Nonhomogeneous Linear SystemsExpected Educational ResultsNonhomogeneous Linear SystemsCompartmental Models in EpidemiologySIR Model (a.k.a., Kermack-McKendrick Model)Investigation 03SEIR ModelCC BY-NC-SA 4.0

**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

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

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

Let be the number of individuals *susceptible* to the disease at time ; let be the number of *infected* individuals at time ; and 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:

where , is the *infection rate*, is the *recovery rate*.

The *epidemiological threshold*, , 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 , then the disease will gradually end; if , then .

**NOTE**: is **not** related to .

**NOTE**: 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, , i.e., each infected person will infect an additional 2.2 individuals, on average.

Some questions to think about:

- Explain why is dependent on .
- Explain why has a negative sign in the formula.
- Explain why has two terms.
- Explain why is dependent only on the variable.

Let be the number of individuals susceptible to the disease at time ; let be the number of individuals exposed to the disease; let be the number of infected individuals at time ; and 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:

where , is the *infection rate*, is the *recovery rate*, is the average incubation period, is the birth rate, and is the death rate.

The *epidemiological threshold*, , 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 , then the disease will gradually end; if , then .

This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License [http://creativecommons.org/licenses/by-nc-sa/4.0/].

**Last Modified**: Thursday, 15 October 2020 6:42 EDT