«لَا عَدْوَى» should not be read as abolishing every causal factor in illness or every protective precaution. The rejected notion is that disease necessarily transmits by an independent power of its own.
Contact with an ill person may be a created cause associated with harm, but it does not act as an independent creator of effects. Thus precaution can coexist with the denial of autonomous and inevitable causation.
Application: prudent use of means does not contradict reliance on God, and reliance on means need not imply belief in their independent power.