2. 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
3. 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).
4. 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).
5. The Kermack-McKendrick model is an SIR Model for the number
of people infected with a contagious illness in a closed
population over time.
6. 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.
7. 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-
𝑑𝑆
𝑑𝑡
= -βSI …(1)
𝑑𝑅
𝑑𝑡
= γI …(2)
But since N= S+I+R,
𝑑𝑆
𝑑𝑡
+
𝑑𝑅
𝑑𝑡
+
𝑑𝐼
𝑑𝑡
= 0
Therefore,
𝑑𝐼
𝑑𝑡
= βSI – γI …(3)
β is the infection rate
γ is the recovery rate
8. 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.
9. (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 monotonically
to a maximum value, and after that, decreases monotonically to
0.
10. 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.