British Medical Bulletin 62:187-199 (2002)
© 2002 The British Council
Mathematical models of vaccination
Almut Scherer and
Angela McLean
Zoology Department, University of Oxford, Oxford, UK
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Abstract
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Mathematical models of epidemics have a long history of contributing
to the understanding of the impact of vaccination programmes.
Simple, one-line models can predict target vaccination coverage
that will eradicate an infectious agent, whilst other questions
require complex simulations of stochastic processes in space
and time. This review introduces some simple ordinary differential
equation models of mass vaccination that can be used to address
important questions about the predicted impact of vaccination
programmes. We show how to calculate the threshold vaccination
coverage rate that will eradicate an infection, explore the
impact of vaccine-induced immunity that wanes through time,
and study the competitive interactions between vaccine susceptible
and vaccine resistant strains of infectious agent.
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Introduction
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One of the very earliest mathematical models in epidemiology
concerned the impact of vaccination. In 1760, Swiss mathematician
Daniel Bernoulli published a study of the predicted impact of
immunization with cowpox upon the expectation of life of the
immunised population
1. Nearly 150 years later, around the time
of the First World War, Ronald Ross produced a series of mathematical
models of the spread of malaria that laid the foundations of
the modern theory of the control of infectious disease
2. Ross's
great advance was to recognise, through the exploration of mathematical
models, that malaria transmission could be prevented through
mosquito control – without removing every last mosquito.
The recognition that disease transmission could be stopped by
control programmes with incomplete coverage had wide-spread
impact on the design of intervention strategies throughout the
20th century
3–5. For vaccination strategies, some of the
simplest questions that arise are: (i) what fraction of the
population must be successfully vaccinated to eradicate the
infectious agent; (ii) what happens if the target coverage for
eradication is not met; (iii) does it matter if vaccine induced
immunity wanes with time; and (iv) what happens if there are
vaccine resistant sub-types? The following sections introduce
mathematical models that address these questions.
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Simple models
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Amplification factors and eradication thresholds
All that is required for the incidence of an infectious disease
to go into decline is that each case should generate, on average,
less than one other case. The number of secondary infections
caused by one infectious individual is often referred to as
the effective reproductive number and denoted
R. Epidemics often
peak and go into decline as
R falls below 1 because the pool
of susceptible individuals has been temporarily exhausted. For
the trajectory of incidence to remain on a downward course until
the agent is eradicated requires that the effective reproductive
rate should remain below 1, even when the number of susceptible
individuals is at its maximum. There are two further amplification
factors that pertain (
Table 1).
R0, the basic reproductive number
is the number of secondary cases caused by one primary case
introduced into a population that is wholly susceptible.
R0p,
the basic reproductive number under vaccination is the number
of secondary cases caused by one primary case introduced into
a population in which a proportion
p have been vaccinated. For
a perfect vaccine that confers life-long protection
 | (1) |
The critical vaccination proportion that will achieve eradication,
pc, is that for which the basic reproductive number under vaccination
is just equal to 1. This yields:
 | (2) |
To calculate numerical values of pc requires estimates of R0 (Table 2), these illustrate how the ease with which an infectious agent can be eradicated varies widely across agents for which cheap safe and effective vaccines are already available. This highlights the issue that the development of such a vaccine, although a necessary prerequisite, is not sufficient to guarantee eradication of an infectious agent.
Post-vaccination dynamics
To study the predicted dynamics of infection after the introduction
of a vaccination programme requires the use of mathematical
models of transmission dynamics. The simplest model that can
be used to study the impact of vaccination keeps track of three
groups of individuals: susceptible, S; infected, I; and recovered
R. The model we study here includes a fourth group; those who
have been vaccinated, V. This refinement allows the investigation
of the impact of waning immunity in the next section (
Fig. 1).
If vaccine induced immunity is life-long, then the equations
of this
SVIR model are:
 | (3) |
 | (4) |
 | (5) |
 | (6) |
Here,
N is the total population size. The transitions described
by each term of the equations of this model are as labelled
and the model's parameters are described in
Table 3.
Figure 2 uses this model to illustrate the predicted impact of vaccination.
The parameters' values are chosen to represent measles in an
industrialised country with
R0 = 11, and epidemics occurring
in biennial cycles. In the pre-vaccine era, the effective reproductive
ratio rises and falls around the value of 1 as the pool of susceptible
individuals is depleted by epidemics of infection and then replenished
by births. At time 6 years, vaccination of a fixed proportion
of new births is introduced. Scenarios modelling the impact
of three different levels of vaccination are presented. The
first achieves coverage of 95%, which is above the critical
proportion for eradication (91% when
R0 = 11). The number of
infections immediately plummets, and no further infections are
seen. The effective reproductive ratio is depressed below 1
after the introduction of vaccination and never rises above
it again. Eradication is achieved. The second scenario represents
the impact of a vaccination programme that reaches high levels
of coverage (85% of all new-borns) which are, nevertheless,
not high enough to lead to eradication of the agent. However,
for the first 15 years after the introduction of vaccination,
it appears as if eradication has been achieved, there are no
infections. Then, suddenly, a new epidemic appears as if from
nowhere. This is an illustration of a phenomenon known as the
‘honeymoon period’. This is the period of very low
incidence that immediately follows the introduction of a non-eradicating
mass vaccination policy. This happens because susceptible individuals
accumulate much more slowly in a vaccinated community. Such
patterns were predicted using mathematical models in the 1980s
6 and have since been observed in communities in Asia, Africa
and South America
7. Honeymoon periods are only predicted to
occur when the newly introduced vaccination programme has coverage
close to the eradication threshold. The third scenario depicted
in
Figure 2 is of vaccination coverage at 70%. Although epidemics
in the era of vaccination are less frequent, there is no obvious
honeymoon period.
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Duration of protection
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Modelling waning immunity
The mathematical model represented in
Equations 3
–6 and
Figure 2 makes the assumption that vaccine-induced protection
is life-long. There is no waning of vaccine-induced immunity.
Until the 1990s, this was a universal assumption of mathematical
models of vaccination. This assumption was routinely made because,
for most of the major vaccines against childhood infectious
disease, it is approximately correct. It is, however, important
to ask about the sensitivity of model predictions to this assumption
8.
The transitions presented in
Figure 1 include the possibility
that vaccinated individuals will eventually pass into the susceptible
class as their vaccine-induced immunity fails. This transition
is trivially included in the equations of the model as a term
V added to
Equation 3 and subtracted from
Equation 4.
The vaccinated basic reproductive number with waning immunity
With this term in place, equilibrium analysis of Equations 3–6 yields a new expression relating the vaccinated reproductive number to the basic reproductive number:
 | (7) |
This is for coverage p with a vaccine that takes in a fraction e of recipients and gives protection that wanes with average duration of protection in a population with average expectation of life µ. This apparently simple equation introduces a second counter-intuitive insight into the impact of vaccination gleaned from mathematical models. The impact of the level of coverage and the ‘take’ of the vaccine upon the vaccinated reproductive number are as one would expect. However, the impact of the duration of immunity is much greater than intuition might lead one to expect. The term µ/(µ + ) in Equation 7 is best interpreted as the fraction of a lifetime for which an individual is protected by a vaccine that gives immunity that wanes at rate in a population with fixed death rate µ. If the expectation of life is 50 years (µ = 0.02), then a vaccine with immunity that wanes at the same rate is only as good as a vaccine that gives protection that does not wane, but only takes in 50% of recipients (Fig. 3).
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Fig. 3 An otherwise perfect vaccine that protects for 50 years on average is only as good as a vaccine that takes in 50% of vaccinated individuals but gives life-long protection.
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Post-vaccination dynamics with waning immunity
Re-arrangement of
Equation 7 yields a new threshold parameter
c, the critical duration of immunity for a given coverage
p,
take
e and basic reproductive number
R0. If vaccine-induced
immunity wanes faster than this critical rate, then eradication
will not be achieved.
 | (8) |
Such a scenario is illustrated in Figure 4. A vaccine that is otherwise perfect (takes in all recipients) and which achieves total coverage yields eradication when the waning rate is below the critical threshold level (line marked < c in Fig. 4). But if the duration of protection is shorter than the critical level, the control programme fails after a time of apparent success. This failure is via a process distinct from that which causes the honeymoon period discussed above, as it is through the accumulation of susceptible individuals who were vaccinated at birth but whose immunity has since waned. It is worth noting that, as emphasised in Figure 3, the two vaccines that give protection that wanes ‘too quickly’ in Figure 4 have very long average duration of protection, at mean duration 25 years and 50 years for the two scenarios labelled >> c and > c, respectively.
In short, the standard assumption that vaccine-induced immunity
will give life-long protection is a very strong assumption indeed.
If it is not true, then many of the calculations about coverage
levels for eradication will turn out to have been over-optimistic.
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The evolution of vaccine resistance
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The evolution of vaccine-resistant strains of infectious agents
is, potentially, a huge problem for their control by vaccination.
Yet, for many infectious diseases, it has been possible to push
them to the verge of extinction without vaccine-escape mutants
arising. A theoretical framework has recently been developed
that allows investigation of why this should be so, what properties
of vaccines allow this situation, and what might happen in situations
where vaccine-resistant mutants do arise. A model of the simplest
situation for multiple strains is presented here. The simplest
situation arises when infection with one strain confers life-long
immunity against all other strains
9. More complex situations
are presented elsewhere in the published literature
9–12.
Modelling infections with total cross immunity
Figure 5 represents the transitions amongst five groups of individuals that must be considered in a model of the vaccine-driven evolution of mutant strains. As before, there are susceptible, vaccinated and recovered individuals. There are two important new features of this model compared with the model in Figure 1. First, two strains of agent exist. Occasionally, infection with the wild-type agent will give rise to a mutant strain. In the absence of vaccination, this mutant is at a selective disadvantage (here modelled as being slightly less infectious to susceptible individuals than the wild-type, ßr < ßs). This means that, in the absence of vaccination, this mutant is at a competitive disadvantage and will rarely, if ever, be seen. However, we assume that this mutant is vaccine resistant, i.e. the vaccine confers stronger protection against the wild-type than it does against the mutant (
r > s). Under these circumstances, it is possible (although not inevitable) that vaccination can shift the competitive balance between the two strains so that, after vaccination, the new vaccine-resistant strain will emerge. The equations of this new model, with the meaning of each transition labelled as before, are as follows:
 | (9) |
 | (10) |
 | (11) |
 | (12) |
 | (13) |
The dynamics of the emergence of vaccine resistance
Figure 6 uses the model defined above to illustrate the sequence
of events that could lead to the vaccination-driven emergence
of a vaccine-resistant strain. Before vaccination is introduced,
only one strain is observed. Although the vaccine-resistant
strain is continuously generated as a mutant form of the circulating
strain, it is less infectious for unvaccinated hosts and is,
therefore, at a competitive disadvantage. It is competitively
excluded by the wild-type strain. At time 6 years, vaccination
is introduced. The vaccination campaign achieves 85% coverage
of all new-born individuals with a vaccine that gives 95% protection
against the only known strain. The immediate impact is dramatic
and there follows a honeymoon period of apparent eradication.
This is ended with an epidemic of the wild-type strain for the
reasons discussed above. Thus, although under this vaccination
regimen the vaccine-resistant strain will eventually have the
competitive advantage, that advantage is not immediately manifest
and indeed is only established after a very large proportion
of the population have been vaccinated. Thus, it is only after
the effective reproductive number for the vaccine-resistant
strain has exceeded that of the wild-type strain that an epidemic
of the vaccine-resistant strain emerges. It is important to
note that the strain emerges not because of a
de novo mutation,
but because the substrate required by the strain (large numbers
of vaccinated hosts) passes a threshold number that gives the
vaccine-resistant strain the competitive advantage.
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Fig. 6 Vaccination can change the competitive balance between two strains. Parameters as in Figure 2, p = 0.85, ßw = 0.0029, ßr = 0.00145, ew = 0.95, er = 0.5, Q = 0.0001.
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Is the emergence of vaccine resistance inevitable?
Although
Figure 6 illustrates an example in which the vaccine-resistant
strain eventually dominates, such emergence is not an inevitable
consequence of vaccination. If the cost of resistance is high
(ß
r << ß
s), the vaccine-resistant
strain will never become competitively superior. This is equally
true if the vaccine is sufficiently broad in specificity, so
that vaccine efficacy against any new strain is only barely
less than efficacy against the existing strain (
r =
s). Alternatively,
low levels of vaccination simply fail to ever generate enough
vaccinated individuals to give competitive dominance to the
vaccine resistant strain.
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Concluding remarks
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We conclude by summarising the responses to the questions posed
in our introduction. The fraction of the population that must
be successfully vaccinated to eradicate an infectious agent
can be expressed in terms of that agent's basic reproductive
number, the amplification factor from one generation to the
next in a wholly susceptible population. Agents with a low basic
reproductive number (
e.g. for smallpox
R0 = 3) have low threshold
coverage levels for eradication. If the target coverage for
eradication is not met, there are some counter-intuitive effects
of vaccination, in particular the honeymoon-period, an interval
of particularly low incidence immediately following the introduction
of a mass vaccination programme. The assumption, inherent in
many models of vaccination, that vaccine induced immunity will
be life-long, has large consequences for the predictions of
such models. If vaccine-induced immunity wanes, the predicted
target coverage for eradication is higher than if immunity is
life-long. Finally, the emergence of vaccine-resistant strains
is not an inevitable consequence of vaccination. If vaccines
have high enough efficacy and cross-reactivity or are targeted
at a small enough section of the population, vaccine-resistant
strains will not be expected ever to gain competitive dominance.
The non-linear nature of host–parasite interactions can lead to non-intuitive responses to apparently straightforward interventions. Mathematical models can act as an aid to our intuition in such circumstances, and, when sufficient data are available, can be used to advise on strategic objectives for vaccination programmes.
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Footnotes
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Correspondence to: Dr Angela McLean, Zoology Department, University
of Oxford, South Parks Road, Oxford OX1 3PS, UK
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Abstract
Introduction
(t) = ßI(t). Infectious individuals recover at rate * to become immune, and natural immunity is assumed to be life-long.
= 26, 
