As in the image below, may someone help us on how to estimate the parameters used on epidemiological mathematical modeling described in systems of non linear differential equations? Any possible method or mathematical software?
You can not do such an estimation without a sample of the population involved. Do you have such a sample?
The attached two papers discuss methods for parameter estimation in stochastic and deterministic dynamic epi modelling
There are at least several methods to do so, and it can be utilized in different types of models. Here's a comparison between exemplary semi-heuristic, and fully automatic methods (it's Open Access so don't worry for subscription):
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But you cannot estimate the exact values, without any reference data. You should think about it, as an advanced interpolation - you have several data series, and try to fit the model to match them. Because of the numerous model parameters in your case, I would highly recommend some MATLAB algorithms. In my experience, the fmincon called from within GlobalSearch procedure is one of the best start points. Alternatively, especially when you are sure, that some of the parameters can be only integers, you can also try your luck with the genetic algorithm (ga procedure). I suggest using GlobalSearch as in so complex model, You should expect at least a few minimums (that's the point you are looking for), so it will be very easy to fall into the trap.
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