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Exploring The Fucntion Of The Sir Model

2255 words - 10 pages

Infectious diseases had major impacts and influences in the human history. Diseases such as Spanish Influenza or the Bubonic Plague have remarkable positions in history. Disease spread models are used to predict outcomes of an epidemic. These models are used to calculate the impact of an infectious disease, funding required for mass vaccinations and data for public health departments. The earliest mathematical model of infectious diseases was created by Daniel Bernoulli in 1766. This model was used to predict the outcome of inoculation against smallpox disease. In the modern world, these models are created using various software programs. The reason why I chose this subject is because I previously worked on some modelling simulations. Also my father is in the healthcare sector, so this topic looked very exciting to me. Predicting outcomes of infectious epidemics may save thousands of lives and millions of dollars. In the healthcare sector, accuracy and reliability is very important. In this project, the work function of the SIR epidemic model and some of its derivatives will be explored along with some theorems about this models. SIR model is the fundamental model of almost all modern epidemic models. SIR model is the most widely used disease spread model in the world. Also it is a simple epidemic model which has mathematics that commensurate with our class.
SIR Model
The model is created by W.O. Kermack and A.G. McKendrick in 1927. SIR Model has three compartments: Susceptible, infected and removed.
S: denotes the number of individuals who are not infected with the disease. These people are vulnerable to catch the disease.
I: represents the number of individuals who have been infected. These people can transfer the disease to people in susceptible category. After the infectious period is finished, they enter the removed compartment.
R: represents the individuals who have been infected and removed the disease from their body. People in this compartment are assured to be immune from the infection and cannot transfer the infection to susceptible people. Removal of the disease is either due to death of vaccination.
β: is the contact rate of the disease. It may also be called as the parameter of infectivity.
γ: is the recovery rate for the infected people. The diagram of the model is given below.
R_0: means the reproductive number. It’s the average number of secondary infections that occur when an infectious person enters a susceptible population.

SIR Equations
The model only works in a closed population. This means there is no emigration from the population or there is no immigration to the population. So the population is fixed. This model can be described by the equations given below.
N=S+I+R
dS/dt= -βSI/N
dI/dt= -βSI/N-γI
dR/dt=γI
γ=1/((Duration of Infectious Period))
β= R_0∙γ
These equations give the S, I or R number of the population in time t. These algorithms are used in the SIR model. Before the disease actually spreads S=N and R+I=0....

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