From wikipedia:
"The germ theory was proposed by Girolamo Fracastoro in 1546, and expanded upon by Marcus von Plenciz in 1762. Such views were held in disdain, however, and Galen's miasma theory remained dominant among scientists and doctors ....
A transitional period began in the late 1850s with the work of Louis Pasteur. This work was later extended by Robert Koch in the 1880s. Viruses were discovered in the 1890s. By the end of the 1880s, the miasma theory was struggling to compete with the germ theory of disease. Eventually, a 'golden era' of bacteriology ensued, during which the theory quickly led to the identification of the actual organisms that cause many diseases."

Jeffrey Amherst lived from 1717 to 1797. Also, there are no letters of his suggesting the use of blankets to spread smallpox.