Kermack, McKendrick, and Kendall Model

     The initial objective of this project is to derive a system of non-linear, ordinary, differential equations to map the spread and containment of an infectious disease. Before deriving any formulas, however, it is absolutely critical that one clearly ascertains the theory and prior research concerning mathematical epidemiological models.

      After conducting extensive research on prior studies, it was found that the Kermack, McKendrick and Kendall model represents both a practical and reputable system of equations for mapping the spread of infectious diseases. This model, which was developed in the early 19th Century, has been used repeatedly in the past to investigate the spread of potent contagions such as the Plague. Although simple, this model is rather complete and effective. It relates the three most elementary variables of a human population subject to an infectious disease; namely, these are the susceptible, infected (and contagious), and immune/deceased/curing sub-populations. It is important to realize, however, that the Kermack, McKendrick model was initially developed to analyze a closed community, in which its overall population remained constant throughout the epidemic. Therefore, the modeled disease cannot be terminal. With some slight modifications, however, this model can be adapted to account for fatalities from the disease. This can be accomplished by incorporating certain death rate characteristics into the equation for the immune sub-population. This will be described in further detail later on.

      The Kermack, McKendrick model allows for the incorporation of an unlimited amount of internal disease and location characteristics. It is these parameters that determine the behavior of the entire system, and are the essence of the experimentation. Hence, their values must be meticulously chosen and tested before implementation.

      The system presented herein utilizes parameters that are comprised of various factors, which ultimately either advance or inhibit the spread of the disease. According to the properties of the Kermack, McKendrick model, this system’s variables must adhere to the following guidelines:

      Our set of equations is based on the Kermack, McKendrick and Kendall model, which states:

• x is the number of susceptible people
• y is the number of infective people circulating within the community
• z is the number of individuals who have been removed from susceptible and infective pool either by isolation, recovery, immunization or death
• b is a grouping of all of the infection-rate factors
• g is a grouping of all of the removal-rate factors.

      The original set of equations implemented for the Kermack, McKendrick model are as follows:

      By adhering to this base model as a conceptual foundation, a more specialized and sophisticated system of equations can be derived and implemented. The ultimate objective is to develop a set of relatively uncomplicated formulas that will indicate the qualitative trends of the disease.

Nate Alford | Peter Chan | Douglas Lockett | Christopher Roblee