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Virulence
ISSN: 2150-5594 (Print) 2150-5608 (Online) Journal homepage: http://www.tandfonline.com/loi/kvir20
Modeling household and community transmission
of Ebola virus disease: epidemic growth, spatial
dynamics and insights for epidemic control
Maria Kiskowski & Gerardo Chowell
To cite this article: Maria Kiskowski & Gerardo Chowell (2015): Modeling household and
community transmission of Ebola virus disease: epidemic growth, spatial dynamics and
insights for epidemic control, Virulence, DOI: 10.1080/21505594.2015.1076613
To link to this article: http://dx.doi.org/10.1080/21505594.2015.1076613
© 2015 The Author(s). Published with
license by Taylor & Francis Group, LLC©
Maria Kiskowski and Gerardo Chowell
Accepted online: 20 Aug 2015.Published
online: 20 Aug 2015.
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Modeling household and community
transmission of Ebola virus disease: epidemic
growth, spatial dynamics and insights for
epidemic control
Maria Kiskowski
1,
*and Gerardo Chowell
2,3
1
Department of Math and Statistics; University South Alabama; Mobile, AL USA;
2
School of Public Health; Georgia State University; Atlanta, GA USA;
3
Fogarty International Center;
National Institutes of Health; Bethesda, MD USA
Keywords: agent-based models, dynamical models, Ebola virus (EBOV), infectious disease dynamics, emergent dynamics,
mathematical epidemiology, reaction diffusion, social networks, waves
Abbreviation: EVD, Ebola virus disease.
The mechanisms behind the sub-exponential growth dynamics of the West Africa Ebola virus disease epidemic
could be related to improved control of the epidemic and the result of reduced disease transmission in spatially
constrained contact structures. An individual-based, stochastic network model is used to model immediate and delayed
epidemic control in the context of social contact networks and investigate the extent to which the relative role of these
factors may be determined during an outbreak. We find that in general, epidemics quickly establish a dynamic
equilibrium of infections in the form of a wave of fixed size and speed traveling through the contact network. Both
greater epidemic control and limited community mixing decrease the size of an infectious wave. However, for a fixed
wave size, epidemic control (in contrast with limited community mixing) results in lower community saturation and a
wave that moves more quickly through the contact network. We also found that the level of epidemic control has a
disproportionately greater reductive effect on larger waves, so that a small wave requires nearly as much epidemic
control as a larger wave to end an epidemic.
Introduction
An unprecedented epidemic of Ebola virus disease (EVD) got
its start in a forested region of Guinea in December 2013 and has
been spreading across Guinea, Sierra Leone, and Liberia for over a
year.
1
Sporadic case importations into Nigeria, Senegal, Mali, and
the United States have generated secondary cases in the range of
zero to only a handful.
2
While prior outbreaks of Ebola have
quickly subsided after a few generations of infections in relatively
isolated communities,
3
this time chains of transmission have been
able to cross countries through a highly mobile population in a
West African region inexperienced with the virus. The factors that
have interacted to trigger this devastating epidemic include: 1)
substantial delays in detecting Ebola outbreaks in the region, facil-
itating several chains of transmission getting a foothold in West
Africa, 2) severely limited public health infrastructure including a
lack of epidemiological surveillance systems, health care settings
with appropriate infection control practices, 3) cultural practices
that promote infection (e.g., touching the body of the deceased),
and 4) resistance of some populations to follow guidelines set by
authorities on how to prevent infection and spread the virus fur-
ther. The number of EVD cases has reached 27341 including
11184 deaths as of June 17, 2015.
2
Toward the final months of
2014, after the peak incidence levels reported in August 2014, the
epidemiological picture of Ebola dramatically improved in West
Africa; the epidemic leveled off in Guinea and Sierra Leone while
Liberia has recently reported a cluster of cases 2 months after hav-
ing been declared Ebola-free in May 2015.
2
While the epidemic
appears to be subsiding, the factors behind the differences in the
spatial-temporal evolution of the epidemic in the most affected
countries are still poorly understood.
The epidemic took off in December 2013 in the district of
Gueckedou, a southern-forested area of Guinea most likely from a
single spillover event originating from an infected bat.
1
The virus
then spread to neighboring Liberia, generating a small wave of
infections from late March to early June 2014, followed by a brief
exponential growth dynamic in national case incidence to about
mid-September 2014. Similarly, reports of EVD cases started to
© Maria Kiskowski and Gerardo Chowell
*Correspondence to: Maria Kiskowski; Email: abyrne@southalabama.edu
Submitted: 04/15/2015; Revised: 07/06/2015; Accepted: 07/21/2015
http://dx.doi.org/10.1080/21505594.2015.1076613
This is an Open Access article distributed under the terms of the Creative Commons Attribution-Non-Commercial License (http://creativecommons.org/licenses/
by-nc/3.0/), which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited. The
moral rights of the named author(s) have been asserted.
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quickly increase in Sierra Leone in mid March 2014. The height of
the epidemic occurred in August 2014, after which the epidemic
significantly declined probably a result of a combination of factors
including improved isolation and treatment capacity, behavior
changes that reduce contact rates in the population, and a reduc-
tion in the time from the onset of symptoms to diagnosis.
3
Mathematical modeling offers a valuable toolkit to compre-
hensively analyze the transmission dynamics of infectious diseases
by developing models that connect the epidemiology of the dis-
ease, the underlying population structure, population behavior,
available public health infrastructure to carry out contact tracing
activities and isolation of infectious individuals, and public health
interventions including education campaigns, social distancing
(e.g., school closures) as well as treatment and vaccination
campaigns.
4
In the context of the 2014 Ebola epidemic, mathe-
matical modeling has provided the opportunity to project trans-
mission scenarios based on limited data,
5,6
assess the risk of
international case importations,
7,8
and evaluate the impact of
control interventions (e.g., construction of Ebola treatment
units).
9-15
Yet, our understanding of the transmission characteris-
tics and the role of control interventions in each of the most
affected countries remain limited. For instance, at the subnational
level, the 2014 epidemic in West Africa can be disaggregated into
asynchronous local epidemics that are characterized largely by
sub-exponential growth that levels off in just a few generations of
the disease.
16
Yet, the mechanisms behind the sub-exponential
growth dynamics are not clearly understood. These dynamics
could reflect a combination of reactive behavior changes, control
interventions or simply a result of disease transmission in spatially
constrained contact structures (e.g., high contact network cluster-
ing).
16
It is critical to better understand the mechanisms and fac-
tors that have shaped the differences in Ebola transmission
dynamics in order to improve our ability to forecast epidemics,
guide cost-effective control strategies in each of the 3 countries
and strengthen preparedness plans to confront future epidemics.
Here we use a relatively simple individual-based stochastic
transmission model previously described by Kiskowski.
17
This
transmission model structures the population into communities
of households to gain insights into the driving mechanisms of
transmission of Ebola in West Africa. We vary properties of the
network, in particular the distribution of contacts within the net-
work to model well-and less-well-mixed populations, while over-
all measures of transmission such as the household and
community reproductive numbers are kept fixed. This requires
that the number of infectious contacts per infected individual
and the probability of infections are fixed as the community size
is varied. Epidemic control with delay is modeled by decreasing
the reproductive numbers based on an external or internal clock.
Using simulations we characterize patterns of the early growth
phase of epidemics as well as the long-term disease dynamics
with and without the role of control interventions.
Ebola Transmission
The Ebola virus is mainly transmitted by direct contact via
body fluids or indirectly via contaminated surfaces. Also, the
virus is most infectious when individuals are very ill or deceased.
3
Consequently, the transmission scope of the EVD tends to be
limited by its mode of transmission. EVD is frequently transmit-
ted among caregivers at home or in health care settings (e.g., rela-
tives, health care workers) and via unsafe burials when funeral
attendants touch the infectious body of the deceased. After an
average incubation period of 10 days (range 2–21 days),
18
indi-
vidual infectiousness increases as the disease progresses when
infectious individuals are likely confined at home or in hospital.
19
The epidemiological picture of the disease often includes nonspe-
cific symptoms such as sudden onset of fever, weakness, vomit-
ing, diarrhea, headache and a sore throat while only a small
fraction of symptomatic individuals exhibit hemorrhagic mani-
festations.
20
Moreover, EVD is one of the most pathogenic
viruses affecting humans.
These observations motivate a higher rate of infection among
close contacts and a lower rate of infection among casual con-
texts. As in the stochastic transmission model previously
described,
17
close contacts with high rates of transmission occur
among members of a household while casual contacts with low
rates of transmission are assumed to occur among members in a
local community.
In this 3-scale network of households within communities
that comprise a larger total population, the community of an
individual reflects a subset of the network for which that individ-
ual has an equiprobable chance of a casual interaction with other
members of the same sub-network. A larger community size cor-
responds to interaction among a larger sub-population and
greater community mixing overall in the population. One of the
authors demonstrated
17
that different community sizes (different
community mixing rates) result in very different epidemic growth
rates. Since the number of infectious contacts per infectious indi-
vidual is assumed constant regardless of the community size, the
probability of interaction with a particular community individual
decreases with community size. Also, the community size does
not affect R0, the rate of spread in a naive population.
Results
Short and long-term dynamics of an epidemic in a network
with household-community structure
We have demonstrated
17
that for a fixed household size H,
varying the community mixing size Cwould result in different
rates of epidemic growth. Figure 1A shows that the growth rate
of cases increases systematically as the community size increases
from CD25 to CD201. While the growth rate appears to tran-
sition from linear to exponential with the increase in community
size, a log-normal plot of the number of cases per time shows
that for all community sizes, the initial growth phase quickly
transitions from exponential to sub-exponential as indicated by
the strong curvature in the cumulative curve shown in Figure 1B.
A leveling off of the number of infectious cases per day to a fixed
constant value shows that for all community sizes, the long-term
dynamic of epidemic growth is linear (Fig. 1C). Results shown
in Figure 1 are for fixed reproductive numbers R
0H
D2, R
0C
D
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0.7; however, a sensitivity analysis confirmed that epidemic
dynamics were qualitatively similar for all pairs of {R
0H
,R
0C
}
that resulted in sustained epidemics
The transition from exponential to sub-exponential growth
can be understood by looking at the average saturation levels of
infected individuals in affected communities. The average satura-
tion of the communities of infected individuals steadily increases
over time and reaches peak levels within 10—15 serial intervals
(Fig. 2A). Saturation levels remain constant at these peak levels.
A long-term dynamic of a constant number of infectious individ-
uals over time (Fig. 1C) and a constant saturation level (Fig. 2A)
suggests that the epidemic achieves a long-term endemic state.
Indeed, the average reproductive number of infectious individu-
als decreases from 2.7 (this is the reproductive number of an indi-
vidual in a na€ıve population) to approximately 1 in the same time
frame of 10-15 serial intervals (Fig. 2B).
Even though there is a large relative dif-
ference in the number of infectious indi-
viduals per day (Fig. 1C), the average
community saturation of infectious
individuals does not strongly depend on
the community size (Fig. 2A).
The long-term dynamics character-
ized by long-term linear growth, con-
stant community saturation levels of
infected individuals and a constant
reproductive number at R
e
D1canbe
understood as an endemic state in
which a wave of infections of fixed
size and velocity passes through
communities. This is demonstrated by
visualizing the “spatial” spread of
the epidemic through the network.
Figure 3A shows the network location
of infectious individuals (as a function
of network distance from the first
infectious individual) over time for a
single epidemic simulation. A wave of infectious individuals
moves through the network with an approximately uniform
velocity. Since the H£Llattice geometry of our network per-
mits 2 waves departing radially from the initialized infectious
individual, the size of a wave (measured, for example, by the
number of infectious individuals per day) as a function of com-
munity size is one half the total number of infectious individu-
als per day observed for that community size (Fig. 1C).
The epidemic wave can also be described from the perspective
of a single community at a fixed location in the network. In a set
of simulations with community size CD101, we show the aver-
age number of infectious individuals in the j
th
community where
jD50 over 100 simulations. As the epidemic passes through
the community, the number of infectious cases increases, reaches
a peak level and then decreases (Fig. 3B).
Figure 1. Within several serial intervals, simulated epidemics transition from exponential to sub-exponential growth in cases. (A) Cumulative Ebola cases,
(B) log-normal plot of cumulative Ebola cases and (C) number of infectious individuals versus simulation day for different community sizes (CD{25, 51,
101, 201}). Each curve shows the average and standard error of the results of 100 simulations seeded with one infectious individuals on the 1
st
day.
Figure 2. The transition from exponential to linear growth in a population with household-community
structure is due to local saturation of communities. (A) Average saturation of the communities of
infected individuals and (B) 14-day running effective reproductive number R
e14
vs. simulation day for
different community sizes (CD{25, 51, 101, 201}). Each curve shows the average and standard error of
the results of 100 simulations seeded with one infectious individuals on the 1
st
day.
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Distinguishing saturation versus
control effects in an epidemic:
epidemic control results in lower
community saturation and epidemic
waves that move faster through the
network
We have demonstrated
17
that the
different growth dynamics of Guinea,
Sierra Leone and Liberia over the 6
months March 22
nd
– August 22
nd
were consistent with different commu-
nity sizes. In Figure 4, we show that
the different growth dynamics over this
time period were consistent with both
different community sizes (Panel A) or
different levels of epidemic control
(Panel B). Panel A shows that the
growth dynamics are consistent with a
community size that varies from CD
27 (reproducing Guinean dynamics) to
CD131 (reproducing Sierra Leonean
dynamics) to CD251 (reproducing
Liberian dynamics), while Panel B
shows that the growth dynamics are
consistent with an epidemic control
that varies from 30% (reproducing
Guinean dynamics) to 10% (reproduc-
ing Sierra Leonean dynamics) to 0%
(reproducing Liberian dynamics) while
the community size is fixed at CD
251.
Since the disparate country dynamics
may be explained by 2 distinct hypothe-
ses, varying community size or varying
epidemic control, we sought to deter-
mine the extent to which these scenarios
may in principle be distinguished. In
particular, we focus on the competing
hypothetical scenarios for Guinea and
ask how it might be determined if the
epidemic growth in Guinea is smaller
than the epidemic growth in Liberia
over the described time period due to
(i) smaller community size (e.g.,, CD
27 for Guinea vs. CD251 for Liberia,
all other parameters equal) or (ii)greater
epidemic control (e.g., b
0
D0.3 for
Guinea versus b
0
D0.0 for Liberia, all
other parameters equal).
Both sets of parameters for the 2
scenarios for Guinea result in steady
state waves that propagate through the
network. Observe that since either (i)
smaller community size or (ii) greater
epidemic control is consistent with the
WHO Ebola case data March 22
nd
–
Figure 3. The endemic state of an epidemic moving through a network with household-community
structure can be understood as a wave progressing at a fixed rate through the network. (A) The net-
work location of all infected individuals versus simulation day in a single simulation (CD101). The net-
work location of an infectious individual is indicated as a darkened pixel at height hDj¡j0
(hDj¡j0is the network distance of an infectious individual at node i;jðÞfrom the initial infected
individual at node i0;j0
ðÞ). (B) The number of infectious cases in the j
th
community vs. simulation day
(jD100, CD101). The curve shows the average and standard error of the results of 100 simulations
seeded with one infectious individuals on the 1
st
day.
Figure 4. Country variation in the 5 month time period (March22nd-August 22nd) of the epidemic is
consistent with either different community structures or different levels of background control. (A)
Cumulative number of Ebola cases (gray curves) for the community size Cand temporal shift for the
first day of the outbreak that provides the best fit of the reported WHO data (filled black circles) from
March 22, 2014 to August 22, 2014 while all other parameters were held constant. For Guinea, Sierra
Leone and Liberia, optimal fits were found for community sizes CD{27,131,251}, respectively while
the temporal shift predicted that the epidemic in each country began (i.e., simulation “Day 1”) on Dec
29
th
, 2013 for Guinea; April 15
th
, 2014 for Sierra Leone and March 5
th
, 2014 for Liberia. The (B) Cumula-
tive number of Ebola cases (gray curves) for the level of epidemic control and temporal shift for the
first day of the outbreak that provides the best fit of the reported WHO data (filled black circles) from
March 22, 2014 to August 22, 2014 while all other parameters were held constant (CD251). For
Guinea, Sierra Leone and Liberia, optimal fits within the nearest 5% reduction were found for epi-
demic control levels b0D0:3;0:1;0:0fg, respectively while the temporal shift predicted that the epi-
demic in each country began (i.e., simulation “Day 1”) on Dec 30
th
, 2013 for Guinea; April 11
th
, 2014
for Sierra Leone and March 5
th
, 2014 for Liberia. Simulation results of cumulative Ebola cases results
are shown as mean §standard error of 100 simulations seeded with one infectious individual on the
1
st
day.
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August 22
nd
, the 2 hypotheses yield a comparable number of
cases over the 180 days and thus have waves that are approxi-
mately the same size over the 180 days. Thus, the 2 scenarios
would not be distinguished by the size of the waves over this time
period. (The steady-state wave size is larger, however, for the case
with greater epidemic control that best matched the WHO Ebola
case data than the case with smaller community size best match-
ing the data).
Although the size of the epidemic waves as measured by the
infectious cases per day are comparable for the 2 scenarios for
Guinea, the predicted community saturation levels do vary. For
the first scenario in which a lower community size is used to fit
the data, the equilibrium community saturation is approximately
49§1% whereas for the second scenario in which a higher epi-
demic control is used to fit the data, the equilibrium community
saturation is approximately 17§1%. This is a consequence of the
observation that the equilibrium community saturation is not
very sensitive to the community size (Figs. 2A, 5A) but decreases
significantly with epidemic control (Fig. 5B).
A consequence that the predicted sizes of the waves are the
same, but that the predicted saturation levels differ, is that the
speed of the waves through the network should vary. Indeed,
the speed of the wave though the network in the case of a smaller
community size (this being the one with higher equilibrium com-
munity saturation) is much slower than the speed of the wave
through the network in the case of higher epidemic control
(Fig. 5C)
Predictions regarding the ending of the epidemic
If the different growth dynamics of Guinea, Sierra Leone and
Liberia are explained by different reproductive control, then
Guinea has a much lower growth rate than Liberia due to a
greater extent of epidemic control. At one extreme, for a
community size of CD251, Liberia would have 0% epidemic
control with Guinea having 30% epidemic control. This is con-
sistent with some analyses in the literature that the Liberian epi-
demic had been consistent with little or no epidemic control over
this time period (21–23).
Whether the difference in Guinean, Sierra Leonean and Liber-
ian dynamics is due to differences in community size or epidemic
control, in either case further epidemic control is required to end
the epidemic. In Figure 6, we investigate the effect of epidemic
control applied to an established epidemic with a fixed delay.
Figure 6A shows that the size of the epidemic wave decreases
with the level of epidemic control applied at 6 months. Even
with 45% epidemic control, there is a small epidemic wave.
(While the epidemic control may be increased even further,
resulting in still smaller waves, as the size of the waves decrease,
with stochastic fluctuations there is a high probability of sponta-
neous extinction.) Figure 6B shows that community saturation
levels decrease systematically with epidemic control, and the epi-
demic reproductive number vs. time in Figure 6C shows that the
reproductive number initially dips (speculatively, since the com-
munity saturation is higher at that time than the new endemic
steady state) and then begins to re-establish at R
e
D1.
If the different growth dynamics of Guinea, Sierra Leone and
Liberia would be explained by different community mixing sizes,
Figure 7A shows the relative effects of different amounts of con-
trol (applied at 6 months) on the steady-state number of infec-
tious cases. While the number of infectious cases per day for
small community sizes is already relatively small, and the number
of infectious cases per day for large community sizes is relatively
very large, this panel shows that small increases in epidemic con-
trol have a large effect for large community sizes. This observa-
tion is further illustrated in Figure 7B, where the effects of 35%
epidemic control are compared for the different community sizes.
Figure 5. Comparing simulation output parameters for epidemics with varying community size or levels of epidemic control. (A) Epidemics with different
community sizes cannot be well distinguished by the equilibrium community saturation. The equilibrium community saturation of infectious individuals
versus simulation day for different community sizes (CD{27,131,251}). (B) Community saturation decreases with increasing levels of epidemic control.
The equilibrium community saturation of infectious individuals vs. simulation day for different levels of epidemic control (b0D0:3;0:1;0:0
fg
) for com-
munity size CD251. (C) Epidemics with waves of comparable magnitude with lower levels of commuity saturation travel faster. The average distance of
infectious individuals from the origin (location of the first infectious individual) verses simulation day for simulation parameters fitting Guinean case data
with either large commuity size and high control (CD251, b
0
D0.3) or lower community size and lower control (CD27, b
0
D0.0). Curves shows the
average and standard error of the results of 100 simulations seeded with one infectious individual on the 1
st
day.
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Discussion
In this paper we have employed a relatively simple stochastic
individual-level transmission model that incorporates transmis-
sion within households and between communities of different
sizes in order to capture the effects of different levels of popula-
tion mixing.
17
Our model is able to successfully capture the qual-
itative patterns of epidemic growth observed in Guinea, Liberia
and Sierra Leone. Specifically, the model yields brief exponential
growth during the first 2-3 generations of infections followed by
sub-exponential epidemic growth during several disease genera-
tions. This is consistent with the local epidemic growth patterns
observed for each of the EVD epidemics in the most affected
countries in West Africa.
6
The sub-exponential growth patterns
provided by our model in the context of the Ebola epidemics in
West Africa (even in the absence of control interventions or
imposed behavior changes) contrasts with the exponential growth
pattern typically derived from transmission models that assume
random mixing of the population.
24
In the absence of control interventions or behavior changes,
our models calibrated to the early growth patterns of EVD in the
3 most affected countries in West Africa yield an endemic state
of disease reflecting a spatial traveling wave of new infections that
moves through the host population over time with a reproduc-
tion number that is asymptotically 1.0. A reproductive number
of approximately 1 indicates a stationary wave; that is neither
shrinking nor growing, since each infected individual on average
infects approximately one additional individual, and is analogous
to the traveling waves of disease that
can be derived from deterministic reac-
tion diffusion models.)
25,26
The 14
th
century Black Death is the flagship
example of a spatially disseminating
wave of disease.
27
Spatial-temporal
profiles consistent with “traveling
waves” of infectious disease have also
been identified for dengue epidemics
in Thailand.
28
and measles epidemics
in the UK.
29
While the reproduction number for
the ongoing Ebola epidemic in West
Africa has been estimated on average
around 2.0 during the early epidemic
growth phase,
4,21-23,30
consistent with
estimates from historical Ebola epi-
demics,
31
the reproduction number
quickly declines after a few generations
of infections perhaps reflecting disease
transmission in a confined/isolated
setting, control interventions, or
Figure 6. The effect of epidemic control during an epidemic with household-community structure on the wave dynamics. The dotted vertical line indi-
cates the simulation day (kD280) when epidemic control is applied. (A) The number of infectious individuals, (B) the average infectious community satu-
ration and (C) the running 2-week reproductive number R
e14
, per simulation day as the level of epidemic control on the k
th
day is varied
(b0D0:00 ¡0:45/for community size CD251. Curves shows the average and standard error of the results of 100 simulations seeded with one infectious
individuals on the 1
st
day.
Figure 7. The effect of epidemic control to reduce the size of an epidemic wave. (A) The equilibrium
number of infectious cases per day (measured on day kD480) versus the level of epidemic control
.b0D0:00 ¡0:45/.(B) The number of infectious cases per day vs. the simulation day as the community
size is varied (CD{27,131,251}) for a fixed level of epidemic control .b0D0:35/. For both panels, in all
simulations, epidemic control is applied on the the k
th
day (kD280). Connected points in (A) and curves
in (B) show the average and standard error of the results of 100 simulations seeded with one infectious
individual on the 1
st
day.
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population behavior changes. It is worth noting that the ongoing
epidemic in West Africa appears to have leveled off in some areas
of Guinea and Sierra Leone, reaching an approximate effective
reproduction number R
e
of about 1.0 in some areas.
32
Our model
provides the additional perspective that an observed reproduction
number R
e
1.0 is indicative of an endemic wave of infection
traveling through the population
In this context, ‘endemic’ means that the infectious wave is in
a steady equilibrium with a fixed size and speed. Our results indi-
cate that the size and speed of the wave depends on the network
properties of the population (e.g., community size) while the
speed of the wave though a community, and net effect on
the community, depends on the transmission characteristics of
the disease (the reproductive numbers, in our case R
0H
and R
0C
,
possibly modulated by control measures). In particular, the size
of the wave (case incidence per day) increases starkly with com-
munity size and the fraction of the community affected increases
with R
0
. These results are consistent with those of,
33
whose spa-
tial model of pathogen spread also resulted in a circular wave of
spread from the pathogen source. They found for this cellular
automata model with no household-community structure, the
contact rate per susceptible also saturated and the resulting uni-
form level of saturation in the wake of the wave depended on the
reproductive number as (1-1/R
0
).
In principle, these results would apply to the progression of a
disease in any network in which otherwise exponential growth
must slow due to spread at a finite rate through the network. In
sub-networks with low motility, such as a set of individuals
within a school or employed at a hospital, regardless of the dis-
ease transmission dynamics, observe that an exponential phase
must always be followed by extinction or a sub-exponential phase
due to saturation effects as exponential growth depletes suscepti-
bles within a small number of generations. The role of long-range
infectious links are to “seed” the epidemic in more distant loca-
tions of the network that supply a new source of susceptibles. A
question of interest for a network model of a given topology is
the threshold fraction of long-range links for which the epidemic
can be expected to grow exponentially rather than linearly.
34-37
Our model provides important insights on the level of control
that would be required to contain Ebola epidemics. Specifically,
findings suggest that a similar level of control effort would be
required to bring the reproduction number below 1.0 in the 3
most affected countries if the transmission dynamics in each of
the 3 countries are driven by different community sizes. This is
somewhat surprising since the disease incidence (i.e., the number
of infectious cases per day, or the size of the wave) increases
starkly with the community size and implies that the size of an
epidemic may not predict how difficult it is to control that epi-
demic. The fact that Liberia has been able to rid itself of the virus
suggests that this population has been able to effectively mitigate
transmission. In contrast, local reports indicate that the Guinean
population has exhibited higher levels of resistance to education
campaigns on how to avoid contracting and disseminating the
virus,
38
which may explain the difficulties in halting transmission
in this country where disease incidence has followed a relatively
steady incidence pattern.
By comparing the predictions of decreasing the community
size versus increasing of epidemic control we sought to determine
the extent to which these competing explanations for a reduced
epidemic growth rate can be distinguished. We found that for a
fixed growth rate (for example, a growth rate matching observed
case data), a greater level of epidemic control with correspond-
ingly larger community size predicts lower community saturation
than a lower level of epidemic control and corresponding smaller
community size. Our model predictions would provide immedi-
ate interpretation if community seroprevalence rates of Ebola
antibodies could be compared in areas of Guinea, Sierra Leone
and Liberia for districts with different epidemic growth rates. In
principle, if these rates were comparable among communities,
then our model results predict that the difference in the district
growth rates is due to differences in their network properties. On
the other hand, if districts with lower epidemic growth rates have
lower seroprevalence rates, this would suggest greater epidemic
control in these areas. Since community size and epidemic con-
trol play a role together in epidemic dynamics, trends in sero-
prevalence rates may be able to discern the relative role of
epidemic control over the course of the epidemic.
Our model is “spatially implicit,”
39
in that the defined dis-
tance between households corresponds to a distance within the
network, and a decreased probability of contact, that only loosely
corresponds, if at all, with spatial distance. For example, in West
African countries, network distances may be shorter between vil-
lages and cities than between villages themselves, though of
course this may not reflect geographic distances. Connectivity
dependencies among villages were studied
40
and found, unsur-
prisingly, to be complex. Seven chiefdoms bordering the Gola
forest in Sierra Leone were well connected to the city of Kenema,
but there were also important lateral dependencies between vil-
lages. In our simulations, the community size varies over an order
of magnitude from 125 to 1255. The community size is the sub-
set of individuals that an infectious individual has an approxi-
mately equiprobable chance of interaction, even if that
probability is very small in large communities. Heuristically, it
may be thought of as the number of people drawn by the market
where the infected individual shops or the number of children
and teachers attending the same school. Small villages may repre-
sent natural upper bounds for community sizes. In a large city in
contrast, certainly a thousand people might be expected to attend
the same market or have children that attend the same school. A
large village or city may be stratified with several partially over-
lapping communities (e.g., corresponding to overlapping school
or market sub-networks, and different ethnic and socioeconomic
groups). The extent of overlap from one community to the next
would be expected to vary in a non-regular way. Our network
model necessarily represents an extreme simplification and the
‘best-fit’ community size in simulations would represent a phe-
nomenological weighted average of a distribution of community
sizes in actual populations. Another limitation of our model is
that it is relatively low-dimensional. Once transmission is estab-
lished, the wave of infection may travel in only 2 directions
within our simplified network. In complex real-world models,
there are in principle no such limits in the dimensionality of the
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network. The low network-connectivity of our lattice is presum-
ably a better description of more small or rural rather than urban
communities, for which there would be more overlap expected
among communities. For example, in a small, rural community,
it would be more likely that the children of parents that work
together also attend the same school. Simulating long-range links
as the seeding of the epidemic to new communities, as done in
Kiskowski et al 2014, increases the complexity of the network.
It is not expected that community size and epidemic control are
constant or even that they vary monotonically over time. The
2014-2015 Ebola virus epidemic can be thought of as a superposi-
tion of asynchronous smaller outbreaks each with potentially dif-
ferent distributions of community sizes and levels of control. In
Kiskowski, a systematic fitting algorithm was able to identify at
least 2 distinct waves in Guinea over the 8 months March 22—
October 15th. The first wave had a relatively low growth rate, for
example consistent with a community size of only 45 individuals,
but a second wave establishing in August had a much faster growth
rate, consistent with a community size of 255 individuals. Simi-
larly, Towers et al observed an increase in the effective reproduc-
tive number of the epidemic at this time when the outbreak in
Guinea spread to Conakry. There is still much that can be learned
by characterizing these individual district level outbreaks.
Our results underscore the importance of incorporating appro-
priate spatial structures into models of infectious disease transmis-
sion. Such considerations may be more important for infectious
diseases that are transmitted via close contact such as Ebola and
HIV.
41
Such population structures used in models could be
designed based on contact tracing data
42-44
or epidemic data of
growth patterns in areas where interventions or population behav-
ior changes are not suspected to have played a significant role such
as the southern forested are of Guinea where the ongoing Ebola
epidemic is suspected to have started back in December 2013.
1
Capturing the appropriate spatial structure in models of disease
transmission is particularly important for epidemic forecasting
because model-based predictions are highly sensitive to assumptions
of contact structure with transmission models that assume homoge-
neous mixing
4,45
predicting the largest epidemic sizes in the absence
of control interventions or behavior changes compared to population
structured models (e.g., age structure, household-community struc-
ture).
10,17,34,46,47
However, more work is needed to elucidate the
dominant factors that have affected the epidemic trajectories in
Guinea, Liberia and Sierra Lone.
16
In addition to contact structure,
other factors that may have played a significant role include the effects
of specific control interventions such as social distancing, increased
hygiene, increased safe burial rate, and improvement in health care
infrastructure (e.g., increased bed capacity).
Methodology: Modeling Transmission Dynamics
of Ebola Virus Disease (Evd)
SEIR network model for distinct transmission within
households vs. communities
We use the 3-scale network based SEIR model described in
detail
17
to study the early transmission dynamics of EVD in
Guinea, Sierra and Liberia. In this model, a hierarchical network
is used to describe the household and community contacts of
individuals within a population. At the smallest scale, individuals
are organized within households of size H. At the second scale,
each household is centered within a community of size Chouse-
holds. Communities are overlapping subsets of a much larger
population of PDHLindividuals, where His the number of
individuals in a household and Lis the total number of
households.
This hierarchical structure is modeled on an H£Llattice so
that each column of the lattice corresponds to a single household;
the i
th
column corresponds to the i
th
household. Two households
h
i
and h
j
on the lattice have a network distance hD|i-j| and they
are in the same community if |i-j|<R
C
, where R
C
is the commu-
nity radius (CD2R
C
C1). The i
th
community is the community
centered at the i
th
household containing households.
hi¡RC;hi¡RcC1;...;hi;hiC1;...hiCRc
fg
Without modification, this model can be viewed as a lattice-
based reaction diffusion model with 2 interaction neighborhoods
defined for each node (i, j): one smaller interaction neighborhood
is the household (a H£1 vertical array corresponding to the
entire column) and larger interaction neighborhood is the com-
munity (a H£Cinteraction neighborhood centered at the j
th
lattice column) (Fig. 8).
SEIR dynamics
Individuals on the lattice are assigned one of 4 states: S
(susceptible), E(exposed), I(infectious) and R(refractory). States
are updated at each time step with the following transition
probabilities:
pS!EðÞDprobability that asusceptible will become exposed
D1¡probability of no exposures from any infected contactsðÞ
D1¡1¡tH
ðÞ
iH:1¡tC
ðÞ
iC
:
where tHand tCare the transmission probabilities within a
household and within the community, respectively, and iHand
iCare the number of infectious household and community con-
tacts in the network, respectively
pE!IðÞDprobability that an exposed individual
becomes infectious D1g;
where gis the average incubation period.
pI!RðÞDprobability thataninfectiousindividual
willbecomerefractory D1λ;
=
where lis the average infectious period.
Transmission in the context of no epidemic control
A susceptible individual becomes infected (exposed) with the
probabilities t
H
and t
C
per household or community infected
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contact per day, respectively. Transmis-
sion can also be described in terms of the
probability that a susceptible will
become exposed when they are within
one of the 2 neighborhoods of an
infected individual. The probability that
each susceptible individual in the house-
hold neighborhood will be exposed per
day is given by the transmission proba-
bility t
H
, and additionally the probability
that each susceptible individual in the
community neighborhood will be
exposed per day is given by the transmis-
sion probability t
C
.
In our model, the reproductive numbers R0Hand R0Ccan be
calculated as the expected number of susceptible individuals
exposed by a single infectious individual that is placed on a lattice
of susceptibles. Given that the expected lifetime of the infectious
individual in days is l, and that the size of the household and
community neighborhoods are H£1 and H£C;respec-
tively, the expected number of exposures caused by the initial
infected individual are:
R0HλtHH¡1ðÞ;R0cλtC.CH¡1/
And thus to define given values of R0Hand R0C, we define the
transmission probabilities:
tH:DR0H
λH¡1ðÞ
;tC:DR0C
λCH¡1ðÞ
Transmission in the Context of Global and Local
Interventions
We define 2 types of epidemic control; external (global) and
internal (local). For both types, epidemic control is modeled as a
percent reduction in transmission probability. For externally
applied epidemic control, the transmission reduction is a reduc-
tion in the transmission rate globally applied to the entire net-
work. For internally applied epidemic control, the transmission
reduction is calculated locally depending on the state of the inter-
action neighborhoods of each infectious individual.
For globally applied control, the household and community
transmission probabilities are reduced by a factor b0
0b01ðÞper timestep.
pS!EðÞDprobability thatasusceptiblewillbecomeexposed
D1¡probabilityofnoexposuresfromanyinfectedcontactsðÞ
D1¡1¡tH1¡b0
ðÞðÞ
iH1¡tC1¡b0
ðÞðÞ
iC
:
For locally applied epidemic control, the household and com-
munity transmission probabilities are maximally reduced by a
factor b00b01ðÞper timestep but the actual reduction bis
calculated for each infectious individual located at the node (i,j)
using the Hill function increasing monotonically from 0 to b0
with qj:
bDb01Cq1=2
qj
p
¡1
Where qjis the total number of infectious or refractory indi-
viduals in the j
th
community. i.e. qjwould be the cumulative
number of infectious (sympotomatic) individuals that a suscepti-
ble community individual has observed. The parameter q12
=
determines the value of qjthat results in bD0:5b0.
Parameter values and initial conditions for simulations
Initially, all individuals (lattice nodes/network vertices) are
susceptible (state S) except for one exposed individual (state I).
In simulations, time steps are discrete and correspond to exactly
one day. The average and standard error of simulation output
parameters (described below) are computed for 100 simulations.
A simulation ends spontaneously when there are no exposed
or infectious individuals remaining in the network, so output
parameters versus time are calculated only for simulations that
have not yet ended. The standard error is calculated as SE Ds
ffiffin
p;
where sis the standard deviation and the number of simulation
nis a non-increasing function of the simulation day (n100/:A
description of the values (or ranges of values) for simulation
parameters is provided in Table 1.InFigure. 4, Ebola case data
is fit by varying either community size (Panel A) or the level of
epidemic control (Panel B) and the calendar date for which the
outbreak is initialized. The best pair of parameters is identified as
the pair that minimized the R-square coefficient of determination
comparing the simulated and Ebola case data.
Output Parameters of Simulations
Average infectious cases, epidemic reproductive number and
community saturation
Simulations track the states of individuals in the network as a
function of the k
th
day. An individual is defined as an Ebola case
when they become infectious (assuming that an individual is not
recognized as a case until they are infectious and that there is no
Figure 8. Two overlapping interaction neighborhoods for an arbitrary node (i,j) when the household
size is HD5 and the community radius R
c
D6. The depicted 5 £26 lattice corresponds to an H£L
population of 130 individuals distributed among 26 households (5 individuals per household). An
arbitrary node is indicated with a ‘’. The household interaction neighborhood for this node is the j
th
column of the lattice indicated in dark gray. The community interaction neighborhood for this node
is the 6 £11 sub-array of the lattice indicated with 2 shades of gray.
www.tandfonline.com 9Virulence
Downloaded by [Georgia State University] at 06:36 29 September 2015
delay in identifying infectious individuals). We calculate the aver-
age cumulative number of Ebola cases by the k
th
day or the aver-
age number of cases per day.
The cumulative effective reproductive number RekðÞis calcu-
lated as a function of the k
th
simulation day as the average num-
ber of infections resulting from all individuals that are refractory
by day k:
RekðÞD
Total#Individuals Exposed by Individuals Refractory by Dayk
Total #Individuals Refractory by Dayk
Note that this calculation is equivalent to:
RekðÞDTotal #Individuals Ever Exposed ¡Individuals Exposed by Individuals Not Yet RefractoryðÞ
Total Individuals Ever Exposed ¡Individuals Not Yet RefractoryðÞ
so that this ratio approaches 1 as a larger and larger fraction of
ever-exposed individuals become refractory (as the serial interval
becomes a smaller and smaller fraction of the days k). We there-
fore define a 2-week running effective reproductive number R
e14
.
The two-week running effective reproductive number is calcu-
lated as a function of the k
th
simulation day as the average num-
ber of infections resultingrom all individuals that have become
refractory in the last 14 days:
Re14 kðÞD
Total Individ uals Exposed by Individuals Refractory
Between Day k ¡13ðÞand Day kðÞ
Total #of Individuals Refractory Between Day k ¡13ðÞand Day kðÞ
The “community saturation” Sof a single infectious individ-
ual located at the node i;jðÞis defined on the k
th
day as the frac-
tion of persons in their community that are no longer susceptible:
Si;j;kðÞD
CH¡1ðÞ¡number of susceptible individuals in
jth community on the kth day
CH¡1
T average community saturation of a simulation on the k
th
day is
the community saturation averaged for all the individuals that
are infected that day. Finally, the average community saturation
averaged over Nsimulations on a given day is the average com-
munity saturation of all infected individuals among the Nsimu-
lations – i.e., the final simulation average is weighted by the
number of individuals that were infectious in each simulation.
Averages on the 1
st
day are always calculated for 100 simulations.
However, simulations die out spontaneously and also as epidem-
ics die out there may be very few infectious cases on a given day.
A gap in the plot of average saturation vs. time may occur on the
k
th
day when there were no infectious cases on the k
th
day. (This
does not necessarily mean the epidemic is over since there may be
incubating individuals.)
Disclosure of Potential Conflicts of Interest
No potential conflicts of interest were disclosed.
Funding
GC acknowledges support from NSF grant #1414374 as part
of the joint NSF-NIH-USDA Ecology and Evolution of Infec-
tious Diseases program, United Kingdom Biotechnology and
Biological Sciences Research Council grant BB/M008894/1,
grant #1R01GM100471-01 from the National Institute of Gen-
eral Medical Sciences (NIGMS) at the National Institutes of
Health, NSF grant 1318788 III: Small: Data Management for
Real-Time Data Driven Epidemic simulation and RAPID NFS
Grants#1518939 and NSF# 1518529, and the Division of Inter-
national Epidemiology and Population Studies, The Fogarty
International Center, NIH.
Table 1. Parameter values used in simulations. This table provides a description of each parameter used in simulations, the value or range that is used, and
the reference source for the value that is used if applicable
Parameter Description Parameter Value (Range) Source
gAverage incubation period 5:3 days (48–51)
lAverage infectious period 5.6 days (48–51)
HHousehold size 5 (52)
CCommunity size 25—251
R0HHousehold reproductive number 2.0 (17) Appendix 3, Fig 1
R0CCommunity reproductive number 0.7 (17) Appendix 3, Fig 1
b0Transmission reduction factor 0—0.75 —
q12
=Infected or susceptible quorum for half-response in Hill equation 0—250 —
pHill equation parameter 3 —
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