# Nonhomogeneous Linear Systems

Author: John J Weber III, PhD Corresponding Textbook Sections:

• Section 9.4 – Differential Operators and the Elimination Method for Systems
• Section 9.4 -- Linear Systems in Nominal Form
• Section 9.5 -- Homogeneous Linear Systems with Constant Coefficients
• Section 9.6 -- Complex Eigenvalues
• Section 9.5 -- Nonhomogeneous Linear Systems with Constant Coefficients

## Expected Educational Results

• Objective 17–1: I can non-homogeneous linear systems using eigenvalues and eigenvectors.

## Nonhomogeneous Linear Systems

### Compartmental Models in Epidemiology

#### SIR Model (a.k.a., Kermack-McKendrick Model)

NOTE: This model was used in TV show Numb3rs in Season 1 Episode 4: https://www.imdb.com/title/tt0663234.

Let $S(t)$ be the number of individuals susceptible to the disease at time $t$; let $I(t)$ be the number of infected individuals at time $t$; and $R(t)$ be the number of individuals who recovered from the disease; then the disease can be modeled by the following system of first-order non-linear odes:

$\dfrac{dS}{dt} = -\dfrac{\beta I S}{N} \\ \dfrac{dI}{dt} = \dfrac{\beta I S}{N} - \dfrac{\gamma I}{N} \\ \dfrac{dR}{dt} = \dfrac{\gamma I}{N}$

where $N=S+I+R$, $\beta$ is the infection rate, $\gamma$ is the recovery rate.

The epidemiological threshold, $\displaystyle R_0=\frac{\beta S}{\gamma}$, is the number of secondary infections caused by a single primary infection; i.e., it determines the number of people infected by contact with a single infected person before his/her death or recovery. If $R_0<1$, then the disease will gradually end; if $R_0>1$, then $\displaystyle \frac{dI}{dt}>0$.

NOTE: $R_0$ is not related to $R$.

NOTE: $R_0$ was frequently referenced in media discussions of the COVID-19 pandemic. An NIH Report [https://www.ncbi.nlm.nih.gov/books/NBK554776/] states that for COVID-19, $R_0=2.2$, i.e., each infected person will infect an additional 2.2 individuals, on average.

#### Investigation 03

1. Explain why $\displaystyle \frac{dI}{dt}$ is dependent on $I$.
2. Explain why $\displaystyle \frac{dI}{dt}$ has a negative sign in the formula.
3. Explain why $\displaystyle \frac{dI}{dt}$ has two terms.
4. Explain why $\displaystyle \frac{dI}{dt}$ is dependent only on the $I$ variable.

### SEIR Model

Let $S(t)$ be the number of individuals susceptible to the disease at time $t$; let $E(t)$ be the number of individuals exposed to the disease; let $I(t)$ be the number of infected individuals at time $t$; and $R(t)$ be the number of individuals who recovered from the disease; then the disease can be modeled by the following system of first-order non-linear odes:

$\dfrac{dS}{dt} = \left(\Lambda - \mu\right) S - \dfrac{\beta I S}{N} \\ \dfrac{dE}{dt} = \dfrac{\beta I S}{N} - (\mu +\alpha)E \\ \dfrac{dI}{dt} = aE - (\gamma + \mu) I \\ \dfrac{dR}{dt} = \gamma I - \mu R$

where $N=S+E+I+R$, $\beta$ is the infection rate, $\gamma$ is the recovery rate, $a^{-1}$ is the average incubation period, $\Lambda$ is the birth rate, and $\mu$ is the death rate.

The epidemiological threshold, $\displaystyle R_0=\frac{a}{a+\mu}\frac{\beta}{\mu+\gamma}$, is the number of secondary infections caused by a single primary infection; i.e., it determines the number of people infected by contact with a single infected person before his/her death or recovery. If $R_0<1$, then the disease will gradually end; if $R_0>1$, then $\displaystyle \frac{dI}{dt}>0$.