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\AtBeginDocument{{\noindent\small {\em
Electronic Journal of Differential Equations},
Vol. 2007(2007), No. 115, pp. 1--12.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu
or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu (login: ftp)}
\thanks{\copyright 2007 Texas State University - San Marcos.}
\vspace{9mm}}
\begin{document}
\title[\hfilneg EJDE-2007/115\hfil An epidemiological model]
{An epidemiological model of Rift Valley fever}
\author[H. D. Gaff, D. M. Hartley, N. P. Leahy\hfil EJDE-2007/115\hfilneg]
{Holly D. Gaff, David M. Hartley, Nicole P. Leahy} % in alphabetical order
\address{Holly D. Graff \newline
College of Health Sciences, Old Dominion University, Norfolk VA
23529, USA}
\email{hgaff@odu.edu}
\address{David. M. Hartley \newline
Georgetown University School of Medicine, Washington, DC 20007,
USA}
\email{hartley@isis.georgetown.edu}
\address{Nicole P. Leahy \newline
Department of Epidemiology and Preventive Medicine\\
University of Maryland School of Medicine\\
Baltimore, MD 21201, USA}
\email{nicole.leahy@jax.org}
\thanks{Submitted October 10, 2006. Published August 22, 2007.}
\subjclass[2000]{34A12, 34D05, 92B05}
\keywords{Rift Valley fever; mosquito-borne disease; livestock disease; \hfill\break\indent
mathematical epidemiology; compartmental model; sensitivity analysis}
\begin{abstract}
We present and explore a novel mathematical model of the epidemiology of
Rift Valley Fever (RVF). RVF is an Old World, mosquito-borne disease
affecting both livestock and humans. The model is an ordinary differential
equation model for two populations of mosquito species, those that can
transmit vertically and those that cannot, and for one livestock population.
We analyze the model to find the stability of the disease-free equlibrium
and test which model parameters affect this stability most significantly.
This model is the basis for future research into the predication of
future outbreaks in the Old World and the assessment of the threat of
introduction into the New World.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\allowdisplaybreaks
\section{Introduction}
Rift Valley fever virus (RVFV; family: Bunyaviridae, genus
\emph{Phlebovirus}) is a mosquito-borne pathogen causing febrile
illness in domestic animals (e.g., sheep, cattle, goats) and
humans. Outbreaks of Rift Valley fever (RVF) are associated with
widespread morbidity and mortality in livestock and morbidity in
humans. Identified in Kenya in 1930~\cite{DaubneyEtAl1930}, RVF is
often considered a disease primarily of sub-Saharan Africa, though
outbreaks occurred in Egypt in 1977 and
1997~\cite{MeeganEtAl1981a,chevalier2004}. Recent translocation to
Saudi Arabia and Yemen
\cite{JuppEtAl2002a,AlAfaleqEtAl2003a,AlHazmiEtAl2003a,MadaniEtAl2003a}
demonstrate the ability of RVFV to invade ecologically diverse
regions. The virus has never been observed in the Western
Hemisphere, and it is feared that introduction could have
significant deleterious impact on human and agricultural health.
In light of the recent North American introduction and rapid
spread of West Nile virus throughout the continent
\cite{Gubler2002a,RappoleEtAl2000a}, it seems prudent to develop a
mathematical model that could enable us to examine the potential
dynamics of RVF should it appear in the Western
Hemisphere~\cite{HouseEtAl1992a}.
In Africa, the disease is spread by a number of mosquito species
to livestock such as cattle, sheep and goats. Some of these
mosquito species are infected only directly through feeding on
infectious livestock, while others species also can be infected at
birth by vertical transmission, i.e.,
mother-to-offspring~\cite{Peters1997a}. RVF in livestock will
cause abortions in pregnant animals and mortality rates as high as
90\% in young animals and 30\% in
adults~\cite{ErasmusCoetzer1981a}. While humans can be infected
with RVF, we restrict our focus in this study to livestock
populations.
\section{The RVF model}
We construct a compartmental, ordinary differential equation
(ODE) model of RVFV transmission based on a simplification of the
picture described above. The model considers two populations of
mosquitoes (one exhibiting vertical transmission and the other
not) and a population of livestock animals with disease-dependent
mortality.
The model is depicted schematically in Figure~\ref{system}. One
population of vectors represent \emph{Aedes} mosquitoes (model
population \#1), which can be infected through either vertically
or via a blood meal from an infectious host (model population
\#2). The other vector population is able to transmit RVFV to
hosts but not to their offspring; here we consider it to be a
population of \emph{Culex} mosquitoes (model population \#3). Once
infectious, mosquito vectors remain infectious for the remainder
of their lifespan. Infection is assumed not to affect mosquito
behavior or longevity significantly. Hosts, which represent
various livestock animals, can become infected when fed upon by
infectious vectors. Hosts may then die from RVFV infection or
recover, whereupon they have lifelong immunity to
reinfection~\cite{Wilson1994a}. Neither age structure nor spatial
effects are incorporated into this model.
Populations contain a number of susceptible ($S_i$), incubating
(infected, but not yet infectious) ($E_i$) and infectious ($I_i$)
individuals, $i=1,2,3$. Infected livestock will either die from
RVFV or will recover with immunity ($R_2$). To reflect the
vertical transmission in the \emph{Aedes} species, compartments
for uninfected ($P_1$) and infected ($Q_1$) eggs are included. As
the \emph{Culex} species cannot transmit RVF vertically, only
uninfected eggs ($P_3$) are included. Adult vectors emerge from
these compartments at the appropriate maturation rates. The size
of each adult mosquito population is $N_i = S_i + E_i + I_i$, for
$i = 1\mbox{ and } 3$. The livestock population is modeled using
a logistic population model with a given carrying capacity, $K_2$.
The total livestock population size is $N_2 = S_2 + E_2 + I_2 +
R_2$.
The system of ODEs representing the populations is given below:
\emph{Aedes} mosquito vectors
\begin{align*} % 1--6
\frac{dP_1}{dt} &= b_1\left(N_1-q_1I_1\right)-\theta_1P_1\\
\frac{dQ_1}{dt} &= b_1q_1I_1-\theta_1Q_1\\
\frac{dS_1}{dt} &= \theta_1P_1-d_1S_1-\frac{\beta_{21}S_1I_2}{N_2}\\
\frac{dE_1}{dt} &= -d_1E_1+\frac{\beta_{21}S_1I_2}{N_2}-\varepsilon_1E_1\\
\frac{dI_1}{dt} &= \theta_1Q_1-d_1I_1+\varepsilon_1E_1\\
\frac{dN_1}{dt} &= (b_1 - d_1) N_1
\end{align*}
Livestock hosts
\begin{align*} %7-11
\frac{dS_2}{dt} &= b_2N_2- \frac{d_2S_2N_2}{K_2}-\frac{\beta_{12}S_2I_1}{N_1}-\frac{\beta_{32}S_2I_3}{N_3}\\
\frac{dE_2}{dt} &= -\frac{d_2E_2N_2}{K_2}+\frac{\beta_{12}S_2I_1}{N_1}+\frac{\beta_{32}S_2I_3}{N_3}-\varepsilon_2E_2\\
\frac{dI_2}{dt} &= -\frac{d_2I_2N_2}{K_2}+\varepsilon_2E_2-\gamma_2I_2-\mu_2I_2\\
\frac{dR_2}{dt} &= -\frac{d_2R_2N_2}{K_2}+\gamma_2I_2\\
\frac{dN_2}{dt} &= N_2 (b_2 - \frac{d_2 N_2}{K_2}) - \mu_2 I_2
\end{align*}
\emph{Culex} mosquito vectors
\begin{align*} %12--16
\frac{dP_3}{dt} &= b_3N_3 - \theta_3P_3\\
\frac{dS_3}{dt} &= \theta_3P_3-d_3S_3-\frac{\beta_{23}S_3I_2}{N_2}\\
\frac{dE_3}{dt} &= -d_3E_3+\frac{\beta_{23}S_3I_2}{N_2}-\varepsilon_3E_3\\
\frac{dI_3}{dt} &= -d_3I_3+\varepsilon_3E_3\\
\frac{dN_3}{dt} &= (b_3 - d_3) N_3,
\end{align*}
where:
\begin{align*}
\beta_{12} &= \text{adequate contact rate: \emph{Aedes} to livestock}\\
\beta_{21} &= \text{adequate contact rate: livestock to \emph{Aedes}}\\
\beta_{23} &= \text{adequate contact rate: livestock to \emph{Culex}}\\
\beta_{32} &= \text{adequate contact rate: \emph{Culex} to livestock}\\
1/d_1&= \text{lifespan of \emph{Aedes} mosquitoes}\\
1/d_2 &= \text{lifespan of livestock animals}\\
1/d_3 &= \text{lifespan of \emph{Culex} mosquitoes}\\
b_1 &= \text{number of \emph{Aedes} eggs laid per day}\\
b_2 &= \text{daily birthrate in livestock}\\
b_3 &= \text{number of \emph{Culex} eggs laid per day}\\
K_2 &= \text{carrying capacity of livestock}\\
1/\varepsilon_1 &= \text{incubation period in \emph{Aedes}} \\
1/\varepsilon_2 &= \text{incubation period in livestock} \\
1/\varepsilon_3 &= \text{incubation period in \emph{Culex}} \\
1/\gamma_2 &= \text{infectiousness period in livestock} \\
\mu_2 &= \text{RVF mortality rate in livestock} \\
q_1 &= \text{transovarial transmission rate in \emph{Aedes}} \\
1/\theta_1 &= \text{development time of \emph{Aedes}}\\
1/\theta_3 &= \text{development time of \emph{Culex}}.
\end{align*}
Approximate parameters values for the model are given in
Table~\ref{paramdef}. Since there are no direct measures for the
adequate contact rates, these values are calculated as
$\beta_{ij} = c_x f_x r_{ij} / g_{x}$, where $x=i$ or
$x=j$ and $i \neq j$ and $x$ is a mosquito population. The value
$c_x$ is the feeding rate per gonotrophic cycle of mosquito
population $x$, $f_x$ is the probability that a mosquito of
population $x$ will feed on livestock, $r_{ij}$ is the rate of
successful RVF transmission per bite from population $i$ to $j$,
and $g_x$ is the length of the gonotrophic cycle in days of
mosquitoes in population~$x$.
We analyzed the resulting model by computing the fundamental
reproduction ratio and sensitivity of model output to variation or
uncertainty in biological parameters. Using numerical simulation
based on parameter estimates obtained from the literature, we have
investigated the expected vector and host species prevalence in
epidemic and endemic situations, as well as the expected risk of
epidemic transmission of introduced into virgin areas.
\section{Stability Analysis}
For epidemiology models, a quantity, $\mathscr{R}_0$, is derived
to assess the stability of the disease free equilibrium.
$\mathscr{R}_0$ represents the number of secondary cases that are
caused by a single infectious case introduced into a completely
susceptible population~\cite{AndersonMay1991a,Heffernan2005}. When
$\mathscr{R}_0<1$, if a disease is introduced, there are
insufficient new cases per case, and the disease cannot invade the
population. When $\mathscr{R}_0>1$, the disease may become
endemic; the greater $\mathscr{R}_{0}$ is above 1, the less likely
stochastic fade out of the disease is to occur. Unlike values of
$\mathscr{R}_0$ for strictly directly-transmitted diseases, the
magnitude of the reproduction ratio does not necessarily scale in
proportion to the intensity of epidemic/epizootic transmission.
It is possible to compute an analytical expression for the basic
reproduction number, $\mathscr{R}_0$, for this model by combining
two previously published
techniques~\cite{LipsitchEtAl1995a,vdDriesscheWatmough2002a}.
Since the model incorporates both vertical and horizontal
transmission, $\mathscr{R}_0$ for the system is the sum of the
$\mathscr{R}_0$ values for each mode of transmission determined
separately~\cite{LipsitchEtAl1995a},
\[
\mathscr{R}_0 = \mathscr{R}_{0,V} + \mathscr{R}_{0,H}.
\]
To compute each component of $\mathscr{R}_0$, we express the model
equations in vector form as the difference between the rate of new
infection in compartment $i$, $\mathscr{F}_i$, and the rate of
transfer between compartment $i$ and all other compartment due to
other processes, $\mathscr{V}_i$~\cite{vdDriesscheWatmough2002a}.
First, we calculate the basic reproduction number for the vertical
transmission route, $\mathscr{R}_{0,V}$. For this case, the only
compartments involved are the infected eggs, exposed adults, and
infectious adults of the \emph{Aedes} population. Thus we have, in
the notation of reference~\cite{vdDriesscheWatmough2002a},
\[
\label{vectorform} \frac{d}{dt}\begin{bmatrix} Q_1\\E_1\\I_1
\end{bmatrix} = \mathscr{F}_V-\mathscr{V}_V =
\begin{bmatrix} 0\\ 0\\ \theta_1Q_1 \end{bmatrix}
- \begin{bmatrix} -b_1q_1I_1+\theta_1Q_1\\ \varepsilon_1E_1+d_1E_1\\
-\varepsilon_1E_1+d_1I_1
\end{bmatrix}.
\]
The corresponding Jacobian matrices about the disease free
equilibrium of the above system are
\[
\label{jacobF} \mathbf{F}_V = \begin{bmatrix} 0&0&0\\ 0&0&0\\
\theta_1&0&0 \end{bmatrix}, \quad \mathbf{V}_V =
\begin{bmatrix} \theta_1 &0& -b_1q_1\\ 0& d_1+\varepsilon_1 &0\\
0 & -\varepsilon_1 & d_1 \end{bmatrix}.
\]
The basic reproduction number for vertical transmission is
calculated as the spectral radius of the next generation matrix,
$\mathbf{F_VV^{-1}_V}$,
\[
\label{vertonly}
\mathscr{R}_{0,V} = \frac{b_1q_1}{d_1}.
\]
Next, we calculate the horizontal transmission basic reproduction
number, $\mathscr{R}_{0,H}$. For this mode of transmission we must
evaluate the exposed and infectious compartments of the
\emph{Aedes}, \emph{Culex} and livestock populations. Disease
related mortality within the livestock population results in a
non-constant livestock population size. To simplify the
calculation of $\mathscr{R}_0$, we transform our system to
consider the percent of the population made up by each
compartment, $x_i = \frac{X_i}{N_i}$, where $X_i$ is a compartment
of population $i$,
\[
\frac{d}{dt}\begin{bmatrix} e_1\\i_1\\e_2\\i_2\\e_3\\i_3
\end{bmatrix} = \mathscr{F}_H -
\mathscr{V}_H = \begin{bmatrix} \beta_{21}s_1i_2\\0\\
\beta_{12}s_2i_1+\beta_{32}s_2i_3\\0\\ \beta_{23}s_3i_2\\0
\end{bmatrix}
- \begin{bmatrix} d_1e_1+\varepsilon_1e_1\\ d_1i_1-\varepsilon_1e_1\\
d_2k_2e_2+\varepsilon_2e_2\\ -\varepsilon_2e_2 +
d_2k_2i_2+\gamma_2i_2+\mu_2i_2\\ d_3e_3+\varepsilon_3e_3\\
d_3i_3-\varepsilon_3e_3 \end{bmatrix},
\]
where $k_2 \equiv \frac{N_2}{K_2}$. As before, we calculate the
matrices $\mathscr{F}_H$ and $\mathscr{V}_H$,
\[
\mathbf{F}_H = \begin{bmatrix} 0 & 0 & 0 & \beta_{21} & 0 & 0 \\ 0
& 0 & 0 & 0 & 0 & 0 \\ 0 & \beta_{12} & 0 & 0 & 0 & \beta_{32} \\
0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & \beta_{23} & 0 & 0 \\ 0 & 0 &
0 & 0 & 0 & 0 \end{bmatrix},
\]
\[
\mathbf{V}_H = \begin{bmatrix} d_1+\varepsilon_1 & 0 & 0 & 0 & 0 & 0 \\
-\varepsilon_1 &
d_1 & 0 & 0 & 0 & 0 \\ 0 & 0 & d_2k_2+\varepsilon_2 & 0 & 0 & 0 \\
0 & 0 & -\varepsilon_2 & d_2k_2+\gamma_2+\mu_2 & 0 & 0 \\ 0 & 0 &
0 & 0 & d_3+\varepsilon_3 & 0 \\ 0 & 0 & 0 & 0 & -\varepsilon_3 &
d_3 \end{bmatrix}.
\]
The spectral radius of $\mathbf{F_HV^{-1}_H}$ results in,
\[
\label{horzonly}
\mathscr{R}_{0,H} = \sqrt{
\frac{\varepsilon_2}{(d_2k_2+\varepsilon_2)
(d_2k_2+\gamma_2+\mu_2)}
\Big(\frac{\varepsilon_1\beta_{12}\beta_{21}}{d_1
(d_1+\varepsilon_1)}
+\frac{\varepsilon_3\beta_{32}\beta_{23}}{d_3(d_3+\varepsilon_3)}
\Big) }.
\]
Thus, we get
\[ %\label{Rnaught}
\mathscr{R}_0
= \frac{b_1q_1}{d_1} + \sqrt{\frac{\varepsilon_2}
{(d_2k_2+\varepsilon_2)(d_2k_2+\gamma_2+\mu_2)}
\Big( \frac{\varepsilon_1\beta_{12}\beta_{21}}{d_1(d_1+\varepsilon_1)}
+\frac{\varepsilon_3\beta_{32}\beta_{23}}{d_3(d_3+\varepsilon_3)} \Big) } .
\]
The first term in the sum corresponds to direct transmission,
i.e., RVFV travels vertically from {\it Aedes} to {\it Aedes}
mosquito, whereas the second term corresponds to indirect (vector
borne) transmission; virus transport between vectors is mediated
by mammalian hosts. This vector-host-vector viral transmission
path is the nature of the square
root~\cite{Deikmann1995,Heffernan2005}.
Biologically, we understand the expression for $\mathscr{R}_0$ as
follows: the $\mathscr{R}_{0,V}$ corresponds to the product of the
mean number of eggs laid over an average floodwater {\it Aedes}
mosquito lifespan ($\frac{b_1}{d_1}$), and the fraction of those
eggs that are infected with RVFV transovarially ($q_1$).
$\mathscr{R}_{0,H}$ is comprised of two parts, corresponding to
the {\it Aedes}-livestock interaction and the {\it
Culex}-livestock interaction. The terms
$\frac{\epsilon_j}{d_j+\epsilon_j}$ represent the probability of
adult {\it Aedes} ($j=1$) or {\it Culex} ($j=3$) mosquitoes
surviving through the extrinsic incubation period to the point
where they can become infectious. Similarly, the term
$\frac{\epsilon_2}{d_2k_2+\epsilon_2}$ corresponds to the
probability that livestock survive to the point where they are
infectious. The $\frac{\beta_{12}}{d_1}$ represents the mean
number of bites {\it Aedes} make throughout the course of their
lifetimes, and similarly for $\frac{\beta_{32}}{d_3}$ in the case
of {\it Culex} mosquitoes. Finally, the mean number of times a
livestock animal is bitten by {\it Aedes} or {\it Culex} species
during the time these vectors are infectious is
$\frac{\beta_{2j}}{d_2k_2 + \gamma_2 + \mu_2}$ for $j=1$ and $3$,
respectively.
\section{Model sensitivity analysis}
Many of the parameters for this model cannot be estimated directly
from existing research. We employed the technique of Latin
hypercube sampling to test the sensitivity of the model to each
input parameter in an approach successfully applied in the past to
many other disease
models~\cite{BlowerDowlatabadi1994a,SanchezBlower1997a,gammack2005}.
Latin hypercube sampling is a stratified sampling technique that
creates sets of parameters by sampling for each parameter
according to a predefined probability distribution. For each
parameter, we assumed a uniform distribution across the ranges
listed in Table~\ref{paramdef}. We then solved the system
numerically using a large set ($n=5000$) of sampled model
parameters. From these results, we calculated a variety of metrics
of model sensitivity including $\mathscr{R}_0$, maximum number of
animals infected, time to reach that maximum and others, to assess
the impact of each parameter on the model results. We used the
partial rank correlation coefficient to assess the significance of
each parameter with respect to each metric. The most significant
parameters were found to be $\beta_{12}, \beta_{21}, \beta_{23},
\beta_{32}$, (adequate contact rates), $\gamma$ (period of
infectiousness in livestock) and $d_3, d_1$ (vector lifespan)
(Table~\ref{prcctable}). Averaging $\mathscr{R}_0$ over all
parameter sets gives a mean of 1.19 (95\% confidence interval:
1.18, 1.21) and a median of 1.11 (Figure~\ref{R0hist}).
$\mathscr{R}_0$ ranged from 0.037 to 3.743.
\section{\label{ModPred} Numerical Simulations}
To explore the behavior of RVF when introduced into a na\"{i}ve
environment, we conducted numerical simulations of an isolated
system (i.e., no immigration or emigration). The model uses a
daily time step and is solved by a fourth order Runge-Kutta
scheme. For each simulation, we start with 1000 susceptible
livestock animals, 1000 \emph{Culex} eggs, 999 \emph{Aedes}
susceptible eggs, 1 \emph{Aedes} infected egg and 1 susceptible
\emph{Aedes} adult mosquito.
To assess the expected vector and host species prevalence in
epidemic and endemic situations, we ran four simulations. For the
first two, we used a relatively high set of values for the
adequate contact rates, $\beta_{ij}$, which would be appropriate
for settings where mosquitoes feed almost exclusively on the
livestock population. The contact rate for the other simulations
were lower, corresponding to settings where there are other
suitable hosts for the mosquito, but these other hosts do not
otherwise influence the dynamics of RVF. Each set of contact rates
were used for a simulation using the higher RVF-associated
mortality of sheep and a simulation using the lower RVF-associated
mortality of cattle.
The percent of livestock infected through time, for specific
simulations, are shown in Figure~\ref{progression}. For these
simulations, we define the ``high set for $\beta$'' as $\beta_{12}
= 0.48$ $\beta_{21} = 0.395$ $\beta_{23} = 0.56$ $\beta_{32} =
0.13$, and ``low set for $\beta$'' as $\beta_{12} = 0.15$
$\beta_{21} = 0.15$ $\beta_{23} = 0.15$ $\beta_{32} = 0.05$. We
also use a case fatality rate of $0.25$ or $0.15$ which gives us
$\mu_2 = 0.0312$ or $\mu_2 = 0.0176$, respectively. For
simulations where $\beta_{ij}$ is high, the initial outbreaks were
sufficiently large that it was necessary to break to y-axis to
demonstrate subsequent outbreaks. Figure~\ref{progression}(a)
shows that with lower estimates for contact rates and the death
rate associated with sheep, after an initial epidemic reaching a
maximum of 0.05\%, the disease dies out for all lifespans.
Figure~\ref{progression}(b) shows that with lower estimates for
contact rates and the death rate associated with cattle, after an
initial epidemic reaching a maximum under 0.13\%, the disease
remains endemic with multiple epidemics prior to a steady state
infection level. The frequency of the subsequent epidemics
reflects the turnover rate of the cattle population.
Figure~\ref{progression}(c) shows that with higher $\beta_{ij}$
values and sheep fatality estimates, after an initial epidemic
reaching over 10\% infected, there are subsequent epidemics with
the final endemic levels of between 0.1 and 0.4\%.
Figure~\ref{progression}(d) shows that with higher $\beta_{ij}$
values and cattle fatality estimates, after an initial epidemic
reaching over 10\% infected, there are subsequent epidemics with
the final endemic levels of between 0.1 and 0.2\%. In all cases,
there is transmission following introduction, albeit at low levels
in the case of the lower $\beta$ values. For all but the lower
$\beta$ with sheep mortality cases, the disease attains a low
level of endemic prevalence after a sequence of epidemics,
suggesting the disease could persist if introduced into an
isolated system.
\section{\label{Imps}Conclusions}
The model presented is a simplified representation of the complex
biology involved in the epidemiology of RVF. As in all models,
much of the value lays in the process of building the model, which
forces researchers to carefully state the many assumptions they
build their thinking upon~\cite{McKenzie2000a}. Relaxation of
model assumptions such as inclusion of age-structure or spatial
variation may demonstrate additional insights. We hope this model
and these results will act as a catalyst to further investigation.
\begin{table}[ht]
\caption{\label{paramdef} Parameters with estimated ranges
for numerical simulations}
\begin{center}
\begin{footnotesize}
\begin{tabular}{|cccc|}\hline
Parameter & (Range) & Units & Reference\\ \hline
$\beta_{12}$ & (0.0021, 0.2762) & 1/day & \cite{CanyonEtAl1999a,HayesEtAl1973a,JonesLloyd1985a,Magnarelli1997a,PrattMoore1993a,TurellEtAl1988a,TurellEtAl1988b}\\
$\beta_{21}$ & (0.0021, 0.2429) & 1/day & \cite{CanyonEtAl1999a,HayesEtAl1973a,JonesLloyd1985a,Magnarelli1997a,McIntoshJupp1981a,PrattMoore1993a,TurellBailey1987a}\\
$\beta_{23}$ & (0.0000, 0.3200) & 1/day & \cite{HayesEtAl1973a,JonesLloyd1985a,Magnarelli1997a,McIntoshJupp1981a,PrattMoore1993a,TurellBailey1987a,WekesaEtAl1997a}\\
$\beta_{32}$ & (0.0000, 0.0960) & 1/day & \cite{HayesEtAl1973a,JonesLloyd1985a,Magnarelli1997a,PrattMoore1993a,WekesaEtAl1997a}\\
$1/d_1$ & (3, 60) & days & \cite{Bates1970a,MooreEtAl1993a,PrattMoore1993a}\\
$1/d_2$ & (360, 3600) & days & \cite{Radostits2001a}\\
$1/d_3$& (3, 60) & days & \cite{Bates1970a,MooreEtAl1993a,PrattMoore1993a}\\
$b_1$ & $d_1$ & 1/day & \\
$b_2$ & $d_2$ & 1/day & \\
$b_3$ & $d_3$ & 1/day & \\
$1/\varepsilon_1$ & (4, 8) & days & \cite{TurellKay1998a}\\
$1/\varepsilon_2$ & (1, 6) & days & \cite{PetersLinthicum1994a}\\
$1/\varepsilon_3$ & (4, 8) & days & \cite{TurellKay1998a}\\
$1/\gamma_2$ & (1, 5) & days & \cite{ErasmusCoetzer1981a}\\
$\mu_2$& (0.025, 0.1) & 1/day & \cite{ErasmusCoetzer1981a,PetersLinthicum1994a}\\
$q_1$ & (0.0, 0.1) &--- & \cite{FreierRosen1987a}\\
$1/\theta_1$ & (5, 15) & days & \cite{PrattMoore1993a}\\
$1/\theta_3$ & (5, 15) & days & \cite{PrattMoore1993a} \\ \hline
\end{tabular}
\end{footnotesize}
\end{center}
\end{table}
\begin{table}[ht]
\caption{\label{prcctable}Results of sensitivity testing using partial rank
correlation coefficients. Results were comparable for all metrics;
only those for $\mathscr{R}_0$ are shown.}
\begin{center}
\begin{tabular}{|ccc|} \hline
parameter & $\mathscr{R}_0$ PRCC & Significance \\ \hline
$\beta_{12}$ & 25.66 & $p<0.001$ \\
$\beta_{21}$ & 26.28 & $p<0.001$ \\
$\beta_{32}$ & 13.21 & $p<0.001$ \\
$\beta_{23}$ & 14.52 & $p<0.001$ \\
$1/\gamma_2$ & -10.55 & $p<0.001$ \\
$1/d_1$ & -11.82 & $p<0.001$ \\
$1/d_3$ & -8.54 & $p<0.001$ \\
$\mu_2$& -2.42 & $p<0.02$ \\ \hline
\end{tabular}
\end{center}
\end{table}
\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.9\textwidth]{fig1} % threeSPblack.eps
\end{center}
\caption{\label{system} Flow diagram of the Rift Valley Fever model }
\end{figure}
\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.8\textwidth]{fig2} % histogram.ps
\end{center}
\caption{\label{R0hist}
Distribution of $\mathscr{R}_0$ values pooling a total of
5000 sets of parameters. The mean is 1.193 (95\% confidence interval: 1.177, 1.209) and a median of 1.113.
The maximum value is 3.743 and then minimum 0.037. }
\end{figure}
\begin{figure}[ht]
\begin{center}
\subfigure[Lower $\beta_{ij}$ and sheep fatality estimates]{
\label{sheeplow}
\includegraphics[width=0.45\textwidth]{fig3a} % SheepLow.eps
}
\subfigure[Lower $\beta_{ij}$ and cattle fatality estimates]{
\label{cowlow}
\includegraphics[width=0.45\textwidth]{fig3b} % CattleLow.eps
}
\subfigure[Higher $\beta_{ij}$ and sheep fatality estimates]{
\label{sheephigh}
\includegraphics[width=0.45\textwidth]{fig3c} % SheepHigh.eps
}
\subfigure[Higher $\beta_{ij}$ and cattle fatality estimates]{
\label{cowhigh}
\includegraphics[width=0.45\textwidth]{fig3d} % CattleHigh2.eps
}
\end{center}
\caption{\label{progression}
Results of numerical simulations for cattle and sheep. Livestock lifespan is
indicated for 10 years (solid line), 5 years (dashed line)
and 2 years (dotted line).}
\end{figure}
\subsection*{Acknowledgements}
We would like to thank C.J. Peters for encouragement to construct,
as well as useful advice on and criticisms of, the model. This
research was supported in part through the Department of Homeland
Security National Center for Foreign Animal and Zoonotic Disease
Defense. The conclusions are those of the authors and not
necessarily those of the sponsor. D.~M.~Hartley is supported by
NIH Career Development Award K25AI58956.
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\end{document}