update vignettes with covid parameter function #71
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update all non-age-structured vignettes with new parameter function
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| \begin{equation} | ||
| I_j(t) = I(0)/2 \exp(\beta t/2), | ||
| \end{equation} |
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I think you need to motivate this a bit more, at least I don't follow your reasoning why both beta and I(0) should be divide by two. If this comes from your simplified version of dI/dt, then I think it might be good to start there and do the integration again. Intuitively, I can see why you would divide I(0), but beta honestly isn't obvious to me as it is an inherent property of the virus and doesn't have anything to do with population size.
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Also, and this might be wrong, but I feel like with split populations, initially, when the population is largely susceptible, is exactly the time when it shouldn't matter that you have two communities because spread of the disease won't yet be limited by the number of susceptible individuals. That point comes later when a good number of people have been infected, and then it depends on how much mixing you have with the other population and whether or not that "supplies" enough susceptibles for the epidemic to grow as quickly as it would in one, well-mixed population
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If anything, for two very weakly connected populations, the peak in the initially infected populations might be reach before the peak of the well-mixed model because there are less people to infect but the peak in the second population will be much later. I think the point to make there is that the peak number of infections in the two communities model, although you will have two instead of one, will be much lower than in a well-mixed model, which is good when thinking about hospital capacity, etc. (I think that is what I write originally in my explanation of this phenomenon).
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Thanks @solveigvandervegt So, it's not beta that halves; it's the susceptible population size that does, resulting in this factor of 2:
dIdt = beta S I
If we start with S=1 this becomes
dIdt = beta I
If we split the population in two then each evolves as:
dIdt = beta / 2 I
You're right though, I need to explicitly show this as it's not clear.
Re: "it shouldn't matter that you have two communities because spread of the disease won't yet be limited by the number of susceptible individuals" -- the number of contacts per unit time is given by S * I. If you halve the population, then each has S * I / 2 -- effectively you are missing out on all the contacts between pop A individuals and pop B individuals. Because of this, the result is an exponential reduction in the growth. Another way to see this is if, instead, you broke the population up into n really small communities: then you'd be missing a massive number of potential contacts. I think you're right that this matters even more when you have more infectives, but even from the start, it represents a significant reduction.
| ## The impact of a fragmented population on rates of transmission | ||
| Aside from reducing flow between subpopulations, fragmenting a population into smaller groups also slows spread for another reason: there are fewer contacts to infect. We now use a straightforward mathematical argument to illustrate this. | ||
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| For simplicity, suppose that the latent period is very short, so that we can neglect it. Also suppose that we consider also a fully mixed population where everyone interacts with everyone else. Our `SEIRD` model then becomes an `SIR` model, with the infectious compartment dynamics determined by: |
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this sentence has also twice
So far, I have updated all non-age-structured vignettes with new parameter function, except for the vaccine one.