The disease that you can contract through contact or inhalation of rat droppings is the Hantavirus also known as Hantavirus Pulmonary disease.
The Hantavirus is a serious respiratory disease which is transmitted from infected Rodents through there Urine, Droppings and even Saliva.
If you do contract the Hantavirus some symptoms that you may experience are muscle aches, Fever, Fatigue, Chills, headaches, dizziness, abdominal pain, Vomiting and Diarrhea.
If you have the Hantavirus you need to seek medical attention since the virus is serious and could lead to death if not treated.
When cleaning up mice or rat droppings always wear a mask and gloves.