Let's take a sequence of points $(y_n,t_n)$ on $Y\times I$ whose limit point is $(y_0,t_0)$ with $y_0\in Y_0$. Then $$\lim_{n\to \infty}G(y_n,t_n)=\sigma\lim_{n\to\infty}F(\sigma^{-1}(y_n),t_n).$$ Since $X$ is proper, $\sigma^{-1}(y_n)$ converges to a point $x_0\in \sigma^{-1}(y_0)\subseteq X_0$. By strongness of $F$, $F(\sigma^{-1}(y_n),t_n)\to (x_0,t_0)$, so $\lim_{n\to \infty}G(y_n,t_n)=(\sigma(x_0),t_0)=(y_0,t_0)$. Therefore $G$ is continuous on $Y_0\times I$. $\Box$
Final remark. One can think of the retraction as describing a flow that sends the vanishing cycles to the singularity on central fiber and send other points to smooth locus $Y_0^{sm}$. In the normal crossing case, the vanishing cycle is the boundary of a tubular neighborhood of $D_1\cap D_2$ (say two components) in $Y$, which is a circle bundle. The retraction is to flow each circle over $y\in D_1\cap D_2$ to the point $y$. To me, this is the topological picture behind the argument.
In addition to the references listed in this answer, one can refer to Clemens' original paper [Cle77] (especially Theorem 5.7) for the strong deformation retract theorem in normal crossing case. Besides, Nicolaescu's notes (e.g. Chapter 14) is also a good introduction to this subject.
[Cle77]: Clemens, C. H.
Degeneration of Kähler manifolds.
Duke Math. J. 44 (1977), no. 2, 215–290.
