The SIR model is 3 EDO's

I am looking to estimate the parameters (b, y) with the method of Gauss Newton

The real data they give me are only the number of infected from day 1 to 14

The system lacks analytical solution

I solve with Euler this gives me the numerical approximation of the system,

Approach the problem of initial values ​​with Cauchy

(DS / dt) = -s * i * b

(DI / dt) = -s * i * b - y * i

(DR / dt) = y * i

Where

S: susceptible persons

I: Infected people

A: People removed

The problem is that when estimating the partial derivatives, I do not understand from which equation to estimate them since it seems logical that I can not do it from dI / dt since it represents the change of the infected with respect to the time osea the derivative thus I think Who should do it from its integral?

My problem is where I make my first partial derivatives and what are they to evaluate the initial matrix?

Any help, bibliographical, correctness, thank you.

regards

Santiago

Just a few quick comments - firstly, you have an 'A' in there which I guess is meant to be an 'R'. And b and y have meanings rather than just being there; b is the transmissivity / infection rate and y is the population's death or immunity rate. Finally, to avoid confusion, are lowercase s, i, and r are the same as uppercase (I think lower case are fraction of population whilst uppercase are absolute values -this would make your setup of partial derivatives incorrect)?

 

Perhaps post all the info ou have been given and maybe someone can help - it is not my cup of tea but others may aid you if you present correct and full information

Assuming your equations are

(dS / dt) = -S * I* b

(dI / dt) = -S * I * b - y * I

(dR / dt) = y * I

and b and y are constants, then, differentiating the first equation a second time,

d^2S/dt^2= -b(dS/dt)I- bS(dI/dt)

Replace "dI/dt" with -S*I*b- y*I from the second equation:

d^S/dt^2= -b(dS/dt)I- bS(-bS- y)I= -b(dS/dt- bS^2- yS)I

we can rewrite the first equation as I= -(dS/dt)/(bS) and putting that into the last equation

d^S/dt^2= b(dS/dt- bS^2- yS)(dS/dt)/bS= (1/S)(dS/dt)^2- S(dS/dt)- dS/dt

That's reduced to a single equation for S but it is a rather badly non-linear equation. I doubt there will be any method for getting an exact solution

for S.