Valid observation model for SIR model

What would be a valid observation model for example for the number of infected people (I_obs) in a SIR model? In this case study, a negative binomial distribution is used with mean equal to the ODE solution obtained from the SIR model (I_ODE) and variance that depends on I_ODE and a noise parameter phi. However, it places probability mass also for I_obs > population_size which doesn’t really make sense as it’s not possible that somehow the measured number of infected people is larger than the population size.

Here are visualizations of the prior predictive distribution for the above case study.
I_prior_pred_dist

Although 0.95 interval seems to be mostly under the population size, there are still draws that are above:

I_prior_pred_draws

EDIT: the first plot is the same as this plot

in Section 2 of the case study, which shows 90% interval.

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A simple binomial observation would be the place to start.

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