Nonhomogeneous Linear Systems

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:

Expected Educational Results

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 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.

Investigation 03

Some questions to think about:

  1. Explain why is dependent on .
  2. Explain why has a negative sign in the formula.
  3. Explain why has two terms.
  4. Explain why is dependent only on the variable.

SEIR Model

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 .

CC BY-NC-SA 4.0

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