How does a virus fall down in static air? If we drop a virus from a height, in static air, will it fall to the ground like a lead ball, a balloon, or like a virus? How will it fall to the bottom? Like a  Brownian particle? It will not float in the air, as its density is higher than air. But still it seems a light particle. If I blow the air in which it's immersed, I can blow it away. From all sides air molecules bump into it. Giving it random pushes. Can we say it's part of the air it's in and as such float in it?
If we imagine ourselves standing beside it, while it lays on a solid structure, and we push it over the edge, what effects the molecules have that fly around on all sides? Will I see it fall down between the speeding molecules, which only give random impulses to all sides of the virus?
 A: The smaller the particle, the less effect gravity has compared with the interactions of other particles.  Viruses are tiny and would (by themselves) fall incredibly slowly in air.  It would be bounced constantly in all directions.  If the air were still, it would trend downward over time, but only on quite long timescales.
Virus transmission doesn't normally consider lone virus interactions with the air because lone virus particles seem to have low viability.  Instead they need to remain within a water droplet.  So it's the size of the water droplet that dominates the path through the air.  Bigger droplets fall faster than smaller droplets.
A: Short answer: It does not.
More involved look:
Particles movement and settling in fluids can be - to rather high accuracy -  modeled through statistical mechanics. Namely, through Laplace-Perrin distribution

And sedimentation length:

This distribution describes how particles immersed in a fluid move stochastically around. Sedimentation length is how far a particle can travel through brownian motion by chance 1/e.
For virii, a fast glance at public documentation and papers gives estimates of mass and size to be m* = 1 femtogram, and diameter of 20nm.
Sedimentation length for virii is thus about 4.2*10^-7 m which is ~20 times larger than their diameter.
Roughly speaking we can say that if sedimentation length is much larger than the size of the particle itself, it stays suspended. This mean that if you arranged a number of virii at the floor, completely still, and a body of air filling the room, completely still too, but all at room temperature, after a while you could expect to encounter some of the virii near the ceiling.
In the real world there are always small disturbances in the air that ensure much larger particles stay suspended too. Also, in the real world we don't really consider free virii as the infectious agent, but small aerosols emitted by people breathing and speaking. The time they spend airborne is heavily dependent on diameter, and follow interesting size- and travel distance distributions.
A: Depending on the size of a particle, its motion in air under gravity can be modeled in two ways:

*

*Air pressure difference on the two sides of the particle. This gives a terminal velocity, which is the fastest speed the particle will fall relative to air.

*Brownian motion caused by the collisions from individual air molecules. Using the Stokes-Einstein relation, a drift speed due to gravity can be calculated.

In either case, if the air is moving faster upwards than the particle is moving downwards relative to air, the particle goes up. Typical average indoor air velocity is on the order of 0.1 m/s, though local velocities near heat sources or movement will be faster.
Individual viruses have a diameter varying from 20 to 300 nm and a  density about twice that of water. Viruses remain infectious much better inside small water droplets called aerosol, which have a size varying from 100 nm to 10 µm.
The transition between these two types of movements can be described by the Knudsen number. In air, Brownian motion starts significantly affecting particle motion below a size of 1 µm.
In the graph below, I've drawn the two speeds up to this transition area. There is a large difference, and in reality this would be blended together over the transitional zone of a few decades of scale.

Aerosols fall in the transitional zone. What this means is that larger aerosol particles will fall at terminal velocity, while smaller ones and individual viruses follow Brownian motion. The latter results in much slower movement speeds and the particles thus remain suspended orders of magnitude longer. This page has some graphs for aerosols in particular.
