IPtables on Linux for mysql server private interface

Looking to make a mysql database server for web-servers.
The server has a public & private interface and I need some assistance writing the rules.

I want to only allow SSH, mysql via private interface, and shutdown the public interface essentially

iptables -nL -v --line-numbers

Allow Public and Private if initiated by server.

iptables -I INPUT -i eth0 -m state --state ESTABLISHED,RELATED -j ACCEPT

Allow Loopback to localhost

iptables -A INPUT -i lo -j ACCEPT

Block all incoming traffic via public interface.

??

Block all SSH via public interface

iptables -A INPUT -i eth0 -p tcp --destination-port 22 -j DROP

Allow SSH via private interface from 1 IP.

iptables -A INPUT -p tcp -s PRIVATE_IP --dport 22 -i eth1 -j ACCEPT

Allow mysql via private interface from specific IP. Reject mysql via eth0 (public)

iptables -A INPUT -i eth1 -p tcp --dport mysql -s PRIVATE_IP -j ACCEPT

iptables -A INPUT -i eth0 -p tcp -m tcp --dport mysql -j REJECT

Allow Ping on eth1 (private), but not eth0 (public)

iptables -A INPUT -i eth1 -p icmp -j ACCEPT
iptables -A INPUT -i eth0 -p icmp -j REJECT

Drop Public connections

iptables -A INPUT -i eth0 -j DROP

Do I have all my bases covered?

Answer

Do I have all my bases covered?

When speaking of a public facing server, one should never think to be ultimately safe; other than that, I would set a DROP policy for the INPUT chain, and after that allow only what is really required.
For example, after a

iptables -P INPUT DROP

it wouldn't be necessary to specify the

iptables -A INPUT -i eth0 -p tcp -m tcp --dport mysql -j REJECT