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How does mass immunisation affect
disease incidence?
Niels G Becker
(with help from Peter Caley)
National Centre for Epidemiology and Population Health
Australian National University
A valuable feature of mathematical models that describe the
transmission of an infectious disease is their ability to
anticipate the likely consequences of interventions, such as
introducing mass vaccination.
This feature is illustrated in these tutorial-style lectures,
by address some specific questions in simplified settings.
Specific Questions
1. How well does immunisation control epidemics?
2. How well does immunisation control endemic transmission?
3. Is it always a good thing to promote vaccination?
4. What is a good strategy to protect a vulnerable group?
Question 1
How well does immunisation control epidemics?
More specifically:
Suppose the infection is absent and everyone is susceptible.
(a) What happens when the infection is imported?
(b) How is this changed when part of the community has
been immunised prior to the importation?
The setting
Assume that the community size (n) is constant over the
duration of the epidemic, and that n is large.
Suppose the infection is transmitted primarily by person-toperson contacts.
For example, measles or a respiratory disease.
Suppose that all n members of community are initially
susceptible to this infection.
At time t = 0, one recently infected individual arrives.
1. What happens?
2. How is ‘what happens’ altered if a fraction of
community members is totally immune?
Assume individuals are homogeneous and mix uniformly
Susceptible
St
infection
Infectious
It
Removed
(immune)
The deterministic SIR epidemic model for this process is
dS t
Rate of change in susceptibles =
dt
dI t
Rate of change in infectives =
dt

  It

 It
St
n
St
n

 It
The solution (found numerically) depends on the initial values
I0 and S0, and on the values of the parameters
, the transmission rate, and
, the recovery rate.
Setting 1
Suppose I0 = 1
n =S0 = 1000
 = 0.3
 = 0.1
What happens ?
The model predicts that the outbreak takes off and the epidemic
is described by
Infectives
Infectives in SIR model
Total number
infected
400
= area under
curve / 10
200
0
0
20
40
60
80
Time (days)
Question: Could something else happen?
100
≈ 941
Setting 2
Suppose I0 = 1 n =S0 = 1000  = 0.3/4  = 0.1
What happens ?
The model predicts that the outbreak peters out and is
described by
Infectives
Infectives in SIR model
1
Total number
infected
= area under
curve / 10
0
0
100
Time (days)
Question: What really happens?
≈4
We have seen these two types of outcomes:
Infectives in SIR model
400
Infectives
Infectives
Infectives in SIR model
200
0
1
0
0
20
40
60
80
100
0
Time (days)
100
Time (days)
This poses the questions:
(a) What determines whether an outbreak takes off?
(b) How large will the outbreak be?
(a) What determines whether an outcome takes off?
dI t
dt

 It
St
n
  It

  St

 I t 
 1 
  n

It increases initially when
 S0
 1
 n
It always decreases when
 S0
 1
 n
Setting 1:
 S0
 3  1
 n
(the outbreak takes off)
Setting 2:
 S0
 .75  1
 n
(it does NOT take off)
 S0
 n
determines whether the outbreak takes off.
(actually, there’s an element of chance)
ASIDE:
R0 
It is

1



is the basic reproduction number.
(rate at which the infective transmits) × (mean duration of the infectious period),
R0 = mean number of individuals a person infects during their
infectious period when everyone they meet is susceptible,
and there is no intervention.
The word basic is used when everyone else is susceptible and no
intervention is in place.
If R0 < 1
there can not be an epidemic.
No intervention is required.
If R0 > 1
an epidemic occurs.
(Can occur)
If R0 > 1 an epidemic is prevented when R0S0 /n <1.
That is, when the susceptible fraction has been reduced to less
than 1/R0 , by immunisation.
(b) How large will the outbreak be ?
Let
s0 = S0 /n = fraction initially susceptible
C∞ = eventual number of cases
c∞ = C∞ /S0 = fraction of initial susceptibles eventually infected
Then
s0 
 ln (1  c  )
R0 c 
where R 0   / 
Heuristic derivation
dS t
dt


0

1 dS t
S t dt
  It
dt
 ln S   ln S 0

 ln (1  c  )
R0 c 


n

 
n
which gives
s0 
St
n


0
I t dt

1
C   


What happens if some community members are immunised?
The initial reproduction number is
Rv 
 S0
 R0 s 0
 n
Illustrate this for Setting 1 I0 = 1, n = 1000,  = 0.3,  = 0.1
Proportion of infections among susceptible individuals
1
0.8
0.6
0.4
0.2
0
s0 
0
 ln (1  c  )
R0 c 
0.2
0.4
0.6
Proportion initially susceptible
Question: What really happens?
0.8
1
Question 2
How well does immunisation control endemic transmission?
More specifically:
Suppose transmission is endemic in the community.
(a) What does this mean?
(b) How is endemic transmission changed when the
community is partially immunised?
(c) What happens to endemic transmission in response to
a pulse of mass vaccination?
An infection is endemic in the community when transmission
persists.
It requires replenishment of susceptibles.
This happens by births, so we add births and deaths.
immunisation
Birth
Susceptible
St
Death
infection
Infectious
It
Death
Recovered
(immune)
Death
Assume no immunisation
dS t
dt
dI t
dt
St
 n   I t
 S t
n
S
  I t t   I t  I t
n
The solution depends substantially on I0 and S0 , but
eventually settles down to steady state endemic transmission.
We determine this state by solving the equations
dS t
dt

0
and
dI t
dt

This gives
sE 
 
 1
R0

and
iE 

R0  1

0
Numerical illustration
n = 1,000,000
R0 = 15 (e.g. measles)
 = 1/(70*365) (life expectancy of 70 years)
 = 1/7
(mean infectious period of 1 week)
sE = 1/15, that is 1,000,000/15 = 66,667 susceptibles
iE ≈ [7/(70*365)]*(1  1/15), that is 256 infectives
In practice the numbers fluctuate around those values,
because of chance fluctuations and seasonal waves driven
by seasonal changes in the transmission rate.
Question:
Would imported infections change this?
What if we immunise a fraction of the newly born infants?
dS t
dt
dI t
dt
Eventually

 (1  v )n   I t

 It
sE 
St
 S t
n
St
  I t  I t
n
 
 1
R0

and
iE 

[(1  v )R 0  1]

Transmission can not be sustained when (1 – v)R0 ≤ 1
The infection is eliminated when the immunity coverage
exceeds 1 – 1/R0.
[Or s ≤ 1/R0.]
Question: Why is sE not affected by the immunisation,
(as long as v ≤ 1 – 1/R0 )?
Question: What happens when iE is small?
Response to enhanced vaccination
Suppose we have endemic transmission (without immunisation)
and have a mass vaccination day.
That is, we immunise a fraction v of susceptibles at t = 0.
s0 
1 v
R0
and
i0 

R 0  1

d it
it
v io

(R 0s t  1)  
at t  0.
dt
 
 
So transmission declines immediately.
How much? And what happens then?
Consider the earlier example:
n = 1,000,000
R0 = 15
 = 1/(70*365)
 = 1/7
Infectives
Here’s what happens if we immunise 1%, namely 667, of the
susceptibles:
500
400
300
200
100
0
0
50
Time (Days)
Question: What really happens?
100
Here’s what happens if we immunise 5%, namely 3333, of
the susceptibles:
Infectives
2000
1500
1000
500
0
0
50
Time (Days)
Question: What really happens?
100
Two of our Specific Questions remain, namely
3. Is it always a good thing to promote vaccination?
4. What is a good strategy to protect a vulnerable group?
We will look at these questions with regard to one simple
model, which we now introduce.
We choose a situation with two types of individual.
One type is more vulnerable to illness, while the other type
contributes more to the transmission.
Practical examples include (a) rubella, and (b) influenza.
First the demography
Partition age into ‘young’ and ‘old’.
People are ‘young’ when they are aged less than c years.
The mortality rate is negligible for the ‘young’, and  for the‘old’.
The total community size is specified by
dN a

dt
0,
a c
 N a , a  c
N0,
a c
Na 
N 0 exp[  (a  c )], a  c
~
N 


0
N a da  N 0c N 0 /   total size of the community
 and N0 are estimated from demographic data
Suppose c = 50*365 = 18250 days and  = 1 / (10*365).
Then the life expectancy is 50+10 = 60 years.
The age distribution is
100
75
50
25
0
0
20
40
With N0 = 100 the community size is 6000.
60 Age 80
Next the transmission model
Consider an SIR model in which the transmission rate is agedependent
We consider only the steady state of transmission.
The steady state force of infection acting on the ‘young’ is , and
that acting on the ‘old’ is ’.
Estimate  and ’ from age-specific surveillance data
(perhaps using incidence data for a period before immunisation).
, the recovery rate, is estimated from disease-specific data
Suppose that
 = 0.0001,
’ = 0.00002,
 = 0.1
The steady state transmission equations are
For a in [0, c)
For a in [c, ∞)
dS a
  S a
dt
dS a
  ( '  )S a
dt
dI a
 S a  I a
dt
dI a
  ' S a  (   )I a
dt
The solution can be found analytically or numerically.
The solution is as follows:
For a in [0, c)
S a  S 0 exp(a )
 exp(a )  exp(a ) 
I a  S 0 

 


For a in [c, ∞)
S a  S 0 exp[c  ( '  )(a  c )]
I a  a long expression which can be approximated by
' S a
1  exp( ' )a 
  '
Proportion infectious at different ages – no vaccination
Proportion infected
0.001
0.0008
0.0006
0.0004
0.0002
0.0000
0
10
20
30
40
50
Age (years)
60
70
80
Proportion susceptible at different ages – no vaccination
Proportion susceptible
1.0
0.8
0.6
0.4
0.2
0.0
0
10
20
30
40
50
Age (years)
60
70
80
 and ’, the forces of infection, change when we change the
vaccination coverage.
In contrast, the rates of making close contacts do not change,
so it is useful to determine the corresponding transmission
rates.
The forces of infection and transmission rates are related by
c
~
~

  YY  I a da / N  O Y  I a da / N
0
c
~
c

~
 '  YO  I a da / N  O O  I a da / N
0
c
Can not determine 4 parameters from 2 equations
Assume proportionate mixing;
i.e. the WAIFW matrix is
~
c
~

    I a da / N   '  I a da / N
0
c
c
~

~
 '   '   I a da / N   '  '  I a da / N
c
0
We find
~
 
N
c

0

I a da  ( /  ' ) I a da
c
and
 '  ' / 
 0.10
 0.51
Question 3
Is it always a good thing to promote vaccination?
More specifically:
Consider a disease with more serious consequences for older
people, but young people transmit more infection.
Practical examples include
(a) rubella, and
(b) influenza.
Any type of immunisation reduces the overall incidence, but
some strategies may actually increase the incidence among
older people, and so increase their risk.
Our parameter values are
c = 50*365 days,  = 1 / (10*365),  = 0.1,
 = 0.51, ’ = 0.10
Now suppose that a fraction v of individuals are vaccinated,
essentially at birth.
Then S0 is reduced from N0 to (1-v)N0.
We first need to find the new expressions for Sa and Ia from the
steady state transmission equations.
Then substitute these in
~
c
~

    I a da / N   '  I a da / N
0
c
~
c

~
 '   '   I a da / N   '  '  I a da / N
0
and solve for the new  and ’.
c
Graph of cases aged over 50 at v relative to v = 0.
Ratio of cases for age>C
1.5
Vaccination at birth
1.0
0.5
0.0
0.0
0.2
0.4
0.6
Vaccination coverage
0.8
1.0
Question 4
What is a good strategy to protect a vulnerable group?
More specifically:
As above, consider a disease which is more serious for older
people and young people transmit more infection.
To protect the old people, is it better
(i) to vaccinate the young, or
(ii) to vaccinate the older people?
Practical examples again include
(a) rubella, and
(b) influenza.
Our parameter values are again
c = 50*365 days,  = 1 / (10*365),  = 0.1,  = 0.51, ’ = 0.10
The above strategy vaccinated a fraction v of individuals at birth.
For comparison, consider a strategy which, instead, vaccinates a
fraction v of individuals as they reach the age of c years.
Then Sc+ is reduced from Sc to (1-v) Sc .
With this change we find the new expressions for Sa and Ia from
the steady state transmission equations and substitute these in
~
c
~

    I a da / N   '  I a da / N
0
c
~
c

~
 '   '   I a da / N   '  '  I a da / N
0
to solve for the new  and ’.
c
Graph of cases aged over 50 at v relative to v = 0.
1.5
Vaccination at birth
Ratio of cases for age>C
Vaccination at C=50 years
1.0
0.5
0.0
0.0
0.2
0.4
0.6
Vaccination coverage
0.8
1.0
Limitations of these deterministic SIR model
• It and St are taken as continuous when they are really
integers. (Of concern when It or St are small)
• They suggest that an outbreak always takes off when
R0 s0 > 1. (Not always the case.)
• They ignore the chance element in transmission.
(Of particular concern when It or St are small, e.g. during
early stages)
The End