Kermack and McKendrick developed one of the first epidemiological models in 1927 to study the transmission dynamics of communicable diseases that confer permanent immunity. Their simple SIR model divides a population into susceptible, infected, and removed compartments. It introduced the basic reproduction number (R0), which determines whether an outbreak occurs - if R0 is less than 1, the disease dies out, whereas if R0 is greater than 1, there is a single outbreak that increases then decreases. This model explained the rapid rise and fall of historical epidemics while making simplifying assumptions about population homogeneity and disease stages.

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

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.

13. THANK YOU