php - Deprecated: mysql_connect() Error Code

My host recently upgrade the PHP version and a certain part of my website now shows the following error:

Deprecated: mysql_connect(): The mysql extension is deprecated and will be removed in the future: use mysqli or PDO instead in url/structure/here on line 49

That is referencing the below code:

function DBConnect() {  
    $this->connectCount ++;
    //echo "$this->connectCount
";


    if ($this->dbType == 'mysql') {
        $dbConnect = mysql_connect($this->dbHost, $this->dbUser, $this->dbPasswd) or die ("MySql Connection Failed: " . mysql_error());
        mysql_select_db($this->dbName, $dbConnect);
    }

    if ($this->dbType == 'postgresql') {
        $dbConnect = pg_connect("host=$this->dbHost port=$this->dbPort dbname=$this->dbName user=$this->dbUser password=$this->dbPasswd") or die ("PostgreSQL Connection Failed: " . pg_errormessage($dbConnect));
        //$dbConnect = pg_pconnect("host=$this->dbHost port=$this->dbPort dbname=$this->dbName user=$this->dbUser password=$this->dbPasswd") or die ("PostgreSQL Connection Failed: " . pg_errormessage($dbConnect));
    }


    return $dbConnect;
}

I'm aware the fact that this is because the current way my site connects to MYSQL is now outdated in the new version of PHP but does anyone know how I would update the above code to make this work?