Specific immunity is a complement to the function of phagocytes and other elements of the innate immune system. In contrast to innate immunity, specific immunity allows for a targeted response against a specific pathogen.Only vertebrates have specific immune responses. Two types of white blood cells called lymphocytes are vital to the specific immune response.Lymphocytes are produced in the bone marrow, and mature into one of several subtypes. The two most common are T cells and B cells. An antigen is a foreign material that triggers a response from T and B cells.on-Specific (Innate) Immunity. The human immune system has two levels of immunity: specific and non-specific immunity. Through non-specific immunity, also called innate immunity, the human body protects itself against foreign material that is perceived to be harmful.