Download Contact rate calculation for a basic epidemic model

Mathematical Biosciences 216 (2008) 56–62
Contents lists available at ScienceDirect
Mathematical Biosciences
journal homepage: www.elsevier.com/locate/mbs
Contact rate calculation for a basic epidemic model
C.J. Rhodes a,b,*, R.M. Anderson b
a
b
Institute for Mathematical Sciences, Imperial College London, 53 Prince’s Gate, Exhibition Road, South Kensington, London SW7 2PG, UK
Infectious Disease Epidemiology Group, Imperial College London, Norfolk Place, London W2 1PG, UK
a r t i c l e
i n f o
Article history:
Received 18 August 2007
Received in revised form 16 July 2008
Accepted 8 August 2008
Available online 22 August 2008
Keywords:
Mass-action
Basic reproductive ratio
Homogeneous mixing
Spatial epidemic model
a b s t r a c t
Mass-action epidemic models are the foundation of the majority of studies of disease dynamics in human
and animal populations. Here, a kinetic model of mobile susceptible and infective individuals in a twodimensional domain is introduced, and an examination of the contact process results in a mass-actionlike term for the generation of new infectives. The conditions under which density dependent and frequency dependent transmission terms emerge are clarified. Moreover, this model suggests that epidemics in large mobile spatially distributed populations can be well described by homogeneously mixing
mass-action models. The analysis generates an analytic formula for the contact rate (b) and the basic
reproductive ratio (R0) of an infectious pathogen, which contains a mixture of demographic and epidemiological parameters. The analytic results are compared with a simulation and are shown to give good
agreement. The simulation permits the exploration of more realistic movement strategies and their consequent effect on epidemic dynamics.
Ó 2008 Elsevier Inc. All rights reserved.
1. Introduction
At the core of any model of a communicable disease epidemic is
a statement of how, in a given population, new infectives are generated by the mixing of uninfected susceptibles with existing infectives. Historically, epidemic models have typically, though not
exclusively, assumed that the rate of increase of new infective
cases occurs in proportion to the product of the number of susceptibles and the number of infectives. This is the well-known ‘massaction’ assumption. These models have underpinned the development of applied epidemiology, and are often used as a basis for
quantitatively assessing disease intervention strategies, such as
vaccination, travel restrictions or culling, for example. They have
been applied to a vast range of human and wildlife diseases [1,2].
McCallum et al. [3] have discussed numerous different assumptions that are commonly used in modelling pathogen transmission
in a variety of human and wildlife populations.
The limitations of mass-action contact rate assumptions have
not escaped the attention of epidemic modellers and, more recently, refinements to such models have sought to introduce heterogeneities that are characteristic of a given epidemic scenario in
order to more adequately reflect the structure inherent within
populations [1]. These heterogeneities serve to modify the frequency of interactions between individuals from that which would
* Corresponding author. Address: Institute for Mathematical Sciences, Imperial
College London, 53 Prince’s Gate, Exhibition Road, South Kensington, London SW7
2PG, UK. Tel.: +44 2075941753; fax: +44 2075940923.
E-mail address: [email protected] (C.J. Rhodes).
0025-5564/$ - see front matter Ó 2008 Elsevier Inc. All rights reserved.
doi:10.1016/j.mbs.2008.08.007
be expected on the basis of a mass-action assumption. This structure is often driven by the fact that populations are dispersed over
geographical space, or that there is social organisation, aggregation
or preferential mixing groups. Much recent work on networkbased models, for example, has sought to better reflect this reality
[4–6]. In the case of spatially dispersed populations the separation
of susceptibles from infectives determines their likelihood of being
infected. By contrast the mass-action assumption implies that each
susceptible and infective are at all times equally accessible to each
other, thereby suggesting that it would be inappropriate to apply
the assumption to a spatial epidemic process.
One of the issues relating to the mass-action assumption is how
the contact rate between individuals scales as population size
changes. In order to clarify this issue de Jong et al. [7] distinguish
two different mass-action assumptions, namely ‘pseudo mass-action’ and ‘true mass-action’. They argue that pseudo mass-action,
where the contact rate scales as bSI (b is the transmission constant,
S is the number of susceptibles and I is the number of infectives) is
not as widely supported by empirical evidence (albeit in data from
vertebrate host infections) as the true mass-action assumption defined by bSI/N, where N is the total population size. Pseudo massaction and true mass-action are also known as density dependent
transmission and frequency dependent transmission, respectively,
and in what follows we will refer to them as such [7]. Much empirical data tends to favour frequency dependent mass-action where
observed reproductive ratios (R0) saturate with increasing population size. Dietz [8] and Schenzle and Dietz [9] have questioned
whether an individual in a population of 106 has many more contacts than an individual in a population of 105. If the number of
57
C.J. Rhodes, R.M. Anderson / Mathematical Biosciences 216 (2008) 56–62
contacts is equal in both populations it is likely that the larger population must occupy a greater area than the smaller, i.e. the overall
densities must be equivalent. Clearly there must be some role
played by the spatial distribution of the population in a given area
when thinking about mass-action assumptions. Also, Antonovics
et al. [10] have shown that density dependent and frequency
dependent transmission represent the extremes of a Holling Type
II response curve, and point out that in many situations the transmission patterns of real diseases often fall somewhere between
these two extremes.
Begon et al. [11] have also discussed the use of frequency and
density dependent transmission terms in epidemic modelling with
the aim of clarifying when these should be used. The model developed below supports their contention that the area occupied by a
population must be maintained constant when considering density
dependent mass-action models.
Given the fundamental questions raised by the role of space and
consideration of contact rates in epidemic models it is of interest to
attempt to calculate the contact rate from first principles using an
ensemble of interacting individuals. Diekmann et al. [12] have stated ‘there is a need for quasi-mechanistic sub-models, especially
for the contact process’. Here, we show how to analytically calculate the contact rate from an underlying model of a finite number
of individuals moving in a two-dimensional domain. It is possible
to see how the mass-action-like terms can arise in a simple spatial
epidemic model of communicable disease that incorporates host
movement. The effect of more realistic host movement strategies
is then investigated in order to assess the applicability of these results to more general situations involving mobile spatially distributed hosts.
Here, our objective is to show how an individual-based approach can give insight into basic epidemiological modelling
assumptions and concepts. In Section 2 the contact process model
is introduced and an analytic expression for the contact rate is derived. It is shown that the contact rate is higher when both infectives and susceptibles are in motion. In Section 3 the results are
extended to include multiple infectives and expressions for the
transmission parameter, b, and basic reproductive ratio, R0, are
presented. Moreover, a frequency dependent mass-action assumption for the rate of generation of new infectives emerges naturally
from this model for populations that maintain their density whilst
changing size. Section 4 compares the analytic results with a simulation and shows how mass-action can accommodate more complex host movement patterns. In Section 5 the results are
summarised and a discussion of how these results might be used
in the context of emerging infectious diseases is given, as well as
indications for future work.
applied to considering spatial epidemic processes. The most widely
studied included reaction-diffusion methods [16], meta-population methods [17] and lattice models [18–20]. Here, we adapt
the Lewis–Pedley [13] and Gerritsen/Strickler [14] models to two
dimensions and explore the dynamics of a communicable disease
process in this system. The approach is similar to that of Koopman
[21] who investigated search strategies in an anti-submarine warfare context. The results presented here give insight into how the
functional form of mass-action assumptions can arise from consideration of a spatial contact process.
The contact process model consists of a population of susceptible individuals that at time t = 0 are randomly distributed over a
two-dimensional plane. This results in a uniform density of susceptible individuals, qs, across the domain. In what follows we work
with population densities, and only consider population numbers
when we consider a defined bounded area in the domain. As illustrated in Fig. 1, each individual moves independently of the others
with a constant straight line velocity v~s with their direction vectors
distributed uniformly in azimuth in the plane. Hence there is no
net change of overall population density in the domain. An infective is introduced and moves through the domain with velocity
v~i . It is assumed that if a susceptible passes within a radius R of
the infective there is a probability p that the susceptible becomes
infected. Here, we constrain R l, where l is the mean inter-particle spacing. Additionally, the length scale of the square plane, C, is
taken to be much larger than l, so that R l C. Fig. 2 shows the
geometry of the interaction. In order for a susceptible to be exposed to an infective during a time period dt the susceptible must
lie within the rectangular-shaped area swept by the infective lying
~ ¼ v~i v~s . The area swept in this
along the direction of the vector w
time period dA = 2Rwdt, where the relative speed is given by
w ¼ ðv2i þ v2s 2vi vs cos /Þ1=2 , where / is the angle between the
velocity vectors.
Given the independent motions of the particles the density of
susceptibles with velocity vectors with directions in the range /
SUS
vs
ϕ
w
2. Contact rate calculation
The rate at which a given individual encounters other members
of a population is a fundamental concern in a variety of ecological
and epidemiological contexts. The study of the predation behaviour of zooplankton has yielded a wealth of insight that can be
adapted to the epidemiological context. Lewis and Pedley [13] have
shown how to calculate the encounter rates between planktonic
predators and prey in a turbulent marine environment. Their work
builds on a model by Gerritsen and Strickler [14] which assumes a
simpler movement behaviour for the organisms; each individual
moves in a straight line trajectory in a three-dimensional volume.
It has been noted that straight line trajectories are optimal for
destructive foragers [15].
In epidemic modelling of communicable disease we are mainly
concerned with two-dimensional domains, which act as a surrogate for real geographic space. A variety of approaches have been
Vi
R
INF
x
y
Fig. 1. Geometry of the interaction between an infective and a susceptible. The
movement of all the individuals takes place in the x, y plane. An infective situated at
the co-ordinate origin moves with velocity vector vi (dashed line) along the
negative y axis, whilst a susceptible moves with velocity vector vs . These vectors are
is the relative velocity vector of the two individuals. The
at an angle u. The vector w
area of hazard for the susceptible (dA = 2Rwdt) is given by the lozenge-shaped area
minus the semicircular end pieces.
58
C.J. Rhodes, R.M. Anderson / Mathematical Biosciences 216 (2008) 56–62
The same result holds if the infective is stationary and the susceptibles are in motion.
This shows that the rate at which new infectives is generated is
higher (around 25%) when both susceptibles and infectives are
moving. Note that the units for Eqs. (6) and (7) are time1, as
expected.
For some diseases there is likely to be a disparity between the
mean speeds of susceptibles and infectives. In rabies infection of
foxes, for example, the infectious phase can be characterised by a
loss of the usual movement constraints imposed by territory and
home range. This is known as ‘furious’ rabies. In this case vi > vs,
and m ? 4vs/vi. Expanding Eq. (5) to first order in m, the rate of
generation of infectives becomes
31
30.8
Contact rate per day
30.6
30.4
30.2
30
29.8
L = inf
29.6
L=20
L = 200
L=10
L=5
29.4
29.2
29
Fig. 2. Contact rates for a given individual as a function of L, the length scale on
which individuals make turns. Contact rate remains independent of step length.
and / + d/ is qsd//2p [22]. Therefore, the number of susceptibles
entering the swept area in dt is dN/ = qs2Rwd/dt/2p. In order to
obtain the total number of susceptibles encountering the infective
it is necessary to integrate over the planar angle /. Hence the number of susceptibles entering the area around the infective bounded
by the radiusR, i.e. the contact rate, CR, is given by:
CR ¼
Z
Z 2p
dN/ qs R 2p
w d/:
¼
dt
p 0
0
ð1Þ
This is equivalent to Eq. (9) in Lewis and Pedley [13] (see also equation A.31 in Lewis and Pedley [13]) and to Eq. (4) in Gerritsen and
Strickler [14]. Substituting for w gives
CR ¼
qs R
p
Z 2p
0
ðv2i þ v2s 2vi vs cos /Þ1=2 d/:
ð2Þ
dI
v2
¼ 2Rpqs vi s :
dt
vi
In many communicable diseases of humans (e.g. influenza), the
infectious period is likely to coincide with a significant reduction
in overall average speed due to the fact that the infective is often
confined at home, so vs > vi, resulting in Eq. (8) again, though with
vi and vs exchanged. In reality mixing patterns are more complex
than this simple analysis suggests and it is not straightforward to
reflect this in models. However, what is clear is that infection can
lead to behavioural changes that alter the mean speed of a host
resulting in a different contact rate.
So far, we have assumed that the susceptible individuals are all
moving at speed vs, and infective individuals are all moving at
speed vi. It is possible that within the population there is a range
of speeds, varying according to a probability distribution function.
In this case, returning to Eq. (4), taking vs = vi = v and assuming that
all individuals are subject to the same speed distribution, the contact rate can be more generally written as
Eq. (2) reduces to
CR ¼
4qs R
p
ðvi þ vs Þ
Z p=2
2
ð1 m sin xÞ
1=2
CR ¼
dx;
ð3Þ
0
where m = 4vsvi/(vs + vi)2. This is an elliptic integral of the second
kind and can be written in standard notation:
CR ¼
4R
p
qs ðvi þ vs ÞEðmÞ:
ð4Þ
Mathematica or pre-calculated tables give values for E(m) [23]. Eq.
(4) gives the rate of contact of a specified individual with all the
others in the population and is equivalent to Eq. (4) in Koopman
[21]. If the specified individual is an infective, and transmits infection to its contacts with probability p, then the rate of production of
new infectives is given by:
dI 4Rp
q ðv þ v ÞEðmÞ:
¼
dt
p s i s
ð5Þ
(1) If the particles are all stationary: vi = vs = 0. It is unlikely
there are other susceptibles within the radius of the infective
(because R l), so no new infective cases are generated.
(2) If the susceptibles travel at the same speed as the infective:
vs = vi = v; m = 1. Therefore, E(1) = 1, and Eq. (5) becomes
dI 8Rp
q v:
¼
dt
p s
ð6Þ
(3) If the susceptibles are stationary and the infective is moving:
vs = 0, vi = v; m = 0. Therefore, E(0) = p/2 and Eq. (5) becomes
dI
¼ 2Rpqs v:
dt
ð7Þ
8Rqs
p
Z
1
vkðvÞ dv:
ð9Þ
0
A candidate distribution speed distribution function k(v) is the Maxwell–Boltzmann (M-B) form [22]. Using this in Eq. (9) gives
CR ¼
8Rqs
p
Z
1
2
v3 keev ;
ð10Þ
0
where k and e are constants that parameterise the M-B distribution.
Evaluating Eq. (10) gives
CR ¼
8Rqs k
:
p 2e2
ð11Þ
¼ k=2e2 , so the
The mean of the M-B distribution is given by v
expression for the contact rate for a population with a distribution
of speeds is given by
CR ¼
Three limiting cases are of interest:
ð8Þ
8Rqs
p
:
v
ð12Þ
should be interpreted as an average speed,
Therefore, the speed v
though the model permits the hosts to have a distribution of speeds.
The M-B speed distribution is appropriate here if the (x, y) components of the individuals’ velocities are Gaussian distributed [22].
This is a reasonable first assumption for a speed distribution in a
large population. Finally, the transmission process described here
is rather simplistic and means that any susceptible that strays within the area bounded by R around an infective will be at equal risk of
becoming infected. More recent work has refined this transmission
mechanism to make it somewhat more realistic by making the risk
proportional to the time spent by the susceptible in crossing the domain, and also to allow for the radial decay of pathogen around an
infective.
C.J. Rhodes, R.M. Anderson / Mathematical Biosciences 216 (2008) 56–62
3. Multiple infectives
The above analysis considered the rate of generation of secondary infectives from a single infective in an otherwise susceptible
population. If many infectives are present, then, in a given area,
A, assuming there is a density of qi infectives in the domain, the
rate of generation of infectives per unit area is given by:
1 dI 8Rpv
q q:
¼
A dt
p s i
ð13Þ
This assumes that the average speeds of the particles are the same,
and that during its infectious lifetime each infective is in contact
with the mean density of susceptibles. Whether this latter assumption will hold will be dependent on the specific movement pattern
of the individuals and how quickly they are moving. For example
straight line motion with high speeds may generate sufficient mixing of infectives and susceptibles to ensure this assumption was
met, whereas a Brownian trajectory with slow moving individuals
might not. This point is explored further in the simulations in Section 4.
In most epidemiological modelling, it is changes to the population sizes that are used to consider the dynamics. Converting Eq.
(13) to population sizes results in:
S I
1 dI 8Rpv
¼
:
A dt
p AA
ð14Þ
And this reduces to
q SI
dI 8Rpv
¼
dt
p N
ð15Þ
59
terms should be used are clarified. Eqs. (15) and (18) correspond
to Eqs. (3) and (2) in Begon et al. [11]. The calculation here also replicates the units of the transmission parameter noted in [11]. As Begon et al. [11] state, the area under consideration must remain
constant when looking at the change of a population over time
and when comparing different population in order for density
dependent mass-action to be applicable and this analytic calculation makes explicit the role of spatial distribution and movement
that is implicit in their presentation. In practice, however, determining whether density changes as populations change in size is
a significant challenge.
The formula for R0 is made up of epidemiological parameters (p,
a), and demographic parameters ðR; v; qÞ. The demographic parameters are fixed for a given population, so the different R0 values for
different diseases in the same communities (e.g. chickenpox, pertussis, influenza, measles) can simply be attributed to their different epidemiological parameters. Given that in humans most nonfatal infectious diseases that confer immunity have a broadly comparable infectious duration (of the order of several days) most of
the empirically observed variation in R0 is due to the efficacy of
transmission of the virus as set by p.
Given the dependence of R0 on population density, there will be
a critical level below which the disease will not be able to establish
itself in the population (irrespective of the numbers of infectives
that are introduced), even though the population is mixing through
motion. For R0 > 1 and the possibility of endemic disease, the population density
q>
pa
8Rpv
:
ð21Þ
or
dI
SI
¼b ;
dt
N
ð16Þ
where
b¼
q
8Rpv
p
ð17Þ
:
Eq. (16) is the frequency dependent mass-action assumption, where
q is the total population density and N is the total population size,
i.e. q = N/A.
Alternatively, Eq. (15) can be stated
dI 8Rpv
¼
SI:
dt
pA
ð18Þ
This yields a density dependent mass-action form dI/dt = bSI; so
b¼
8Rpv
:
pA
ð19Þ
By using these two different values of the contact rate (Eqs. (17) and
(19)) in the expressions for the reproductive ratios in Appendix A
(assuming the infectives recover to a condition of lifetime immunity
after a mean time period a1) it follows that the reproductive ratio
for the epidemic in this model is given by:
R0 ¼
q
8Rpv
pa
:
ð20Þ
From Eq. (20), it is evident that if a population changes size whilst
keeping its overall density constant then the reproductive ratio of
the infection, R0, will be independent of the population size, so a frequency dependent mass-action description with a contact rate
determined by Eq. (17) is apposite. Alternatively, if a changing population is constrained within a fixed area, A, then the reproductive
ratio becomes dependent up on N (because q = N/A) so a density
dependent mass-action description with a contact rate determined
by Eq. (19) is appropriate. In this way, the circumstances under
which frequency dependent and density dependent transmission
4. Comparison with simulation
It is possible to simulate the model described above in order to
validate the analytic results and to introduce more realistic host
movement strategies thereby permitting investigation of the effect
that these have on the resulting epidemic dynamics. The contact
rate between individuals is assessed first, followed by the simulation of an epidemic process. It is shown that basic mass-action can
accommodate complex movement trajectories, and this is discussed in terms of the length scales that relate, respectively, to
the whole population and to the individual.
4.1. Contact rates
A collection of N individuals are distributed randomly with a
uniform density over a two-dimensional domain of size L* L*.
Each is assigned a speed v, and direction of travel, h (drawn from
a uniform distribution in azimuth), along which they continue to
travel. Periodic boundary conditions are applied, so individuals
leaving one side of the domain are re-introduced at the same edge
position along the opposite side. A given individual is taken to have
made contact with another individual if they pass within a radius R
of each other at some time during their travel. This simulation represents the essentials of the contact process described in Section 2.
In order to simplify the presentation it is easiest to work in real
units with reference to a specific scenario. The simulation has been
implemented using a population N = 3000, L* = 1000 m (giving
q = N/L*2 = 3 103 m2), R = 2 m, and v = 0.023 m/s. These figures
correspond to an urban population density of 3000 humans per
square kilometre, a contact radius R of 2 m, and each individual
travels approximately two kilometres per day. The time-step for
the simulation is set at 10 s, so each individual travels approximately 1% of the contact radius length scale (2 cm) each time step.
This gives an accurate reconstruction of the area swept by each
individual in its journey across the domain. In what follows it
60
C.J. Rhodes, R.M. Anderson / Mathematical Biosciences 216 (2008) 56–62
should be noted that R l, where l is the mean spacing between
individuals in the domain (for q = 3 103/m2 the inter-individual
spacing l 18 m), hence we are in the dilute limit. Also, in the simulations presented here all individuals have the same speed,
though, as shown in Section 2, the individuals can have a range
of speeds and the contact rate is in proportion to the mean speed
in the population.
Eq. (12) gives the contact rate for a specified individual in the
population. Using the values stated above, the contact rate for each
individual is 30.36 contacts per day. Using the simulation, it is
found that the contact rate is 30.22 ± 0.23 (±1SD) contacts per
day. Other simulations (results not shown here) demonstrate that
the contact rate scales as expected when R, v and q are varied, indicating that the simulation is representative of the basic underlying
contact process.
Both the contact process model and the simulation imply that
each individual moves independently along its own straight line
trajectory. In reality individuals within populations (both terrestrial animal and human) exhibit somewhat more complex patterns
of movement [24–27]. It is straightforward to introduce randomly
directed turns to each simulation trajectory after each individual
has moved a given distance. Each individual’s trajectory is now a
random walk with a fixed step length, L. Fig. 2 shows the effect that
turning has on the contact rate. The rate at which an individual
makes contact with others in the population is essentially independent of the step length. This applies at length scales shorter than
the mean inter-individual spacing l, and down to the contact length
scale set by R.
4.2. Epidemic model
It is straightforward to adapt the simulation described above to
represent a basic compartmental susceptible-infective-recovered
(SIR) model [1,16] and to investigate the extent to which the
mass-action assumption can represent the epidemic dynamics in
a spatially explicit mobile population. Here, we compare the simulation against a conventional epidemic model.
In the simulation each individual begins in a susceptible state.
An infective individual is introduced to the population and disease
spreads as the population mixes by contact between infectives and
susceptibles. Any susceptible individuals that, whilst following its
trajectory, moves to within a distance R of an infective becomes
infective themselves with a probability p. After a time period set
by a1 infectives recover and remain immune to further infection.
Given that the population level is fixed, and there is no injection of
susceptibles by a birth process, the infection will spread through
the population before dying out.
Fig. 3 shows the number of infectives as a function of time for
an epidemic based on a frequency dependent mass-action [7] SIR
model. The model equations are those in Appendix A (A1-A3).
Number of infectives
2500
2000
1500
1000
500
0
0
1
2
3
4
5
6
7
8
9
Time (days)
Fig. 3. Epidemic curves for the individual-based simulation and an SIR model. Dots
are the simulation and the solid line is the homogeneous SIR model.
To replicate measles in an urban population we take
b = 3.0 day1, N = 3000, a = 0.2 day1. The reproductive ratio, R0,
is given by b/a, so in this case R0 = 15 which is typical for measles
in large cities. Overlaid on this is in Fig. 3 is the epidemic curve for
a single realisation of the individual-based spatial model from the
simulation, using the same parameters as in the SIR model. This
demonstrates that the analytic expressions for the basic reproductive ratio are good and that a mass-action-based epidemic model
can account for epidemic dynamics in a mobile spatially distributed population. (A stochastic simulation of the SIR model equations also matches the epidemic curves given by the
homogeneous equations and the individual-based simulation.) In
the simulation, the time of the peak of the epidemic will be different for different spatial arrangements of the individuals at the
start. This is because the single index case can be a varying distance from the first susceptible it infects and variability in the
establishment of the epidemic when infectives are low at the start
of the epidemic. Once the epidemic is established fluctuations are
minimal due to the large reproductive ratio. In the differential
equation SIR model the infection process always begins at t = 0.
Here, and in what follows, the SIR model time series is translated
in time slightly in order to permit comparison with the
simulations.
4.3. Effect of movement patterns on epidemic dynamics
So far we have provided a particular host movement pattern
which leads to epidemic dynamics that can be described by
mass-action mixing and with a contact rate that can be calculated
analytically. Whilst permitting a distribution of host speeds, it assumes straight line trajectories for the hosts. Whilst this pattern is
effective in generating homogeneous SIR dynamics it could be argued that realistic patterns of host movement are more complex
than straight lines. Typical movement patterns generally involve
changes of direction, and boundaries and home-range territories
may influence how a host moves around its environment. In the
ocean, movement patterns of plankton and pathogenic viruses
are determined by the turbulent flow field.
To simulate this each host now undertakes a random walk with
step lengths of a given length L. Each particle travels in a distance L
in a straight line and then it makes a change of direction by randomly selecting a new direction. Above it was shown that this
has no effect on the contact rate experience by a given individual.
Here, the effect on epidemic dynamics is investigated.
There are two relevant length scales to be considered; the mean
inter-host spacing, l, and the length of the straight line segments of
the random walk, L. The inter-host spacing is determined by the
population density and, as stated above, corresponds to 18 m in
this simulation. It has been shown in Fig. 4 that for L ? 1 the simulation and a homogeneous SIR model are in good agreement.
Fig. 4 shows the effect of reducing L. As host turning becomes more
frequent (as L reduces) the simulation continues to exhibit SIR
behaviour and can be approximated by a homogeneous SIR model
with basic reproductive ratio R0 = 15. This remains true at least until L = 50 m. When L = 30 m, small departures from the SIR model
are evident, particularly around the peak of the epidemic. Once
L = 20 m, the SIR model (still with a basic reproductive ratio of
15) starts to show significant departures. Simulation for L = 10 m
is shown in Fig. 4d. Whilst the dynamics can still be approximated
by the homogeneous SIR model, now an effective reproductive ratio R0 ’ 10 provides a better fit. However, it is clear that in this regime of frequent turning there are noticeable departures from the
SIR epidemic curve, consequently the homogeneous model is becomes a less effective description.
From this, it appears that until L drops below 2l the simulation
and SIR homogeneous model are in agreement. For step lengths
C.J. Rhodes, R.M. Anderson / Mathematical Biosciences 216 (2008) 56–62
Number of infectives
a
2500
2000
1500
1000
500
0
0
1
2
3
4
5
6
7
8
9
6
7
8
9
Time (days)
Number of infectives
b 2500
2000
1500
1000
500
0
0
1
2
3
4
5
Time (days)
Number of infectives
c
2500
Rather, the susceptible background density is slightly depleted by
the presence of other nearby infectives thereby reducing the potential for each infective to pass on infection to susceptibles during
its finite lifetime. As the step length for the hosts is reduced this
correlation effect increases and the effective R0 for the epidemic reduces. The mean square separation of two infected Brownian walkers scales as t1/2, whereas for straight line motion (L ? 1) they will
separate at a rate proportional to t. The impact of the presence of
spatial correlation resulting from a particle birth–death process
has been analysed by Young et al. [28] and Martin [29]. In the epidemic model the total particle number is conserved, though the
infection process (which is analogous to the birth process in the
Young/Martin model) can only occur in the spatial proximity of another infective. Likewise, recovery from infection (which is analogous to the death process in the Young/Martin model) can occur
at any point in space, independent of other individuals. For very
small values of L the random walk begin to equate to the addition
of a diffusion term to the dynamical model.
The determinant of whether a mass-action-based model will be
appropriate is the ratio of the length scales, d = L/l. If d > 2 then
mass-action will be applicable and the dynamics will be insensitive
to L, whereas for d < 2 the epidemic curve and the reproductive ratio
that is used to describe it will be dependent on L. In practice, the value of this ratio will be host-dependent. For animals that inhabit
home-ranges or defined territories, or are contained within farms
for example, they are likely to show d < 2. By contrast humans
undertake journeys for many reasons (to work, school, recreation
etc) so their mixing patterns are likely to show d > 1, hence the success of mass-action approaches in accounting for communicable disease epidemics, particularly in urban human populations [17,29,30].
2000
1500
5. Discussion
1000
The functional form of many epidemic models depends upon
mass-action assumptions. Here, a simple model of individuals
moving with a probabilistically distributed velocity components
in a two-dimensional domain, subject to an epidemic process, is
introduced in order to clarify disease transmission patterns in mobile spatially distributed populations. Specifically, for populations
that maintain a constant density, the form of mass-action is that
termed frequency dependent mass-action [7], whereas if a fixed
area with a fluctuating population is considered, density dependent mass-action is relevant. The model makes explicit the role
of particle movement and population density in determining the
basic reproductive ratio for pathogens. Moreover, it provides a kinetic sub-model for Begon et al.’s [11] clarification of the role of
population size, density and area in determining disease transmission. In practice it is probably as difficult to estimate host velocities
(and how these change with infectious status) as it is to estimate
the corresponding contact rate (b) in a given outbreak, and there
is no suggestion that should be attempted. Rather, the utility of
the approach described here is that provides an analytic insight
into how population dispersion and movement affect epidemic
dynamics.
The model explains why mass-action models work well in mobile populations that have a relatively uniform population density.
Despite the fact that real human populations (e.g. large urban centres) are distributed over geographic space, the action of host
movement results in a mixing that serves to generate mass-action-like behaviour. Mixing must be sufficient to ensure that the
local density of susceptibles in the vicinity of an infective reflects
the overall susceptible density. Problems may arise in the use of
mass-action assumptions where there are significant departures
from uniformity of population density in the region under
consideration.
500
0
0
1
2
3
4
5
6
7
8
9
6
7
8
9
Time (days)
d 2500
Number of infectives
61
2000
1500
1000
500
0
0
1
2
3
4
5
Time (days)
Fig. 4. Epidemic curves for varying step length L. (a) Simulation for L = 50 and the
epidemic curve for the SIR model with R0 = 15. (b) Simulation for L = 30 and the SIR
model with R0 = 15. (c) Simulation for L = 20 and the SIR model with R0 = 15. (d)
Simulation for L = 10 and the epidemic curve for the SIR model with R0 = 10. In each
case the dots are the simulation and the solid line is the homogeneous SIR model.
shorter than this the epidemic peak occurs progressively later and
the epidemic curve broadens, suggesting that the effective reproductive ratio is lower in this regime. For such short step lengths
there is a tendency to increased spatial correlation between infectives, so those who are infected do not experience the average
background density of susceptibles during their infective lifetime.
62
C.J. Rhodes, R.M. Anderson / Mathematical Biosciences 216 (2008) 56–62
In reality the movement of individuals across geographical
space is of course more complex than straight line trajectories.
Much of contemporary epidemiology is concerned with trying to
establish the rules (or equations of motion) that describe host
movement in a concise manner and using these to underpin epidemic models. However, the simple model presented here suggests
that mass-action-based models are (probably under most circumstances) capable of capturing epidemic dynamics in spatially distributed mobile populations. Specifically it has been shown that
the requirement for continuous straight line trajectories in the
model presented here can be relaxed and more complex trajectories introduced. Assuming that the individuals can make changes
to their direction of movement (with a uniform distribution for
the turn angle) the argument leading to Eq. (1) still holds and the
results presented here will still apply. In this case, the factor that
is the strongest determinant of epidemic dynamics will be population density. The specific way hosts move around will have much
less of an impact and basic homogeneous modelling will give a
good account of the epidemic dynamics. However, if the rate of
turning is sufficiently frequent there is the possibility that some
new contacts will be with previously contacted individuals, leading
to the development of significant spatial correlations between
infectives. In this case it is likely that mass-action may not emerge
from the model, and the specific pattern of movement (e.g. Levy
flight, Brownian) will influence the pattern of spread. Future work
will investigate this is more detail.
Of the terms that go into determining the basic reproductive ratio (Eq. (20)) the available points of intervention to reduce R0
(thereby encouraging the elimination of the disease) are the population density and the mean speed. Therefore, as noted by Haydon
et al. [30] vaccination strategies in both wildlife and human diseases must have a spatial perspective and should be aimed at
reducing susceptible densities, rather than simply focussing on
absolute numbers. Also, in relation to human diseases, mechanisms such as travel restrictions or workplace/school closures
. Alternatively, curfews that restrict populacould serve to reduce v
tion movement on certain days would also serve to reduce the
mean speed. In the case of new emergent infectious disease of humans, in the absence of a suitable vaccine, only travel restrictions
can form the basis for an intervention programme.
Acknowledgments
This work is supported by the Research Councils of the United
Kingdom and Imperial College London. The authors thank the
reviewers for their insightful comments on the original
manuscript.
Appendix A
The choice of density dependent mass-action or frequency
dependent mass-action influences how the reproductive ratio
relates to the population size. Take a simple susceptible-infective-recovered epidemic model (SIR) of a non-fatal communicable
disease that confers lifelong immunity following an infectious period of mean duration a1. The following equations describe the
time evolution using the density dependent mass-action
assumption:
dS
¼ bSI;
dt
dI
¼ bSI aI;
dt
dR
¼ aI:
dt
ðA:1Þ
ðA:2Þ
ðA:3Þ
The reproductive ratio R0 is defined as the number of secondary
infections caused by an infective in a largely susceptible population.
For the epidemic to materialise dI/dt > 0, leading to R0 = bN/a, where
N ffi S at the start of the epidemic. Consequently the reproductive
ratio scales with the population size. Repeating the calculation
using the frequency dependent mass-action assumption (i.e. with
bSI/N in Eqs. (A.1) and (A.2)) gives R0 = b/a, hence the reproductive
ratio is independent of the population size.
References
[1] R.M. Anderson, R.M. May, Infectious Diseases of Humans: Dynamics and
Control, Oxford University, Oxford, 1991.
[2] B.T. Grenfell, A.P. Dobson (Eds.), Ecology of Infectious Diseases in Natural
Populations, Publications of the Newton Institute, Cambridge University,
Cambridge, 1995.
[3] H. McCallum, N. Barlow, J. Hone, How should pathogen transmission be
modelled?, Trends Ecol Evol. 16 (2001) 295.
[4] M.J. Keeling, The effects of local spatial structure on epidemiological invasions,
Proc. R. Soc. B 266 (1999) 859.
[5] I.Z. Kiss, D.M. Green, R.R. Kao, The network of sheep movements within Great
Britain: network properties and their implication for disease spread, J. Roy. Soc.
Interface 3 (2006) 669.
[6] J. Truscott, T. Garske, I. Chis-Ster, J. Guitain, D. Pfeiffer, L. Snow, J. Wilesmith,
N.M. Ferguson, A. Ghani, Control of a highly pathogenic H5N1 avian flu
influenza outbreak in the GB poulty flock, Proc. Roy. Soc. B 274 (2007) 2287.
[7] M.C.M. de Jong, O. Diekmann, H. Heesterbeek, How does transmission of
infection depend on population size?, in: D. Mollison (Ed.), Epidemic Models:
Their Structure and Relation to Data, Publications of the Newton Institute,
Cambridge University, Cambridge, 1995.
[8] K. Dietz, Overall population patterns in the transmission cycle of infectious
agents, in: R.M. Anderson, R.M. May (Eds.), Population Biology of Infectious
Diseases, Springer, Berlin, 1982, p. 87.
[9] D. Schenzle, K. Dietz, Critical population sizes for endemic viral transmission,
in: W. Fricke, E. Hinz (Eds.), Raumliche Persistenz und Diffusion von
Krankheiten, Heidelberger geographische Arbeiten 83, 1987, p. 31.
[10] J. Antonovics, Y. Iwasa, M.P. Hassell, A generalised model of parasitoid,
venereal, and vector-based transmission processes, Am. Nat. 145 (1995) 661.
[11] M. Begon, M. Bennett, R.G. Bowers, N.P. French, S.M. Hazel, J. Turner, A
clarification of transmission terms in host–microparasite models: numbers,
densities and areas, Epidemiol. Infect. 129 (2002) 147.
[12] O. Diekmann, H. Heesterbeek, H. Metz, The legacy of Kermack and McKendrick,
in: D. Mollison (Ed.), Epidemic Models: Their Structure and Relation to Data,
Publications of the Newton Institute, Cambridge University, Cambridge, 1995.
[13] D.M. Lewis, T.J. Pedley, Planktonic contact rates in homogeneous isotropic
turbulence: theoretical predictions and kinematic simulations, J. Theor. Biol.
205 (2000) 377.
[14] J. Gerritsen, J.R. Strickler, Encounter probabilities and community structure in
zooplankton: a mathematical model, J. Fish. Res. Brd. Can. 34 (1977) 73.
[15] G.M. Viswanathan, S.V. Buldyrev, S. Havlin, M.G.E. Da Luz, E.P. Raposo, H.E.
Stanley, Optimising the success of random searches, Nature 401 (1999) 911.
[16] J.D. Murray, Mathematical Biology, second ed., Springer, Berlin, 1993.
[17] B.T. Grenfell, B.M. Bolker, A. Kleczkowski, Seasonality and extinction in chaotic
metapopulations, Proc. R. Soc. B 259 (1995) 97.
[18] R. Durrett, S.A. Levin, The importance of being discrete (and spatial), Theor.
Popul. Biol. 46 (1994) 363.
[19] R. Durrett, S.A. Levin, Stochastic spatial models: a user’s guide to ecological
applications, Philos. Trans. R. Soc. Lond. B343 (1994) 329.
[20] C.J. Rhodes, R.M. Anderson, Power laws governing epidemics in isolated
populations, Nature 381 (1996) 600.
[21] B.O. Koopman, The theory of search. I. Kinematic bases, Oper. Res. 4 (1956)
324.
[22] F.W. Sears, G.L. Salinger, Thermodynamics, Kinetic Theory and Statistical
Thermodynamics, third ed., Addison-Wesley, Reading, 1975.
[23] E. Jahnke, F. Emde, Tables of Functions: With Formulae and Curves, Dover
Publications, 1986.
[24] R.P.D. Atkinson, C.J. Rhodes, D.W. Macdonald, R.M. Anderson, Scale-free
dynamics in the movement patterns of jackals, Oikos 98 (2002) 134.
[25] T. Brockmann, L. Hufnagel, T. Geisel, The scaling laws of human travel, Nature
439 (2006) 462.
[26] A.M. Reynolds, Co-operative random Levy flight searches and the flight pattern
of honey bees, Phys. Lett. A354 (2006) 384.
[27] M.C. Gonzalez, C.A. Hidalgo, A.-L. Barabasi, Understanding individual human
mobility patterns, Nature 453 (2008) 779.
[28] W.R. Young, A.J. Roberts, G. Stuhne, Reproductive pair correlations and
clustering of organisms, Nature 412 (2001) 328.
[29] A.P. Martin, A Malthusian curb on spatial structure in micro-organism
populations, J. Theor. Biol. 230 (2004) 343.
[30] D.T. Haydon, D.A. Randall, L. Matthews, D.L. Knobel, L.A. Tallents, M.B.
Gravenor, S.D. Williams, J.P. Pollinger, S. Cleaveland, M.E.J. Woolhouse, C.
Sillero-Zubiri, J. Marino, D.W. Macdonald, M.K. Laurensen, Low-coverage
vaccination strategies for the conservation of endangered species, Nature
443 (2006) 692.