Well I thought that was where this was going but what is f(x)? The other confusion I see is that infection growth is mostly function of time rather than a function of our behaviour, population density and other factors. If I can use the analogy of credit card interest, the way to avoid the exponential growth of that debt is to pay off the balance or at least part of it or maybe reorganize debt! Everybody needs to adjust their behavior and there has to serious thought of punishing those who refuse.
I agree that infection rates are generally independent of population behaviors. Reviewing past data shows that implementation of rigorous lock down measures has had little or no impact on the infection or death rates associated with this virus. There appears to be a general opinion that the lock down measure in April/May caused the decline in infections, i.e. flattened the curve, when in fact infections and deaths were already falling and following a seasonal downward trend that is always associated with viruses of this nature.
It is difficult to get a grip on an argument that lock downs did not have a positive affect on controlling the virus, when there are many confounding factors such as societal, economic, dietary, ethnicity, latitude and seasonality, which all have significant impacts on outcomes, especially when comparing one or two countries. However, if you look at the data on the following link you will see that numerous countries that have implemented very stringent lock down, have not fared any better than countries with far less stringent or no lock downs at all.
https://app.powerbi.com/view?r=eyJrIjoiMGVjYjhkMjMtMzhjMy00OWRkLWJlNWItNjM0NzI0NjhiNTlkIiwidCI6IjlkZWYwNTBlLTExMDUtNDk1ZC1iNzUzLWRhOGRiZTc5MGVmNyJ9I'm not suggesting that we should ignore all guidance and have a free for all. I agree that social distancing, hand washing and selective wearing of face masks are beneficial behaviors that will provide some protection to us older folks and there should be more consideration about how people are to be held accountable for their behavior. However, the concern I have is determining how far the behavioral restrictions should go. There is ever mounting evidence that the number of deaths due to COVID during this seasonal increase or "second wave" is far less than the deaths attributed to cancer, stroke, heart attack patients who have not received prompt attention due to the perceived need to protect our health care system and increased suicide rates.
Therefore, before sever measures are implemented, such as a lock down, there has to be a full and balanced accounting of the relative immediate risks pose to society and societies ability to continue to function and protect its citizens from all forms of harm in the following years. I do not think I have seen any government offering this type of analysis and basing their decisions on a properly formulated cost benefit risk assessment. Maybe they have behind the scenes, but I cannot find any indication of this logical process, which is rather frightening.