classical_Liberal wrote: ↑Thu Apr 23, 2020 9:25 pm
Thanks. This is what I assumed. Can someone with a mathematical background answer this...
Not really, ... well sort of. There are three different types of models I'm aware of. The IHME is a phenomenological model. Such a model is curve fitting US numbers to parametric curves based on outbreaks elsewhere. IOW, not knowing/postulating the exact dynamics, one assumes that it's everywhere the same and then go with that. If one has access to a lot of computational power, it's also possible to make agent-based models. These are similar to computer games like e.g. The Sims in that there's a model for different types of individuals, how they meet, how they interact, etc. One would have to specify these agents to match reality closely. The third type is the SIR-type models. They look like this,
dS/dt = -beta*I*S
dI/dt = +beta*I*S-gamma*I
dR/dt = +gamma*I,
where S+I+R=1 are the fractions of susceptible, infected, and recovered, respectively; beta indicates the likelihood of a susceptible person transferring to infected status based on their overlap (simple random statistics of drawing an I and an S from a population); and gamma is the removal parameter from the infected group going to either recovered or dead or isolated.
If you start with a virgin population, S~1. The second equation then reduces to dI/dt = +beta*I-gamma*I. It immediately follows that the reproduction number or the R0 that everybody is talking about is R0=beta/gamma because it's the ratio between how many an I person can infect in a given time before they are "removed" via gamma. IOW, if the beta rate (1/person^2/s) is 5 times higher than the gamma rate (1/person/s), then R0 is 5 (per person).
Social distancing affects beta and isolation affects gamma. Both therefore control R0. If immunity is a thing, then S will reduce over time and this will reduce I*S. If immunity is NOT a thing, then the equations become more like this:
dS/dt = -beta*I*S+alpha*R
dI/dt = +beta*I*S-gamma*I
dR/dt = +gamma*I-alpha*R
as recovered people get reintroduced into the susceptible group. alpha is affected both by natural loss of immunity and also by isolated people going outside to meeting other people again.
Add: If immunity holds for a while, you're need another variable to track that. Ditto if people can be infected for a while before they get infecting. This framework can be expanded ad nauseum and would ultimately converge on an agent-based model (although those are more likely to be formulate with discrete numbers instead of differential equations.)
Solving the first set of equations is relatively easy and somewhat robust. Solving the latter is a royal mess and can easily get dynamic (second wave). One would need a very good model for alpha, beta, and gamma because the results would be heavily dependent on the time evolution of those. This is also why the predictions of adaptive systems tend to come out less pessimistic than initially assumed. My rule of thumb is to take the current worst case and divide by half after and only after a majority of people take it seriously. Why half ... just because.
As a modeller, one would assume the structure of these equations and then try to fit the time evolution on actual observations. This will tell us what the parameters are and from this we can derive things like R0. This method crashes and burns IFF the assumed structure is NOT the correct structure because it is either wrong or overly simplistic. Conversely, if it is too complex, it becomes possible to fit anything but doing so also means that any conclusion about variables is suspect. This is why parsimony is practically a religion for data modelling. There are statistical methods for calculating precision based on how many variables one applies. I forget what they're called, but adding in extra variables is heavily penalized.
