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?

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-to-person 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?

The setting

infectionSusceptibleSt

InfectiousIt

Removed(immune)

tt

tt

tt

t

InS

IdtdI

nS

IdtdS

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.

Assume individuals are homogeneous and mix uniformly

Rate of change in susceptibles =

Rate of change in infectives =

The deterministic SIR epidemic model for this process is

Setting 1Suppose 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

Total number infected

= area under curve / 10

≈ 941

Question: Could something else happen?

Infectives in SIR model

0

200

400

0 20 40 60 80 100Time (days)

Infe

ctiv

es

Setting 2Suppose I0 = 1 n =S0 = 1000 = 0.3/4 =

0.1What happens ?The model predicts that the outbreak peters out and is described by

Total number infected

= area under curve / 10

≈ 4

Question: What really happens?

Infectives in SIR model

0

1

0 100Time (days)

Infe

ctiv

es

This poses the questions:

(a) What determines whether an outbreak

takes off?

(b) How large will the outbreak be?

Infectives in SIR model

0

1

0 100Time (days)

Infe

ctiv

es

Infectives in SIR model

0

200

400

0 20 40 60 80 100Time (days)

Infe

ctiv

es

We have seen these two types of outcomes:

Setting 1: (the outbreak takes off)

130 nS

175.0 nS

1 0 nS

Setting 2: (it does NOT take off)

It increases initially when

It always decreases when

1 0 nS

1

nS

IInS

IdtdI t

ttt

tt

Setting 1: (the outbreak takes off)

130 nS

175.0 nS

1 0 nS

Setting 2: (it does NOT take off)

It increases initially when

It always decreases when

1 0 nS

nS0

determines whether the outbreak takes off.

(actually, there’s an element of chance)

(a) What determines whether an outcome takes off?

If R0 < 1 there can not be an epidemic. No intervention is required.

If R0 > 1 an epidemic occurs. (Can occur)

1

0

R is the basic reproduction number. It is (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.

ASIDE:

/ where )1(ln

00

0

RcR

cs

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

cRc

s

w

Cn

SS

dtIn

dtdtdS

S

nS

IdtdS

tt

t

tt

t

00

0

00

)1(ln

gives hich

1

lnln

1

Heuristic derivation

0

0.20.4

0.60.8

1

0 0.2 0.4 0.6 0.8 1Proportion initially susceptible

What happens if some community members are immunised?

The initial reproduction number is

Illustrate this for Setting 1 I0 = 1, n = 1000, = 0.3, = 0.1

000 sR

nS

Rv

Proportion of infections among susceptible individuals

)1(ln

0

0

cRc

s

Question: What really happens?

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?

immunisation

infectionSusceptibleSt

InfectiousIt

Recovered(immune)

Death

Death

Death

Birth

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.

Assume no immunisation

ttt

tt

tt

tt

IInS

IdtdI

SnS

IndtdS

0 and 0 dtdI

dtdS tt

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

This gives

1 and R1 0

0

Ris EE

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?

ttt

tt

tt

tt

IInS

IdtdI

SnS

InvdtdS

)1(

]1)1[( and R1 0

0

Rvis EE

Eventually

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.

1 and 1 00

00

Ri

Rv

s

.0at )1( 0

tiv

sRi

dtid o

ttt

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

Here’s what happens if we immunise 1%, namely 667, of the susceptibles:

0

100

200

300

400

500

0 50 100

Time (Days)

Infe

ctiv

es

Question: What really happens?

Here’s what happens if we immunise 5%, namely 3333, of the susceptibles:

Question: What really happens?

0

500

1000

1500

2000

0 50 100

Time (Days)

Infe

ctiv

es

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

community the of size total /~

ca )],(exp[

,

ca ,

,0

00 0

0

0

NcNdaNN

caN

caNN

N

ca

dtdN

a

a

a

a

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

0

25

50

75

100

0 20 40 60 80Age

With N0 = 100 the community size is 6000.

Next the transmission model

Consider an SIR model in which the transmission rate is age-dependent

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, ∞)

The solution can be found analytically or numerically.

aa S

dtdS aa

a ISdtdI

aa S

dtdS

)'( aaa IS

dtdI

)('

The solution is as follows:

For a in [0, c)

For a in [c, ∞)

)exp(0 aSSa

)])('(exp[0 cacSSa

aS

I aa )'(exp1

''

by edapproximat be can which expression long a

)exp()exp(

0

aaSI a

Proportion infectious at different ages – no vaccination

Age (years)

Pro

port

ion

infe

cted

0 10 20 30 40 50 60 70 80

0.0000

0.0002

0.0004

0.0006

0.0008

0.001

Age (years)

Pro

port

ion

su

sce

ptib

le

0 10 20 30 40 50 60 70 80

0.0

0.2

0.4

0.6

0.8

1.0

Proportion susceptible at different ages – no vaccination

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

NdaINdaI

NdaINdaI

c aOO

c

aYO

c aOY

c

aYY

~/

~/'

~/

~/

0

0

Can not determine 4 parameters from 2 equations

Assume proportionate mixing; i.e. the WAIFW matrix is

We find

and

NdaINdaI

NdaINdaI

c a

c

a

c a

c

a

~/''

~/''

~/'

~/

0

0

0.51)'/(

~

0

c

c aa daIdaI

N

0.10/''

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 arec = 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

NdaINdaI

NdaINdaI

c a

c

a

c a

c

a

~/''

~/''

~/'

~/

0

0

and solve for the new and ’.

Vaccination coverage

Ra

tio o

f cas

es

for

age

>C

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.5

1.0

1.5Vaccination at birth

Graph of cases aged over 50 at v relative to v = 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 againc = 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

NdaINdaI

NdaINdaI

c a

c

a

c a

c

a

~/''

~/''

~/'

~/

0

0

to solve for the new and ’.

Vaccination coverage

Ra

tio o

f cas

es

for

age

>C

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.5

1.0

1.5Vaccination at birthVaccination at C=50 years

Graph of cases aged over 50 at v relative to v = 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 EndThe End