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Let's say we have a mathematical model, like say the SIR model. Now if I collect the spread of say covid in my locality and collect real world data. What are the ways in which I can find the best parameters in the SIR model, so that these parameters are a best fit to the data I collected? Like is there any solid technique like parameter estimation for probability distributions
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It's model-dependent. For some models,  there are standard ways of calculating the parameters. For other models,  you may have to try some kind of numeric minimisation method to find the best-fitting parameters.
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For most distributions, one calculates an "unbiased estimator" of a parameter in order to find the best fitting one. I do not know the SIR model well, but I would start by searching up how the R0 parameter is defined, and how it is estimated!
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The SIR model describes the rates of an equivalent birth death type process (using a continuum approximation). The statistics of births and deaths are each then poisson. However you need to account for observation noise which may be dependent on many different things.

Using the stochastic model and observation model you can develop a likelihood function
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