I only have goats, rabbits, chickens, ducks, cats and fish.
Yours is a most interesting and complex question. The simple answer is that healthy animals do not transmit disease.
If you have a sick animal, the answer has many variables.
In southern China there are places where they thought it was clever to design a farm where the feces of one animal was fed to another. This is clever only if there is no disease or disease forming organisms in the whole system. Otherwise it enhances the opportunity for transmission and viral mutation.
The opposite is that when clean barnyard practices are adhered to, (i.e. the soil is not burdened with excrement, clean food, clean water, good housing, etc.) in a farm where there is not a lot of new animals passing through, there is little risk of disease transmission.
I may offend pig people, but there is a reason they are called 'pigs'. Pigs may be a bit more difficult to keep clean than goats.