[Solved] Nonlinear-control Homework1 On a nonlinear infection model

Reference: Can the COVID-19 epidemic be managed on the basis of daily data? by Francesco Casella, Dipartimento di Elettronica, Informazione e Bioingegneria Politecnico di Milano, Italy.

The standard and basic SIR model is used to model the population dynamics of some infection as COVID-19. It considers a population of N individuals, consisting in three compartments:

• S is the amount of Safe individuals, not yet infected and Susceptible to become infected. Their population will decrease if the Safe patients have the opportunity to encounter Infected patients. This sub-population does not include patients which were infected but have Recovered from the infection.
• I represents the amount of Infected patients. Their population will increase when some S individuals get infected, and it decreases when Infected people Recover from the disease.
• R is the amount of population which was infected, but has Recovered from the infection and is healthy again.

Note that by definition N = S + I + R.

The dynamics is thus described by the differential equations

(1)

(2)

(3)

The amount of newly infected patient is proportional to both S and I. The parameter is a virulence factor which depends on drugs (if any), on vaccine and eventually on political regulations on the mobility of the population. Thus, is considered to be the control input. is the inverse of the constant duration of the disease for an individual.

QUESTION 1: Among the three variables S, I and R how many are independent, or transcendent with respect to the field of real numbers ?

ANSWER 1 : Only two variables, say S and I. In fact, we got the algebraic relation R = NSI. Furthermore, there is no way to get an algebraic relation (over real number) involving the two remaining variables S and I. The latter are thus independent, or transcendent with respect to the field of real numbers.

We may drop equation (3) above to get the well-posed nonlinear state space representation:

(4)

(5)

1

Define further the two output equations

QUESTION 2: Check whether or not the two outputs Y₁ an Y₂ are differentially algebraically dependent or transcendent with respect to the field of real numbers. I.e., does there exist a (possibly nonlinear) differential equation F(Y₁,Y₂,Y₁,Y₂,) = 0 which does not involve the control input ?

Recall that N and are constant real numbers and my be involve in such a differential equation.

ANSWER 2:.