If infected cells die and release more virus, why does an infection not eat a hole in the tissue?

There are two further constraints worth adding.

Viruses are picky about which cells they can enter. Infection requires a specific receptor on the cell surface, and often other factors inside. A respiratory virus can infect certain cells in the airway lining and simply cannot enter most other cell types. So the infection is confined to a subset of cells even within the tissue it reaches — the "nearby cells" in your model are frequently not susceptible at all.

Not every infected cell bursts. Some release particles gradually while surviving, some are killed by the immune system before producing much, and many produce defective particles that infect nothing. The yield per cell in practice is far below the theoretical maximum.

So the exponential you are imagining is damped at every step.

It is also worth saying that your model is not wrong — it is describing what happens when the containment fails, and those cases exist.

Some infections do produce visible destruction: blistering lesions where cells are killed in a patch, and severe infections where large areas of tissue are damaged. Severe respiratory illness involves exactly the process you describe, at scale, in tissue that cannot be replaced fast enough.

And there is an important twist: a great deal of the damage in severe infections is caused by the immune response rather than the virus. Inflammation is destructive by design, and when it runs hard in delicate tissue, the collateral damage can exceed what the pathogen would have done alone.

So the reason ordinary infections do not leave holes is that containment usually works, and the visible cases are the ones where it did not.

Your model is a reasonable first approximation and it is missing the two things that dominate the outcome.

The immune response is fast and local. Infected cells detect they are infected and immediately signal it — releasing molecules that put neighbouring cells into an antiviral state before the virus reaches them. So the expanding front you imagine runs into tissue that has already been warned and is much harder to infect. That response begins within hours, long before any antibodies exist.

The tissue is renewing constantly. The surfaces most infections attack — airway lining, gut lining — replace themselves rapidly as a matter of routine. Cells are dying and being replaced there continuously with or without a virus. Losing a scattering of them is absorbed by a process already running at high speed.

Between those two, the infection is contained and the damage is repaired concurrently rather than accumulating.

One evolutionary note that completes the picture: a virus that destroys its host tissue rapidly is generally not doing well by its own standards.

Transmission requires a host who is up and about, breathing on people, for as long as possible. Killing the local tissue quickly shortens that window. There is real selective pressure on many pathogens towards being contained enough to keep the host functional — which is not benevolence, it is arithmetic about how many further hosts get infected.