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We have some very nice discussions about SIR model fitting at CV. As I explore the model with different parameters, I have some questions on the definitions of susceptible and recovered population. And how to align the mode with real world scenario.

Specifically, we always see some plot like this, where green, red, and blue represent S,I,R respectively.

enter image description here

My biggest confusion is that, In real world, the green curve and the blue curve should far away from each other, i.e, most diseases will never infect everyone in all population eventually.

Therefore I changed some parameters to make some thing like this happen.

enter image description here

Then, I think it is not realistic because $R0$ is almost equal to $1.0$.

Here is my question: let's use covid19 as example, we know this virus will not affect everyone in a given country (we draw this conclusion from the observation from ourself, we know most of our friends and families are not infected, well hopefully). And the $R0$ of covid19 is not close to $1.0$.

Then there are conflicts here, and how can we fix it?

Haitao Du
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  • - If you do not want to resort to more complex implementations of SIR then you could simply change the size of the initial susceptible population which will make less people sick. - You could also scale the sick infectious people or make two different classes of infectious people (asymptomatic and symptomatic), which can actually be solved with the same equations if you do some transformations. - The real world scenario are observations of something different. The SIR model does not fit this and you need to add some patch. (The simplest would be to scale the outcome according to some factor). – Sextus Empiricus Apr 27 '20 at 10:48
  • in a SIR model the R0 is constant all the time, which is not a good fit to reality. because of mitigation and awareness in the population, the risk of infection changes and should be represented by a dynamic parameter Rt (effective reproduction number). – Edgar Apr 27 '20 at 10:55
  • @Edgar in a simple SIR model the *basic* reproduction number, $R_0$, is constant, but the *effective* reproduction number is still a dynamic parameter $$R_t = R_0 \frac{S(t)}{N}$$ But you are right that there is more to the dynamics of the effective reproduction than just this factor $S(t)/N$. Normally we will see a drop because of other reasons. And if those reasons are not put into the model, then the SIR model will put all of this drop into the $R_0$ being close to 1. – Sextus Empiricus Apr 27 '20 at 18:05
  • @sextus that's what I meant – Edgar Apr 27 '20 at 18:07
  • @SextusEmpiricus so you mean we should set much smaller S instead of some countries's population? For example if 1 country has 1 milion people, we just say S=100K that other 90% are will not be affected? (say they are super rich and live in isolated islands) ... – Haitao Du Apr 27 '20 at 18:10
  • @SextusEmpiricus I think in real world this S will be changing all the time, for example if some lock-down in place, that 70% of the people are following the rules, then this S will be dramatically reduced. – Haitao Du Apr 27 '20 at 18:12
  • @HaitaoDu the simple SIR model, which is a simple (non-spatial, non-network) compartment model, makes little sense regarding the mechanism of the spread of covid-19 (the assumptions that allow you to use a compartment model do not hold at this large scale). However, you can still use a SIR model as an empirical model (instead of a mechanistic model), but then you should not fix parameters like number of susceptible at the start. That is because the parameters are not realistic (the SIR model is not a mechanistic model, but a empirical model where the parameters do not make any physical sense). – Sextus Empiricus Apr 27 '20 at 18:18
  • I agree that the epidemiological parameters should not be considered fixed and change in time. An example is the spatial model [here](https://stats.stackexchange.com/a/461455/164061) which has a change in the $\beta$ parameter after some time. In that model the $S$ is not changed but a more natural/mechanistic way to model it is applied by considering more compartments (in that case it is agent based but you could do the same in bulk by dividing S into two different compartments: a compartment that follows rules and a compartment that doesn't). – Sextus Empiricus Apr 27 '20 at 18:22

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