The appearance of epidemics in early Islamic history may be attributed in part to the cyclical recurrences of plague in the Middle East following the Plague of Justinian, beginning in A. D. 541. (Plague in Early Islamic History by Michael W. Dols, 1974)

This is in addition to the well documented influence of the bubonic plague and the less ubiquitous Bombay plague. The above link describes historians' ability to recreate (to some extent) the epidemiological history of plagues in the Umaiyad period. The presence of these plagues prompted medical, religious, and legal explanations and instructions. These were instructions for citizens of nations as much as they were religious doctrine for adherents of faiths.

The study goes on to propose that plague was a large factor in why the center of political power shifted from the Byzantine empire under Justinian to more northern European regions. According to the author, this is due to northern Europeans being relatively unaffected by the pervasive diseases of the time.

So, to answer your question, plague control tactics were absolutely in place. These tactics came from many sources, not just religious ones. Finally, if the author was correct, plague control tactics of the time were not enough to protect Middle Eastern populations from the extent of damage that disease would go on to inflict, partially contributing to the destabilization of the region. Arabs (and by extension, many Muslims) were able to resist some of the effects of disease due to their nomadic and semi-nomadic way of life adjacent to the empire.

In the classical world, at least, it was not. There are two basic prerequisites for a quarantine policy: a basic understanding of contagion, and civic or regional governments willing and able to shut down the lifeblood of trade. The Greeks and Romans had only a vague sense of the first, and wholly lacked the second.

There was always something supernatural about epidemics. Greek literature, famously, begins with Apollo showering arrows of pestilence on the Greeks at Troy; and well over a millennium later, Procopius could describe the Plague of Justinian in these terms:

"Apparitions of supernatural beings in human guise of every description were seen by many persons, and those who encountered them thought that they were struck by the man they had met in this or that part of the body, as it happened, and immediately upon seeing this apparition they were seized also by the disease." (History of the Wars, 2.22)

The idea that plagues reflected divine displeasure was complemented, but never fully displaced, by the "miasma" theory of infection, first associated with the School of Hippocrates. This explained pestilence as a product of tiny particles (sometimes called "seeds") emanating from the stars, swamps, or corpses. These particles, if inhaled, were thought to cause pestilence. The terrible Antonine Plague, for example, was said to have arisen when:

"a pestilential vapor arose in a temple of Apollo from a golden casket which a soldier had accidentally cut open...spread thence over Parthia and the whole world" (SHA, Verus 8.2)

Though unclear on how plagues spread, the Greeks and Romans were acutely aware that proximity to a sick person increased one's chances of contracting disease. This is clear as early as Thucydides' famous description of the Plague of Athens, which notes that doctors tended to die first since they were close to the sick (2.47.4) and that people were afraid to visit houses the plague had touched (2.51.5).

But the sense that plagues were (somehow) contagious was not matched by the ability or will to impose an official quarantine. Although city governments in the classical world were strong, they seem to have regarded plagues as an inevitability, or at least as a lesser evil than the privation and shortages that would follow a shutdown of trade. The wealthiest citizens sometimes fled to escape sickness (the great Roman doctor Galen for example, left Rome when the Antonine Plague arrived, and the emperors themselves did the same a few years later when the plague struck a detachment of troops in Aquilea). But official restrictions on movement were, to the best of my knowledge, never imposed.