Model for Estimating Acute Health Impacts
from Consumption of Contaminated Drinking Water
Regan Murray1; James Uber2; and Robert Janke3
Abstract: Disease transmission models predict the spread of disease over time through susceptible, infected, and recovered populations,
and are commonly used to design public health intervention strategies. A modified disease model is linked to flow and transport models
for water distribution systems in order to predict the health risks associated with use of contaminated water. The proposed framework
provides information about the spatial and temporal distribution of health risks in distribution systems and is useful for understanding the
vulnerability of drinking water systems to contamination events, as well as for designing public health and water utility strategies to
reduce risks.
DOI: 10.1061/(ASCE)0733-9496(2006) 132:4(293)
CE Database subject headings: Estimation; Public health; Potable water; Water quality.
Introduction
Contamination of drinking water distribution systems can result
from cross-connections with nonpotable water, permeation and
leaching of pipes, chemical reactions and microbial growth within
pipes, or intentional acts of contamination. Hydraulic and water
quality models can be used to model the fate and transport of
contaminants within utility-specific distribution system networks
(Rossman 2000; Uber et al. 2004a). Recently, new methods have
been developed to allow for modeling of multiple interacting spe-
cies in flow (Shang et al. 2004), for example, the full dynamics
between chlorine and an organic contaminant. By linking flow
and transport models to dynamic models for disease, a framework
is presented for estimating the spatial distribution of health risks
associated with ingestion of contaminated drinking water.
Contamination warning systems, sometimes referred to as
early warning systems, have recently been proposed as a promis-
ing approach for reducing the risks associated with the intentional
contamination of drinking water systems (USEPA 2005). Con-
tamination warning systems use continuous online contaminant
detectors or water quality sensors to detect potential contamina-
tion events and provide an early warning of potential health risks.
The technology involved is early in its development, and costs are
high; thus, there is a strong incentive to limit the number of sen-
1 Research Scientist, U.S. Environmental Protection Agency, 26 W.
Martin Luther King Dr. (MS 163), Cincinnati, OH 45268. E-mail:
murray.regan@ epa.gov
2Professor, Dept. of Civil and Environmental Engineering, Univ. of
Cincinnati, Cincinnati OH 45221. E-mail: jim.uber@uc.edu
3Research Scientist, U.S. Environmental Protection Agency, 26 W.
Martin Luther King Dr. (MS 163), Cincinnati, OH 45268. E-mail:
janke.robert@epa.gov
sors and locate them optimally. Most sensor location methods rely
on spatial estimates of risk associated with potential contamina-
tion events [see, for example, Berry et al. (2005) and Uber et al.
(2004b)]. The risk assessment approach described in this paper
will provide a more comprehensive framework for estimating
health effects associated with contamination events.
This framework is also useful for designing effective public
health and water utility intervention strategies. By simulating
the spatial and temporal health risks associated with consumption
of contaminated drinking water, for instance, one can identify
the locations of exposed populations in need of public health
treatment. The same tools are useful for assessing the value of
hydraulic control options for isolating or flushing contaminated
water, as well as the potential for treating the water in situ. In
addition, such tools should be useful for planning and preparing
for contamination events and also for real-time planning of re-
sponse actions.
Disease models have already been applied to waterborne dis-
ease outbreaks (defined as diseases that can be traced back to
water by epidemiological evidence); see, for example, Eisenberg
et al. (1998). The U.S. Centers for Disease Control and Preven-
tion (CDC) along with the USEPA have collected data on water-
borne disease outbreaks since 1971. In 2001-2002, 31 outbreaks
were reported, causing illness among approximately 1,000 per-
sons and resulting in seven deaths. Recent outbreaks in the United
States were linked to Cryptosporidium, Giardia, E. Coli, Salmo-
nella, Campylobacter jejuni, Legionella, and Noroviruses, among
others (Blackburn et al. 2004).
The framework presented in this paper includes the capability
of tracking waterborne disease outbreaks spatially. This is espe-
cially important because of the difficulty of linking disease out-
breaks back to water sources. A CDC report found that reported
water borne illnesses "probably represent only a small proportion
of all illnesses associated with waterborne-disease agents" (CDC
1990). Most illnesses resulting from ingestion of chemicals are
not collected by this system, nor are diseases resulting from long-
term exposure to low levels of contaminants. Many small-scale
outbreaks are probably not reported. Moreover, because of the
difficulty in linking public health events to drinking water
sources, the number of outbreaks reported is likely conservative.
-------
The modeling tools presented in this paper allow one to simulate
both the contamination event in the water system and the result-
ing health impacts in a population, improving the capability of
linking a public health event back to the water supply.
This paper links hydraulic models for water flow through dis-
tribution systems to models for estimating health impacts in order
to predict the spread of disease over time in a population using
contaminated water. The focus is 011 biological agents, such as
bacteria and viruses, but some of the methodology would also be
appropriate for chemical agents. In Section 2, the recommended
approach for biological agents is examined, as well as the ac-
cepted risk assessment paradigm for chemicals, whereas Section 3
describes a disease transmission model in detail. In section 4 the
infectivity rate that links the hydraulic models to the disease mod-
els is described in detail, and in Section 5 the common methods
used to predict flow and transport in distribution systems are dis-
cussed. In Section 6 some additional assumptions and simplifica-
tions to the model are presented. Finally, the new framework is
used to study a particular contamination example in Section 7.
There are many potential applications for this framework; how-
ever, the focus of this paper is on developing the methodology.
Discussion of Methods for Quantifying Health Risks
Dynamic Disease Model
The model used in this paper is derived from the general dynamic
model proposed by Anderson and May (1991) and is similar to
the model used by Chick et al. (2003). The model describes the
spread of disease through a population of susceptible persons (S),
infected but not symptomatic persons (/), diseased (infected and
symptomatic) persons (D), recovered and immune persons (R),
and those impacted fatally from disease (F). The disease can be
transmitted person to person or from drinking water. The dynamic
model is given by the following equations:
dS
— = yR- \S (1)
at
— = \S-oI (2)
dt
— = (tI - (a + v)D (3)
— = vD — yR (4)
dt
Generally, attempts to quantify the risk associated with consum-
ing contaminated water have used static models that determine
the probability of individual illness based on a single exposure.
This approach follows the accepted paradigm for chemical risk
assessment. The dose and the dose-response curve—the amount
of contaminant consumed and the probability of a given health
response—are the main criteria used to determine health impacts.
Such an approach has been used to estimate health risks associ-
ated with long-term exposure to low levels of toxic chemicals
(Risk 1983).
The same information is important for assessing risks associ-
ated with exposure to viruses, bacteria, and protozoans. However,
biological risk assessment requires the consideration of additional
factors such as multiple infectivity paths (person to person, envi-
ronment to person), possible secondary transmission paths (per-
son to environment to person, environment to person to person),
immunity to disease, microbial incubation periods, and the poten-
tial for asymptomatic carriers of disease. See Haas et al. (1999)
for a complete treatment of this topic.
Recently, the use of dynamic disease transmission models has
been introduced into the biological health risk assessment process
(Eisenberg et al. 2002). Disease progression models predict the
temporal spread of disease through subgroups of a population,
such as susceptible, infected, and recovered subpopulations.
While traditional risk assessment approaches use data-driven
models, disease transmission models include physically based pa-
rameters that describe the disease process. Such models may also
account for the population dynamics of pathogens within the host
or the natural reservoir. Disease models have been used to assess
health risks, to differentiate epidemics from endemic disease
cycles, and to design treatment and control strategies for diseases,
such as vaccination programs (Anderson and May 1991). Similar
models have been used to estimate the spread of disease follow-
ing a bio terrorism attack and to optimize public health interven-
tion strategies (Barrett et al. 2005).
— = a D (5)
dl
where 5"(xi,t) = number of susceptible persons at time t;
l(xi,t) = number of latently infected but not symptomatic persons
at time t: D(xi,f) = number of diseased (infected and symptomatic)
at time t, R(xi,t)= number of recovered and immune at time t;
F(xt, t) = number of fatalities due to disease al time t; and
X(x, ,f) = force of infection at time t and is discussed in more detail
in Section 6.
The parameters in the model are:
• r=pcr capita recovery rate (mean duration of illness is 1/u);
• a=rate al which hosts move from / to D (mean latency period
is 1 /
-------
Infectivity Rate
In the disease model, Eqs. (l)-(5), X is the force of infection or,
more specifically, the per capita rate of acquisition of infection.
In general, for any route of transmission, X can be written as
the product of the rate of exposure to the pathogen and the prob-
ability of infection given that exposure. In this paper, infection
can be transmitted in two ways, from close person to person con-
tact, or from ingestion of contaminated drinking water. Therefore,
let k=kP+kw where the subscripts P and W refer to transmission
from people or water. For an infectious disease transmitted person
to person, \P is assumed to have the form
Xp(x,-,f) = 0(/(x;, f) + D(Xi,t)) (6)
where (3 = transmission parameter dependent on the disease-
inducing organism and many environmental and societal factors.
For instance, (3 may depend on personal hygiene, a person's age,
and the seasonal variability in microbial behavior. In this form,
the disease can be transmitted between persons near to each other
spatially (i.e., located at Node xt). Note that Eq. (6) could be
modified to reflect the mobility of a population that interacts with
people some distance away from Node x;. Note also that \P may
vary significantly from node to node because the population den-
sity of infected persons varies spatially as well as the likelihood
of exposure.
The infectivity rate resulting from the consumption of con-
taminated drinking water is related to the amount of water con-
sumed and the amount of contaminant in the water. Let d{xt,t) be
the cumulative dose of a contaminant received by the population
at Node x; and let r(x;,f) be the corresponding probability of
infection given dose d. The dose d can be calculated from flow
and transport simulations, as described in the next section. The
probability of infection r describes the percent of the population
responding to a given dose, which is essentially a dose-response
relationship as a function of space and time, r(xi,t) = r(d(xi,ij).
The infectivity rate can be modeled as
drS0 drddS0 f S(t)
Aw= = or r(x;,f) = I Xw(x;,t) di (7)
w dt S dddt S J 0 s0 '
where dd/ <9f=rate of exposure to the pathogen; and dxl dd=pro-
bability of infection given that exposure. S0=S(xi,0)= total num-
ber of customers at Node x; that may be susceptible to infection.
Therefore, \w at time t is the rate of new infections rS0 at time t
per total number of susceptibles S at time t. This formulation of
\w is a generalization of that used by Chick et al. (2001).
For many contaminants, dose-response curves are available in
the literature that were developed by matching experimental data
with an understanding of disease kinetics in the human body. A
dose-response curve generally has a sigmoidal shape, and an ex-
ample of one is given by the curve shown in Fig. 1. Along the
horizontal axis is the dose in number of organisms, and along the
vertical axis is the percent of persons expected to become infected
by the disease at a specific dose. Note that this dose-response
curve corresponds to an ID50 dose (the dose at which 50% of the
population would be infected) of 100,000 organisms.
Exposure to contaminated drinking water is possible from sev-
eral routes, including ingestion, inhalation, and dermal contact.
Ingestion is the focus of this paper; however, the model can be
modified to incorporate other exposure routes as well. The cumu-
lative dose of a contaminant ingested by the population at x; at
time t is calculated according to
Doso-Response Curve (or Biological
u
-------
The commonly used models for flow and transport in distribution
systems will be combined with the disease model to predict the
spatial distribution of health effects.
In drinking water distribution systems, the direction and mag-
nitude of flow is determined by time-varying demands and the
complex, looped geometry of the pipe network. Hydraulic models
that include pipe lengths and diameters, pipe connectivity, and
network operations are needed in order to predict time-dependent
average water velocity in pipes. Most hydraulic models solve the
flow equations in a pipe network by conserving mass at the pipe
ends (nodes or junctions) and conserving energy in the pipes.
EPANET (Rossman 2000) is a publicly available software pack-
age for simulating flow through pipes and forms the basis for
many other commercially available software packages. For a
complete treatment of pipe hydraulics, including the specific mass
and energy balance equations in pipe networks, see Walski et al.
(2003).
Using network hydraulic models, concentrations of chemicals
or pathogens can be estimated spatially and temporally. Pathogens
can be assumed to flow with the water in distribution system
pipes, react with the chlorine residual, adhere to biolilms on pipe
surfaces, and grow and decay according to natural processes. A
general one-dimensional model for pathogen fate and transport in
a distribution system is as follows:
dW dW
— + v(x.t)— =fl(W) + (b-a)W-gl(W,C) (9)
dt dx
where W= W(x,t)~concentration of the contaminant in the water.
The concentration is advected with the average water velocity
v(x,t). Molecular diffusion is neglected, because mixing by tur-
bulent flows is dominant in distribution system mains. The micro-
bial concentration grows with rate b, dies with rate a, increases
according to the source term fu and reacts with chlorine C, ac-
cording to the reaction function gl. Note that b is assumed
to be a constant, but in reality may depend on such factors as
nutrient and substrate concentrations, temperature, and pH. The
constant a depends on natural death rates of the pathogen. The
source tcrm/i represents the addition of the pathogen to the water
from human sources, such as shedding or contamination. If the
reaction dynamics with chlorine or other disinfectants are known,
the chlorine concentration can also be modeled by
dC . dC
V + vM — =/2(C) - g2(W,C)
dt dx
(10)
In Eq. (10), the chlorine concentration increases with the source
term f2 and decreases as it reacts with the microbial agent and
other organic and inorganic compounds according to the reaction
term g2. Eqs. (9) and (10) describe conceptually how pathogen-
chlorine interactions could be modeled within a pipe network to
support the spatio-temporal estimation of acute health impacts.
For an example, sec Prop at o and Uber (2004).
disease is proportional to the recovery rate, v() = ct, meaning that
those in the disease state move at the same rate to both categories.
The final assumption is related to the transition time between
disease slates. In the disease model, Eqs. (l)-(5), the transition
time is exponentially distributed with a mean time between in-
fected and diseased states of 1/cr and a mean time between the
diseased and fatal states of 1/v. If both of these transition times
arc assumed to always be constant, the model can be rewritten as
a time delay model. For a discussion on arbitrary transition
times in disease models, see Van den Driessche (2002). The new
model is
dS dr
~ = - S()(*;) —
dt dt
dl / x dr
— = S0(xi)—
dt dt
dD . x dr
— =S0{x%) —
at at
¦ .Sq(a*j)
dr
dt
-SqU,)
dF dr
TrftSoWa
(11)
(12)
(13)
(14)
This model can be solved exactly to obtain the following
solutions:
/(x,.,f) =
Sixij) = S0(x)(1 - r(x;,f))
S0(x)r(x;,f) if 0 < f <
S0(x)(r(xi,i) r(x{,t- l J)) if t > fCT
Dixj.t) =
0 if 0 < t < t„
S0(x,) r(xif t - tj if t,T < t < t„
S0(x) (r(xh t - t„) - r(xiy t - tv) ) if 1 > tv
F(xi,t) = -
0
if 0 < t < t„
QS0(x,)(r(xi,t - t,,)) if t> tv
where t„= I/ct^latency period; and f^l/v^disease duration. In
this model, the number of susceptible persons decays linearly
with the response function r. (Note, however, that r is not a con-
stant but varies in space and lime.) In turn, the number of infected
persons increases linearly with rate r until after the latency period
t„, when infected persons start showing symptoms and move into
the diseased stage. After the latency period, the number of dis-
eased persons grows linearly until after the duration of the illness,
at which time a fraction of the diseased persons moves into the
fatality stage. This simple model is solved for a specific example
in the next section.
Assumptions and Model Simplifications
In this section, several assumptions are made in order to simplify
the disease model and examine its basic dynamics in more detail.
First, if the pathogen is not communicable but is transmitted only
via the drinking water, then k = kw. Second, if a person who has
recovered from disease cannot reenter the susceptible state, then
7=0 (i.e., after infection a person either gains permanent immu-
nity or dies). Third, it is assumed that the death rate from the
Case Study: Biological Contamination Event
An example case study is presented in which the health impacts
of a contamination event are calculated according to the disease
model, and the times for effective intervention are considered as
well as the benefit of intervention. In this example, a large quan-
tity of a pathogen is introduced at one particular location in a
specific drinking water distribution system. The operations and
hydraulics of this system are known, and the water velocity,
v(xiyt), pressure head, h(xiyt), and pathogen concentration in the
-------
i oiu ~Jijfhiliun «>f onj.inivms
(_>
30
CL
20
10
I
1
I
I
Y
:l
:l
n
: I
Infected
Diseased
Fatalities
100
200 300 400
Time in Hours
500
600
Fig. 3. Infected, diseased, and fatally impacted populations over time
at one location in distribution system.
-------
Disease Progression in Total Population
— Infected
¦-Diseased
•••Fatalities
I
I
i
1
t
1
l
I
i
I
l ,
A
: 1
: \
V
Variation in Total Infections with Volume of
Contaminant Introduced
200 300 400
Time in Hours
500
600
Fig. 4. Disease progression over time in total population
Fig. 3 shows the disease progression through the population at
the same node shown in Fig. 2. The four curves show the number
of susceptible persons, the number of infected persons, the num-
ber of diseased persons, and the number fatally impacted. The
slope of the infections curve is directly related to the infectivity
rate or the response function r, which encapsulates all the infor-
mation about the hydraulics of the contamination event. The sus-
ceptible population quickly becomes infected and drops off to a
very small asymptotic number. The infected population grows
rapidly, sustains itself as the disease is latent (for one week), and
then drops quickly as the infected persons transition into the dis-
eased stage. Similarly, the diseased (symptomatic) population
grows rapidly, sustains itself for the duration of the illness (one
week), then a proportion of the diseased population recovers,
while the remaining die. A similar set of curves could be drawn
for each of the approximately 2,000 nodes in the network; how-
ever, each set of curves would be unique depending on the prox-
imity to the location of contaminant introduction and the flow and
transport dynamics near the node.
Fig. 4 shows the disease progression throughout the entire
population (summed over all the nodes). Over the entire popula-
tion served by the water system, a total of 25% of the population
became infected after consuming contaminated water. It is inter-
esting to note that the shape of these curves can change dramati-
cally with different parameters. For some diseases, the latency
period and disease duration are not equal; therefore, the / and I)
50 100 150
Volume of Contaminant
200
Fig, 6. Sensitivity of infections to volume of contaminant introduced
curves have different shapes. Figs. 5-7 show how the cumulative
infections vary with three parameters: (1) the time of day of con-
taminant introduction; (2) the volume of contaminant introduced;
and (3) the slope of the dose-response curve. In this example, the
number of infections changes with the time of day of contaminant
introduction, through the change is less than 1% of the total popu-
lation. The number of infections increases with the volume of
contaminant introduced, though it is not a linear relationship. Fi-
nally, as the slope of the dose-response curve increases and the
curve steepens, the number of infections increases. Given that the
data used to generate dose-response curves is often sparse and
sometimes conflicting, this represents a source of great uncer-
tainly in the model.
Information from Fig. 4 could be used by decision makers to
plan public health and utility intervention strategies. In this case,
the biological agent has a one-week latency period during which
people would not yet be symptomatic or aware of the illness. This
is a long period, during which a drinking water contamination
warning system could provide the first detection of the incident.
Following detection of a contamination incident, the utility could
identify the contaminant through laboratory analysis and provide
information to the public health sector about which neighbor-
hoods were likely exposed to the specific contaminant, thereby
informing the public health process. The data in Fig. 4 also
shows, however, that the utility would need to detect and respond
very rapidly in order to prevent exposure. Indeed, within 20 h of
the contamination event, more than 50% of the exposures have
already occurred. The modeling framework can also be used to
compare the costs and benefits of various intervention strategies.
Variation in Total Infections with the Time of Day
of Contaminant Introduction
0 3 6 9 12 15 18 21 24
Time of Day of Contaminant Introduction
Fig, 5. Sensitivity of infections to time of day of contaminant
introduction
Variation in Total Infections with the Slope of the
Dose-Response Curve
27.5
1.00E+08 1.00E+07 1.00E+06 1.00E+05 1.00E+04
Tau in Dose-Reqionse Curve (related to dope)
Fig. 7. S ensitivity of infections to slope of dose-response curve
-------
Conclusions
This paper presents a new framework for estimating the spatial
and temporal distribution of health impacts resulting from inges-
tion of contaminated drinking water. The example in this paper
shows that the method is not restricted to small problems but can
be applied even to large drinking water systems. Moreover, the
method is flexible enough to accommodate most types of diseases
that could be transmitted through water. The model could be ex-
tended to incorporate exposure to contaminated water through
dermal and inhalation routes.
Though the focus of this paper is on describing the models and
methods in detail, there are many important and useful applica-
tions that can be studied in future papers. This framework can be
applied to both accidental and intentional contamination sce-
narios. Given the necessary parameter values for the health im-
pacts of contaminants, the framework could be used to estimate
the potential health risks of accidental backfiows and intrusion
events. Combined with flow information calculated in EPANET,
the economic impacts of contamination events could be estimated
(including public health costs and water utility cleanup and recov-
ery costs). In addition, the public health costs and benefits of
control options such as flushing and superch 1 orination could be
examined.
In understanding the threat of intentional contamination of
drinking water, this framework provides several useful tools.
First, the number of infections or fatalities could be used as a
metric in determining the optimal number and location of con-
tamination warning system sensors (Berry et al. 2005; IJber et al.
2004b). Many methods for locating sensors attempt to minimize
quantities such as the volume of contaminated water; however,
the number of infections may be a more accurate reflection of
public risk. In addition, the disease model could allow decision
makers to determine the contaminant-specific time available for
effective public health intervention strategies, such as vaccina-
tion, treatment, and "Do Not Drink" or "Do Not Use" orders.
Finally, given the spatial estimates of health risk, decision makers
could identify and prioritize populations and regions in the most
urgent need of public health intervention.
References
Anderson, R. M., and May, R. M. (1991). Infectious diseases of humans:
Dynamics and control, Oxford University Press. Oxford, U.K.
Barrett, C. L., Eubank, S. G., and Smith, J. P. (2005). If smallpox strikes
Portland .... Scientific American. San Francisco.
Berry, I., Fleischer, L., Hart, W., Phillips, €., and Watson, J. P. (2004).
"Sensor placement in municipal water networks." J. Water Resour.
Plan. Manage., 131(3), 237-243.
Blackburn. B., et al. (2004). "Surveillance for waterborne-disease
outbreaks—United States, 2001-2002." MMWR surveillance summa-
ries, 53(No. SS-9).
Centers for Disease Control and Prevention (CDC). (1990). MMWR,
39(No. SS-1).
Chick, S. E., Koopman, J. S., Soorapanth, S., and Brown, M. E. (2001).
"Infection transmission system models for microbial risk assessment."
Sci. Total Environ., 21 A, 197-207.
Chick, S. E., Soorapanth, S., and Koopman, J. S. (2003). "Inferring in-
fection transmission parameters that influence water treatment deci-
sions." Manage. Sci., 49(7), 920-935.
Codeco, C. T. (2001). "Endemic and epidemic dynamics of cholera: The
role of the aquatic reservoir." BMC infectious diseases, 1:1.
Colford, J. M., Eisenberg. D. ML, Eisenberg, .1. N. S., Scott. J., and Seller.
J. A. (2003). "A dynamic model to assess microbial health risks as-
sociated with beneficial uses of biosolids—Phase 1." Rep. 98-REM-l,
Water Environment Research Foundation, Alexandria. Va.
Eisenberg, J. N. S., Seto, E. Y. W., and Colford, J. M. (1998). "An analy-
sis of the Milwaukee Cryptosporidium outbreak based on a dynamic
model of disease transmission." Epidemiology, 9, 255-263.
Eisenberg, J. N. S., Brookhart, M. A., Rice, G., Brown, M., and Colford,
J. M. (2002). "Disease transmission models for public health decision
making: Analysis of epidemic and endemic conditions caused by
waterborne pathogens." Environ. Health Perspect., 110(8), 783-790.
Haas, C. N., Rose, J. B., and Gerba, C. P. (1999). Quantitative microbial
risk assessment, Wiley, New York.
Propato, M., and Uber, J. (2004). "Vulnerability of water distribution
systems to pathogen intrusion: How effective is a disinfection re-
sidual?" Environ. Sci. Technol., 38(13), 3713-3723.
Rossman, L. (2000). The EPANET 2 user's manual, U.S. Environmental
Protection Agency, Washington, D.C.
Uber, J., Janke, R., and Murray, R. (2004a). "Use of systems analysis to
assess and minimize water security risks." J. Contemp. Water Res.
Edit.. 129, 34-40.
Uber, J.. Janke, R., Murray, R.. and Myer, P. (2004b). "Agreedy heuristic
method for locating water quality sensors in distribution systems."
Proc., ASCE/EWRl Congress, ASCE, Reston, Va., 1-9.
Uber, J., Shang, E, and Rossmann, L. (2004c). "Extensions to EPANET
for fate and transport of multiple interacting chemical or biological
components." Proc., ASCE/EWRl Congress, ASCE, Reston, Va.
U.S. Environmental Protection Agency (USEPA). (2005). "Technologies
and techniques for early warning systems to monitor and evaluate
drinking water quality: A state-of-the-art review." EPA600/R-05/156,
(http://www.epa.gov/nhsrc/pubs/reportEWS 120105.pdf) (March 27.
2006).
Van den Driessche, P. (2002). "Time delay in epidemic models." Math-
ematical approaches for emerging and re-emerging infectious dis-
eases: An introduction, Springer, New York, 119-128.
Walski, T. M., Chase, D. V, Savic, D. A., Grayman, W., Beckwith, S.,
and Koelle, E. (2003). Advanced water distribution modeling and
management, Ilaested, Waterbury, Conn.
------- |