This is a question about SIR or SEIR models and understanding the trajectory of the covid infections in Florida and Texas in September 2021. This is a genuinely academic question, and not meant to be political in any way. I am not sure this is the best SE site for this question, but it is definitely the best one to ask questions about SIR models, so let me give it a try.
I have been watching the rapid rise in covid cases in Florida and Texas due to the Delta variant of Covid over the past 6 weeks or so. As we know, the governor in Florida and Texas were quite opposed to imposing any restrictions for masking or encouraging vaccinations, etc. That is just what I am seeing in the new coverage.
Now SIR models are used to model epidemic disease spread. So if we have a mixing population with no additional restrictions, a vanilla SIR model would predict continued transmission of the illness until everyone is either recovered, infected, or dead. That is the very toy and simple result of what an SIR model would provide.
Now, I am trying to understand what is causing the deceleration in cases in Florida and Texas, given that these states don't seem to be taking the actions that would diminish the transmission rate for the delta strain. That is the question?
It could be that despite the governor's actions in both states, the local restrictions are having an impact. Though I have not heard of any locally imposed lockdown or such–I figured the governor would need to order that. Or is it that people are just changing their own behavior to increase distancing or not going to restaurants again,etc? Or are the governors doing more than is covered on the news?
Again, I know that SIR models are far from predictive models, but I was just trying to understand how these real life waves of infection can end when restrictions are seemingly not imposed.
Again, my question is simply academic. Thanks.
Best Answer
It's not true that a vanilla SIR predicts "continued transmission of the illness until everyone is either recovered, infected, or dead", even "if we have a mixing population with no additional restrictions". In fact, in all scenarios where the number of susceptible individuals is initially positive in an SIR model, that number always remains greater than some strictly positive lower bound, to which it decreases monotonically as $\ t\rightarrow\infty\ $. While the SIR model is so crude that I don't expect it to do much more than reproduce some of the more prominent features of how some epidemics behave, it does at least seem to me to do that quite nicely.
If the proportion of non-susceptible (i.e. infected or recovered) individuals in the population is substantially less than the model's so-called herd immunity threshold, then the number of infected individuals will rise steeply at the start, and continue to do so until the proportion of non-susceptible individuals approaches the herd immunity threshold. The rate of increase in the number of infected individuals will then drop off, and that number will reach a maximum when the proportion of non-susceptible individuals passes through the herd immunity threshold. The number of infected individuals will thereafter decrease continually. Although the proportion of susceptible individuals also decreases continually, it always approaches a strictly positive limit $\ s_\infty\ $, say, as $\ t\rightarrow\infty\ $. When it's sufficiently close to that limit, the number of infected individuals will be proportional to $\ e^{-\big(\gamma-\beta s_\infty\big)t}\ $, where $\ \gamma\ $ and $\ \beta\ $ are two parameters of the model, which are always positive, and $\ \gamma>\beta s_\infty\ $. Thus, the number of infected individuals tends to zero at an exponential rate as $\ t\rightarrow\infty\ $.
The answer to your question, therefore, is that the number of infected individuals (i.e. the number of active cases) is now decreasing in Texas and Florida because the proportions of immune individuals there have now passed the herd immunity thresholds currently pertaining in those two states. The failure to mandate the wearing of masks, or impose restrictions on movements, doesn't eliminate the possibility of herd immunity, it merely makes its threshold larger than what it would have been if those measures had instead been $\text{introduced}\,.^1$
The SIR model is not very well suited to modelling a COVID-19 epidemic because it assumes a constant total population, and doesn't make proper allowance for the acquisition of immunity through vaccination. The differential equations of the standard SIR model are \begin{align} \frac{ds}{dt}&=-\beta si\\ \frac{di}{dt}&=\beta si-\gamma i\\ \frac{dr}{dt}&=\gamma i \end{align} where $\ s(t), i(t)\ $ and $\ r(t) \ $ are the proportions of the population still susceptible to, currently infected by, or having recovered from, the disease at time $\ t\ $, respectively, $\ \beta>0\ $ is a transmission rate constant, and $\ \gamma>0\ $ a recovery rate constant.
In the figure below, the number of active cases in Florida over the period July $6$ to September $25$ (blue curve) is compared with the values of $\ i(t)\ $ "predicted" by the above model (pink curve) for $\ \gamma=0.0661$, $\ \beta=$$3.995\gamma\approx$$0.2641\ $, $\ i(0)=$$\frac{128,136}{21,538,187}\ $, $\ r(0)=$$\frac{12,248,593}{21,538,187}\ $ and $\ s(0)=$$1-i(0)-r(0)=$$\frac{9161458}{21,538,187}\ $. The denominator of $\ i(0), r(0)\ $ and $\ s(0)\ $ here is the total population of Florida, as recorded in the census of April 1, 2020, the numerator of $\ i(0)\ $ is the number of active cases of COVID-19 in Florida on July $6$, as recorded by the COVID-19 worldometer, and the numerator of $\ r(0)\ $ is the total accumulated number of cases (from the same source), as of July $6$, minus the number of active cases, plus the number of individuals doubly vaccinated by July $6$, as recorded by GitHub. The values of $\ \gamma\ $ and $\ \beta\ $ were chosen to make the height and location of the peak in $\ i(t)\ $ coincide with those of the actual number of active cases. The "predicted" herd immunity threshold is $\ 1-\frac{\gamma}{\beta}=0.75\ $, corresponding to a susceptible population of around $\ 5.4\times10^6\ $. According to the records cited above, the actual number of individuals who had never been infected with the disease, or been fully vaccinated by September 16, the day when the number of active cases reached a peak, was around $\ 6.1\times10^6\ $. The number who had never been infected with the disease, or even received one dose of vaccine by that date was around $\ 3.2\times10^6\ $.
Plugging the above parameter values and initial conditions into WolframAlfa's SIR simulator confirms the above results, and tells us that the limiting value, $\ s_\infty\ $, of the susceptible proportion of the population "predicted" by the SIR model is approximately $\ \frac{2,697,213}{21,538,187}\approx0.125\ $.
A nice online tutorial on the SIR model, by David Smith and Lang Moore, is available on the website of the Mathematical Association of America.
$\,^1$ At least, according to some authorities. As Lutz Lehmann's comment below suggests, the effectiveness of masks and social distancing is a subject of some dispute.