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\pagestyle{myheadings} \markright{Appropriate Models
for Infectious Diseases, Wearing et al.}
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\section*{Supporting information}
{\it Classic $SEIR$ epidemic}\\
In the traditional $SEIR$ framework, individuals in the population are
classified according to their infectious status: Susceptible, Exposed
(infected but not infectious), Infectious and Recovered. In the case
of a new infection for which the population has no prior immunity, the
population starts out at the disease-free equilibrium
$(S=N,E=0,I=0,R=0)$, where $N$ is the total population size, and the
dynamics are determined by the following equations describing the
rates of change of each of the four classes.
\begin{eqnarray*}
\frac{dS}{dt} &=& -\frac{\beta SI}{N} \\
\frac{dE}{dt} &=& \frac{\beta SI}{N}-\sigma E \\
\frac{dI}{dt} &=& \sigma E - \gamma I \\
\frac{dR}{dt} &=& \gamma I \;.
\end{eqnarray*}
After an infectious individual is introduced, and if the basic
reproductive ratio, $R_0=\beta/\gamma$, is greater than 1, the
infection sweeps through the population until too few contacts are
between susceptible and infected individuals to sustain further
transmission. If we neglect background birth and death processes, then
the population eventually reaches the following state ($S=S_\infty$,
$E=0$, $I=0$, $R=N-S_\infty$), where $S_\infty$ is defined as the
solution to $R_0(S_\infty/N-1)=\ln(S_\infty/N)$ for large $N$.
To incorporate gamma-distributed latent and infectious periods,
the exposed and infectious classes are divided into $m$ and $n$
subclasses, respectively. To ensure that the average time spent in the
exposed class is still $1/\sigma$ and in the infectious class
$1/\gamma$, the rate of movement between the subclasses is defined as
$m\sigma$ and $n\gamma$, respectively. This is equivalent to assuming
the following probability density functions for the latent
($p_E(t)$) and infectious ($p_I(t)$) periods:
\[p_E(t)= \frac{(m\sigma)^m e^{-m \sigma t} t^{m-1}}{(m-1)!}\]
\[p_I(t)= \frac{(n\gamma)^n e^{-n \gamma t} t^{n-1}}{(n-1)!}\;.\]
Straightforward calculation of the Jacobian matrix for this model results in the
following characteristic equation for the eigenvalues of the
disease-free equilibrium:
\begin{equation*}
\label{eq:chareq} \lambda (\lambda + \gamma n)^n \left[\lambda
(\lambda+\sigma m)^m - R_0 \gamma (\sigma m)^m \left(1-
\left(\frac{\lambda}{\gamma n} +1 \right)^{-n}\right) \right]= 0 \;.
\end{equation*}
Since we are interested in the dominant positive eigenvalue, only the
expression in the square brackets is relevant. Setting this equal to
zero and rearranging in terms of $R_0$ gives equation (2) in
the main text.
Some special cases include when both latent and infectious periods
are exponentially distributed ($n=m=1$), in which case we have:
\begin{equation*}
\label{eq:R0exp}
R_0 = \left(\frac{\lambda}{\sigma} + 1\right)
\left(\frac{\lambda}{\gamma} +1 \right)\;.
\end{equation*}
If the latent and infectious periods have a fixed length ($n, m
\to \infty$), then:
\begin{equation*}
\label{eq:R0delta}
R_0 = \frac{\lambda \exp \left(\displaystyle
\frac{\lambda}{\sigma}\right)}
{\gamma \left(1- \exp \left(\displaystyle
-\frac{\lambda}{\gamma}\right)\right)}\;.
\end{equation*}
And, if there is no latent period ($1/\sigma = 0$):
\begin{equation}
\label{eq:R0nolatent}
R_0 = \frac{\lambda} {\gamma \left(1- \left(\displaystyle
\frac{\lambda}{\gamma n} +1 \right)^{-n}\right)},\;
\end{equation}
which is always less than the value given by the full expression for
$R_0$ (equation (2) in the main text).
{\it Contact tracing and isolation model}\\
To incorporate contact tracing and isolation into the standard $SEIR$
model, we assume that the traditional transmission parameter, $\beta$,
is simply the product of the number of contacts per unit time, $k$,
and the probability of transmitting the disease, $b$. Thus, a
fraction $q$ of those who had contact with an infectious and
symptomatic individual ($I_S$) (but did not contract the infection)
will be removed to the quarantined susceptible class, $S_Q$, where
they will spend exactly $\tau_Q$ days. An identical fraction of newly
exposed individuals are also quarantined. Isolation of newly
infectious cases occurs with a delay of $\tau_D$ days, which
represents a period when infecteds are infectious but asymptomatic or
undetectable ($I_A$), at a daily rate of $d_I$. Vital rates are
assumed to be negligible. The latent period is
gamma distributed with $m$ classes and average length 1/$\sigma$ and
the infectious period is gamma distributed with $n$ classes and
average length 1/$\gamma$ ($>\tau_D$).
\noindent
The model equations are given by:
\begin{eqnarray*}
\frac{dS}{dt} &=& -\frac{(kbI(t)+qk(1-b)I_S(t))S(t)}{N} +
\frac{qk(1-b)S(t-\tau_Q)I_S(t-\tau_Q)}{N}\\
\frac{dS_Q}{dt} &=& \frac{qk(1-b)S(t)I_S(t)}{N} -
\frac{qk(1-b)S(t-\tau_Q)I_S(t-\tau_Q)}{N}\\
\frac{dE_1}{dt}&=& \frac{kb(I(t)-qI_S(t))S(t)}{N}-m\sigma E_1(t) \\
\frac{dE_i}{dt}&=& m\sigma E_{i-1}(t)-m\sigma E_i(t), \quad i=2,...,m\\
\frac{dI_{A,1}}{dt}&=& m\sigma E_m(t) -n\gamma I_{A,1}(t) - P_{I,1}(t)\\
\frac{dI_{A,i}}{dt}&=& n\gamma I_{A,i-1}(t)-n\gamma I_{A,i}(t) -
P_{I,i}(t), \quad i=2,...,n\\
\frac{dI_{S,1}}{dt}&=& P_{I,1}(t) - (n\gamma+d_I)I_{S,1}(t)\\
\frac{dI_{S,i}}{dt}&=& P_{I,i}(t) + n\gamma I_{S,i-1}(t)-(n\gamma+d_I)I_{S,i}(t),
\quad i=2,...,n\\
\frac{dQ}{dt} &=& \frac{qkb S(t)I_S(t)}{N} + d_I I_S(t)\\
\frac{dR}{dt} &=& n\gamma (I_{A,n}(t)+I_{S,n}(t)) \;,
\end{eqnarray*}
where
\[
P_{I,i}(t)=m\sigma E_m(t-\tau_D) \exp(-n\gamma \tau_D)
\frac{(n\gamma \tau_D)^{i-1}}{(i-1)\!}\;
\]
is the probability that an infectious individual is still in the
infectious subclass $i$ after a fixed delay of $\tau_D$ days (the
onset of symptoms). Also, $I_A=\sum^n_{i=1} I_{A,i}$,
$I_S=\sum^n_{i=1} I_{S,i}$ and $I=I_A+I_S$. Note that here $R$
represents those that recovered before they could be
isolated/quarantined. As the epidemic dies out, $Q+R$ will represent
the total number of recovered individuals, since $Q$ keeps track of
all those infected individuals who are quarantined or isolated, and
effectively removed from the infectious population. $N$ is the total
population size. We do not adjust $N$ to discount those in quarantine
when calculating the contact frequency since we want to assume that
the level of mixing remains the same following interventions.
We note that, accounting for the isolation of infecteds with a delay $\tau_D$,
the average infectious period for the exponentially-distributed model
($n=1$) is
\begin{equation} \label{eqS:avexp}
\frac{1-\exp(-\gamma \tau_D)}{\gamma} +
\frac{\exp(-(\gamma+d_I)\tau_D)}{\gamma + d_I}
\quad \in \left[\frac{1-\exp(-\gamma \tau_D)}{\gamma},\frac{1}{\gamma}\right]
\quad \forall d_I\;,
\end{equation}
whereas for the fixed period model ($n \to \infty$) it is
\begin{equation} \label{eqS:avfixed}
\tau_D+\frac{1}{d_I} \left(1-\exp(-d_I (1/\gamma-\tau_D)) \right)
\quad \in \left[\tau_D,\frac{1}{\gamma}\right]
\quad \forall d_I\;.
\end{equation}
Assuming that $1/\gamma >\tau_D$,
\eqref{eqS:avexp}$<$\eqref{eqS:avfixed} for $d_I>0$.
\end{document}