Download Models of Infectious Diseases in Spatially Heterogeneous

Bulletin of Mathematical Biology (2001) 63, 547-571
doi:10.1006/bulm.2001.0231
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Models of Infectious Diseases in Spatially Heterogeneous
Environments
DIEGO J. RODRÍGUEZ∗ AND LOURDES TORRES-SORANDO
Instituto de Zoologı́a Tropical,
Universidad Central de Venezuela,
Apartado 47058, Caracas 1041-A,
Venezuela
E-mail: [email protected]
Most models of dynamics of infectious diseases have assumed homogeneous mixing in the host population. However, it is increasingly recognized that heterogeneity can arise through many processes. It is then important to consider the existence
of subpopulations of hosts, and that the contact rate within subpopulations is different than that between subpopulations. We study models with hosts distributed in
subpopulations as a consequence of spatial partitioning. Two types of models are
considered. In the first one there is direct transmission. The second one is a model
of dynamics of a mosquito-borne disease, with indirect transmission, and applicable to malaria. The contact between subpopulations is achieved through the visits
of hosts. Two types of visit are considered: one in which the visit time is independent of the distance travelled, and a second one in which visit time decreases with
distance. There are two types of spatial arrangement: one dimensional, and two
dimensional. Conditions for the establishment of the disease are obtained. Results
indicate that the disease becomes established with greater difficulty when the degree of spatial partition increases, and when visit time decreases. In addition, when
visit time decreases with distance, the establishment of the disease is more difficult
when the spatial arrangement is one dimensional than when it is two dimensional.
The results indicate the importance of knowing the spatial distribution and mobility
patterns to understand the dynamics of infectious diseases. The consequences of
these results for the design of public health policies are discussed.
c 2001 Society for Mathematical Biology
1.
I NTRODUCTION
The majority of models of the dynamics of infectious diseases have assumed
the existence of populations with homogeneous mixing (Anderson and May, 1992;
Grenfell and Dobson, 1995). These models consider that all individuals have the
same contact rate with each other. However, it has been increasingly recognized
that this is not the case in many real populations. For example, in some venereal
diseases humans within some social groups have higher contact rates than those
∗ Author to whom correspondence should be addressed.
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c 2001 Society for Mathematical Biology
548
D. J. Rodrı́guez and L. Torres-Sorando
between groups (Lajmanovich and Yorke, 1976), in malaria contact rates can be
age dependent (Dietz, 1988), and in measles hosts within a locality and age group
have more contact than those among localities or age groups (Sattenspiel and Dietz,
1995). It is therefore important to develop mathematical models which consider
such heterogeneities.
In ecological studies there have been important developments in the last two
decades as a consequence of the recognition that the spatial distribution of natural
populations is not homogeneous, and consists of subpopulations. These developments have given origin to the theory of metapopulations (Gilpin and Hanski,
1991; Hanski and Gilpin, 1997; Hanski, 1999), and a large body of mathematical
models which take into account such spatial heterogeneity (Tilman and Kareiva,
1997; Bascompté and Sole, 1998). Elements of this theory have been applied to
the study of infectious diseases (Grenfell and Harwood, 1997; Hess, 1996; Thrall
and Burdon, 1997)
In recent years some authors have developed models of dynamics of infectious
diseases in which hosts are divided into subpopulations. The majority of these
studies have concluded that with heterogeneity the conditions for the establishment
of the disease are different from those when there is homogeneous mixing (Lajmanovich and Yorke, 1976; Hethcote, 1978; Nold, 1980; Anderson and May, 1992;
Hethcote and Van Ark, 1987; Hasibeder and Dye, 1988; Sattenspiel and Simon,
1988; Becker and Dietz, 1995; Sattenspiel and Dietz, 1995). In the present work we
study models in which the spatial distribution of hosts is partitioned, giving origin
to subpopulations. The contact rates between groups depend on the mobility patterns of individuals. Accordingly, the larger the mobility the higher the contact rate
between any two subpopulations will be . Two types of diseases are studied. In one
the transmission is direct, and in the other there is indirect transmission exemplified
with malaria. In the latter case it is considered that only humans, and not the vector,
can move between localities. We find conditions for the establishment for various
regimes of host mobility, and the level and type of partition of the environment.
In a previous work (Torres-Sorando and Rodrı́guez, 1997) we analysed a model
of malaria in an environment divided in two patches, and found conditions for the
establishment of the disease. In the present article we generalize these results.
Studies of this type in malaria are urgently needed. Malaria is a very important infectious disease. It was eradicated in the 1960s in many parts of the world,
but it reappeared and today is considered as an emergent (Levins et al., 1994) or
resurgent (Gratz, 1999) disease. About 1 to 3 million persons die each year from
malaria, most of them infants or very young children, approximately 300 million
people are infected with one or more species of parasite, and more than 2 billion
(nearly 40% of the world’s population) live in malaria endemic parts of the world
(Collins and Paskewitz, 1995). In addition, some authors have stressed the importance of human geographical distribution and migration patterns for the understanding of the epidemiology of malaria (Bailey, 1982; Sifontes, 1985; Prothero,
1991; Sandia-Mago, 1994).
Infectious Diseases in Heterogeneous Environments
2.
2.1.
549
T HE M ODELS
Spatially homogeneous environment.
2.1.1. Direct transmission. Let us assume a disease with direct transmission.
N is the total number of humans, X (t) is the number of infected individuals at time
t, g is per capita rate of recovery such that 1/g is the duration of the disease, and b
is the transmission rate per susceptible individual and per infected individual. Then
d X (t)
= b(N − X (t))X (t) − g X (t).
dt
(1)
This equation corresponds to an SIS model.
2.1.2. Indirect transmission: the malaria model of Ross–Macdonald. Now
consider the indirect transmission of malaria. The classic model of malaria dynamics, after the work of Ross (1911), Macdonald (1957), and Aron and May (1982)
takes the following form:
α
β (N − X (t))Y (t) − g X (t)
N
α
dY (t)
=
γ (M − Y (t))X (t) − mY (t),
dt
N
d X (t)
=
dt
where M is the total number of mosquitoes, Y (t) is the number of infected
mosquitoes at time t, and m is per capita death rate of mosquitoes. α is the number
of hosts bitten by a mosquito per unit time, β is the proportion of infectious bites
on humans that produce infection, and γ is the proportion of bites of susceptible
mosquitoes on infected humans that produce an infection.
The previous formulation assumes that the number of bites per host is proportional to the density of mosquitoes. This is a very important assumption that very
few workers have tried to test. Figure 1 presents experimental evidence which
supports that assumption. The data in Fig. 1 have been taken from Rubio-Palis
et al. (1992), who performed experiments with two human baits exposed for one
night at each one of three locations. In each location the number of mosquitoes
belonging specifically to three species, Anopheles nuneztovari, An. albitarsis and
An. oswaldoi, were recorded. The number of bites by each species for each person
was also recorded over a 12 hour period. Figure 1 strongly suggests that the above
assumption of proportionality is a logical one.
As a mosquito has a limited capacity of biting humans it could be hypothesized
that α depends on N as a type II functional response, as shown in Fig. 2. This
type of dependence is observed in the functional response of a predator to the
density of a prey when the predator needs a time period, termed the handling time,
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D. J. Rodrı́guez and L. Torres-Sorando
No. bites/person/night
450
(a)
400
350
nuneztovari
300
250
albitarsis
200
oswaldoi
150
100
50
0
0
5000
10000
15000
20000
25000
Mosquito density
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(b)
No. bites/person/night
30
25
20
albitarsis
15
oswaldoi
10
5
0
0
500
1000
1500
2000
2500
Mosquito density
Figure 1. The number of bites per person as a function of mosquito density. In (a) the data
of the three mosquito species is shown. In (b) we show, at a smaller scale, the data for the
two mosquito species with lower biting rates. Data taken from Rubio-Palis et al. (1992).
to deal with each prey (Begon et al., 1996). As the prey density increases the
capture of prey also increases but approaches a plateau, due to saturation of the
time unit with the handling of captured prey. It is possible that if the time needed
to process a blood meal is equivalent to a handling time, the curve of Fig. 2 really
occurs.
If the number of humans relative to the number of mosquitoes is low, which
according to the data of Rubio-Palis et al. (1992) seems to be a logical assumption,
we can consider that α is proportional to N , that is to say α ≈ δ N , with δ constant.
Then (α/N )β ≈ δβ, and (α/N )γ ≈ δγ . For simplicity it is assumed that β = γ ,
551
Rate of humans bitten per
mosquito, α
Infectious Diseases in Heterogeneous Environments
No. of humans, N
Figure 2. Hypothetical relationship between the rate of humans bitten by a mosquito, α,
and the number of hosts, N .
then δβ = δγ = b. We then arrive at
d X (t)
= b(N − X (t))Y (t) − g X (t)
dt
dY (t)
= b(M − Y (t))X (t) − mY (t).
dt
(2)
There is no immunity in our models. We also assume that changes in the total
density of humans or mosquitoes are negligible. This means that either per capita
natality and mortality rates are equal or these rates are extremely low in comparison
with the rates of contagion and recovery of the disease.
2.2. Spatially heterogeneous environment. Now we consider that the habitat is
partitioned in k localities. Ni (t) and X i (t) are the numbers of humans and infected
humans, respectively, at time t in locality i(i = 1, . . . , k). In the malaria model
the parameters Mi (t) and Yi (t) are also included which represent the number of
mosquitoes and the number of infected mosquitoes, respectively, at time t in locality i(i = 1, . . . , k).
(a) Patterns of contact between localities. We consider three patterns of contact
between localities.
(a.1) No contact between localities.
Directly transmitted disease
Note that Ni (t) = Ni , so we have
d X i (t)
= b(Ni − X i (t))X i (t) − g X i (t),
dt
with i = 1, . . . , k.
(3)
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D. J. Rodrı́guez and L. Torres-Sorando
Malaria
For this model, given that Ni (t) = Ni and Mi (t) = Mi , we have
d X i (t)
= b(Ni − X i (t))Yi (t) − g X i (t)
dt
dYi (t)
= b(Mi − Yi (t))X i (t) − mYi (t),
dt
(4)
with i = 1, . . . , k.
(a.2) Visitation between localities. Now we assume that a fraction vi j of the time
devoted by humans to reside in locality i per unit time, is devoted to visit locality
j (i = j; i, j = 1, . . . , k). After the visit these humans return to their locality of
origin. Let V = {vi j }, with vii = 1.
We are then assuming that hosts have a constant residence time, during which
they stay in their locality of origin. During the rest of the time unit there are
visitations to other localities. But total visitation time need not be a constant. It
is just the sum of visiting times, each one of which depends on the distance of
the locality being visited from the locality of origin. This scenario is applicable to
human hosts, who are able to plan their time schedule.
Directly transmitted disease
Note that in this case Ni (t) = Ni . Then the equations are
d X i (t)
= b(Ni − X i (t))X i (t) − g X i (t)
dt
+ b(Ni − X i (t))
v ji X j (t) + b(Ni − X i (t))
vi j X j (t)
j=i
(5)
j=i
with i = 1, . . . , k.
Malaria
Note that in the case of malaria also Mi (t) = Mi . Then, given that humans can
travel but mosquitoes cannot, the equations are
d X i (t)
vi j Y j (t)
= b(Ni − X i (t))Yi (t) − g X i (t) + b(Ni − X i (t))
dt
j=i
dYi (t)
v ji X j (t),
= b(Mi − Yi (t))X i (t) − mYi (t) + b(Mi − Yi (t))
dt
j=i
(6)
with i = 1, . . . , k.
(b) Patterns of spatial array. Two patterns of spatial array were considered. The
first is a unidimensional one [Fig. 3(a)], in which the k localities are arranged as
Infectious Diseases in Heterogeneous Environments
553
(a)
1
2
3
4
5
(b)
1
2
3
4
5
6
7
8
9
Figure 3. (a) Unidimensional array with k = 5. (b) Bidimensional array with k = 9. Each
locality is denoted by a number. The arrows indicate the direction of the movement of
individuals from locality 2 to other localities.
cells in a row. This array can simulate the spatial distribution of localities along a
coast. The second array is a bidimensional one [Fig. 3(b)], in which the k localities
are arranged in a rectangle whose four sides are of the same length or number of
cells. This bidimensional array can simulate a general spatial distribution on a land
surface. The cells are numbered as exemplified in Fig. 3.
(c) Patterns of distance effects on contact between localities.
(c.1) No effect of distance. In this case vi j = v for visitation (i = j; i, j =
1, . . . , k). This means that the magnitude of contact between localities is constant,
no matter how far apart they are.
(c.2) The contact between localities decreases with distance. Assume that v
is the probability of visiting a locality one distance unit away per unit time, and
vi j = v ei j , with ei j being the number of distance units between localities i and j
(i = j; i, j = 1, . . . , k).
There is a matrix E = {ei j }. For the unidimensional case
ei j =
(i − j)2 .
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D. J. Rodrı́guez and L. Torres-Sorando
[See Fig. 1(a)]. For example, for k = 5 we have
0
1

E = 2

3
4
1
0
1
2
3
2
1
0
1
2
3
2
1
0
1
4
3

2.

1
0
For the bidimensional case the elements of the matrix E have
√ for
√ to be obtained
each k. For example for k = 9, e12 = 1, e13 = 2, e15 = 12 + 12 = 2 [See
Fig. 1(b)], and we can write the following:

E1
E =  E2
E3
E2
E1
E2

E3
E2  ,
E1
where

√0

E1 = √1
4
√
√4

E3 = √5
8
√ √ 
1 √4
1,
√0
1 0
√ √ 
√5 √8

√4 √5 .
5
4
√
√1

E2 = √2
5
√
√2
√1
2
√ 
√5

√2 ,
1
and
Obviously we are not considering edge effects which can arise in a grid with a
finite number of patches.
3.
A NALYSES
It is assumed that in heterogeneous habitats the total number of individuals (humans and mosquitoes) are initially evenly distributed. Note also that the patterns
of movement between localities we consider do not produce any net change in the
number of individuals per locality. So the number of humans per locality will be
N /k, and that of mosquitoes M/k. Then the models can be written as follows. For
no contact between localities, the directly transmitted disease model becomes
d X i (t)
N
=b
− X i (t) X i (t) − g X i (t).
dt
k
(7)
In equation (7) we are assuming that the number of new infectious individuals
appearing per unit time is proportional to the number of infected individuals and
Infectious Diseases in Heterogeneous Environments
555
also to the number of susceptible individuals. The malaria model becomes
d X i (t)
N
=b
− X i (t) Yi (t) − g X i (t)
dt
k
dYi (t)
M
=b
− Yi (t) X i (t) − mYi (t),
dt
k
(8)
with i = 1, . . . , k.
For the case of visitation, the expression for the directly transmitted disease becomes
N
d X i (t)
N
=b
− X i (t) X i (t) − g X i (t) + b
− X i (t)
(vi j + v ji )X j (t).
dt
k
k
j=i
(9)
In (9) we are assuming that individuals from patch i visiting patch j only have
contact with individuals from j, and not with other individuals which are visiting
j and come from other patches. This could be the case for certain sexually transmitted diseases in which travelling individuals only contact resident individuals in
other patches. And for the malaria model the equations are
d X i (t)
N
N
=b
− X i (t) Yi (t) − g X i (t) + b
− X i (t)
vi j Y j (t)
dt
k
k
j=i
M
dYi (t)
M
v ji X j (t), (10)
=b
− Yi (t) X i (t) − mYi (t) + b
− Yi (t)
dt
k
k
j=i
with i = 1, . . . , k.
For all the models the conditions for invasibility of the disease were obtained
with the eigenvalues of the jacobian matrix of the differential equations system
evaluated at the trivial equilibrium which corresponds to the absence of the disease.
If the largest eigenvalue of such a jacobian matrix has a positive real part then the
disease can invade. Otherwise it will not become established.
4.
R ESULTS
The analyses to find the conditions for invasibility in the models are detailed in
the Appendix. For all the models it was possible to find analytically a mathematical
expression for such a condition, except for the case of visitation with the effects
of distance acting on the mobility patterns. In this case, however, it was possible
to show that
√in the parameter space of bN /g vs k for the directly transmitted disease, and b N M/(gm) vs k for malaria, there is a curve that separates the regions
556
D. J. Rodrı́guez and L. Torres-Sorando
Table 1. Conditions for the disease to become established in the directly transmitted disease
models.
Model
Condition
Homogeneous environment
b Ng > 1
Heterogeneous environment without
connection
b Ng > k
Heterogeneous environment with
visitation and no effect of distance
k
b Ng > 1+2(k−1)ν
Heterogeneous environment with
visitation and effect of distance
b Ng > h(k, ν)
b Ng > k(1−ν)
1+3ν
(k 1, one dimen.)
of invasibility and non-invasibility. Such a curve was found numerically, and an
analytical approximation to this curve was also found.
Table 1 shows the conditions for invasibility for the models with direct transmission, and Table 2 those for the models of malaria. Figures 4 and 5 show graphically these conditions. Each one of the curves in Figs 4 and 5 was obtained in
the following way. For a given pair of values of v and k, we found numerically
the combination of the other parameters for which the dominant eigenvalue of the
jacobian matrix at the trivial equilibrium was closest to zero. Given a value of v,
for small values of k the time spent by a human visiting other localities per unit
time is smaller than the time spent in his locality per unit time. These situations are
biologically reasonable, and are denoted by solid circles. By increasing the value
of k we reach a point where total visitation time per time unit in humans is larger
than the time of residence in the locality of origin per unit time. These last situations are denoted with dashes, and, although they cannot be considered biologically
possible, are shown in order to better understand the mathematical behavior of the
curves.
As expected, the curves raise with k. That is, the establishment of the disease is
more restricted as the environment is more partitioned. This is due to the decrease
in the effective density for contagion as the degree of partition increases. It is also
observed that, when there is a distance effect on the visit time, the conditions for
establishment are more restricted in the one-dimensional case than in that with
two dimensions. This is because, other things being equal, for a given number of
localities a two-dimensional array implies a greater degree of contact than a onedimensional array.
All curves increase with k approaching a plateau. This implies that the effectiveness of a control policy which restricts the movements between portions of a host
population will also reach a plateau, beyond which new increases in the degree of
partition will have a negligible effect.
Infectious Diseases in Heterogeneous Environments
557
Table 2. Conditions for the disease to become established in the malaria models.
Model
Condition
M >1
b Ngm
Ross–Macdonald
(Homogeneous environment)
Heterogeneous environment without
connection
b
Heterogeneous environment with
visitation and no effect of distance
b
NM >
k
gm
1+(k−1)ν
Heterogeneous environment with
visitation and effect of distance
b
b
NM > k
gm
N M > h(k, ν)
gm
N M > k(1−ν)
gm
1+ν
(k 1, one dim.)
Figures 4 and 5 also indicate that the approximations of formulas (A18) and (A24),
although supposedly only good for large values of k, work well even for small values of the number of localities. We can say that these approximations work for
models of an environment with at least five localities.
5.
D ISCUSSION
In the direct transmission model we have assumed that b is a parameter. This
means that the rate of encounters of an individual is proportional to the density. Our
results depend critically on this assumption.
Although authors such as
Hethcote and Van Ark (1987) cite empirical data which throw doubts on the validity of the above assumption, in the epidemiological literature there are many experimental studies [references in Anderson and May (1992)] which indicate than
our assumption is a logical one.
In the model of malaria our results also depend on the assumption that b is a
constant, as a consequence of assuming that α is proportional to N . To see this
consider an extreme alternative scenario in which α is a constant independent of N .
In this case it can be shown that curves equivalent to those of Fig. 5 are dramatically
changed, and that the smaller k is, the higher the quotient M/N has to be to allow
for the establishment of the disease.
A fundamental assumption of the model of malaria is that the number of bites
per host is proportional to the density of mosquitoes. Figure 1 gives evidence for
the validity of this assumption. An interesting point arising from Fig. 1 is that the
data of the three mosquito species seems to fall on the same straight line. This
means that a mechanism common to the three species governs the dependence on
the number of bites on the mosquito density. Another important assumption is that
558
D. J. Rodrı́guez and L. Torres-Sorando
ONE DIMENSION
TWO DIMENSIONS
bN/g
bN/g
No Distance effect
40
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15
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0
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0
No Distance effect
0
0.01
0.05
0.1
0.3
1
0
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10
15
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30
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40
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0
0.01
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0.1
0.3
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0
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0.1
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1
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Distance effect
40
0
0.01
0.5
0.1
0.3
1
0
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25
30
35
40
Number of localities
Figure 4. Families of parameters for which the directly transmitted disease can be established. Invasion of the disease is possible for parameter values over the specific curve. At
the right-most extreme of each curve there is a number which indicates the value of the
basic visitation time unit fraction, v. The part of the curve with solid circles corresponds
to biologically reasonable situations in which the sum of all the time an individual devotes
to visitation is smaller than the residence time. The rest of the curve, marked with dashed
lines, corresponds to biologically unreasonable situations, where the total visitation time is
higher than the residence time. This last part is shown in order to provide an understanding of the mathematical behavior of the curve. In one dimension with the distance effect
applied the solid lines correspond to the approximation (A18).
α depends linearly on N . This, of course, is only valid for low enough numbers of
humans per mosquito.
The study of the dynamics of infectious diseases with heterogeneous contact
rates as they arise, for example, with the presence of a partitioned environment,
has been addressed by many authors, the majority of whom have analysed deterministic models. One of the first works is that of Rushton and Mautner (1955),
who studied a system with direct transmission and identical subpopulations, and
could solve analytically the differential equation systems for an initial condition
where only one of the localities has infected individuals. Other approaches have
studied stochastic models (Watson, 1972; Becker et al., 1995; Becker and Dietz,
1995; Becker and Hall, 1996; Becker and Starczak, 1997; Ball, 1999).
Our study of the conditions for the establishment of the disease is based on the
calculation of the dominant eigenvalue of the jacobian matrix of the dynamical
system evaluated at the trivial equilibrium. This method, although not specifically stated in some cases, has been used by previous authors with models of heterogeneous contact rates with direct transmission (Lajmanovich and Yorke, 1976;
Hethcote, 1978; Post et al., 1983; May and Anderson, 1984; Hethcote and Thieme,
1985; Travis and Lenhart, 1987; Hethcote and Van Ark, 1987; Sattenspiel and
Infectious Diseases in Heterogeneous Environments
ONE DIMENSION
TWO DIMENSIONS
b[NM/(gm)]1/2
No Distance effect
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0
No Distance effect
0
0.01
0.05
0.1
0.3
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5
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25
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0
0
0.01
0.05
0.1
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1
0
5
10
b[NM/(gm)]1/2
Distance effect
0
0.01
0.05
0.1
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559
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0
0
0.01
0.05
0.1
0.3
1
0
5
10
15
20
25
30
35
40
Number of localities
Figure 5. Families of parameters for which malaria can be established. Invasion of malaria
is possible for parameter values above the specific curve. At the right-most extreme of
each curve there is a number which indicates the value of the basic unit fraction visitation
time, v. The part of the curve with solid circles corresponds to biologically reasonable
situations in which the sum of all the time an individual devotes to visitation is smaller than
the residence time. The rest of the curve, with dashed lines, corresponds to biologically
unreasonable situations, where the total visitation time is higher than the residence time.
This last part is shown to provide a better understanding of the mathematical behavior of
the curve. In one dimension with the distance effect applied the solid lines correspond to
the approximation (A24).
Simon 1988, Andreasen and Christiansen, 1989). Some of them also perform a
rigorous mathematical analysis to prove that a positive real part of that dominant
eigenvalue also insures the existence of a unique nontrivial equilibrium. In our
models, for cases without the distance effect in the movements between localities
it is very simple to show this. For cases with the distance effect it is much more
difficult to make this analysis. But, given the qualitative characteristics of these
systems, it is very likely that invasibility in them also insures the existence of a
unique nontrivial equilibrium.
Some models have had particularly successful applications. For example Rvachev
and Longini (1985), Longini et al. (1986) and Longini (1988) have developed a
simulation model of the geographical spread of influenza that nicely predicts the
existing data.
May and Anderson (1984) found that, when there are heterogeneous contact
rates, an optimal policy implies the immunization of a fraction of the population
lower than that to immunize when, other things being equal, there is homogeneous
mixing. Travis and Lenhart (1987) have applied this result to discuss immunization
policies for smallpox. Our results also imply that with homogeneous mixing, that
is no partition of the environment, the fraction of susceptible hosts to immunize
in order to avoid establishment of the disease will always be higher than that with
560
D. J. Rodrı́guez and L. Torres-Sorando
heterogeneous contact rates, that is with any level of partition.
Other authors have given methods to calculate the basic rate of reproduction of
the disease when hosts are distributed in subpopulations (Diekmann et al., 1990;
De Jong et al., 1994).
Dye and Hasibeder (1986) and Hasibeder and Dye (1988) have proposed a model
of mosquito-borne disease dynamics, where humans and mosquitoes have a patchy
distribution. They found that the patchiness of human distribution, but not that
of the mosquito, increases the likelihood of establishment of the disease. At least
one of two factors in the model given by these authors can explain the difference
between their results and ours. First the patches of humans in their model do not
coincide with those of mosquitoes, and there can be preferences of mosquitoes of
one patch for humans in another specific patch. Secondly, the value of α in their
model is constant, so a smaller number of humans as a consequence of patchiness
can increase the contagion.
Some models with heterogeneous contact rates consider specific patterns of contact, applicable to cases of hosts with a patchy distribution which move between
patches with total (Sattenspiel and Simon, 1988) or partial (Sattenspiel and
Dietz 1995) return per time unit. The model of Sattenspiel and Dietz (1995) has
been useful to aid the understanding of the spatial spread of influenza in aboriginal communities of Central Canada during the world epidemic of 1918 and 1919
(Sattenspiel and Herring, 1998,; Sattenspiel et al., 2000). Our pattern of movement
within the hosts has total return. This would be applicable to hosts with fidelity to
their living patch, for example, humans that live in specific localities, or animals
with a territory that spend some time of the day foraging in other patches.
In the malaria model with distance effects, when there is a one-dimensional spatial array the disease establishes with more difficulty than in the case with a twodimensional array. This could explain why in Venezuela the incidence of malaria
in central regions is higher than in the coastal regions (Torres-Sorando, 1998).
The number of parameters in the models we have studied has been kept to a
minimum. In addition, all the parameters are easy to interpret from a biological
point of view, and also to estimate with common ecological methods. This makes
the models easy to apply.
Our results clearly indicate that the probability of invasion of a disease is sensitive to variation in the pattern of mobility, and suggest that it is important to gather
information on such patterns. With information like that in Figs 4 and 5 it can be
possible to know the degree of mobility restriction and the degree of fragmentation
to apply in order to avoid establishment of the disease.
ACKNOWLEDGEMENTS
We are grateful to J Jiménez, M G Basáñez, R De Nóbrega, M E Grillet and
Y Rangel for helpful comments, to L Sattenspiel and K Dietz for providing us
Infectious Diseases in Heterogeneous Environments
561
with useful literature, and to Y Rosal for technical assistance. L TorresSorando thanks Fundayacucho for a doctoral fellowship. We thank Consejo de
Desarrollo Cientı́fico y Humanı́stico of Universidad Central de Venezuela, whose
Grants 03.31.3473.95 and 03.31.4336.99 to D J Rodrı́guez supported the work.
A.
A PPENDIX
A.1. Spatially homogeneous environment.
Directly transmitted disease. Equation (1) can be written as follows
dX
= F(X ).
dt
(A1)
The trivial equilibrium of (A1) is X ∗ = 0. The disease can be established when
∂F
> 0,
∂ X X =0
which translates to
bN
> 0.
g
(A2)
Malaria
System (2) can be written as follows:
dX
= F(X, Y )
dt
dY
= G(X, Y ).
dt
(A3)
This system has two equilibria, which are
∗ X1
0
=
,
∗
0
Y1
and
X 2∗
Y2∗
=
(b2 N M − gm)/[b(Mb + g)]
(b2 N M − gm)/[b(N b + m)]
.
For the disease to become established we need the trivial equilibrium to be unstable. This is equivalent to saying that the jacobian matrix of (A3) evaluated at such
equilibrium has its largest eigenvalue of positive real part. That matrix is


∂F
∂F
∂X
∂Y 

A=
,
∂G
∂G 
0
∂X
∂Y
0
562
D. J. Rodrı́guez and L. Torres-Sorando
and its eigenvalues are roots of the characteristic equation
|A − λI| = 0.
(A4)
The two roots are always real. The dominant eigenvalue is
−(g + m) +
(g + m)2 − 4(gm − b2 N M)
,
2
which will be positive when
(A5)
b
NM
> 1.
gm
(A6)
It can also be shown that when the trivial equilibrium is unstable the nontrivial one
is stable, and vice versa.
A.2. Spatially heterogeneous environments. Before analysing the invasibility
conditions of the models with spatial heterogeneity we will establish some fundamental results of matrix algebra.
T HEOREM A.1. Let Z be a square matrix such that
Z=
Z1
Z3
Z2
Z4
,
where Z1 and Z4 are square matrices. If Z1 is nonsingular then
|Z| = |Z1 ||Z4 − Z3 Z−1
1 Z2 |.
Also, if Z4 is nonsingular then
|Z| = |Z4 ||Z1 − Z2 Z−1
4 Z3 |.
The proof of this theorem is very simple, and can be found in Searle (1982).
T HEOREM A.2. Let D = {di j } be a square matrix of order k, such that di j = d1
if i = j, and di j = d2 if i = j. Then
|D| = (d1 − d2 )k−1 (d1 + (k − 1)d2 ).
To prove this theorem we use induction. For k = 2 we have
|D| = d12 − d22 = (d1 − d2 )(d1 + d2 ).
Infectious Diseases in Heterogeneous Environments
563
So the theorem is true for k = 2. Now let us assume that the theorem is true for a
particular k = n. For k = n + 1 the matrix D can be partitioned as follows:

d1
  d2

 ·

d2
( d2
d  
2
 d2  
  
.
·

d2 n
( d1 )
d2 · d2 
d1 · d2 

·
·
d2 · d1 nxn
d2 · d2 )n
Now applying Theorem A.1 to evaluate the determinant of the previous matrix,
and given that we assumed the theorem to be true for k = n, such a determinant is
equal to
(d1 − d2 )k (d1 + kd2 ).
And this indicates that the theorem is true for k = n + 1, which completes the
proof.
T HEOREM A.3. Let H be a square matrix of order l, being l even, such that
H=
P Q
R S
.
Submatrices P, Q, R and S are square of order l/2. Their elements are pi j = p if
i = j, and 0 if i = j; qi j = q1 if i = j, and q2 if i = j; ri j = r1 if i = j, and r2 if
i = j; si j = s if i = j, and 0 if i = j. Then
|H| = (sp − (r1 − r2 )(q1 − q2 ))
l
2 −1
sp − (r1 − r2 )(q1 − q2 )
l2
l
+ (r2 q1 + r1 q2 − 2r2 q2 ) + r2 q2
2
4
.
To prove this theorem we first apply Theorem A.1, which says that
|H| = |S||P − QS−1 R|,
and then evaluate this determinant applying Theorem A.2.
A.2.1.
No contact between localities.
Directly transmitted disease
System (9) can be written as follows:
d Xi
= Fi (X 1 , X 2 , . . . , X k ),
dt
(A7)
564
D. J. Rodrı́guez and L. Torres-Sorando
with i = 1, . . . , k. The trivial equilibrium is
 X∗ 
1
0
 ·  ·
  

e =  ·  =  · ,
  

·
·
∗
Xk
0
which is a vector of k entries. The disease will become established if the largest
eigenvalue of the matrix B = {bi j }, with bi j = ∂ Fi /∂ X j , evaluated at e, has a
positive real part. This means that there is invasibility if the largest root of the
characteristic equation
|B − λI| = 0
(A8)
has a positive real part. It is easily shown that the k eigenvalues are equal to
So the invasibility condition is
bN
> k.
g
bN
k
− g.
(A9)
Malaria
System (10) can be written as follows:
d Xi
= Fi (X 1 , X 2 , . . . , X k , Y1 , Y2 , . . . , Yk )
dt
dYi
= G i (X 1 , X 2 , . . . , X k , Y1 , Y2 , . . . , Yk ),
dt
(A10)
with i = 1, . . . , k. The trivial equilibrium is
  
0
X 1∗
 ·  ·
  

 ·  ·
  

 ·  ·
 ∗  
 X  0
k 
 
t=
 Y∗  = 0,
 1   
 ·  ·
  

 ·  ·
  

 ·  ·
0
Yk∗

which is a vector of 2k entries. The disease becomes established when the largest
eigenvalue of the jacobian matrix
C=
P Q
R S
t
Infectious Diseases in Heterogeneous Environments
565
with pi j = ∂ Fi /∂ X j , qi j = ∂ Fi /∂Y j , ri j = ∂G i /∂ X j , and si j = ∂G i /∂Y j , has a
positive real part, that is to say if the largest root of the characteristic equation
|C − λI| = 0
(A11)
has a positive real part. Evaluating C and applying Theorem A.1 we can say
that (A11) is equivalent to
b2 N M
(λ + g)(λ + m) −
k2
k
= 0.
Then the k eigenvalues of C come in k equal pairs. In each pair the two eigenvalues
are real and the largest one is
−(g + m) +
(g + m)2 − 4 gm −
2
which will be positive if
b2 N M
k2
,
(A12)
b
NM
> k,
gm
(A13)
and this is the condition for invasibility.
A.2.2.
Visitation.
A directly transmitted disease without the distance effect
System (9) with vi j = v (i = j; i, j = 1, . . . , k) can be written as (A7), and
applying the same steps starting with (A7), we evaluate the jacobian matrix at the
trivial equilibrium B. Applying Theorem A.2 it can be shown that the roots of the
characteristic equation are solutions of
2bN
bN
−g−λ−
v
k
k
k−1 2bN
bN
− g − λ + (k − 1)
v = 0.
k
k
Then the largest eigenvalue of B is (bN /k)[2(k − 1)v + 1] − g, and the condition
for invasibility is
bN
k
>
.
(A14)
g
1 + 2(k − 1)v
Directly transmitted disease with the distance effect
System (9) with vi j = v ei j (i = j; i, j = 1, . . . , k) can be written as (A7), in
which case the jacobian matrix at the trivial equilibrium, given that V is symmetric,
becomes
bN
B=
(2V − I) − gI.
(A15)
k
566
D. J. Rodrı́guez and L. Torres-Sorando
It follows that if ρ is an eigenvalue of V, then (bN /k)(2ρ − 1)− g is an eigenvalue
of B. And if ρ is the largest eigenvalue of V, then (bN /k)(2ρ − 1)−g is the largest
eigenvalue of B. As ρ is going to depend on v and k, let f (k, v) = k/(2ρ − 1).
Then the condition for invasibility is
bN
> f (k, v).
g
(A16)
It was not possible to find analytically the eigenvalues of V when there are distance
effects. As a consequence the explicit form of f (k, v) could not be found. Then
the frontier of the graph bN /g vs k over which the invasion occurs was found
numerically, and is shown in Fig. 4.
However, for the case of one dimension it is possible to find an approximation to
f (k, v) for large values of k. For one dimension

1
 v
 2
 v
 3
 v
V=
 ·

 ·

 ·
v k−1
v
1
v
v2
·
·
·
v k−2
v2
v
1
v
·
·
·
v k−3
v3
v2
v
1
·
·
·
v k−4
·
·
·
·
·
·
·
·
·
·
·
·
· · ·

v k−1
v k−2 

v k−3 

v k−4 
.
· 

· 

· 
1
(A17)
Working on the results of Berlin and Kac (1952), May (1974) found that, for k 1,
the largest eigenvalue of (A17) tends to (1 + v)/(1 − v). Then, for k 1, the
condition for invasibility can be approximated to
k(1 − v)
bN
>
.
g
1 + 3v
(A18)
Malaria without distance effects
Assume system (10) with vi j = v (i = j; i, j = 1, . . . , k). Expressing (10) in the
same way as (A10), and applying the same steps starting with (A10), by evaluating
the jacobian matrix at the trivial equilibrium and applying Theorem A.3 it can be
shown that the roots of the characteristic equation are solutions of
b2 N M
λ + (g + m)λ + gm −
(1 − v)2
k2
b2 N M
2
[1 + (k − 1)v] = 0.
−
k2
2
k−1 λ2 + (g + m)λ + gm
(A19)
The 2k roots of (A19) come in k pairs. Each one of the first k − 1 pairs are equal to
the two roots of the parabola in the first bracket. The remaining pair is composed of
Infectious Diseases in Heterogeneous Environments
567
the two roots of the parabola of the second bracket, one of which is the largest
eigenvalue and will be positive when
b
NM
k
>
.
gm
1 + (k − 1)v
(A20)
Then (A20) is the condition for invasibility.
Malaria with distance effects
Writing (10) with vi j = v ei j (i = j; i, j = 1, . . . , k), and expressing it as (A10),
we evaluate the jacobian matrix at the trivial equilibrium C. Given that V is symmetric and applying Theorem A.1 it can be shown that the roots of the characteristic
equation are solutions of
2
b N M/k 2
V2 = 0,
|−(g + λ)I|−(m + λ)I +
g+λ
or
2
−(g + λ)(m + λ)I + b N M V2 = 0.
2
k
(A21)
Now let ρ = (g + λ)(m + λ). The values of ρ are the eigenvalues of the matrix
(b2 N M/k 2 )V2 . As this matrix is symmetric all its roots ρ are real. For each ρ
there are two values of λ which satisfy (A21), and are the roots of the equation
ρ = (g + λ)(m + λ). Let ρ be the root for which one of the two corresponding λ
values is the maximum eigenvalue of C. This maximum eigenvalue is
−(g + m) +
(g + m)2 − 4(gm − ρ)
2
and will always be real. It will be also positive if
gm > ρ.
Let
ρ=
(A22)
b2 N M
,
h 2 (k, v)
being h(k, v) some unknown function of k and v. Then (A22) becomes
b
NM
> h(k, v).
gm
(A23)
Condition (A23) allows
√ us to say that invasion occurs for a parameter family over
a frontier in a graph b N M/(gm) vs k for a particular value of v.
568
D. J. Rodrı́guez and L. Torres-Sorando
As it is not possible to find analytically the eigenvalues of V, the explicit form
of h(k, v) cannot be found. So the frontier above referred was found numerically,
and is depicted in Fig. 5. However, for one dimension and k 1, similarly as was
stated in the analogous situation for the directly transmitted disease, it is possible to
find approximations for the eigenvalues of V (May, 1974). These approximations
are (1 + v)/(1 − v) for the maximum eigenvalue, and (1 − v)/(1 + v) for the
minimum eigenvalue. So all eigenvalues of V are positive, and then the maximum
eigenvalue of V2 is (1 + v)2 /(1 − v)2 . All this allows us to say that, for k 1 in
the one-dimensional case the condition for invasibility is
b
NM
k(1 − v)
>
.
gm
1+v
(A24)
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Received xx Month 2000 and accepted xx Month 2000