An Epidemic model is a simplified means of describing the transmission of communicable disease through individuals.
The outbreak and spread of disease has been questioned and studied for many years. The ability to make predictions about diseases could enable scientists to evaluate inoculation or isolation plans and may have a significant effect on the mortality rate of a particular epidemic. The modeling of infectious diseases is a tool which has been used to study the mechanisms by which diseases spread, to predict the future course of an outbreak and to evaluate strategies to control an epidemic (Daley & Gani, 2005).
The first scientist who systematically tried to quantify causes of death was John Graunt in his book Natural and Political Observations made upon the Bills of Mortality, in 1662. The bills he studied were listings of numbers and causes of deaths published weekly. Graunt’s analysis of causes of death is considered the beginning of the “theory of competing risks” which according to Daley and Gani (Daley & Gani, 2005, p. 2) is “a theory that is now well established among modern epidemiologists”.
The earliest account of mathematical modeling of spread of disease was carried out in 1766 by Daniel Bernoulli. Trained as a physician, Bernoulli created a mathematical model to defend the practice of inoculating against smallpox (Hethcote, 2000). The calculations from this model showed that universal inoculation against smallpox would increase the life expectancy from 26 years 7 months to 29 years 9 months (Bernoulli & Blower, 2004).
Following Bernoulli, other physicians contributed to modern mathematical epidemiology. Among the most acclaimed of these were A. G. McKendrick and W. O. Kermack, whose paper A Contribution to the Mathematical Theory of Epidemics was published in 1927. A simple deterministic (compartmental) model was formulated in this paper. The model was successful in predicting the behavior of outbreaks very similar to that observed in many recorded epidemics (Brauer & Castillo-Chavez, 2001).
The SIR / SIRS Models
In 1927, W. O. Kermack and A. G. McKendrick created a model in which they considered a fixed population with only three compartments, susceptible: S(t), infected, I(t), and recovered, R(t).The compartments used for this model consist of three classes:
- S(t) is used to represent the number of individuals not yet infected with the disease at time t, or those susceptible to the disease
- I(t) denotes the number of individuals who have been infected with the disease and are capable of spreading the disease to those in the susceptible category
- R(t) is the compartment used for those individuals who have been infected and then recovered from the disease. Those in this category are not able to be infected again or to transmit the infection to others.
The SIRS Model is simply an extension of the SIR model as we will see from its construction. The only difference is that it allows members of the recovered class to be free of infection and rejoin the susceptible class.
See also: https://en.wikipedia.org/wiki/Epidemic_model
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