How to set up a Virtual VPS Server under Laravel 4.2?

Question

Laravel

I have a problem setting up a server. The site used to run on CodeIgniter 2.2 and everything worked fine.
I reprogrammed the site under Laravel 4.2, of course everything works for me on localhost, but as soon as I make a clone git clone https://...on the server, then I run composer install --no-devit, I have a problem, the site cannot start due to an ERROR 500 error .
Maybe someone knows what the problem is? Here is the access.log and error.log :
Access.log :

xxx.xxx.xxx.xxx - - [15/Mar/2015:10:54:43 +0100] "GET / HTTP/1.1" 403 500

Error.log :

[Sun Mar 15 10:54:25.798769 2015] [autoindex:error] [pid 25276] [client xxx.xxx.xxx.xxx:54465] AH01276: Cannot serve directory /data/web/example.com/public/: No matching DirectoryIndex (/public/index.php) found, and server-generated directory index forbidden by Options directive

I think the settings are also needed to understand what's going on:
public/.htaccess

DirectorySlash Off
Options +FollowSymlinks -Indexes -MultiViews
AddDefaultCharset utf-8
DirectoryIndex /public/index.php

<IfModule mod_rewrite.c>
        RewriteEngine On
        RewriteBase /

        # Redirect Trailing Slashes...
        #RewriteRule ^(.*)/$ /$1 [L,R=301]

        # Handle Front Controller...
        RewriteCond %{REQUEST_FILENAME} !-f
        RewriteCond %{REQUEST_FILENAME} !-d
        #RewriteRule ^ index.php [L]
        RewriteRule ^(.*)$ /public/$1 [L,QSA]
</IfModule>

and /etc/apache2/sites-enabled/example.com.conf :

<VirtualHost *:80>
        DocumentRoot /data/web/example.com/public
        ServerName example.com
        ServerAlias www.example.com
        Alias /phpmyadmin /usr/share/phpmyadmin
        Alias /phppgadmin /usr/share/phppgadmin
        Alias /webftp /data/web/webftp
        ErrorLog /data/web/example.com/logs/error_log
        CustomLog /data/web/example.com/logs/access_log common
</VirtualHost>
<VirtualHost *:80>
        VirtualDocumentRoot /data/web/example.com/sub/%-3
        ServerName example.com
        ServerAlias *.example.com
        ErrorLog /data/web/example.com/logs/error_log
        CustomLog /data/web/example.com/logs/access_log common
</VirtualHost>

<Directory /data/web/example.com>
        Options FollowSymLinks
        AllowOverride All
        Require all granted
</Directory>

Alias /example.com/ /data/web/example.com/public/

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