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In analyzing the plague, I am going to use the Kermack and McKendrick models which is an SIR model. For a clearer understanding, I have used an epidemic curve of this great plague. The Kermack and McKendrick model explains the total number of people infected by the disease within a given period. The SIR model of Kermack and McKendrick depicts an exponential rise, turnover, and later decline. These characteristics are what make the article described below. Although the analysis encourages the understanding of epidemics through mathematical modeling, some features of the epidemic curve are not predicted in the SIR model, for instance, the jagged features in the curve. This, however, does not lessen the essence of SIR model in the epidemic analysis. This is because the features not captured by the SIR model with a considerable mathematical effort the fine details can be accounted for. This is done by replacing Kermack and McKendrick differential equations with other equations, such as the stochastic.

The Kermack-McKendrick model is a SIR Modeling the number of individuals infected with contagious diseases within a closed population for a certain period. This model was proposed to explain the quick fluctuation in the number of infected persons observed in epidemics, like the great plague of 1965-66 in London and the measles of New York. The assumptions under this model is that; 1) We have a fixed population size, (meaning that there are zero deaths due to diseases and natural calamities and that new births are held constant.), 2). The incubation period of the infectious agent is instantaneous, 3). The period of the infection is the same as the period of the disease. 4). There is a homogeneous population, regardless of age, race, spatial, economic, or social structure.

Analysis of a less contentious epidemic, such as the measles, whose infecting virus is well known, will give a clearer understanding of the model. The disease affects the respiratory tract and is transmitted through the air. Red spots that are easily identifiable characterizing the disease and thus parents easily identify and take their children to the doctor. This means that many measles cases are reported to the doctor, thus making data available for modeling. In modeling this epidemic we can get the data for the number of deaths during the measles in New York of 1962. A 17-month period just as that of the London plague is used in the analysis.

Stay Connected The model has three nonlinear differential equations;

t = Time,    S (t) = Number of susceptible people,   I (t) = number of infected people. R (t) =  represents the population size of persons who have recovered and also developed the immunity.  The epidemiological threshold determines the evolution of the above equations and it is represented by:

Where R is the number of new infections caused by a primary infection, i.e. it calculates the number of new infections through the contact of an infected person before the person recovers or dies. When R < 1  means that each person, who contracts the epidemic, will infect less than 1 person before he dies or before he recovers, when R > 1 it means that any person who contracts the disease will infect more than one person, and in the end it will lead to the widespread of the disease.

The figure below shows the weakly mortality rate for a week during the great plague in London. On a longer period, complex patterns of plague epidemics are characteristic that includes extinctions and re-emergencies. The longer period trends have not been graphed in this article, as they cannot be explained by the use of the basic SIR model even where stochastic processes have been used in the formulation. A major supposition made in this analysis is that plague is the human spread forgetting the fact that there is a reservoir of the sickness with rodents. In this regard, they can continue to spread for years with little effort to notice it ending up re-emerging explosively. The inclusion of exclusion of the rodents in the model has sparked a debate as to whether the causal vector to these plagues is same pathogenic organism.

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When the total population is N, and everybody is susceptible i.e.; when, a newly infected person can possibly infect  number of people. The derivative  is the ratio of change in S within a time divided by the time interval, where the limit approaches zero. When the time is to the derivative, it can be obtained by  and then calculate the number of susceptible when the period is  in the near future as. In the same breath, the number of deaths can also be approximated by the following formula

The above two equations provide an approximation of the SIR model. If we may analyze the infection of the disease, we assume that the main infection rate was  and the reproduction number is R=18 and that we assume the total population to be 7.5 million people. We assume that 1 infection came into town , everybody in the city is prone to the disease; we obtain a curve similar to the above. The curve is upward sloping, because of the many number of susceptible people within reach, the rate of growth and spread of the pathogen is very fast, there was no immediate medical attention for the victims, and there was a good bleeding ground for the pathogens in the poor squalid lives of the poor people in the city.

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