kermack & mc kendrick epidemic threshold theorem

Kermack & McKendrick Epidemic Threshold Theorem- BY SMRUTI MOKAL MSC. BIOSTATISTICS AND DEMOGRAPHY IIPS, MUMBAI.

SOURCES

A generalization of the Kermack and McKendrick deterministic epidemic model- Capasso and Serio, mathematical biosciences 1978.

Wolfram Mathworld

The mathematics of infectious diseases – Lenka Bubniakova

Some History…

McKendrick was a physician commissioned by the English Army to India.

McKendrick became involved with in the study of epidemic diseases using mathematical models through the direct encouragement of Sir Ronald Ross who was also a physician.

His simple epidemic model was published in a joint paper with Kermack (Kermack and McKendrick, 1927).

It involved the study of the transmission dynamics of a communicable disease that provide permanent immunity after recovery.

Their model was used to study single epizootic outbreaks.

Their mathematical work led to the first widely recognized threshold theorem in epidemiology.

It was proposed to explain the rapid rise and fall in the number of infected patients observed in epidemics such as the plague (London 1665-1666, Bombay 1906) and cholera (London 1865).

The Kermack-McKendrick model is an SIR Model for the number of people infected with a contagious illness in a closed population over time.

Assumptions

The population size is fixed (i.e., no births, deaths due to disease, or deaths by natural causes).

Incubation period of the infectious agent is instantaneous.

Duration of infectivity is same as length of the disease A completely homogeneous population with no age,

spatial, or social structure.

This simple model is formulated for a population of N being divided into three dis-joint subpopulation– the susceptible class S, the infective Class I and the Removed class R. The model consists of a system of ordinary differential equations- = -βSI …(1) = γI …(2)But since N= S+I+R, + + = 0Therefore, = βSI – γI …(3)

β is the infection rateγ is the recovery rate

The key value governing these equations is the so called epidemiological threshold,

R0 = βS/γ

The quantity R0 defined above is referred to as the basic reproduction number of the SIR model, accounting for the average number of new infections that a single infectious individual can cause during the infection life time.

(i) When R0 = βS/γ <1, there is no outbreak of the disease in the sense that the population of the infectious class decreases monotonically to 0.

(ii) When R0 = βS/γ >1, there will be a single outbreak of the disease in the sense that firstly increases monotonicallyto a maximum value, and after that, decreases monotonically to 0.

From the summary of the model, we know that the disease dynamics of this model is very clear: the disease either dies out quickly without causing new infectious, or experiences a single outbreak before dying out.

This model does not include demographic structure and is suitable for describing those diseases that suddenly develop in a community and then disappear without infecting the entire community.

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