In two words, human behavior. Restrictions in most places have not been like a binary on-off switch, but a matter of varying degree, and people do behave differently over time regardless of restrictions ("pandemic fatigue"). I think this is the most important effect overall for causing regular/repeated waves. But there are also many other effects of varying level of importance and depending on the region and timeframe, which we should explore as well.My other point was, if we understand so well how these viruses spread, why do we have these large waves? Most have been living with restrictions for more than a year, and we can hardly explain those waves by assuming that there were times when people simply ignored the restrictions and recommendations and other times when people were compliant. The favourite answer seems to be mutations, but it seems to me as an oversimplification.
First, why repeated waves? A good way to think about it is by analogy to oscillations. An oscillation happens if when a quantity is big, there is a mechanism to reduce it, and when the quantity is small, there is a mechanism to increase it. And that's exactly what happens with human behavior during a pandemic.
When the effective reproduction number R is greater than 1, the virus spreads exponentially. This leads to a faster and faster rate of new cases, until eventually it cannot be ignored any further and society takes action. Either government steps in to limit social activities, or people become more careful of their own will. (Often both.) If the increased caution and reduced interaction manages to push R below 1, then new cases decline exponentially. Then when the rate of new cases falls low enough (especially in comparison to where it had been), then people gradually begin to relax, and more restrictions get lifted.
But even if restrictions stay the same, people tend to become less careful, even if they are not aware of it. This has often been called "pandemic fatigue". If R had been only slightly less than 1, then it does not take much change in behavior to get it above 1 again, and this triggers a new wave of exponential growth until people are forced to react again. The cycle repeats, until either the virus is eradicated or an effective immunity is reached. (And if mutations continue to occur and force us to do repeated vaccinations, which appears increasingly likely, then this cycle might go on for a long time to come, just like with influenza.)
It is interesting to examine this effect in different regions. Let's look at Washington State in particular. Here are the rates of new cases (per day per 100k people), and the reconstructed curve for R(t) based on the cases and hospitalizations:
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Seasonal effect? Virus mutation effect? Random chance? I don't think so. I think this is mostly the above effect of human/societal behavior. In Washington State, we get a wave, then we react to it, then it takes about 4 months for us to 'get tired of it' and act in a way slightly closer to normal. And it's such a small change in R that causes it, that most aren't even necessarily aware of it. Even as we keep masks on and avoid large groups (especially indoors), people are on average just becoming slightly less careful over time, and that's all it takes. The effect creeps up over time.
Seasonal effects:
I think the seasonal effect is real, in the sense that when the weather is colder, more people stay inside longer, which is where it is easier for the virus to spread through the air. This is why the pandemic appears to be stronger in each hemisphere's winter. But it cannot be the only or even most important effect, because there are waves in many regions despite the season. I think this effect just makes waves easier to occur or larger than they would be otherwise (basically like a small increase to R on top of anything else that happens.) And it really is human behavior in response to the weather, more so than the weather affecting the virus directly (because we now know infection outdoors or through surfaces is not the dominant means of transmission.)
Holidays and Religious events:
I think these play a huge role for some waves in many countries. In the US, and even in the above figures for Washington, you can see two separate peaks in the third wave. The first peak is Thanksgiving, the second peak is Christmas and New Years. Both peaks involve a huge amount of travel and family getting together, which is a great way to spread the virus. The American public was warned about this repeatedly with every major holiday of 2020, yet so many people traveled and socialized anyway. Every time, the event was followed shortly after by a surge in cases and hospitalizations and deaths. Totally sad. Totally avoidable.
Holidays and religious gatherings also explain a number of major waves in Israel, according to a friend there. And I'm sure we can find many more examples.
Mutations:
There is now no doubt that more infectious variants have caused more severe waves. One of the most powerful demonstrations of this is to look at the fraction of sequenced cases that belong to a particular clade over time:
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I think the severity of the current wave in India is largely due to more infectious variants combined with poor ability of their government to control it. (Rather, they did an outstanding job of controlling it throughout most of 2020, but then they relaxed, and because the early restrictions caused such economic harm it is now even harder to going back to controlling it, especially in the face of the more infectious variants.) The rate of growth is much more rapid for a given number of cases, indicating a larger effective R. Sequencing there also shows several variants that are more infectious than the original, including the UK variant. There is also a new variant with two mutations to the spike protein, the exact effects of which remain to be seen.
Another problem in India: they have fallen far behind much of the rest of the world for vaccinations per capita.