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The coevolution of parasites and their hosts has both general biological interest and practical implications in agricultural, veterinary and medical fields. Surprisingly, most medical, parasitological and ecological texts dismiss the subject with unsupported statements to the effect that ‘successful’ parasite species evolve to be harmless to their hosts. Recently, however, several people have explored theoretical aspects of the population genetics of host-parasite associations; these authors conclude that such associations may be responsible for much of the genetic diversity found within natural populations, from blood group polymorphisms (Haldane, 1949) to protein polymorphisms in general (Clarke, 1975, 1976) and to histocompatibility systems (Duncan, Wakeland & Klein, 1980). It has also been argued that pathogens may constitute the selective force responsible for the evolution and maintenance of sexual reproduction in animal and plant species (Jaenike, 1978; Hamilton, 1980, 1981, 1982; Bremermann, 1980).
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Parasitology
(1982), 85, 411^426
411
With 2
figures
in
the text
Coevolution
of
hosts and parasites
R. M. ANDERSON1 and
R.
M. MAY2
1 Department
of
Pure and Applied Biology, Imperial
College,
London University,
London SW1
2BB
2 Department of Biology, Princeton University, Princeton, New Jersey 08544
{Accepted 30 April 1982)
INTRODUCTION
The coevolution of parasites and their hosts has both general biological interest
and practical implications
in
agricultural, veterinary and medical fields. Surpris-
ingly, most medical, parasitological and ecological texts dismiss the subject with
unsupported statements
to the
effect that
'
successful' parasite species evolve
to
be harmless
to
their hosts. Recently, however, several people have explored
theoretical aspects
of
the population genetics
of
host^-parasite associations; these
authors conclude that such associations may be responsible for much of the genetic
diversity found within natural populations, from blood group polymorphisms
(Haldane, 1949)
to
protein polymorphisms
in
general (Clarke, 1975, 1976) and
to
histocompatibility systems (Duncan, Wakeland & Klein, 1980).
It
has also been
argued that pathogens
may
constitute
the
selective force responsible
for the
evolution
and
maintenance
of
sexual reproduction
in
animal
and
plant species
(Jaenike, 1978; Hamilton, 1980, 1981, 1982; Bremermann, 1980).
The present paper aims to take
a
line that
is
somewhat more empirical than most
of the previous theoretical work. Defining parasites broadly
to
include viruses,
bacteria, protozoans
and
helminths,
we
observe that
the
virulence
of a
parasite
(the rate at which it induces host mortality)
is
usually coupled with the transmission
rate and with
the
time taken
to
recover
by
those hosts
for
whom
the
infection
is
not lethal. Specifically,
in
mice, men
and
other vertebrates (Fenner & Ratcliffe,
1965;
Burnet & White, 1972) and
in
many invertebrates (Maramorosch & Shope,
1975;
Anderson & May, 1981) low virulence is generally associated with effective
immunological
or
non-specific responses which tend
to
suppress pathogen replica-
tion, with
a
concomitant reduction
in
transmissibility. Using data
for the
epidemiological parameters characterizing
the
various grades
of
myxoma virus
infecting rabbits
in
Australia, we show how
in
this particular case virulence may
be expected
to
evolve
to an
intermediate value;
the
analysis appears
to
accord
with the observed facts. Other examples are discussed
in a
more qualitative way.
In general,
we
conclude that
the
complicated interplay between virulence
and
transmissibility of parasites
leaves
room for many revolutionary paths to be followed,
with many endpoints.
As
a
preliminary,
it
seems useful
to
present
a
brief overview and attempt
at a
synthesis
of
the diverse, scattered
and
growing body
of
theoretical literature
on
the coevolution
of
hosts
and
their pathogens.
412 R. M. ANDERSON AND R. M. MAY
EXPLICITLY GENETIC MODELS
One class of models deals explicitly with the population genetics of the host
species, with the genetic structure of the parasite population being handled
in
various ways.
In
these models, the fitness
of
the different host genotypes are
explicitly or implicitly frequency dependent. Density-dependent fitness effects, on
the other hand, do not usually enter; that
is,
the interplay between epidemiological
processes and population densities are not usually accounted for.
There
is
a substantial literature, largely relating to plant breeding, in which one
specifies the fitness of various host genotypes when exposed to various parasite
genotypes, and then studies the ensuing dynamical behaviour of the host and
parasite gene frequencies (Mode, 1958; Person, 1966; Yu, 1972; Leonard, 1977;
Lewis, 1981a, b; for biologically orientated reviews, see Van der Plank, 1975 and
Day, 1974). Thus, for example, under the usual 'gene-for-gene' assumption, a one
locus/two allele system in
a
diploid host may interact with
a
diallelic locus in
a
haploid parasite; the fitness WAa of the Aa host genotype will be a weighted sum
over the fitness wAa
$
for this host when attacked by parasites of the genotype
i
(weighted by the gene frequencies of the haploid parasite genotypes i), and so on.
The equilibrium and stability properties of such systems can be studied, once
explicit assumptions are made about the magnitudes
of
the fitness constants
(.wAa,i)
and the
like.
Although straightforward in principle, such dynamical studies
can become very messy by virtue of the proliferation of parameters. These systems
are clearly capable of yielding stable polymorphisms or stably cyclic oscillations
between resistant and susceptible hosts and virulent and avirulent parasites; they
can yield chaotic fluctuations
in
gene frequency. As Lewis (1981a) has recently
emphasized, however, such polymorphisms do not automatically ensue: 'The
intuitive arguments are sufficient
to
guarantee the existence of
a
polymorphic
equilibrium, but not to guarantee its stability. There are conditions under which
such equilibria may be stable; however,
it
seems that there are bounds on how
different the virulent and avirulent reactions can be if there is to be stability.'
This style of analysis has the merit that the genetics of both host and parasite
are treated explicity. Epidemiological factors, on the other hand, are not really
considered:
if
two or more strains of parasite are present,
it
is usually assumed
that all hosts are infected, with the fraction of
hosts
infected by each strain being
in simple proportion to the relative abundance of that strain (which, for a haploid
parasite, means in proportion to the parasite gene frequencies). At best,
it
may
be assumed that
a
constant fraction, x, of the host population escapes infection;
this can introduce significant additional complications (Lewis, 1981a; Yu, 1972).
But x is treated as one more phenomenological constant rather than as a dynamic
variable determined by the interaction between host and pathogen populations.
In most
of
the models just discussed, whether framed
as
difference
or as
differential equations,
it
is assumed that host and parasite generations tick over
on much the same time scale. This
is a
reasonable approximation
for
some
interesting plant—parasite systems. The parasites of many animals, however, cycle
through many generations in a single generation of the host, and are thus rapidly
evolving compared with their hosts.
In
this event,
it
makes sense to study the
genetic structure of the host population, subsuming the genetical dynamics of the
Coevolution
of
hosts
and
parasites
413
parasites in frequency-dependent fitness functions for each genotype of host. One
way to do this would be to take the kind of' gene-for-gene' analysis outlined above,
and reduce the dimensionality of the system of equations by assuming the parasite
dynamics operate on a faster time scale than the host dynamics, so that parasite
gene frequencies are always at the equilibrium appropriate to the (slower changing)
host gene frequencies. In this way one would have equations for the host gene
frequencies, in which all the fitness functions were frequency dependent. To our
knowledge, no one has adopted this approach.
A
second approach (Hamilton, 1980;
1981;
Clarke, 1976; see below) is simply to make some ad
hoc
assumption about
the frequency-dependent fitness functions of the various host genotypes
still
without any explicit epidemiological analysis - and explore the possible range of
dynamical behaviour.
A third approach is to let conventional epidemiological assumptions, of
one
kind
or another, dictate the form of the frequency-dependent fitnesses of the host
genotypes (Gillespie, 1975; Kemper, 1982; Anderson & May, 1982a). Gillespie's
(1975) pioneering study initially assumes a population of haploid hosts, and
explores the convential metaphor of one locus with two alleles: individuals of the
a genotype are resistant to some particular
disease,
but pay a cost in having a lower
fitness (by a factor
1
s)
than the susceptible individuals who do not become
infected; individuals of the A genotype are susceptible to the disease, and those
individuals who actually contract the disease have their fitness decreased to
1
t,
which is lower than the fitness of the resistant individuals (i.e. / > s). Gillespie
assumes the disease spreads through each generation in an epidemic fashion, in
a manner described by the standard equations of Kermack
&
McKendrick (1927)
(see also Kendall, 1956 and Bailey, 1975):
(la)
(16)
dZ/dt = vY. (lc)
Here X, Y, Z are the number of hosts in a given generation of the susceptible
population that are as yet uninfected, infected and recovered (and thereby
immune),
respectively. The coefficient
/?
measures the transmission rate (and may
itself depend on the magnitude of the host population), and v is the recovery rate.
From these equations, it can be shown that the fraction, /, of the susceptible hosts
that are affected by the disease is given implicity, for these haploid hosts, by the
relatlon:
7=1- exV(-lpN/NT). (2)
Here NT is the threshold host density, NT = fi/v, below which the disease cannot
be maintained, N is the total host population density, and p is the frequency of
the allele A (so that pN is the population of susceptible hosts). It follows that the
fitnesses of resistant and of susceptible hosts, Ws respectively, are
•WR
= l-s, (3a)
Ws=\-It.
(3
6)
The fitness Ws
is
frequency dependent, in a way that depends on
the
epidemiological
assumptions that are embodied in equation (2). Gillespie (1975) subsequently
414 R. M.
ANDERSON
AND R. M. MAY
indicates how the analysis and broad conclusions extend to a diploid population
in which the heterozygotes Aa are equivalent either to the susceptible AA
individuals {A dominant) or to the resistant aa individuals
{a
dominant).
Gillespie's work shows that a stable polymorphism, with both susceptible and
resistant genotypes (both A and a alleles) present, will ensue for a particular range
of values of the fitness values s and t, and of the magnitude of the host population
in relation to the threshold population, N/NT. If the fitness cost of resistance, s,
is relatively small, then the population will evolve toward all being resistant; if
the fitness cost of contracting the infection t, is sufficiently small relative to s
(t
-> s),
or if N is close to (or below!) NT, resistance will not be present.
Gillespie's analysis can readily be extended to encompass diseases that are stably
endemic, rather than sweeping in epidemics through each generation. Kemper
(1982) has given a thorough treatment of the case of
endemic
infective agents that
do not confer immunity (roughly corresponding, for example, to gonorrhea
(Yorke, Hethcote & Nold, 1978) or to many viral infections of insects (Anderson
& May, 1981)). The dynamics of this system is described by equation (1), except
that there is no immune class Z and recovery is directly back into the uninfected
but susceptible class X. For a haploid host, as above, the equilibrium fraction, /,
of the susceptible hosts that are infected is then
I=l-(N
T
/pN).
(4)
The definitions of
p,
N and NT, and the subsequent analysis, are as outlined above,
except that equation (4) replaces equation (2) in determining the frequency-
dependent fitness function Ws of equation (36). Like Gillespie, Kemper concludes
that the selective pressure of the parasite produces a 'stable polymorphic
equilibrium as long as the fitness of the immunity-producing allele is neither too
large not too small in comparison to the fitness of a diseased individual'.
For an endemic infection that does induce immunity (for example, measles,
rubella or pertussis), the calculation of/ is slightly more complicated; the fraction
of susceptibles in any given cohort decreases with age (see, for example, Anderson
& May, 19826). For estimating the fitness of susceptible genotypes, equation (3),
the fraction of the susceptible population that has acquired infection, I, is given
approximately by (Anderson & May, 1982a)
I=l-exp[-(Tc/L)(pN-NT)/NT]. (5)
Here Tc is the cohort generation time (May, 1976), L is the average life-expectancy
and p, N and NT are as before. Again, the conclusions broadly accord with those
of Gillespie and Kemper (Anderson & May, 1982a).
Very generally, these analyses make it plain that, were selective pressures always
to favour the evolution of 'harmless' or 'avirulent' parasites (so that t -> 0), we
would not expect to find polymorphisms in host susceptibility (or resistance)
associated with such infectious agents. This prediction does not accord with the
available empirical evidence, since variability in host susceptibility to infection by
a specific pathogen appears to be the rule rather than the exception.
As mentioned above, a different approach is to investigate the genetical
dynamics of a host population in which various kinds of frequency dependence are
assigned to the fitness of the different host genotypes. Clarke (1976) has argued
Coevolviion
of
hosts and
parasites
415
that, in general, a variety of plausible biological mechanisms could result in a rare
genotype enjoying
a
selective advantage over commoner host genotypes in the
presence of parasites or predators. Numerical exploration of models embodying
these ideas showed that stable, or cyclic, or chaotic polymorphism could result,
depending on the strength of the frequency-dependent advantage that accrued to
rarer genotypes. Clarke (1975, 1976) has suggested that many protein polymorp-
hisms may be maintained
in
this way. Subsequent authors have given formal
explications of the way cycles and chaos arise
in
these nonlinear systems (for
example, Oster, Ipaktchi & Rocklin, 1976; May, 1979), leading to complex and
non-steady genetic systems. The essential ideas here go back to Pimentel's (1968)
'genetic feedback' and to Haldane (1949).
Jaenike
(1978) has gone
further, making the tentative suggestion that frequency-
dependent aspects of the host-parasite association may help explain the evolution
and maintenance of sex in host populations. In an important series of recent papers,
Hamilton (1980, 1981, 1982; Hamilton
&
Zuk (1982)) has explored these kinds of
models for the genetical dynamics of host populations. He shows that provided
the frequency dependence associated with resistance and susceptibility to parasites
is sufficiently intense, such models can generate complex cycles
in
host gene
frequency, and that 'in certain states of
cycling
sexual species easily obtain higher
long-term geometric mean fitness than any competing monotypic asexual species
or mixture of such' (Hamilton, 1980). Hamilton urges the bold notion that the
widely discussed selective disadvantages of sex (Williams, 1975; Maynard Smith,
1978) may typically
be
outweighed
by
the advantages sexual recombination
confers under the sort of strong frequency-dependent selective forces that the
genetic interplay between host and parasites may often induce. Several testable
hypotheses follow from these
ideas
(Hamilton,
1982).
In particular, gene frequencies
at
a
polymorphic locus should exhibit systematic changes from generation
to
generation if the polymorphism is maintained by Hamilton's cycling. We observe
that some support
for
this conjecture
is
provided
by
genetic studies
of
gene
frequency changes in populations of wild mice (Berry, 1980). Combining seven
surveys of blood parasites in North American passerines, Hamilton
&
Zuk (1982)
showed that among these bird species there is significant correlation between the
incidence of chronic blood infections and striking sexual display (specifically, male
'brightness', female 'brightness' and male song).
MODELS BASED ON EPIDEMIOLOGY AND POPULATION INTERACTIONS
Other models
seek to give
a relatively accurate account of the density dependence
and epidemiology of the interaction between hosts and parasites, without retaining
the explicit genetics. These models examine the dynamic interactions between
a
host population and populations of different strains of a parasite; the genetics is
crudely implicit in, for instance, the varying rates of transmission and virulence
of different strains of the parasite.
Levin & Pimentel (1981) have recently used this approach
to
examine the
coexistence
or
otherwise
of a
host population with two different strains
of a
pathogen: one of the strains is more virulent than the other, inducing a mortality
rate,
ax, which is greater than that due to the less virulent strain a2 (ax> a2);
416 R. M.
ANDERSON
AND R. M. MAY
both strains have identical transmissibility (susceptible individuals, on contact
with an infected individual, acquire the infection at a
'
transmission rate' ft in both
cases),
but the more virulent strains can 'take over' individuals already infected
with the less virulent strain (at a per capita transmission rate afi). Denoting the
populations of hosts that are susceptible, infected with strain 1, and infected with
strain 2 by X, Yx and Y2, respectively, Levin & Pimentel (1981) described the
dynamics of this system by the set of differential equations
dX/dt = aN-pX
Yx-fiX
Y2
-
bX,
(6)
dY1/dt = 0XY1
+
vftY1Y2-(a1
+
b)Yl, (7)
d YJdt = pX
Y2
-
<rfiY1
Y2 -
+
b)
Y2. (8)
Here a is the per capita birth rate (assumed to be unaffected by infection), b the
per capita death rate in the absence of infection, and the total population is
N = X+ Yt + Y2. It is assumed that both strains of the infection are lethal, so that,
once infected, no hosts recover. This kind of model differs from conventional
epidemiological ones in that the total host population is a dynamical variable,
which may or may not be regulated to a stable equilibrium value by the infection;
traditional studies (for example, Bailey, 1975) assume N to be a constant,
determined by other factors. Despite their simplicity, equations of this general type
have been shown to give a good fit to data for endemic infections that regulate
experimental populations of laboratory mice (Anderson & May, 1979), and to give
plausible explanations for the population dynamics of associations between foxes
and rabies in Europe (Anderson, Jackson, May
&
Smith, 1981) and between various
arthropods and viral or protozoan parasites (Anderson & May, 1981).
Analysing the system of equations (6)-(8), Levin
&
Pimentel (1981) showed that
if one strain has significantly greater virulence in relation to its transmission
advantage, it will not persist; conversely, if the pathogenicity
<xx
is not significantly
greater than a2 while the transmission advantage is substantial (a relatively large),
the more virulent strain will win. For an intermediate range of the ratio aja^ in
relation to the transmission advantage enjoyed by the more virulent strain, the
two strains can co-exist.
Levin & Pimentel's (1981) study can be generalized in a variety of ways, and
the essentials of their conclusions remain intact. Thus it is not neccessary to assume
that the more virulent strain can infect hosts bearing the less virulent strain, but
that the reciprocal process is impossible; it is only necessary to assume that
crfi
is the net excess of infections 2
> 1
over 1
2. The actual data for myxomatosis
(see,
for example, the work of Saunders, (1980)) suggest that individuals, once
infected with one strain, do not acquire infection with another strain. Levin &
Pimentel's general analysis can, however, be preserved by considering the more
virulent strain (virulence aj to have a higher transmission rate, /?1; than that, fi2,
of the less virulent strain (virulence a2). In this case, if the disease is considered
always to be lethal, the strain with the lower value of (aj + 6)//?j will always win.
But if recovery is possible, or if other realistic complications are admitted, a range
of coexistence is possible, corresponding to the two strains having roughly
comparable values of the overall ratio between pathogenicity and transmissibility
(Anderson & May, 1982a).
Coevblution
of
hosts
and parasites
417
Bremermann (1980)
has
also used models
of
this type
to
examine
the
general
way
in
which epidemiological parameters
-
such
as
transmission rate
/?,
virulence
a
and
recovery rate v
-
are likely
to
evolve
in
response
to the
selective pressures
exerted on host and on parasite populations. As in all such studies
of'
evolutionarily
stable strategies' (ESS; Maynard Smith & Price, 1973),
the
underlying genetics
is eschewed
in
pursuit
of a
more transparent
but
less rigorous analysis. Bremer-
mann's (1980) approach parallels that which we bring to bear on the myxomatosis-
rabbit
and
other data, below.
Bremermann (1980) uses
the
basic
set of
equations introduced
by
Anderson &
May (1979):
dX/dt
=
aN-bX-pXY +
yZ,
(9)
=
fiXY-(a + b +
v)Y, (10)
(11)
Here
X, Y, Z
are
the
number
of
susceptible, infected
and
recovered-and-immune
hosts, respectively;
N =
X+Y+Z.
The
disease-induced mortality rate
a, per
capita birth rate a, disease-free mortality rate
b,
recovery rate v, and transmission
rate
y?
are all as
defined previously,
and y is the
rate
of
loss
of
immunity
(y = 0
if immunity
is
lifelong;
y = oo if
there
is no
immunity,
so
that recovered
individuals are again susceptible). This population grows exponentially
at
the rate
(a-b)
in the
absence
of
the disease. Once
the
disease becomes established,
it
will
regulate the host population
to a
stable equilibrium
if
the virulence
is
sufficiently
high (specifically
if a >
[a-b][ix+v/(b
+ y)])
and
will slow
the
rate
of
exponential
growth otherwise (Anderson & May, 1979). Following May & Anderson (1979),
Bremermann (1980) observed that the fitness of the parasite is increased by having
large /?,
and
small
<x
and v,
whereas
the
host fitness
is
increased
by
having small
/?,
small
a and
large
v.
Were these parameters
not
inextricably linked
by the
biological processes whereby virulence, recovery rate
and
production
of
trans-
mission stages of the infective agent are intertwined, the ESS would clearly favour
a -»
0
(although
the
countervailing interests
of
host
and
parasites with regard
to
/?
and v
would tend
to
drive them
to
some intermediate value,
or
into cycles).
Insofar
as
there
is a
basis
for the
common view that 'successful parasites
and
pathogens are harmless', this
is it. But
the biologically based interlinkage among
a, /? and
v
invalidates this simple argument.
Bremermann (1980) explored some particular host-parasite systems, paying
special attention
to the
role
of
polymorphism
in
maintaining immunological
defenses. He was led, by this different route, to
a
central conclusion broadly similar
to Hamilton's, namely that
the
main reason
for
sexual reproduction
is
that,
through recombination,
it
maintains polymorphisms which
are
'essential
in
preventing pathogens from adaptively breaking through immunological host
resistance' (Bremermann, 1980).
In
arriving
at
this conclusion, however, Bremer-
mann used
the
mathematical models
in a
purely metaphorical
way.
GENERAL DISCUSSION: BASIC REPRODUCTIVE RATE OF A PARASITE
An ambitious project
is to
meld
the
models described
in the
preceding
two
sections,
to
get sets of epidemiologically detailed equations like equations (6)-(ll)
418
R. M.
ANDERSON
AND R. M. MAY
for each individual genotype
of