Week 4 part 2 (Stability analysis of an SIR model)

[@kcheerc29]

this is the best tool out for learning stability analysis. thank you.

[@hashansingh6911]

Thank you so much!

[@ahexcuseme6936]

Thanks, for the video, sir.

[@justineoninika]

After following this now I can do anything concerning stability

[@abbiebackers2085]

thank you so much for this video! sooooo helpful for my final year project!!!. Absolute lifesaver

[@adoringangel6781]

Hello, I'm doing a research on stability analysis on HIV/aids and i need some help with my project. Please can you help me ? I need help really urgently since my presentation is tomorrow.

[@mathinnam]

Thank you this series has been soooooo helpful!!!

[@asifhassan737]

Thank you for your help ❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤

[@olusojivictor9553]

I’m working with a SEIR model, can you help me out on that please? I’m getting really confused.

[@hannatulchinsky8975]

What paper did you find this model from?

[@Daniel_Maxin]

No particular paper. The classical SIR model which is the one in the video without the "alpha" term is in nearly all textbooks on the topic as an introductory model. If you really need to cite it you can just pick any textbook that contains it. The wikipedia page has a list of citations for it en.wikipedia.org/wiki/Compartmental_models_in_epidemiology#The_SIR_model_without_vital_dynamics. The addition of the "alpha" terms was done by me just to add variation of these models for the purpose of teaching but I am sure this is not new. Most contemporary papers deal with models that have more features and terms than the ones chosen in the textbooks.

[@aldoyactayo05]

Why do you find roots only for I´=0 and not for S´=0?

[@Daniel_Maxin]

I do use both I'=0 and S'=0. These two form a system of two equations and two unknowns (S and I) that we solve by substitution. I start with I'=0 and then substitute in S'=0 (at 5:55). I could have just as well started with S'=0 but it is usually easier to start with I'=0 because it factors.

[@aldoyactayo05]

@@Daniel_Maxinthank you. But what I mean It´s that at minute 5:55 you use roots I=0 and S=r/lambda from equation I´=0. But you don´t get and use roots (I and S) from S´=0. That´s my question. Thank you very much

[@Daniel_Maxin]

@@aldoyactayo05 S'=0 does not give you roots for I and S directly because it does not factor. It only gives you I in terms of S or S in terms of I. BUt if you really want to start with S'=0 you eventually get the same equilibria. Suppose I start with S'=0 and solve I in terms of S. I will get I=alpha(N-S)/(alpha+lambda*S). Notice the denominator is positive because I am looking for positive roots only (since there is no negative population). Now I replace I with alpha(N-S)/(alpha+lambda*S) in the second equation (I'=0) which, in the factored form, now looks like (lambda*S-r)*(alpha(N-S))/(alpha+lambda*S)=0. Equating each factor to zero we obtain S=r/lambda and S=N. If you replace S with r/lambda in the substitution alpha(N-S)/(alpha+lambda*S) you will get S=alpha(lambda*N-r)/(lambda(r+alpha)). If you replace S with N in the same substitution we get I=0. So the same equilibria that we obtained if we started with I'=0 instead. So because this is a system, you use one equation to substitute into the other. We normally use I'=0 simply because it factors and it makes it easier to solve for the equilibria.

[@aldoyactayo05]

@@Daniel_Maxin thank you very much, very clear, very kind. I´m watching your videos. Thanks again.