Why is "wie" used in the first sentence and not "dass" and then "dass" is used in the second sentence and not "wie"?
Die Infektionsweise bezieht sich auf die Weise, wie die Infektion von einer Person zur anderen übertragen wird.
Dies gelang Yntema auf die Weise, daß er das verladene Gewicht aller Produktarten mit Hilfe von Kosten-Äquivalenzziffern addierbar machte.
The subordinate clause in die Weise, wie tells you the question:
wie die Infektion von einer Person zur anderen übertragen wird.
(how the infection is transmitted from one person to another).
"Wie" is used as a question word here. After you hear this sentence, you know the question, but not the answer. The subordinate clause is literally a question, just in a different word order:
Wie wird die Infektion von einer Person zur anderen übertragen?
(How is the infection transmitted from one person to another?)
The dass-clause, however, in "auf die Weise, dass" tells you what Yntema did to solve a question that must have been described before (refered to bei "dies" in your example). In other words, it doesn't pose a question but reports something that happened.
... dass er das verladene Gewicht aller Produktarten mit Hilfe von Kosten-Äquivalenzziffern addierbar machte.
... that he made the loaded weight of all product types additive with the help of cost equivalence numbers.
The implicit claim here is that "making the loaded weight of all product types additive with the help of cost equivalence numbers" is a way of solving the problem.
The wie in the first sentence is equivalent to how. The way of infection is the way, how the infection is carried from one person to another.
The dass in the second sentence is a difficult and pretty ugly way of expressing the meaning. The dass is equivalent to a indem (by). Yntema managed to do this by making the loaded weight of all types of products addable with the help of cost-equivalence figures.
