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\title[Nosocomial Transmission in the SARS Outbreak]
{Critical Role of Nosocomial Transmission in the Toronto SARS Outbreak}
\label{title}
\author[G. F. Webb,
M. J. Blaser, H. Zhu, S. Ardal and J. Wu]{}
\subjclass{92D30} \keywords{severe acute respiratory syndrome (SARS),
coronavirus, epidemiology, mathematical model, nonlinear dynamics,
containment, quarantine.}
\email{glenn.f.webb@vanderbilt.edu}
\email{Martin.Blaser@msnyuhealth.org}
\email{huaiping@mathstat.yorku.ca}
\email{Sten@Cehip.org}
\email{wujh@mathstat.yorku.ca}
\begin{document}
\maketitle
\centerline{\scshape Glenn F. Webb}
\medskip
{\footnotesize \centerline{Department of Mathematics, Vanderbilt University}
\centerline{1326 Stevenson Center, Nashville, TN 37240-0001 } }
\medskip
\centerline{\scshape Martin J. Blaser}
\medskip
{\footnotesize \centerline{Department of Medicine, New York
University School of Medicine}
\centerline{OBV A606, 550 First Avenue, New York, NY 10016 } }
\medskip
\centerline{\scshape Huaiping Zhu}
\medskip
{\footnotesize \centerline{Laboratory for Industrial and
Applied Mathematics, Department of Mathematics and Statistics}
\centerline{York University, 4700 Keele Street, Toronto, ON M3J 1P3, Canada } }
\medskip
\centerline{\scshape Sten Ardal}
\medskip
{\footnotesize \centerline{Central East Health
Information Partnership }
\centerline{Box 159, 4950 Yonge Street, Suite 610, Toronto, ON M2N 6K1, Canada } }
\medskip
\centerline{\scshape Jianhong Wu}
\medskip
{\footnotesize \centerline{
Laboratory for Industrial and Applied Mathematics, Department of
Mathematics and Statistics, }
\centerline{York University, 4700 Keele Street, Toronto, ON M3J 1P3, Canada} }
\medskip
\centerline{(Communicated by Denise Kirschner)}
\medskip
\begin{abstract}
We develop a compartmental mathematical model to address the role of
hospitals in severe acute respiratory syndrome (SARS) transmission
dynamics, which partially explains the heterogeneity of the epidemic.
Comparison of the effects of two major policies, strict hospital infection
control procedures and community-wide quarantine measures, implemented in
Toronto two weeks into the initial outbreak, shows that their combination
is the key to short-term containment and that quarantine is the key to
long-term containment.
\end{abstract}
\section{Introduction}\label{s1}
One of the salient features of the outbreak of severe acute respiratory
syndrome (SARS) in the Greater Toronto Area (GTA) is the role of the
hospital in transmission. Of 144 early patients, 111 (77\%) were exposed
to SARS in the hospital setting; of these, 73 patients (51\%) were
health-care workers, including nurses, respiratory therapists, physicians,
radiology and electrocardiogram technicians, housekeepers, clerical staff,
security personnel, paramedics, and research assistants \cite{Booth03}.
The high risk of transmission within the health-care setting has a
significant impact on the conduct of public-health interventions in the
continuing SARS epidemic \cite{Booth03,Schabas03} and potentially for
other emerging respiratory diseases.
To examine the SARS outbreak in GTA, we develop a compartmental model
dividing the entire population into classes of susceptibles, exposed,
infectives, hospitalized, and removed and subclasses representing the
general public and individuals in the hospital setting including
health-care workers and patients (HCWP). The model reflects the extremely
intense exposure of HCWP to infected individuals prior to awareness of
SARS by the medical community; their heightened risk continued until
adequate precautions were fully operant in hospitals. In two recent
analyses of SARS epidemiology \cite{LR03,Lipsitch03}, all members of the
public were considered as one class, despite evident heterogeneity in
transmission, and the rapid initial spread of SARS in Vietnam, Hong Kong,
and Canada in hospital wards \cite{Dye03}. In our analysis, the secondary
infection induced by a hospitalized patient for the HCWP ($R_0\approx
4.5$) is much larger than the secondary infection induced by an average
infective for the general public ($R_0\approx 1.6$) during the first two
weeks of the SARS outbreak in GTA. These secondary infection rates
decreased when hospital infection control procedures and community-wide
quarantine measures were introduced.
\section{Mathematical models and analysis of dynamics}
Models were developed to correspond to the two stages of the SARS outbreak
in GTA: pre-(Model I) and intra-(Model II) quarantine. Model I consists
of the following compartments: {\it Susceptibles $S$} (individuals not yet
infected); {\it Exposed $E$} (susceptibles who have become infected and
are not yet infectious); {\it Infectives $I$} (exposed individuals who
have become infected and can spread the SARS coronavirus); {\it Removed
$R$} (individuals who have been exposed or infective and who are no
longer considered to be susceptible); and {\it Hospitalized $U$}
(infectives who are in the immediate environment of HCWP; these
individuals are not considered to pose any risk to the general public, but
may infect HCWP). For each class, subindices $g$ and $h$ represent general
public and HCWP, respectively.
\begin{figure}[ht]
\begin{center}
{\includegraphics[angle=0, width=10cm]{figure/scheme.eps}}
\caption{Schematic of SARS transmission when the total population is
divided into the categories of the general public (g) and HCWP (h) who
have direct contact with hospitalized infected patients. Each category is
divided into classes of susceptibles (S), exposed (E), infectives (I), and
hospitalized (U). In Model I, no quarantine (or special hospital
infection control procedures) is implemented and the subclasses of
infectives $(I^Q_g, I^Q_h)$ under quarantine do not exist. Circles $R$
indicate all individuals removed from the infective and hospitalized
classes.} \label{fig.scheme}\end{center}\end{figure}
Model I consists of 8 coupled nonlinear differential equations describing
the transfer of individuals from one compartment to another
(Fig.~\ref{fig.scheme}):
\begin{equation}\label{model1}
\left\{
\begin{array}{lllllllll}
&\frac{d}{dt}S_g(t)=-a_gS_g(t)\Big(I_g(t)+I_h(t)\Big)\\
\\
&\frac{d}{dt} S_h(t)=-a_hS_h(t)\Big(I_g(t)+I_h(t)\Big)
-a_uS_h(t)\Big(U_h(t)+U_g(t)\Big)\\
\\
&\frac{d}{dt} E_g(t)=a_gS_g(t)\Big(I_g(t)+I_h(t)\Big)-b_gE_g(t)\\
\\
&\frac{d}{dt} E_h(t)=a_hS_h(t)\Big(I_g(t)+I_h(t)\Big)
+a_uS_h(t)\Big(U_h(t)+U_g(t)\Big)-b_hE_h(t)\\
\\
&\frac{d}{dt} I_g(t)=b_gE_g(t)-c_gI_g(t)-r_gI_g(t)\\
\\
&\frac{d}{dt} I_h(t)=b_hE_h(t)-c_hI_h(t)-r_hI_h(t)\\
\\
&\frac{d}{dt} U_g(t)=r_gI_g(t)-\epsilon _g U_g(t)\\
\\
&\frac{d}{dt} U_h(t)=r_hI_h(t)-\epsilon _hU_h(t)
\end{array}
\right.
\end{equation}
where $a_g$, $a_h$, and $a_u$ are the transmission coefficients for the
general public and HCWP infectives, and for hospitalized infectives for
HCWP, respectively; $b_g$ and $b_h$ are the transmission coefficients for
exposed individuals to the infective class; $c_g$ and $c_h$ are the
transmission coefficients for infective individuals to the removed class,
and $r_g$ and $r_h$ are the transition coefficients for infectives to
hospitalization. The transition coefficients for the removed class are
$\epsilon _g$ and $\epsilon _h$, reflecting the effectiveness of
treatments. The second equation in (\ref{model1}) describes the additional
risk of HCWP resulting from their direct contact with SARS patients in the
health-care setting.
The time scale considered in Model I (\ref{model1}) is short enough that
all demographic details can be ignored. As a result, each of the total
populations in the subclasses $I_g(t)$, $I_h(t)$, $E_g(t)$, $E_h(t)$,
$U_g(t)$, and $U_h(t)$ eventually approaches zero, and explicit formulae
can be obtained to calculate the total numbers of infected and
hospitalized infectives during the entire course of the infection (see
Appendix for more details).
Although simulations based on Model I correlate with the actual data in
GTA \cite{CA_prof_e} for the first two weeks of the outbreak (Fig. 2), the
model predicted a much larger number of infectives subsequently. On March
31 (two weeks after the first cases), Ontario declared a provincial
emergency to contain the spread of SARS and introduced public-health
measures, including extensive contact tracing, isolation of suspect and
probable cases, and voluntary home quarantine for asymptomatic contacts.
Hospital infection control procedures also were enforced. Clinicians,
initially unaware of the communicability of the SARS coronavirus, had used
positive-pressure ventilation methods to alleviate respiratory symptoms,
inadvertently augmenting dispersion of contagious droplets. Such
treatments were stopped, negative-pressure rooms were used, and face
shields were introduced after masking alone proved insufficient. These
more stringent hospital measures may have been important in controlling
the spread of SARS in GTA \cite{CDC} as well as elsewhere
\cite{LR03,Lee03}, and are reflected in Model II by significant reduction
of the coefficients $a_h$ and $a_u$.
In Model II (2), $S, E, I, R$, and $U$ represent the same groups as in
Model I (Fig. 1). We introduce a second category of infectives $I^Q$
representing infected individuals quarantined before they are admitted
into (or never admitted to) the hospital. They pose no risk to HCWP and a
low risk to the general public. Model II reads
\begin{equation}\label{model2}
\left\{
\begin{array}{llllllllllllllllll}
&\frac{d}{dt}S_g(t)=-a_gS_g(t)\Big(I_g(t)+I_h(t)\Big)\\
\\
&\frac{d}{dt}S_h(t)
=-a_hS_h(t)\Big(I_g(t)+I_h(t)\Big)-a_uS_h(t)\Big(U_h(t)+U_g(t)\Big)\\
\\
&\frac{d}{dt}E_g(t)
=a_gS_g(t)\Big(I_g(t)+I_h(t)\Big)-q_gb_gE_g(t)-(1-q_g)b_gE_g(t)\\
\\
&\frac{d}{dt} E_h(t)
=a_hS_h(t)\Big(I_g(t)+I_h(t)\Big)
+a_uS_h(t)\Big(U_h(t)+U_g(t)\Big)\\
&\qquad\qquad\quad -q_hb_hE_h(t)-(1-q_h)b_hE_h(t)\\
\\
&\frac{d}{dt} I^Q_g(t)
=q_gb_gE_g(t)-c_gI^Q_g(t)-r_gI^Q_g(t)\\
\\
&\frac{d}{dt} I^Q_h(t)
=q_hb_hE_h(t)-c_hI^Q_h(t)-r_hI^Q_h(t)\\
\\
&\frac{d}{dt} I_g(t)
=(1-q_g)b_gE_g(t)-c_gI_g(t)-r_gI_g(t)\\
\\
&\frac{d}{dt} I_h(t)
=(1-q_h)b_hE_h(t)-c_hI_h(t)-r_hI_h(t)\\
\\
&\frac{d}{dt} U_g(t)=r_g(I_g(t)+I^Q_g(t))-\epsilon _g U_g(t)\\
\\
&\frac{d}{dt} U_h(t)=r_h(I_h(t)+I^Q_h(t))-\epsilon _hU_h(t)
\end{array}
\right.
\end{equation}
where parameters $q_g$ and $q_h$ are the fractions of exposed general
public and HCWP that have been quarantined, respectively.
Note that in the case where $q_g=q_h=0$, $I^Q_g=I^Q_h=0$ if their initial
values are zero; thus Model II reduces to Model I. The rates of transfer
from the $E$ class to $I$ class are described by new equations on
$I^Q_g(t)$ and $I_h^Q(t)$ to reflect the effect of quarantine measures
(see Appendix).
This model allows analysis of the dependence of the total number of
infected and hospitalized individuals $X(t)$ at time $t$ on the parameters
$a_g$, $a_h$, $a_u$, $q_g$, and $q_h$. This model also provides an
explicit formula for the lowest possible ultimate number of infected and
hospitalized ($X(\infty)$) when hospital infection control measures and
quarantine measures are strictly enforced, and for the most conservative
estimation of the quarantine fraction for $X(\infty)$ to fall below a
specified level (see Appendix for details).
\section{ Results}
The parameters and initial conditions for the simulations are based on the
1996 census adjusted by 1999 intercensus estimates for the year 2003. The
total population in GTA (by PHU) is the following: PEEL: 1,107,504; CITY
OF TORONTO: 2,620,228; DURHAM: 544,069; HALTON: 398,592; and YORK REGION:
778,295. Also, according to the statistics from Health Canada
\cite{CA_prof_e}, we have the initial population data in
Table~\ref{tab.1}, where the initial $t=0$ corresponds to March 18.
\begin{table}[ht]\begin{center}
\begin{tabular}{|
c|c|c|c|c|c|c|c|}
\hline
$S_g(0)$ & $S_h(0)$ & $E_g(0)$ & $E_h(0)$ &
$I_g(0)$ & $I_h(0)$ & $U_g(0)$ & $U_h(0)$ \\
\hline
5, 443, 104 &3, 000 & 3 & 12 & 1 & 2 & 6 & 1 \\
\hline
\end{tabular}
\caption{Initial population data for Model I, $t=0$ corresponds to March
18.}\label{tab.1}
\end{center}
\end{table}
Although it is difficult to estimate $a_h, a_g$ and $a_u$, we know that
$a_g\le a_h<$
100 days; (b)-(d) depict $I_g$, $I_h$, and $U_g$ as functions of
the time, and illustrate multiple peaks of infection.}
\end{center}\label{fig.4}
\end{figure}
A key issue in the GTA SARS epidemic, and for future outbreaks, is
identifying which of these two policies was most effective in containing
the outbreak. Further simulations of Model II show that hospital control
measures must be strict to contain the virus. The results of the
simulations in Fig. 3 (pre-March 30 values $a_h=100a_g$ and $a_u=1000a_g$
changed to $a_u=a_h$ $=10a_g$) and Fig. 4(a) (no change post-March 30) are
sharply different. In the latter case, even when the quarantine fraction
is high ($q_g=0.8$ and $q_h=0.9$) but nosocomial transmission rates remain
as for Model I, the total hospitalized and removed individuals on May 8th
would be 1324, the outbreak would last $>$ 100 days, and about 3, 000
individuals would be either hospitalized or removed (Fig. 4(a)). The
multiple local maxima indicating possible multiple peaks should hospital
control procedures not be taken (Fig. 4(b)-(d)), mirror the actual
patterns in other regions: an initial peak reflective of early exposures
mostly to HCWP followed by a second peak that includes both those exposed
through hospital and general population contacts \cite{LR03}. Following
this second peak, with improved control measures, the number of new cases
declines as the outbreak comes under control ($R_0<1$).
\begin{figure}[ht]
\begin{center}
\subfigure[ 40 days]
{\includegraphics[angle=0, width=6.3cm]{figure/SARS.July.7.Fig.5a.eps}}
\subfigure[ 150 days]
{\includegraphics[angle=0, width=6.3cm]{figure/SARS.July.7.Fig.5b.eps}}
\caption{The number $X(t)$ of infected and hospitalized at (a): $40$
days, (b): $150$ days, after the first two weeks (t = 0 = March 30) as a
function of the hospital transmission parameter $a_u$ vs. the fraction
quarantined $q = q_g = q_h$. Substantially reducing {\it either} $a_u$
{\it or} increasing $q$ significantly is sufficient to reduce the number
of cases to low levels at $40$ days. At $140$ days reducing hospital
transmission to $0$ without quarantine results in $\approx 25,000$ cases,
whereas increasing the hospital transmission parameter to $a_u = 100 a_g$
and the quarantine fraction to $.5$ results in only $\approx 3,000$ cases.
To contain the epidemic over time, quarantine of exposed individuals is
necessary, because $R_{0} > 1$ for the general public.}
\end{center}\label{fig.anal}
\end{figure}
The simulations of Model II indicate that both hospital infection control
and quarantine measures are important in containing the epidemic. In Fig.
5, we set $q_g = q_h = q$, and all other parameters and initial values are
as in Model I, except that $a_h= a_g$. The total number $X(t)$ of infected
and hospitalized cases are functions of $a_u$ and $q$. By 40 days,
hospital control and quarantine measures, if sufficiently strong,
substantially reduce $X(t)$ (Fig. 5(a)), but quarantine measures are
essential for long-term (150 days) containment of the epidemic (Fig.
5(b)). If enhanced measures for hospital infection control and quarantine
are relaxed prematurely, the number of hospitalized individuals continue
to diminish for several days, and then rise again (Fig. 6). The reported
SARS cases in GTA on May 23 illustrate the consequences of premature
relaxation of infection control measures. \vskip .3cm
\begin{figure}[ht]
\begin{center}
\subfigure[$(t, U_g(t))$]
{\includegraphics[angle=-90,width=5.5cm]{figure/sars_q3_Ug.eps}}
\subfigure[$(t, U_h(t))$]
{\includegraphics[angle=-90, width=5.5cm]{figure/sars_q3_Uh.eps}}
\caption{This simulation shows the consequences of relaxation of hospital
infection control measures and quarantine measures. The simulation starts
on May 9 ($t=0$, using the initial data obtained from Model II) and under
the assumption that parameters $a_h$ and $a_u$ are increased to
$a_h=100a_g$ and $a_u=500a_g$ and $q_g$ and $q_h$ are decreased to
$q_g=20\%$ and $q_h=40\%$, respectively. The total number of the removed
and hospitalized individuals will increase by 32 after 20 days on May 29.}
%\label{fig.sq3}
\end{center}\label{fig.6}
\end{figure}
\section{Discussion} In total, our simulations show that the
combination of moderate quarantine but strict hospital infection control
procedures was the key to the containment of SARS in GTA; increasing the
effectiveness of quarantine $>$ 85\% did not significantly reduce the
total number of hospitalized and removed individuals \cite{Dwosh03}.
Conversely, without strict hospital control procedures, the outbreak
duration will be significantly longer ($>$ 100 days), with multiple peaks
of infection during the entire course and affecting greater numbers of
individuals (Fig. 4).
Our considerations here also may be relevant to the spread of other
respiratory infections, including pneumonic plague or other emerging
respiratory infections. Hospitals have been amplifiers of diseases that
involve the general public [such as influenza (respiratory), and
enteropathogenic {\it E. coli} (enteric)]; our models should be useful to
address issues related to their control. The epidemiology of SARS has been
complex with rapid spread in some areas (e.g., Beijing) but not others,
(e.g. Shanghai) despite introduction of the causative agent, as well as
the control of the epidemic in many localities without continuous
community spread \cite{who}. These patterns are more consistent with
secondary infection in the health-care setting being much greater than in
the general community, with an overall $R_0>1$ \cite{LR03}. The ability of
relatively modest quarantine measures to decrease $R_0<1$ in the general
community suggests a pathogen that is not well-adapted to human hosts, and
implies strong selection for transmission. Thus, stringent control
measures in hospital settings-the major amplifiers of transmission-are
needed to minimize the risk for pandemic spread of SARS.
%\newpage
\section{Appendix: Analysis of dynamics for Models I and II }
In this section, we give the analysis of dynamics for Models I and II. In
particularly, we give the optional quarantine fractions to control the
total number of hospitalized and removed individuals below a specified
level.
\subsection{Total numbers of infected and hospitalized infectives,
without quarantine}
If $U_g(0)=U_h(0)=0$, then
$$
\begin{array}{lll}
&\displaystyle{\lim _{t\to \infty}(I_g(t), I_h(t), E_g(t), E_h(t), U_g(t),
U_h(t))
=(0,0,0,0,0,0)},\\
\\
&\displaystyle{
\lim _{t\to \infty} S_g(t)=S_g(0)e^{-a_g(I_g^\infty+I_h^\infty)}},\\
\\
&\displaystyle{\lim _{t\to \infty} S_h(t)
=S_h(0)e^{-a_h(I_g^\infty+I_h^\infty) -a_u(U_g^\infty+U_h^\infty)}}
\end{array}
$$
where $ I_g^\infty =\int ^\infty _0I_g(s)ds,\ \ U_g^\infty =\int ^\infty
_0U_g(s)ds, \ \ I_h^\infty =\int ^\infty _0I_h(s)ds,\ \ U_h^\infty =\int
^\infty _0U_h(s)ds $ describe the total numbers of infected and
hospitalized infectives during the entire course of infection, and these
values are determined as follows:
$$
\left\{
\begin{array}{lll}
&U_g^\infty =\frac{r_g}{\epsilon _g}I_g^\infty, \,\,\,\, U_h^\infty
=\frac{r_h}{\epsilon _h}I_h^\infty, \\
\\
&S_g(0)+E_g(0)+I_g(0)
=S_g(0)e^{-a_g(I_g^\infty +I_h^\infty
)}+(c_g+\epsilon_g)I_g^\infty,\\
\\
&S_h(0)+E_h(0)+I_h(0)
=S_h(0)e^{-a_h(I_g^\infty+I_h^\infty) -a_u(U_g^\infty+U_h^\infty)}
+(c_h+\epsilon_h)I_g^\infty.\\
\end{array}
\right.
$$
\subsection{ Total numbers of infected and hospitalized infectives,
with quarantine}
For Model II, we have
$$
\begin{array}{lll}
&\displaystyle{\lim _{t\to \infty}
(I_g(t), I_h(t), I_g^Q(t), I_h^Q(t), E_g(t), E_h(t), U_g(t), U_h(t))
=(0,0,0,0,0,0,0,0)},\\
\\
&\displaystyle{
S_g(\infty ):=\lim _{t\to \infty} S_g(t)
=S_g(0)e^{-a_g(I_g^\infty+I_h^\infty)}},\\
\\
&\displaystyle{S_h(\infty ):=\lim _{t\to \infty}
S_h(t)=S_h(0)e^{-a_h(I_g^\infty+I_h^\infty) -a_u(U_g^\infty+U_h^\infty)}}
\end{array}
$$
where
$$
\left\{
\begin{array}{ll}
&I_g^\infty =\int ^\infty _0I_g(s)ds, I_g^{Q\infty} =\int ^\infty
_0I_g^Q(t)dt, E_g^\infty =\int ^\infty _0E_g(t)dt,
U_g^\infty =\int ^\infty _0U_g(s)ds,\\
\\
&I_h^\infty =\int ^\infty _0I_h(s)ds, I_h^{Q\infty}=\int _0^\infty
I_h^Q(t)dt, E_h^\infty =\int _0^\infty E_h(t)dt, U_h^\infty =\int ^\infty
_0U_h(s)ds
\end{array}
\right.
$$
and these parameters are determined by
$$
\left\{
\begin{array}{llllll}
&U_g^\infty =\frac{r_g}{\epsilon _g}(I_g^\infty+I_g^{Q\infty})
+\frac{1}{\epsilon _g}U_g(0),\\
\\
&E_g^\infty =\frac{1}{b_g}E_g(0)
+\frac{1}{b_g}S_g(0)[1-e^{-a_g(I_g^\infty +I_h^\infty )}],\\
\\
&(c_g + r_g)I_g^\infty - I_g(0) - (1 -q_g)b_gE_g^\infty = 0,\\
\\
&(c_h + r_h)I_h^\infty - I_h(0) - (1 - q_h)b_hE_h^\infty = 0,\\
\\
&(c_g + r_g)I_g^{Q\infty} - q_ggb_ggE_g^\infty - I_g^{Q}(0) = 0,\\
\\
&(c_h + r_h)I_h^{Q\infty} - q_hb_hE_h^\infty - I_h^{Q}(0) = 0.
\end{array}
\right.
$$
\subsection{Optional quarantine fractions}
The dependence of $X(\infty )$ on the parameters $(a_h, a_u, q_g, q_h)$
can be calculated. Since
$$
\dot X=\dot U_g+\dot U_h+c_g(I_g^Q+I_h^Q)
+(c_gI_g+c_hI_h)+\epsilon _gU_g+\epsilon _hU_h\\
$$
then $ X(\infty )=X(0)+(c_g+r_g)(I_g^\infty
+I_g^{Q\infty})+(c_h+r_h)(I_h+I_h^{Q\infty}).$
For the sake of simplicity, we let $ q=q_g=q_h, b=b_g=b_h,
\epsilon=\epsilon _g=\epsilon _h, r=r_g=r_h, \alpha =c_g+r_g=c_h+r_h.$
Denote by $ I(0)=I_g(0)+I_h(0), U(0)=U_g(0)+U_h(0), E(0)=E_g(0)+E_h(0), $
and
$
I^\infty =I_g^\infty+I_h^\infty , I^{Q\infty
}=I_g^{Q\infty}+I_h^{Q\infty}, \ U^\infty =U_g^\infty +U_h^\infty ,
E^\infty =E_g^\infty +E_h^\infty$ and recall that $ I_g^Q(0)=I_h^Q(0)=0$,
we get
$$
\begin{array}{lllll}
\alpha I^\infty&=&I(0)+(1-q)bE^\infty,\\
\alpha I^{Q\infty}&=&qbE^\infty,\\
U^\infty&=& \frac{r}{\epsilon}(I^\infty
+I^{Q\infty})+\frac{1}{\epsilon}U(0) = \frac{r}{\epsilon \alpha
}(I(0)+bE^\infty )+\frac{1}{\epsilon}U(0). \end{array}
$$
Therefore,
$$
\begin{array}{lllll}
bE^\infty &=%E(0)+S_g(0)(1-e^{-a_gI^\infty})
%+S_h(0)(1-e^{-a_hI^\infty -a_uU^\infty })\\
E(0) +S_g(0)(1-e^{-\frac{a_g}{\alpha}I(0)})
e^{-\frac{a_g}{\alpha}(1-q)bE^\infty }\\
&\qquad +S_h(0)(1-e^{-\frac{a_h}{\alpha}I(0)}
e^{-\frac{a_u}{\epsilon\alpha}rI(0)-\frac{a_u}{\epsilon }U(0)}
e^{-\frac{a_h}{\alpha}(1-q)bE^\infty } e^{-\frac{a_u}{\epsilon \alpha
}rbE^\infty}).
\end{array}
$$
It is straightforward to show that the above equation always has a unique
positive solution $Y=bE^\infty $, and hence when all others are fixed,
$bE^\infty =f(a_h, a_u, q)$ is a function of $a_h, a_u$, and $q$. This is
an increasing function of either $a_u$ or $a_h$, when all others are
fixed, and a decreasing function of $q$, when all others are fixed. The
same is true for $X(\infty)$, since $X(\infty )=X(0)+I(0)+bE^\infty $.
The above equation also allows calculations of the minimal quarantine
fraction to control the total number of hospitalized and removed
individuals below a specified level. As an illustration, we note that
there clearly is a limit to what can be achieved by both quarantine and
hospital control procedures. In the best possible case, in which $a_u=0$
and $q=1$,
$$
bE^\infty=E(0)+S_g(0)(1-e^{-\frac{a_g}{\alpha}I(0)})
+S_h(0)(1-e^{-\frac{a_g}{\alpha}I(0)})=
E(0)+S(0)(1-e^{-\frac{a}{\alpha}I(0)})
$$
and
$$
X(\infty )=X(0)+E(0)+I(0)+S(0)(1-e^{-\frac{a}{\alpha}I(0)}):=X(best).
$$
Assuming the hospital control procedures are so strictly enforced that
$a_u=0$, and we want to control the outbreak so that $X\le X^*$, a given
number. Note that necessarily, $X^*\ge X(best)$. Let $Y^*=X^*-X(0)-I(0)$.
Then, we can find $q^*\in [0, 1]$ so that
$$
Y^*= E(0)+S_g(0)(1-e^{-\frac{a}{\alpha}I(0)}e^{-\frac{a}{\alpha}(1-q)Y^*})
+S_h(0)(1-e^{-\frac{a}{\alpha}I(0)} e^{-\frac{a}{\alpha}(1-q)Y^*}).
$$
Thus, $X(\infty )\le X^*$ if and only if $q\ge q^*$.
\section*{Acknowledgments}
We thank MITACS and all members of the MITACS project "Transmission
Dynamics and Spatial Spread of Infectious Diseases: Modelling, Prediction
and Control" team for their support and discussions. This work was
partially supported by NSERC (H. Zhu and J. Wu), CRC (J. Wu), Ontario
Ministry of Health and Long Term Care (S. Ardal), NSF (G. Webb), and
R01GM63270 from NIH (M. Blaser and G. Webb).
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\medskip
\medskip
Received on Dec. 7, 2003. Revised on Jan. 6, 2004.\\
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