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Theoretical Population Biology TP1323
Theoretical Population Biology 52, 101118 (1997)
Epidemic Thresholds and Vaccination in
a Lattice Model of Disease Spread
C. J. Rhodes and R. M. Anderson
Centre for the Epidemiology of Infectious Disease,Department of Zoology,
University of Oxford,South Parks Road,Oxford OX13PS,United Kingdom
Received December 1, 1995
We use a lattice-based epidemic model to study the spatial and temporal rates of disease
spread in a spatially distributed host population. The prevalence of the disease in the popula-
tion is studied as well as the spread of infection about a point source of infection. In particular,
two distinct critical population densities are identified. The first relates to the minimum
population density for a epidemic to occur, whilst the second is the minimum population
density for long-term persistence to occur. Vaccination regimes are introduced that are used
to measure the impact of spatially and nonspatially dependent intervention strategies. Specifi-
cally we show how a ring of vaccinated susceptibles, of sufficient thickness, can halt the
spread of infection across space. ]1997 Academic Press
1. INTRODUCTION
Recently there has been renewed interest in the effect of
spatial heterogeneity in modelling dynamic processes in
ecology, epidemiology, and evolutionary biology. Due to
increases in computational power this important feature
of most real interacting populations can be addressed
using a diverse variety of techniques.
The first point of departure for many epidemiological
studies is a set of deterministic mass-action-based
differential equations incorporating the major epidemi-
ological features of the disease in question (Anderson
and May, 1991; Bailey, 1975). An early investigation of
the likely impact of spatial heterogeneity on the vaccina-
tion requirement for the eradication of a canonical
infectious disease is by May and Anderson (1984).
They considered a host population that predominantly
inhabited one large city with the remaining fraction
evenly distributed amongst a number of small villages.
With the reasonable assumption that intragroup
transmission is higher than intergroup transmission, an
optimally applied immunisation program to eradicate
the infection can result in fewer individuals being
vaccinated than would be estimated by assuming the
population mixes homogeneously. The corollary to this
is that a uniformly applied immunisation schedule will
require more vaccinations than the homogeneous case.
One of the most thoroughly studied diseases is measles
virus infection in humans. The basic SEIR equations
which are often used in quantitative discussions of
measles infection can be enhanced by the addition of
more important realistic features such as age-structured
transmission rates (Schenzle, 1984). Corresponding
stochastic implementations of this program of work,
using Monte Carlo simulation techniques, have also been
extensively studied (Grenfell, 1992; Olsen et al, 1988;
Olsen and Schaffer, 1990). Generalisation of both the
deterministic and stochastic models to allow for spatial
heterogeneity result in meta-population models. The
population is divided amongst a number of distinct sites
with a conventional SEIR process within each site, as
well as a coupling term allowing influence proportional
to the magnitude of infection at the other sites (Bolker
and Grenfell, 1995). Most of this corpus of work has been
aimed at reconciling observed persistence and temporal
dynamic patterns as observed in real measles data sets.
However, the conventional mass-action assumption is
retained for the within-patch dunamics and there is as yet
Article No. TP971323
101 0040-580997 K25.00
Copyright ]1997 by Academic Press
All rights of reproduction in any form reserved.
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no a-priori way of assigning the coupling magnitudes
between patches.
A second widely used method to account for a spatially
dispersed poulation is the diffusive approach. Population
movement is accounted for by the addition of diffusion
terms to the conventional coupled differential equation
systems. Such models have been used to study likely
spread of rabies in the U.K. (Murray et al., 1986) and the
actual spread of the Black Death (Noble, 1974) across
medieval Europe. Any likely impact of demographic
stochasticity cannot be assessed as these are continuum
models.
A more recent trend in modelling spatial effects has
been the use of lattice methods. In these systems the
spatial distribution of the population is explicitly
represented by the use of a discrete set of (usually square)
lattice points. A set of rules then defines the range of
possible interactions between the individuals on the lat-
tice and their range of movement, if any. Pacala, Hassell,
and May (1990) and Hassel, Commins, and May (1991)
used this approach to investigate spatial hostparasitoid
interactions. The resulting dynamical behaviour is very
rich, with the spantaneous emergence of ordered
phases, consisting of spiral waves and regular ``crystal''
arrangements, as well as chaotic patterns. From a
population biology perspective the important conclusion
that can be drawn is that the combination of a spatially
heterogenous environment and local dispersion can
stabilise otherwise unstable hostparasite arrangements.
A related study is that of Rand et al. (1995) on a
generic hostpathogen dies out and, if mutation of the
transmissibility is allowed, evolution of transmissibility
towards this critical value occurs. This appears to be a
novel consequence of using a spatially extended host
population.
In the area of evolutionary biology Nowak and
May (1992) introduced a lattice implementation of the
Prisoner's Dilemma approach, conventionally used to
study the emergence of cooperative behaviour. As in the
other lattice methods the appearance of spatially ordered
structures or chaos is a feature.
More general spatial models in epidemiology and
ecology, using interacting particle systems, were studied
initially by Mollison (1977, 1991) in order to introduce
the stochasticity and nonlinearity inherent in dynamical
processes occurring in the natural world. More recently
this approach has been developed by Cox and Durrett
(1988), Durrett and Neuhauser (1991, 1994), and
Durrett and Levin (1994a, 1994b). These models are
similar in spirit to our approach and additionally they
are able to obtain some analytic results for spatial
contact processes in some instances. Durrett and Levin
(1994a) compare the use of mean field approaches (i.e.,
coupled ordinary differential equations), patch models,
reaction diffusion models, and interacting particle
models, as applied to three examples of spatially dis-
tributed interacting populations, and make thorough
and illuminating comparisons between the results
obtained for each of the different methods. Durrett
(1995) has recently provided the most up-to-date
summary of spatial contact processes as applied to the
spread of epidemics and this is the individual-based
system about which most is now known.
In this paper we discuss results obtained using a lattice
implementation of a simple epidemic process in a
uniform, spatially distributed, and mobile population.
Particularly, we are interested in the conditions for an
epidemic to occur and the impact of a variety of vaccina-
tion schedules on the endemic fixed point. Previously we
have studied similar lattice models and their relation to
mean-field formulations (Rhodes and Anderson, 1996a,
and 1996b). As discussed below, the lattice approach
provides a model for epidemic spread where the host
population exists in integral units ande each individual
interacts with those other individuals in its immediate
neighbourhood. Individuals may be susceptible,infective,
or recovered and birthdeath processes can take
place with the desired frequency. Disease transmission
occurs by the stochastic infection of a susceptible by a
neighbouring infective, and spread takes place when
infectives mix by diffusion amongst susceptibles. These
are all features of disease spread that are in tune with our
intuitive understanding of how epidemics of simple
communicable diseases occur in the real world and are
what gives lattice models their appeal as a method for
studying epidemiological dynamics.
2. THE MODEL
In the lattice epidemic model the host population
inhabits a square (L_L) grid of points usually taken to
be of dimension 100_100. During a given time-step each
individual is free to move from its current node to any of
the eight nearest neighbour nodes which happen to be
uninhabited, or it can remain where it is. Each of the
possible movements, or the possibility of remaining
stationary, occurs with equal probability. A set of trans-
ition rules then determines the nature of the interaction
of susceptible individuals with any other infective
individuals who may be at any of the four nearest
neighbour lattice points. We have in mind the modelling
of a simple nonfatal communicable disease in a popula-
tion which remains constant in size.
102 Rhodes and Anderson
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The probability that a susceptible becomes infective
when adjacent to an infective is a variable, p, set at the
start of a simulation (in more sophisticated models this
transmission parameter can be made to vary in a time-
dependent way, but we do not consider that possibility
here). The population density, C, is kept low (r0.2) in
order that overcrowding and the inability of individuals
to make a move does not start to affect the dynamics. To
mirror the epidemiological situation we wish to model,
each member of the host population is either susceptible,
infective, or recovered. The susceptibles remain in the
susceptible class until such a time as they become
infected. Once in the infected class individuals may
recover from infection with a probability p
r
; thus they
spend an average time span of 1p
r
as an infective. From
the infective phase the individuals move into the
recovered phase. After recovery, individuals can move
back into the susceptible phase with probability p
s
,
where p
s
<p
r
. This is a simple device to simulate
birthdeath processes whilst keeping the population
density constant. In our simulations we have set p
r
=0.1
and p
s
=0.05; so the life-time of the individuals on the
lattice is longer than the time spent as an infective, but
not so long as to make the simulation times
unmanageably time-consuming and it allows us to see
the effect of replenishment of the susceptible class. It is a
straightforward matter to include an exposed class of
individual in the model, namely those who have been
infected with the disease but who are not yet capable of
transmitting the disease to other susceptibles. Although
the inclusion of an exposed class would add a measure of
epidemiological realism to our model, we do not do so
here as it introduces another parameter (an average
incubation period) and increases stochastic effects for a
given population size (as the total population would be
divided amongst four, as opposed to three, classes).
In overall structure the model is similar to that of
Boccara and Cheong (1992) who used a lattice approach
to show the importance of population mixing in disease
spread. A related epidemic model, using a static host
population, is discussed in Johansen (1994, 1996). Peri-
odic boundary conditions are imposed on the lattice and
at each time step the numbers of susceptibles, infectives,
and recovereds are recorded. This lattice formulation
does not impose any fragmentation of the population
into patches and at each time-step every individual
is interacting with only its local neighbourhood of
individuals. The infection process generates a spatially
localised increase in disease correlation which is
dissipated over time by the motion of the infectives.
Propagation of the disease relies on the physical mixing
of infectives into regions of susceptible density.
Additionally, the model has a straightforward mean-
field limit. Following Boccara and Cheong (1992) the
mean-field equations are
S (t+1)=S (t)&S (t)[1&(1&pI (t))
4
]+bR (t)(1)
I (t+1)=I (t)+S (t)[1&(1&pI (t))
4
]&aI (t)(2)
R(t+1)=R (t)+aI (t)&bR (t), (3)
where
S (t)=7
i,j
S
i,j
(t)L
2
I (t)=7
i,j
I
i,j
(t)L
2
R (t)=7
i,j
R
i,j
(t)L
2
and a
&1
=p
r
,b
&1
=p
s
.
At equilibrium
S (t+1)=S (t)=S *
I (t+1)=I (t)=I *
R (t+1)=R (t)=R *.
Solving for I * (for I small) we find
S *= a
np (4)
I *=C&a4p
(1+ab)(5)
R *=a(C&anp)
a+b.(6)
In the simulation we can recover agreement with these
equations by randomly redistributing the entire popula-
tion over the lattice after each time-step, as this effectively
destroys any buildup of localised clustering of infection.
Expanding about the fixed point S (t)=S *(t)+$(t)
and I (t)=I *(t)+'(t), we can establish the stability of
the fixed point S *=Cand I *=0. We are led to the
characteristic equation
(*
1
+b)(npC&a&*
2
)=0. (7)
Therefore the eigenvalues are *
1
=&band *
2
=npC&a.
Stability requires that C<anp; otherwise, above this
critical population density a saddle point results in an
endemic level of infection with I *{0. We have shown
how by gradually increasing the hopping range of the
population on the lattice (Rhodes and Anderson, 1966b)
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it is possible to smoothly interpolate from the fixed point
of a simulation to the fixed point of the mean-field equa-
tions, clearly demonstrating the effect of increased mixing
and spatial heterogeneity on the prevalence of disease in
a population.
At the start of a coventional simulation a number of
susceptibles are randomly scattered over the lattice
nodes. A single Infective ``seed'' is then placed at the
centre of the lattice to initiate the course of infection in
the population. We can monitor the total number of
infectives as a function of time as well as the spatial
distribution of the infectives.
3. EPIDEMIC TIME SERIES
Because of the stochastic nature of the simulation pro-
cess each realisation of an infectious time-series will be
different. Consequently we take the mean and variance of
a large number of simulations and see how these quan-
tities stabilise as the ensemble of simulations increases in
number. At each time step we have a mean number of
FIG. 1. Time series of the mean number of Infectives in a lattice epidemic simulation. We use p=0.5, p
r
=0.1, and p
s
=0.05 and a lattice size
L=100. The population density C= 0.1. Ensemble averages over 100, 250, 400, and 500 simulations are shown. All subsequent simulations, the results
of which are shown in the figures below, use these same parameters for ease of comparison.
infectives on the lattice and an associated variance. In
Fig. 1 a typical time-series for the mean number of infec-
tives is shown. The population density is 0.1. An average
over 100, 250, 400, and 500 separate simulations was
taken to obtain the four profiles in the figure. Above 400
simulations the mean has stabilised and increasing the
number of runs any further does very little to add any
increase in the numerical accuracy of the result. Figure 2
shows the associated square-root of the variance as a
function of time for each of the graphs in Fig. 1. Again,
this quantity stabilises to a well defined limit above 400
simulations. Whilst we cannot predict the outcome of an
individual realisation of a simulation, the mean and
variance at least give us statistical estimators of the likely
infective population size as a function of time. From this
approach we can begin to investigate whether a given
host population density is likely to maintain the disease
in an endemic state. From this we infer that the popula-
tion density of 0.1 is below the critical population
threshold. As when using any stochastic model it is
necessary to bear in mind the expected size of variance of
the infective population when estiating quantities such as
the time-to-extinction or the critical community size.
104 Rhodes and Anderson
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FIG. 2. The square root of the variance for the simulations in Fig. 1.
FIG. 3. The coefficient of variation as a function of time obtained from Fig. 1 and Fig. 2.
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FIG. 4. The mean number of infectives as a function of time for a population density C=0.2.
FIG. 5. The coefficient of variation relating to Fig. 4.
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As the magnitude of the variance is related to the infec-
tive population size it is useful to look at the coefficient
of variation, that is CV(t)=-[Var(t)]Ave(t). In Fig. 3
CV(t) as a function of tis plotted. Clearly this quantity
diverges and is >1 \t>70. After 70 time-steps the
standard deviation is greater than the mean which
implies that some realisation are dying out. The implica-
tion of this is that we will see fade-out of the infection in
the population.
In Fig. 1 the disease is unable to maintain itself in the
host population. If we increase the population density on
the lattice above a certain threshold the disease becomes
endemic. Figure 4 represents this situation and as before
we plot the mean of the infective population as a function
of time for ensembles of up to 500 simulations. The host
density on this case is 0.2, corresponding to 2000
individuals. The disease appears to maintain itself in the
population, with CV(t)<1 \t, as is clearly shown in
Fig. 5.
This simple spatial model of epidemic spread is
capable of reproducing the intutively understood idea
that as a population increases in density it is more
capable of sustaining endemic infection. A criterion for
persistence would appear to be CV(t)<1 \t. For the
FIG. 6. Phase space plot for the lattice epidemic model showing the evolution to the endemic fixed point and the extinct fixed point.
parameters used in the above simulations this occurs for
a host population density of 0.14.
It is possible to generate a rough estimate of this
population threshold from the rate of decline of infectives
in the subendemic situation. From the stability analysis
above we know that perturbations decay at a rate
$(t)&$(0)e
&(a&npC)t
,(8)
so if t
12
is the time taken for the number of infectives to
decline by a factor of 2, we can estimate the ``effective''
numberr of neighbours, n
eff
, as a phenomenological
parameter,
n
eff
=n&a
pC&ln 2
pCt
12
(9)
and the threshold for persistence will be given by
C=a
pn
eff
. (10)
From Fig. 1 the estimated t
12
is around 50 days
which gives n
eff
=1.72, implying that the threshold for
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persistence is a population of approximately 1200
individuals. Clearly this calculation, relying as it does on
n
eff
being weakly dependent on C, provides only a rough
estimate of the critical population threshold. In the next
section we will discuss population thresholds in more
detail.
Our results can be compared with those obtained in
the lattice model of Johansen (1994, 1996), where infec-
tion is introduced into a stationary spatially distributed
host population. There oscillation in the numbers of
infectives is seen, resulting from the emergence of
organised coherent motion of fronts across the lattice,
followed by regrowth of susceptibles after the fronts have
passed. In our case, movement of the population across
the lattice would appear to destroy any tendency to
oscillation, resulting in a stable fixed point rather than a
noisy limit cycle as Johansen observes. In the limit of
infinite lattice size the oscillations observed by Johansen
disappear. At most animal and human populations are
mobile lattice simulations suggest that any oscillations
that are observed in disease incidence probabily
arise from seasonal effects or other forms of host
heterogeneity.
Figure 6 is a phase-space plot for three population
densities showing the emergence of the endemic fixed
point. It is a fixed point in the sense that for any given set
of model parameters, L,p,p
r
,p
s
, and S
o
, the system will
evolve to the same endemic level, regardless of the num-
ber of infectives, I
o
, initally present on the lattice at the
start of the simulation. Each trajectory is made up of a
sequence of plots of the mean infectiove and susceptible
populations after 500 simulations. We have to bear in
mind that each of the points along each of the trajectories
should be surrounded by an ``error ellipse'' whose major
and minor axes correspond to the variances of the infec-
tive and susceptible populations.
4. EPIDEMIC SPREAD
In models that utilise a spatially distributed popula-
tion the formation of wave-fronts can occur under the
right conditions. For example, the model of a rabies
epidemic in the event of a U.K. outbreak by Murray et al.
(1986) demonstrates the propagation of an epizootic
front through the susceptible fox population. Recent data
for the spread of rabies across continental Europe
(Anderson et al., 1981) shows a frontlike progress, where
propagation is faster in regions of higher fox density.
Clearly such spatially explicit models have great scope
for investigating the effect of various intervention
strategies, be that involving culling or vaccination.
Similarly, in the model of Noble (1974) for the spread of
plagne in human populations, the rate of progress of the
disease can be estimated from epidemiologically derived
parameters yielding acceptable agreement with the
historic record of the epidemic's progress after its intro-
duction into southern Europe in 1347. The epidemic
simulation model we have used is rather less directly
applicable than the coupled differential equation
approach in that, at present, estimations of epidemiologi-
cal parameters and their relation to variables in the
simulation has not been solved uniquely. However, the
model does show many of the expected properties of
epidemic spread through a spatially distributed popula-
tion and we are nevertheless able to gain some insight.
The formation of wave-fronts in the various classes of
cellular automata is itself a matter of some general
interest (Schonfisch, 1995), although here we confine our
discussion to disease spread.
In the above section we concluded that there is a cer-
tain threshold density of host population above which an
infection can remain endemic. To investigate how this
arises it is necessary to look at the spatial dynamics of the
infection. If we place a seed of infection at the centre of a
lattice populated wholly by susceptible hosts then,
broadly speaking, three possible outcomes may arise for
the time-course of the disease. The simplest outcome is
that the disease in incapable of spreading into the bulk of
the population and, over a time-scale of the average
duration of infection for a host, the seed of infection
simply fades out. Alternatively, the population density
might be high enough for the seed of infection to generate
a positive number of secondary infectives, so at some
time t>0 we have I>I
o
. This is generally what is meant
by an epidemic threshold. However, it is possible that
these secondaries are unable to sustain the chain of infec-
tion and these, too, will ultimately fade out, even though
the infection may have spread out some distance into the
population. A third possibility is that the population den-
sity is sufficient to maintain the chain of infection and a
``front'' of infection appears to move out concentrically
from the source of infection at the centre of the lattice. It
continues to move radially outwards until it reaches the
edge of the lattice. What happens at this stage is very
much dependent on the choice of boundary conditions.
For simplicity we have used periodic boundary condi-
tions where opposite sides of the lattice are mapped onto
each other.
In Fig. 7 we show the time series for the infection
where the density of population, 0.05, is too low for the
epidemic to occur and I<I
o
for t>t
o
. Figure 8 shows
the radial density of infectives on the lattice for t=30,
t=50, and t=70 time-steps. The abscissa is the radial
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FIG. 7. Time series of number of infectives for a population density of C=0.05 (i.e., <S
c1
). The dots indicate the three times at which the radial
distribution of infectives are measured. An ensemble average over 500 simulations is shown.
FIG. 8. The radial density distribution of infectives for a population density of C= 0.05 (i.e., <S
c1
). Three times-steps, corresponding to the radial
distribution of Infectives at t=30, t=50, and t=70, are shown.
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FIG. 9. Time series of number of infectives for a population density of C= 0.08 (i.e., >S
c1
). An ensemble average over 500 simulations is shown.
FIG. 10. The radial density distribution of infectives for a population density of C=0.08 (i.e., >S
c1
). Three time-steps are shown.
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FIG. 11. Time series of number of infectives for a population density of C= 0.2 (i.e., >S
c2
). An ensemble average over 500 simulations is shown.
FIG. 12. The radial density distribution of infectives for a population density of C= 0.2 (i.e., >S
c2
). Three times-steps are shown.
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distance from the centre of the lattice, in units of lattice
spacing. The ordinate is the average density of infectives to
be found at that radial distance. At time t=0 the seed
infective effectively forms a delta function at the centre of
the lattice and for all t>0 the amplitude continues to fall.
The host population density is too low to sustain the
progress of a front of infection and the disease dies out long
before any infectives appear near the edge of the lattice.
As the susceptible population density increases we
expect to see the formation of some secondary cases of
infection and thus cross the epidemic threshold. Empiri-
cal observation suggests that for the parameter choice
indicated in the caption this epidemic threshold is
crossed at S
o
=S
cl
r0.065. Figure 9 corresponds to
S
o
=1.23S
cl
and clearly there is a short-term increase in
the number of infectives. From the plot of radial density
of infectives as a function of time in Fig. 10 we see that,
although a distinct front o9f infection has moved into the
susceptible population it has eventually dissipated before
reaching the lattice edge.
Further increases of susceptible population density on
the lattice will expand the radial distance that the wave
propagates. A second threshold, S
c2
, is reached above
which the front reaches the lattice edge and long-term
persistence ensues. Figure 11 is the total number of infec-
tives as a function of time for a population density of 0.2,
and an epidemic leading to endemic disease ensues. In
Fig. 12 is plotted the radial density of infectives for
this same population density. A front is moving with
approximately constant velocity into the susceptible
population and increasing in amplitude as it does so until
it eventually hits the edge of the lattice. I(t)>I
o
for this
phase of the simulation. For the parameters used in our
simulation, S
c2
r0.14.
This leaves us with the conclusion that there are two
regimes in which this class of lattice model is useful. If we
have a situation where a point source of infection appears
in a community of susceptible hosts then it is most
straightforward to talk about those epidemics that finish
before any infectives reach the boundary of the lattice.
Thus, the susceptible population threshold S
c1
is the
most fundamental epidemic threshold quantity. Second,
if we accept the limitations that periodic or reflecting
boundary conditions impose and think of our population
as a closed community, the resulting fixed points can
then be regarded as being a reasonable representation of
disease persistence in that community. In this second
case, unlike the first, any attempt at a spatial investiga-
tion will not give too much insight as persistence effects
will be a result of the choice of boundary conditions; the
population threshold S
c2
will be the most important
epidemic threshold parameter.
5. VACCINATION STRATEGIES
One of the central motivations for developing models
of epidemic disease is so that an estimate of the likely
impact of any vaccination strategy can be assessed. In
our model, as we are not considering age-dependent or
seasonal effects, it is sufficient to determine some critical
fraction of susceptibles to be vaccinated in order to
eradicate the infection. Due to the spatial nature of the
model we can look at the spatial distribution of vaccina-
tion coverage. Vaccination involves moving a chosen
susceptible directly into the recovered class. We are
primarily interested in studying the reduction of the
endemic disease intensity by vaccination as would be
applied to human populations and to some animal pop-
ulations. The simplistic mixing patterns that occur in our
model due to movement of individuals on the lattice
means that our vaccination schedules and conclusions
should not be interpreted as directly addressing disease
eradication programmes in human communities. Much
greater social and spatial heterogeneity would be
required in the model in order to do this. As in any com-
putationalo simulation study of this sort there are myriad
epidemiological scenarios that can be considered, so
inevitably we must select a subset of cases for examina-
tion. Also, our investigation is circumscribed by the fact
that we do not, as yet, have a comprehensive working
phenomenology for the spatial dynamics of the model.
As noted above, the model can be used in two regimes.
First (Case 1), we can consider the endemic state to
represent a fixed point of the dynamics of the disease
model and thus determine the fraction of susceptible
hosts who have to be instantaneously vaccinated at every
time-step in order for complete eradication to occur. An
alternative strategy is to adopt a ``pulsed'' vaccination
campaign. At temporally space intervals a predetermined
proportion of the susceptible population is vaccinated. In
neither the continuous nor the pulsed vaccination
schedule is the spatial dependence of the susceptibles
taken into account when vaccination is administered; our
only concern is the proportion of susceptibles to be vac-
cinated.
Second (Case 2), we can consider a point source of
infection appearing at the centre of the lattice which sets
up a propagating front of infection which moves radially
outwards to the boundary. A sensible approach in this
instance is to vaccinate only those (or a proportion of
those) susceptibles found within an annulus centered on
the source of the outbreak, creating a ring-like region of
suppressed susceptible density. This forms a ``fire-break,''
the purpose of which is to dissipate the ``energy'' of the
infective wave-front so that it cannot propagate into the
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susceptible population beyond. In this case the spatial
location of susceptibles as well as the proportion to be
vaccinated are of concern here. Note also that the vac-
cinated ring is not a static entity; it begins to disperse due
to the intrinsic mixing effect due to the motion of the
individuals that made it up at the time of formation.
From the time the vaccinated ring is constituted to the
time the front reaches it, the density of recovereds in the
ring will have reduced slightly. One can imagine for
extreme choices of parameters a situation where the den-
sity of the ring has reduced to such an extent as to render
it ineffective at epidemic control when the front is finally
incident upon it. If we were to model a real epidemic con-
tainment scenario, where resources are finite, we might
have to consider the effect of using a second or perhaps
a third concentric vaccination ring, where each ring
reduces the intensity of the epidemic. Because of com-
putational limitations we have not attempted to model
such a strategy yet. Instead, we maintain the density of
vaccination ring by repeatedly vaccinating a proportion
of the susceptibles found in the ring at that time. This
avoids any complications due to ring dispersal or
depletion.
FIG. 13. Vaccination of population density of C=0.2 with 1 0,30, and 5 0of susceptibles vaccinated every time-step. Ensemble averages over
500 simulations is shown.
Case 1. Figure 13 shows the number of infectives as
a function of time for a population density of 0.2. A vac-
cination regime is introduced at t=250. At each subse-
quent time-step a proportion of the available susceptibles
currently on the lattice are vaccinated. Three schedules
are shown, 1 0,30, and 50of available susceptibles
are vaccinated at each time-step. For 10vaccination the
model simply moves to a lower endemic fixed point. On
the other hand, vaccination of 50of available suscep-
tibles eliminates the infection. Simulations show that the
threshold proportion of vaccinations to eliminate the
disease is approximately 30. This number will vary as
other parameters in the simulation are changed, but most
interestingly, as the population density is increased
(keeping the epidemiological parameters constant)
simulations show that the vaccination proportion for the
elimination of infection must be increased. Greater
population crowding makes for easier disease spread,
hence, the greater the proportion to be vaccinated.
As discussed above, a pulsed vaccination strategy can
also be considered. Figure 14 illustrates the time-series of
infectives when such a regime is introduced. There are
numerous combinations of time-interval and vaccination
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FIG. 14. Pulsed vaccination campaign; 50 0and 100 0of susceptibles vaccinated every 25 time-steps. Ensemble averages over 500 simulations
is shown.
proportions that could be tried in such an experiment.
The illustration shows 500and 100 0of susceptibles
being vaccinated every 25 time-steps. Extensive simula-
tions show that the threshold for elimination is such that
approximately 900of susceptibles should be vaccinated
in each pulse. Interestingly, for our parameters chosen
here, it would seem that there is an upper critical time
interval (of the order of 30 time-steps) for vaccination to
lead to eradication. Vaccinating 1000of susceptibles
every 40 time-steps could never lead to elimination of
infection. As the population density increases this critical
time interval gets shorter. Also, the closer in time the
pulses of vaccination occur the less the proportion of
susceptibles who have to be vaccinated.
Case 2. A single infective is placed at the centre of a
lattice with a host population density of 0.2. Without vac-
cination we would expect a front of infective density to
move outwards to the edge of the lattice as illustrated in
Fig. 12. The vaccination strategy this time involves main-
taining over time a vaccinated proportion of the suscep-
tibles which are found in a circular ring about the focus
of infection from the moment the infection takes hold at
the centre of the lattice. In our vaccination scenario we
confine out attention to the simplest possible case;
assuming we can vaccinate all the susceptibles in the ring,
is the ring sufficient to halt the spread of the disease
across the lattice, and what is the minimum thickness of
the ring required to do this? A population of 2000
individuals are placed on the lattice and an infected index
case is introduced at the centre. A vaccination ring is set
up with inner radius of 15 lattice spacings and the outer
radius can be set as required. Figure 15 shows the ring of
vaccinated cases around the focus of infection with an
inner radius of 15 and an outer radius of 25 at the
beginning and towards the end of a typical simulation.
This clearly illustrates how the ring of vaccination seeks
to isolate those susceptibles outside the ring from the
infectives within. We have performed a series of simula-
tions for vaccination rings of increasing thickness, and
the radial density of infectives as a function of time are
shown in Fig. 16. For thin rings the front is able to move
relatively unimpeded across the lattice and it is possible
to transfer infective density into the pool of susceptibles
beyong the ring. As the thickness increases it becomes
increasingly difficult for infectives to cross the barrier in
any number and eventually there reaches a critical mini-
mum width of the order of 10 lattice spacings above
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FIG. 15. A pictorial representation of the lattice. Susceptibles are
indicated by crosses, infectives by squares, and the recovereds by
circles. The top panel shows the location of the ring of vaccinated
individuals soon after formation and the lower panel shows the
situation some 60 tipe-steps into a typical simulation. Note that all the
infectives are contained within the ring of vaccination.
FIG. 16. Radial density of infectives at time t=30, t=50, and
t=70 timesteps. Vaccination rings of increasing thickness are used. The
position of the inner and outer limit of the rings are shown by the verti-
cal lines on the radial axis. The top panel has a vaccination ring of
width-2 lattice spacings; the middle panel has a width-5 lattice spacings
and the bottom panel has a vaccination ring of width-10 lattice springs.
In contrast, Fig. 12 shows what happens when the spread is unimpeded
by any zone of vaccination.
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which there is no further progress of the front of infection
once it hits the ring. For vaccination rings greater than 10
lattice spacings all the infective density is confined to the
space within the region that has been vaccinated for as
long as the ring is maintained.
These simulations suggest that ring vaccination
around known foci of infection can be an effective
method for reducing disease incidence when we have a
mobile spatially distributed population. Providing the
ring is maintained at sufficient density, the mobility of the
infectives is no guarantee that the epidemic front can
transcend the barrier. Clearly, though, in human com-
munities, particularly in the developed world, individuals
can be highly mobile and ring vaccination is not likely to
work well, because a vaccination ring of enormous
width would be required, effectively resulting in a mass-
vaccination policy. In the past, most studies of spatial
vaccination strategies have used continuum-based
epidemiological models with additional diffusion terms.
Although it is sometimes possible to obtain analytical
results for the required width of vaccination zones, there
is a problem with the appearance of unfeasibly small
residual regions of infective density which can generate
further epizootics long after the initial wave has passed,
which can lead to erroneous interpretations of observed
dynamics. Our approach can be seen to eliminate this
problem. By developing the individual-based lattice
model it should be possible to circumvent problems of
this sort and investigate vaccination strategies within a
framework that allows for spatial distribution of the host
population and its ability to move through mixing.
6. DISCUSSION
We have used a lattice-based SIR epidemic model to
study the spatio-temporal evolution of infection in a
spatially distributed host population. It is one amongst a
number of different models introduced recently that tries
to account for the effect of spatial heterogeneity on
epidemic spread. As yet we have no a-priori method for
calculating this threshold in the lattice model. Also, as
the model is stochastic, we have presented means and
variances of an ensemble of simulations, giving an idea of
the number required to obtain a stable mean and
variance for a given population density. Empirical
investigation of the model indicates that that persistence
is possible when CV(t)<1 \t. The simulations above are
the simplest representation of the epidemic process.
Development of the model, by adding further
refinements, such as age-structured contact rates,
subgroups with different mobilities, inhomogeneously
distributed population (the list could go on, reflecting
whatever epidemiological situation we wished to model)
would greatly add to the appeal of this approach. Before
these additions are made, however, an understanding of
the basic processes at work in this class of model is
required.
To investigate the spatial dynamics of the model we
tested the response of the model to a point source of
infection placed at the centre of the lattice. For suf-
ficiently high population densities, a front of infection
could be seen moving radially outward from the source
of infection. To the accuracy obtainable in the simula-
tions the velocity of the front appears to be constant. For
lower host population densities the the front of infectives
dissipates before reaching the boundaries of the lattice.
Two distinct regimes of behaviour are evident in the
model. In the first, a front of infection can be seen to
propagate radially outward from a point source of infec-
tion at the cen tre of the lattice. For low population den-
sities the propagation dissipates before reaching the edge
of the lattice. For sufficiently high densities the front
reaches the edge of the lattice and long-term persistence
ensues. For the interval of time before the front reaches
the lattice edge a coherent spatial distribution of infec-
tives can be seen. The uniform preepidemic density of
susceptibles exists ahead of the front and suppressed
susceptible population density is evident behind it. In the
second, long term persistence occurs by virtue of the
fact that we have periodic boundary conditions and a
replenishment of fresh susceptibles at a suitable rate. There
is no overall spatial organisation relating to this state.
A natural extension of the model is to include vaccina-
tion of susceptibles. There exist clear thresholds for the
proportion of susceptibles that have to be vaccinated in
both continuous and pulsed vaccination strategies. A
spatially explicit ring-vaccination strategy showed the
effect of the front of infection impinging upon a region of
low susceptible density and how this obstructs the move-
ment of infective density into regions beyond the ring. In
this model complete protection of the susceptible popula-
tion beyong the ring seems to be afforded.
The similations show many of the aspects associated
with disease spread, although we have got no further on
the issue of parameter estimation. Clearly, if better
parameter estimation becomes available for spatial dis-
ease processes then incorporating those parameters into
this kind of lattice model might prove useful in assessing
the usefulness of the sort of vaccination strategies con-
sidered above. Also, as stressed by Durrett and Levin
(1994b), despite the fact that a lattice model of inter-
acting individuals naturally incorporates spatial distribu-
tion and population discreteness effects, quantitative
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predictions are, on the whole, impossible to obtain. We
can only really expect to see the emergence of a
qualitative understanding of the role of each of the
variables in the model. Although still at an early stage in
its development, spatial interacting particle models show
much promise as an approach to studying the dynamics
of disease in spatially distributed host populations.
However, using a related lattice-based approach we have
recently been able to account for the dynamics of measles
epidemics in small isolated populations (Rhodes and
Anderson, 1996c).
The lattice epidemic model is quite general and flexible
and could be applied to a variety of animal epizootics
too. For rabies would be probably be a suitable case to
model as the animals tend to remain within home ranges
and good spatial and temporal data is available for U.K.
urban foxes (Smith and Harris, 1991). A good model of
spread would be required in the event of outbreak in the
U.K. in order to quantify the effect of various contain-
ment strategies. In that case we are likely to see a front of
infection spreading out from the focus of infection
(probably in a large urban centre), with a requirement
for cullingvaccination intervention in a circle outside the
wave-front. As understanding of the lattice and interact-
ing particle system approaches increases this epizootic
systems might be a useful place to investigate realistic
parameterisation. Additionally, using our model frame-
work, it would be straightforward (if it were found to be
necessary) to introduce animal motion parametererised
by Levy statistics as has recently been observed in radio-
trackingt experiments (Viswanathan et al., 1996). The
presence of such patterns of motion would have a signifi-
cant impact on the spread of communicable disease in
those an imal communities where it was observed.
ACKNOWLEDGMENTS
The authors than the Wellcome Trust for kindly providing research
support. We also thank Dr. C. Neuhauser of the Department of
Ecology and Evolutionary Biology, Princeton University, for useful
discussions and the referees for many helpful comments.
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