The outbreak of swine flu that was first detected in Mexico was declared a global epidemic on ju

the outbreak of swine flu 是主语，

that was first detected in Mexico 是修饰主语的定语从句。

was declared a global epidemic on June 11,2009, 这句话是谓语部分

所以这句话主干就应该是：

The outbreak of swine flu （that was first detected in Mexico） was declared a global epidemic on june 11,2009.