text:
\section{Introduction}

Network theory is a powerful tool that can be applied in many disciplines.
In this framework, real systems such as the power grid, the brain, and societies are represented by a network \cite{barabasi2016network},
which is a graph composed of nodes and links that represent the interaction between nodes.
Many researchers use network theory to study the spreading of an epidemic in order to predict its evolution and to implement strategies to decrease
its impact in healthy populations \cite{newman2002spread}. Diseases like Ebola \cite{faye2015chains}, H1N1 \cite{eastwood2010responses}, and the novel coronavirus COVID-19 \cite{world2020coronavirus} spread not only domestically, but also from one country to another, mainly through air transportation \cite{gardner2013global}. These international airports are bridge nodes, which establish connections between more than one community.
In this work, we explore how bridge nodes affect the disease spreading.

The most used model that reproduces the final state of nonrecurrent epidemics is the
Susceptible-Infected-Recovered (SIR) model \cite{bailey1975mathematical,anderson1992infectious,newman2002spread}. In this model a
susceptible individual (S) in contact with an infected one (I) gets
infected with probability $q$ at each time step. An infected individual recovers (R) after $t_r$ time steps since it was infected. Once an
individual is recovered, it does not play any role in the spreading. In
this model the fraction of recovered individuals $R$ is the order
parameter of a continuous phase transition with a control parameter
$T=1-(1-q)^{t_r}$, where $T$ is the effective probability of
infection denoted as the transmissibility.
It is known that there exists a critical value $T_c$ that separates a nonepidemic phase from an epidemic phase, so that in the thermodynamic limit $R=0$ for $T\le T_c$, and $R>0$ for $T>T_c$ \cite{cohen2010complex,newman2010networks}.
It was shown \cite{grassberger1983critical,newman2010networks,stauffer2018introduction} that the final state of the SIR model can be mapped into link percolation, due to the fact that infecting through a link in SIR is equivalent to
occupying a link in link percolation,
and thus the final state of SIR can be solved using percolation tools.
Each realization of the final state of SIR is one cluster in link percolation, and an epidemic corresponds to the giant component (GC) in link percolation, which is distinguished from outbreaks (corresponding to the finite clusters) by a threshold for the cluster size $s_c$ \cite{lagorio2009effects}.
In random complex networks it is worthwhile to find exact solutions
for the main magnitudes of the final state of the SIR model using the
generating function formalism.  In this approach two generating
functions are used \cite{newman2001random,callaway2000network}.  One of them is the generating function of the degree distribution $G_0(x)=\sum_k P(k) x^k$, where $P(k)$ is the degree distribution with $k_{\min} \leq k \leq k_{\max}$, and $k_{\min}$ and $k_{\max}$ are
the minimum and maximum degree respectively. The other is the generating function of the excess degree distribution $G_1(x)=\sum_k k P(k)/\langle k \rangle x^{k-1}$, where $\langle k \rangle$ is the average degree of the network.
In the SIR model for \emph{isolated} networks, the probability $f_{\infty}$ that a branch of infected nodes reach the infinity for a given transmissibility $T$ satisfies the self-consistent equation
$f_{\infty} =
1-G_1(1-T f_{\infty})$ \cite{braunstein2007optimal,newman2010networks}.
Note that $G_1(1-T f_{\infty})$ is the probability that following a random chosen
link, which leads to a node, the branch of infection does not reach the infinity through its $(k-1)$ outgoing links.
The fraction of recovered individuals $R$, which is equivalent to the fraction $P_{\infty}$ of nodes belonging to the GC in link percolation, is given by $R=
1-G_0(1-T f_{\infty})$ \cite{braunstein2007optimal,newman2010networks}, since $G_0(1-T f_{\infty})$ is the probability that a
random chosen node can not reach the infinity with infected nodes through any of its $k$ links. The critical value of the transmissibility is $T_c=1/(\kappa-1)$, where $\kappa=\langle k^2
\rangle/\langle k \rangle$ is the branching factor, and $\langle k^2
\rangle$ is the second moment of the degree distribution \cite{lagorio2011quarantine,buono2014epidemics}.
For Erd\"os-R\'enyi (ER) networks \cite{erdos1959random}, the degree follows a Poisson distribution $P(k)=\langle k \rangle^k e^{-\langle k \rangle}/k!\,$, and thus $T_c=1/\langle k \rangle$.
Around criticality $T_c$, many physical quantities behave as power laws, e.g., $P(s) \sim s^{-\tau+1}\exp(-s/s_{\max})$, where $P(s)$ is the probability to find a cluster of size $s$, $s_{\max}\sim |T-T_c|^{-1/\sigma}$ is the largest finite cluster size, and the fraction of recovered $R\sim |T-T_c|^{\beta}$ \cite{newman2010networks,cohen2002percolation}.

Before the last decade, researchers concentrated on studying these processes in isolated networks \cite{lagorio2011quarantine,pastor2001epidemic}.
However, real networks are rarely isolated \cite{gao2012networks,kenett2014network}. For example, each country has its own transportation network, and those networks from different countries are connected into a larger network due to international transportation. Also, different communities of people can hold different opinions, but their opinions can exchange through influencers. Thus it is more realistic to consider systems composed of many networks, which are called a network of networks (NON) \cite{buldyrev2010catastrophic,gao2011robustness,kenett2014network,gao2013percolation,kryven2019bond,kivela2014multilayer}.
A case of NON is a system composed of several communities (or layers), where a fraction of nodes $r$ from each layer are bridge nodes which are connected to bridge nodes from other communities through $k^b$ bridge links.
Bridge nodes, which can represent airports connecting countries, may have a huge impact on the system because they can influence individuals in other communities.
As these kinds of nodes are few compared to the number of nodes inside a community, it is reasonable to study these problems in the limit $r\rightarrow 0$.

In Ref.~\cite{dong2018resilience},
the authors studied node percolation in two ER communities with an ER distribution of bridge links, and studied the behavior of $R$ with $r$, in the limit $r\rightarrow 0$, with the constraint $r\langle k^b \rangle=\text{constant}$.
They found, using scaling relations, that $R \propto r^{1/\epsilon}$, where $r$ was associated to an external field.\footnote[2]{They used $\delta$ instead of $\epsilon$ since this behavior is analogous to the relation $M \propto H^{1/\delta}$ between the magnetization $M$ and the external field $H$ in the Ising model \cite{reynolds1977ghost}.}
In Ref.~\cite{valdez2018role}, the authors extended this result to an SIR model and also studied the dynamics. In the final state for $r\rightarrow 0$, they found the same value of the exponent as in Ref.~\cite{dong2018resilience}, and explained it from a geometrical point of view. In their interpretation, the GC was formed by finite clusters in both communities connected through bridges links at $T=T_c(r=0)$. Thus the exponent $\epsilon$ was associated with the exponent $\tau$ of the finite cluster size distribution, which allowed them to derive this exponent theoretically and obtained $\epsilon=1/(\tau-2)$.
Note that due to the constraint $r\langle k^b \rangle = \text{constant}$, the average external connectivity $\langle k^b \rangle$ diverges as $r\rightarrow 0$. In addition, they used the same transmissibility $T$ along intra- and interlinks.

However, from a realistic point of view, the fraction of bridge nodes and
the average external connectivity do not have to be related, and building a large number of connections for one node is practically expensive,
so it is unrealistic to study the case when $\langle k^b \rangle \rightarrow \infty$.
On the other hand, the interaction mechanisms are in general different for internal links than those for bridge links, and strategies like cutting international flights can be used to reduce the disease spreading, so the transmissibility along internal links and bridge links can be very different. In this paper, we use a more realistic approach in which the fraction of
bridge nodes $r$ and the average external connectivity $\langle k^b
\rangle$ are independent, and the transmissibility along bridges links
$T^b$ is different from the internal transmissiblity $T^I$.  When $r$ is small, we find very rich behaviors of $R$ with $r$, many of which are power laws $R\propto r^{1/\epsilon}$, depending on the values of $T^b$ and $T^I$. In these regions the exponent $\epsilon$ follows different functions of the exponents in the finite cluster size distributions.
Our theoretical results are in very good agreement with simulations.

\section{Model}
In our model we consider a system composed of two communities $A$ and $B$,
with degree distributions $P^A(k)$ and $P^B(k)$.
The communities are connected through a fraction $r$ of bridges nodes with degree distribution $P^b(k)$.
The transmissibility within each community is $T^I$ and the transmissibility along bridge links is $T^b$.
To reduce the number of parameters, we will assume that both communities have the same degree distribution, i.e., $P^A(k)=P^B(k) \equiv P(k)$.

Using the generation function formalism, the self-consistent equations of the system are given by
\begin{eqnarray}
\label{eq_f}
f &=& (1-r)\left[1-G_1(1-T^I f)\right] + r\left[1-G_1(1-T^I f)G_0^b(1-T^bf^b)\right],\\
\label{eq_fb}
f^b &=& 1-G_1^b(1-T^bf^b)G_0(1-T^I f),
\end{eqnarray}
where $f$ is the probability to expand a branch to the infinity through an internal link, $f^b$ is the probability to expand a branch to  the infinity through a bridge link, and $G_0(\cdot)$, $G_1(\cdot)$, $G_0^b(\cdot)$, and $G_1^b(\cdot)$ are the generating functions of the degree and excess degree distributions for internal and bridge links, respectively. The
first term on the RHS of Eq.~(\ref{eq_f}) is the contribution of nonbridge nodes that
transmit only internally, while the second term is the contribution of
bridge nodes which transmit both internally with $T^I$ and to the other
community through bridge links with $T^b$. Thus the fraction of recovered nodes of the system and the fraction of recovered nodes of bridge nodes are given by
\begin{eqnarray}
\label{eq_r}  R & = &(1-r)\left[1-G_0(1-T^I  f)\right] + r\left[1-G_0(1-T^I f)G_0^b(1-T^bf^b)\right], \\
\label{eq_rb}  R^b &= &\left[1-G_0(1-T^I f)G_0^b(1-T^bf^b)\right].
\end{eqnarray}

\begin{figure}[h!]
\begin{minipage}[t]{0.49\linewidth}\centering
\includegraphics[width=7cm]{{./figures/RvsTv_Ta0.25_diffr}.eps}
\medskip
\centerline{(a)}
\end{minipage}\hfill
\begin{minipage}[t]{0.49\linewidth}\centering
\includegraphics[width=7cm]{{./figures/RvsTv_Ta0.2_diffr}.eps}
\medskip
\centerline{(b)}
\end{minipage}
\caption{$R$ as a function of $T^b$ when both internal links and bridge links are ER networks, with $\langle k \rangle=4$ and $\langle k^b \rangle=10$, respectively. Theoretical solutions (dark blue solid lines for $r=0.1$ and light orange solid lines for $r=0.01$) are compared with stochastic simulation results of SIR in the final state (dark blue circles for $r=0.1$ and light orange squares for $r=0.01$), for (a) $T^I=0.25$ and (b) $T^I=0.2$. For the simulations, system sizes $N_A=N_B=10^5$, $k_{\min}=0$, $k_{\max}=100$, $s_c=200$, and are averaged over $10^3$ realizations.}
\label{fig_rvstb}
\end{figure}

In Fig.~\ref{fig_rvstb} we show the fraction of the recovered $R$ as a
function of $T^b$ for different values of $T^I$ and $r$, for a system where both internal links and bridge links follow an ER degree
distribution, with $\langle k \rangle=4$, and $\langle k^b \rangle=10$.
The solid lines show the numerical solutions obtained from Eqs.~(\ref{eq_f})-(\ref{eq_r}), and the square and circle symbols are the results from the SIR stochastic simulations. We can see that the theory agrees very well with the simulation results, and thus we will mainly use theoretical solutions hereafter.
Both theoretical solutions and simulation results in Fig.~\ref{fig_rvstb} show a critical value of $T^b$ that depends on $T^I$ and $r$.
The system is in a nonepidemic phase, with $R=0$, when $T^b \le T^b_c$, and is in an epidemic phase with a finite positive $R$ when $T^b > T^b_c$.
This is due to the fact that the self-consistent
Eqs.~(\ref{eq_f}) and (\ref{eq_fb}) have only one solution $f=f^b=0$ when $T^b \le T^b_c$, and a nontrivial physical solution exists only when $T^b>T^b_c$.

The theoretical value of $T^b_c$ can be obtained by solving
$|J-I|_{f,f^b=0}=0$, where $|\cdot|$ is the determinant, $J$ is the Jacobian matrix, and $I$ is the identity. Note that the elements of the
Jacobian matrix are given by $J_{i,j}= \left.\frac{\partial f_i}{\partial f_j}\right|_{f,f^b=0}$, where each of $f_i$ and $f_j$ represents $f$ or $f^b$. Thus explicitly $|J-I|_{f,f^b=0}=0$ can be written as
\begin{equation}\label{eq_critical}
    \begin{vmatrix}
      T^I(\kappa-1)-1 & rT^{b}_{c}\langle k^b\rangle \\[1ex]
      T^I\langle k \rangle & T^{b}_{c}(\kappa^b-1)-1
    \end{vmatrix}
    = 0.
\end{equation}
So $T^{b}_{c}$ is given by
\begin{equation}
\label{eq_tbc}
T^b_c =
\frac{T^I(\kappa-1)-1}{(T^I\left(\kappa-1\right)-1)(\kappa^b-1)-rT^I\langle
  k\rangle\langle k^b\rangle}.
\end{equation}

In the equation above, $T^b_c$ has physical meaning only when $T^I \le 1/(\kappa-1)$ (see Appendix \ref{app_sec_critical} for details).
This implies that any strategy that reduces the transmissibility between communities will prevent a macroscopic number of infected nodes only if the internal transmissibility is below the critical value for an isolated community.
We can see from Eq.~(\ref{eq_tbc}) that, as $r$ approaches $0$, the critical value $T^b_c(r \rightarrow 0)= 1/(\kappa^b-1)$.

\section{Different Regimes: Asymptotic Behaviors}

\begin{figure}[h!]
\centering
\includegraphics[width=10cm]{./figures/regimes.eps}
\caption{Phase diagram for two ER communities connected by ER bridge links, with $\langle k \rangle=4$ and $\langle k^b \rangle=10$, respectively. The blue areas are the nonepidemic phases ($R=0$) for $r=0.5,0.1,0.01,0.001$, and the white area is the epidemic phase with $R>0$. As $r\rightarrow 0$, the nonepidemic phase expands and tends to be a rectangle.}
\label{fig_regimes}
\end{figure}

In Fig.~\ref{fig_regimes} we show the phase diagram for two ER communities connected by ER bridge links with $\langle k \rangle=4$ and $\langle k^b \rangle=10$, for different values of $r$.
As $r\rightarrow 0$, the nonepidemic phase tends to be a rectangle.
The boundaries of the rectangle are
$T^I = T^I_c=1/(\kappa-1)$ and $T^b=T^b_c(r\rightarrow 0)=1/(\kappa^b-1)$, which split the whole space into several regimes, where the relation between $R$ and $r$ follows different behaviors asymptotically.
In this section, we derive the asymptotic behavior of $R$ versus $r$, i.e., as $r\rightarrow 0$, by first looking at how $R$ depends on $(rR^b)$, and then how $R^b$ depends on $R$.

\begin{figure}[h!]
\begin{minipage}[t]{0.49\linewidth}\centering
\includegraphics[width=6cm]{./figures/topology_1.eps}
\medskip
\centerline{(a)}
\end{minipage}\hfill
\begin{minipage}[t]{0.49\linewidth}\centering
\includegraphics[width=6cm]{./figures/topology_2.eps}
\medskip
\centerline{(b)}
\end{minipage}
\caption{An illustration of how finite clusters of recovered nodes in each community (circled in green) are connected due to clusters by bridge links (circled in blue), and thus there exists a GC of recovered nodes in the entire system. (a) Only finite clusters exist in each community and for bridge links. (b) Only finite clusters exist in each community but a giant component exists for bridge links. All recovered nodes are plotted in green, except for patient zero, which is plotted in black, and all nodes that are never infected are plotted in gray. Links through which the disease is transmitted are plotted in green, while links that fail to transmit the disease are plotted in gray. Squares denote bridge nodes, and circles denote internal nodes.}
\label{fig_topology}
\end{figure}

When $T^I\le T^I_c$, there are only finite clusters of recovered nodes within each community.
However, in the epidemic phase, these finite clusters in each community are connected due to bridge links,
and thus form a GC of recovered nodes in the entire system, as illustrated in Fig.~\ref{fig_topology}.
Using the mapping between the SIR model and link percolation,
as any node in each community has a probability $r$ to be a bridge node, and each
bridge node has a probability $R^b$ to be recovered, a finite cluster
of size $s$ has a probability
$1-(1-r R^b)^s$ to have at least one recovered bridge node, and thus
belong to the GC of recovered.
Thus,
the size of the GC as $r \rightarrow 0$ is given by
\begin{equation}
\label{eq_rsum}
R = 1-\sum_{s=1}^\infty P(s)(1-r R^b)^s,
\end{equation}
where $P(s) \sim s^{-\tau+1}\exp(-s/s_{\max})$ is the probability of a finite cluster of size $s$ within a community, $\tau$ is the Fisher exponent of each community, and the largest finite cluster size $s_{\max}\sim |T^I-T^I_c|^{-1/\sigma}$. Then we can derive the behavior of $R$ with $r$ for $T^I$ below, or equal to the critical internal transmissibility.

At the critical value $T^I=T^I_c=1/(\kappa-1)$ so that $s_{\max}$ diverges and thus $P(s) \sim s^{-\tau+1}$, Eq.~(\ref{eq_rsum}) can be simplified into $R\propto (rR^b)^{\tau-2}$ [see Eqs.~(\ref{app_eq_r})~and~(\ref{app_eq_rat}) in Appendix \ref{app_sec_exponents} for details].
This is due to the fact that the average number of infected bridge nodes in each finite cluster of a community depends on the topology of the community, and thus depends on $\tau$.
When $T^I < T^I_c$, Eq.~(\ref{eq_rsum}) can be reduced to $R\propto rR^b$, due to the finite $s_{\max}$ [see Eqs.~(\ref{app_eq_r})~and~(\ref{app_eq_rbelow}) in Appendix \ref{app_sec_exponents} for details].
This is intuitive since as $T^I$ is so small that each finite cluster of a community has very few bridge nodes, then the number of nodes in the GC will be proportional to the number of bridge nodes in the GC [as in Fig.~\ref{fig_topology} (a)].

When $\kappa^b<\infty$, which is always the case in reality, we need to explore the behavior of $R^b$ as well.
For each cluster connected through bridge links,
each bridge node has a probability $1-G_0(1-T^If)$ to be connected to the GC through internal links.
So as $r \rightarrow 0$, a finite cluster of bridge nodes of size $s$ has a probability $1-[G_0(1-T^If)]^s$ to belong to the recovered bridge nodes, and thus
\begin{equation}
\label{eq_rbsum}
R^b = 1-\sum_{s}^\infty P^b(s)[G_0(1-T^If)]^s,
\end{equation}
where $P^b(s) \sim s^{-\tau^b+1}\exp(-s/s^b_{\max})$ is the probability of a finite cluster of size $s$ connected by bridge links, $\tau^b$ is the Fisher exponent of bridge links, and $s^b_{\max}\sim |T^b-T^b_c(r\rightarrow 0)|^{-1/\sigma^b}$ is the largest finite cluster size of bridge links.
From Eq.~(\ref{eq_r}) we know that $R\approx 1-G_0(1-T^If)$ as $r\rightarrow 0$, so
\begin{equation}
\label{eq_rbsumapprox}
R^b \approx 1-\sum_{s}^\infty P^b(s)(1-R)^s.
\end{equation}

At the critical value $T^b=T^b_c(r\rightarrow 0)=1/(\kappa^b-1)$,
$s^b_{\max}$ diverges, and thus $R^b\propto R^{\tau^b-2}$ [see Eq.~(\ref{app_eq_rbat}) in Appendix \ref{app_sec_exponents} for details].
When $T^b < T^b_c(r\rightarrow 0)$,
$R^b\propto R$ since $s^b_{\max}<\infty$ [see Eq.~(\ref{app_eq_rbbelow}) in Appendix \ref{app_sec_exponents} for details].
When $T^b>T^b_c(r\rightarrow 0)$,
most bridge nodes are connected into one big cluster through bridge links, so Eqs.~(\ref{eq_rbsum})~and~(\ref{eq_rbsumapprox}) do not apply and
$R^b$ is not a power law of $R$ [see Fig.~\ref{fig_topology} (b)].

In summary,
\begin{equation}
\label{eq_rvsrrb}
R \propto
\begin{cases}
rR^b, & \text{if } T^I<1/(\kappa-1) \\
(rR^b)^{\tau-2}, & \text{if } T^I=1/(\kappa-1) \\
\text{not a power law of } (rR^b), & \text{if } T^I>1/(\kappa-1)
\end{cases}
,
\end{equation}
\begin{equation}
\label{eq_rbvsr}
R^b \propto
\begin{cases}
R, & \text{if } T^b<1/(\kappa^b-1) \\
R^{\tau^b-2}, & \text{if } T^b=1/(\kappa^b-1) \\
\text{not a power law of } R, & \text{if } T^b>1/(\kappa^b-1)
\end{cases}
.
\end{equation}

\begin{table}[htb]
\centering
\begin{tabular}{c c c c c c c}
\hline
\hline
  &\hspace*{5ex}& $T^b<\frac{1}{\kappa^b-1}$ &\hspace*{5ex}& $T^b=\frac{1}{\kappa^b-1}$ &\hspace*{5ex}& $T^b>\frac{1}{\kappa^b-1}$ \\ [1ex]
\hline
$T^I<\frac{1}{\kappa-1}$ && \begin{tabular}{@{}c@{}}$\varnothing$\vspace{-1ex} \\ (Regime I) \end{tabular} && \begin{tabular}{@{}c@{}}$\epsilon=1-(\tau^b-2)$\vspace{-1ex} \\ (Regime II) \end{tabular} && \begin{tabular}{@{}c@{}}$\epsilon=1$\vspace{-1ex} \\ (Regime III) \end{tabular}\\ [4ex]
$T^I=\frac{1}{\kappa-1}$ &&
\begin{tabular}{@{}c@{}}$\epsilon=\frac{1}{\tau-2}-1$\vspace{-1ex} \\ (Regime IV) \end{tabular} &&
\begin{tabular}{@{}c@{}}$\epsilon=\frac{1}{\tau-2}-(\tau^b-2)$\vspace{-1ex} \\ (Regime V) \end{tabular} &&
\begin{tabular}{@{}c@{}}$\epsilon=\frac{1}{\tau-2}$\vspace{-1ex} \\ (Regime VI) \end{tabular} \\ [4ex]
$T^I>\frac{1}{\kappa-1}$ && \begin{tabular}{@{}c@{}}$\varnothing$\vspace{-1ex} \\ (Regime VII) \end{tabular} && \begin{tabular}{@{}c@{}}$\varnothing$\vspace{-1ex} \\ (Regime VII) \end{tabular} && \begin{tabular}{@{}c@{}}$\varnothing$\vspace{-1ex} \\ (Regime VII) \end{tabular} \\ [1ex]
\hline
\hline
\end{tabular}
\caption{Asymptotic power-law behaviors of $R$ with $r$ in different regimes. The exponent $\epsilon$ in $R\propto r^{1/\epsilon}$ is independent of the specific values of $\kappa$ or $\kappa^b$, but varies with the regimes where the combination of $T^I$ and $T^b$ falls in. $\varnothing$ means there is no power-law relation in that regime.}
\label{tb_exponents}
\end{table}

Combining Eqs.~(\ref{eq_rvsrrb})~and~(\ref{eq_rbvsr}), we obtain the asymptotic power-law behaviors of $R$ with $r$ in many regimes. Different values of $\epsilon$ in the relation $R \propto r^{1/\epsilon}$ are summarized in Table~\ref{tb_exponents}.
As an example,
when both communities and the bridge links are all ER networks, we have $\tau = \tau^b = 5/2$, and in the limit $\kappa^b \rightarrow \infty$ so that $T^b>1/(\kappa^b-1)$ all the time, we obtain the same exponents that were found in Refs.~\cite{dong2018resilience,valdez2018role},
in which $r \langle k^b \rangle = \text{constant}$,
and $r \rightarrow 0$.
Note that the results in Table~\ref{tb_exponents} apply to networks with any degree distributions,
i.e., either internal or bridge links or both can be homogeneous or heterogeneous.
Also,
a similar methodology can be applied to a system when the two communities have different degree distributions,
i.e., $P^A(k)\ne P^B(k)$,
and we can still correctly predict the asymptotic power-law relations between $R$ and $r$ for all regimes (see Appendix \ref{app_sec_panepb} for details).

\begin{figure}[h!]
\begin{minipage}[t]{0.33\linewidth}\centering
\includegraphics[width=5cm]{{./figures/theory_R_vs_r_T_i_0.2_T_b_0.1}.eps}
\medskip
\end{minipage}\hfill
\begin{minipage}[t]{0.33\linewidth}\centering
\includegraphics[width=5cm]{{./figures/theory_R_vs_r_T_i_0.2_T_b_0.2}.eps}
\medskip
\end{minipage}\hfill
\begin{minipage}[t]{0.33\linewidth}\centering
\includegraphics[width=5cm]{{./figures/theory_R_vs_r_T_i_0.25_T_b_0.05}.eps}
\medskip
\end{minipage}\hfill
\vspace{1.5\baselineskip}
\begin{minipage}[t]{0.49\linewidth}\centering
\includegraphics[width=5cm]{{./figures/theory_R_vs_r_T_i_0.25_T_b_0.1}.eps}
\medskip
\end{minipage}\hfill
\begin{minipage}[t]{0.49\linewidth}\centering
\includegraphics[width=5cm]{{./figures/theory_R_vs_r_T_i_0.25_T_b_0.2}.eps}
\medskip
\end{minipage}
\caption{$R$ as a function of $r$ for different regimes where a power law exists: (a) Regime II: $T^I=0.2$, $T^b=0.1$; (b) Regime III: $T^I=0.2$, $T^b=0.2$; (c) Regime IV: $T^I=0.25$, $T^b=0.05$; (d) Regime V: $T^I=0.25$, $T^b=0.1$; and (e) Regime VI: $T^I=0.25$, $T^b=0.2$. Both internal links and bridge links are ER networks, with $\langle k \rangle=4$ and $\langle k^b \rangle=10$, respectively. In each regime, numerical solutions of Eqs.~(\ref{eq_f})-(\ref{eq_r}) are plotted in black solid lines, and a dashed line is drawn with the slope predicted by Table~\ref{tb_exponents}.}
\label{fig_exponents}
\end{figure}

In Fig.~\ref{fig_exponents} we show the numerical solutions of Eqs.~(\ref{eq_f})-(\ref{eq_r}) with the log-log plot of $R$ with $r$ for small $r$ in different regimes, for a system of two ER communities connected by ER bridge links, and thus $\tau=\tau^b=5/2$.
In each regime, we plot a dashed line with the slope predicted by the theory (see Table~\ref{tb_exponents}). We can see that our predictions are in good agreement with the numerical results. In Appendix \ref{app_sec_compare}, we also compared simulations using link percolation with numerical solutions. We use link percolation mapping instead of SIR to simulate the final state because the former is much less time-consuming for big system sizes. The simulation results agree well with theoretical solutions, except for some finite-size effects when $r$ is very small (see Appendix \ref{app_sec_compare} for details).

When a highly infectious epidemic occurs, one of the first strategies used by many countries is to shut down some international airports. Those international airports serve
as bridge nodes in the whole system of global transportation, so shutting them down is essentially reducing the percentage of bridge nodes $r$.
Meanwhile, international flights are cut for those airports that are still open, which mitigates the disease spreading by reducing the transmissibility along bridge links.
Also, social distancing strategies like staying at home as long as it is possible or wearing facial masks if having to go outside reduce the chance of face-to-face infection, which is utilized by most countries as another strategy to reduce both $T^I$ and $T^b$.

As can easily be seen from our results in this section, strategies like shutting down international airports are not as effective in some regimes as in others.
In those regimes with a smaller $\epsilon$, shutting down international airports to reduce $r$ will significantly reduce the fraction of recovered $R$, while in regimes with a larger $\epsilon$, $R$ will be reduced only slightly.
This helps us to decide what kind of strategies we are supposed to use to control disease spreading effectively; i.e., shutting down international airports had better be combined with strategies to reduce $T^I$ and $T^b$, so that it falls in a regime with a small $\epsilon$.

\section{Crossovers when $T^I\lesssim T^I_c$}

In Table \ref{tb_exponents}, we can see that the asymptotic values of the exponent $\epsilon$ change abruptly between regimes. However, in this section, we are going to show that, for $T^I \lesssim T^I_c$, the system behaves in the same way as $T^I=T^I_c$  for a relatively large value of $r$, but changes to its asymptotic behavior continuously as $r$ decreases.

From a percolation point of view, a finite cluster belongs to the GC if it contains recovered bridge nodes (with an overall percentage of $r R^b$).
As $r R^b$ decreases from $1$, the GC starts to lose some finite clusters so that $R$ also decreases from $1$.
When $r$ is not too small, finite clusters of \emph{smaller sizes} are more likely to be detached from the GC.
Note that the probability of a cluster of size $s$ is $P(s) \sim s^{-\tau+1}\exp(-s/s_{\max})$ for $T^I<T^I_c$ and $P(s) \sim s^{-\tau+1}$ for $T^I=T^I_c$, which are the same for smaller cluster sizes, so the behaviors of $R$ versus $r$ for different values of $T^I \lesssim T^I_c$ are the same when $r$ is not small enough.
The distribution $P(s)$ starts to differ when the GC starts to lose relatively large finite clusters, i.e., when $s$ is comparable to $s_{\max}$, and a crossover is going to show up.
Denote $r^*$ as where the crossover occurs, $R^*$ and $R^{b*}$ as the fraction of recovered nodes and bridge nodes at the crossover respectively, we have $r^* R^{b*}\sim 1/s_{\max}$ (see Appendix \ref{app_sec_exponents}).

\begin{figure}[h!]
\begin{minipage}[t]{0.33\linewidth}\centering
\includegraphics[width=5cm]{{./figures/crossover_R_vs_r_T_b_0.05}.eps}
\medskip
\centerline{(a)}
\end{minipage}\hfill
\begin{minipage}[t]{0.33\linewidth}\centering
\includegraphics[width=5cm]{{./figures/crossover_R_vs_r_T_b_0.1}.eps}
\medskip
\centerline{(b)}
\end{minipage}\hfill
\begin{minipage}[t]{0.33\linewidth}\centering
\includegraphics[width=5cm]{{./figures/crossover_R_vs_r_T_b_0.2}.eps}
\medskip
\centerline{(c)}
\end{minipage}
\caption{
Theoretical solutions of $R$ as a function of $r$ when $T^I\lesssim T^I_c$, i.e., $T^I=0.248,0.2485,0.249,0.2495,0.2499$ (from light blue to dark blue solid lines) with (a) $T^b=0.05$, (b) $T^b=0.1$, (c) $T^b=0.2$. Both internal links and bridge links are ER networks, with $\langle k \rangle=4$ and $\langle k^b \rangle=10$, respectively. Black dashed lines are drawn with the slope predicted by Table~\ref{tb_exponents} for different regimes.}
\label{fig_crossover}
\end{figure}

In Fig.~\ref{fig_crossover} we use different values of $T^I \lesssim T^I_c=1/(\kappa-1)$ to show the behavior of $R$ versus $r$ near the critical point, when $T^b$ is below, equal to, and above $T^b_c(r\rightarrow 0)$.
We can see a crossover, where the exponent $\epsilon$ is the same as the $\epsilon$ for $T^I = T^I_c$ when $r$ is not small enough (i.e., $r^*\ll r \ll 1$), and has the same $\epsilon$ as the asymptotic one for $T^I < T^I_c$ when $r$ is extremely small (i.e., $r\ll r^*$).
For example, when $T^b<T^b_c(r\rightarrow 0)=1/(\kappa^b-1)$, i.e., $T^b=0.05$, which is shown in Fig.~\ref{fig_crossover} (a), we can see the power-law behavior $R\propto r^{1/\epsilon}$ with $\epsilon=1$ (as in regime IV) when $r$ is relatively large,
but it is in the nonepidemic phase with $R=0$ (as in regime I) as $r\rightarrow 0$.
When $T^b=T^b_c(r\rightarrow 0)=1/(\kappa^b-1)$, i.e., $T^b=0.1$, which is shown in Fig.~\ref{fig_crossover} (b), the exponent $\epsilon$ changes from $3/2$ (as in regime V) to $1/2$ (as in regime II) as $r\rightarrow 0$.
When $T^b>T^b_c(r\rightarrow 0)$, e.g., $T^b=0.2$, which is shown in Fig.~\ref{fig_crossover} (c), the exponent $\epsilon$ changes from $2$ (as in regime VI) to $1$ (as in regime III) as $r\rightarrow 0$.

As was mentioned above, since the crossover occurs when $r^* R^{b*}\sim 1/s_{\max}$, and $s_{\max}$ depends on the internal transmissibility $T^I$, the values of $r^*$ and $R^*$ also depend on $T^I$. To be more explicit, they follow power laws of the difference between $T^I$ and its critical value, i.e., $r^*\sim|T^I-T^I_c|^{\beta_r}$, and $R^*\sim|T^I-T^I_c|^{\beta_R}$.
The values of $\beta_r$ and $\beta_R$ can be derived as the following.

Combined with the criteria $1/s_{\max} \sim r^*R^{b*}$, and considering the relation between $R^b$ and $R$ as in Eq.~(\ref{eq_rbvsr}), we will get
\begin{equation}
\label{eq_smaxvsrr}
1/s_{\max} \sim
\begin{cases}
r^*R^*, & \text{if } T^b<1/(\kappa^b-1) \\
r^*(R^*)^{\tau^b-2}, & \text{if } T^b=1/(\kappa^b-1) \\
r^*, & \text{if } T^b>1/(\kappa^b-1)
\end{cases}
.
\end{equation}
Since curves with different values of $T^I$ overlap for a relatively large $r$ (which is also verified in Fig.~\ref{fig_crossover}), we also have the relation $R^* \propto (r^*)^{1/\epsilon}$, where $\epsilon$ is the one for $T^I=1/(\kappa-1)$, respectively. If we combine $R^* \propto (r^*)^{1/\epsilon}$ with Eq.~(\ref{eq_smaxvsrr}), and knowing that $s_{\max}\sim |T^I-T^I_c|^{-1/\sigma}$, we obtain that
\begin{equation}
\label{eq_rstar}
r^* \sim
\begin{cases}
|T^I-T^I_c|^{(3-\tau)/\sigma} \sim |T^I-T^I_c|^\gamma, & \text{if } T^b<1/(\kappa^b-1) \\
|T^I-T^I_c|^{(1-(\tau-2)(\tau^b-2))/\sigma} \sim |T^I-T^I_c|^{(\tau^b-2)\gamma+(3-\tau^b)/\sigma}, & \text{if } T^b=1/(\kappa^b-1) \\
|T^I-T^I_c|^{1/\sigma}, & \text{if } T^b>1/(\kappa^b-1)
\end{cases}
.
\end{equation}
In any region, the crossover point $r^*$ goes to $0$ as $T^I$ approaches $T^I_c$, so we do not see a crossover unless $T^I$ is below but very close to $T^I_c$.

Knowing that the mean finite cluster size $\langle s \rangle\sim |T^I-T^I_c|^{-\gamma}$, and the largest finite cluster size $s_{\max}\sim |T^I-T^I_c|^{-1/\sigma}$,
Eq.~(\ref{eq_rstar}) can also be written as
\begin{equation}
\label{eq_rstarsimplified}
1/r^* \sim
\begin{cases}
\langle s \rangle, & \text{if } T^b<1/(\kappa^b-1) \\
\langle s \rangle^{\tau^b-2} {s_{\max}}^{3-\tau^b}, & \text{if } T^b=1/(\kappa^b-1) \\
s_{\max}, & \text{if } T^b>1/(\kappa^b-1)
\end{cases}
,
\end{equation}
and for all three regions of $T^b$, $R^* \sim |T^I-T^I_c|^{(\tau-2)/\sigma} \sim |T^I-T^I_c|^\beta$,
whose exponent is the same as the one in $R\propto |T-T_c|^\beta$ for an isolated network.

\begin{comment}
\textcolor{blue}{Intuitively, $1/r$ is the average number of nodes a bridge node is allowed to be connected to, which has to be larger than the finite cluster sizes of all bridge nodes that are in the GC, in order for the finite clusters of different bridge nodes not to overlap, and thus the number of nodes in the GC is proportional to the number of bridge nodes in the GC.
When $T^b<T^b_c(r\rightarrow 0)$, only a very small proportion $R^b\ll 1$ of bridge nodes are in the GC, so their largest finite cluster size is highly likely to be the average finite cluster size $\langle s\rangle$. When $T^b>T^b_c(r\rightarrow 0)$ and $R^b\sim 1$ so that almost all bridge nodes are in the GC, $1/r$ has to be larger than the largest size of almost all finite clusters, which means $1/r>s_{\max}$.}
\end{comment}

The scaling relation between $R$ and $r$ around the critical internal transmissibility ($T^I\lesssim T^I_c$) can then be written as
\begin{equation}
\label{eq_scaling}
R = R^*\,F\left(\frac{r}{r^*}\right),
\end{equation}
where $F(x)$ is given by $F(x)\sim x^{1/\epsilon}$ and
\begin{equation}
\label{eq_deltaprime}
\epsilon =
\begin{cases}
\epsilon (T^I<T^I_c), & \text{if } x \ll 1 \\
\epsilon (T^I=T^I_c), & \text{if } x \gg 1
\end{cases}
.
\end{equation}
(See Table \ref{tb_exponents} for values of $\epsilon$ for different values of $T^b$.)

\begin{figure}[h!]
\begin{minipage}[t]{0.33\linewidth}\centering
\includegraphics[width=5cm]{{./figures/crossover_R_vs_r_T_b_0.05_collapsed}.eps}
\medskip
\centerline{(a)}
\end{minipage}\hfill
\begin{minipage}[t]{0.33\linewidth}\centering
\includegraphics[width=5cm]{{./figures/crossover_R_vs_r_T_b_0.1_collapsed}.eps}
\medskip
\centerline{(b)}
\end{minipage}\hfill
\begin{minipage}[t]{0.33\linewidth}\centering
\includegraphics[width=5cm]{{./figures/crossover_R_vs_r_T_b_0.2_collapsed}.eps}
\medskip
\centerline{(c)}
\end{minipage}
\caption{Theoretical solutions of $R/R^*$ as a function of $r/r^*$, where $R^*=|T^I-T^I_c|^{\beta_R}$ and $r^*=|T^I-T^I_c|^{\beta_r}$ when $T^I\lesssim T^I_c$, i.e., $T^I=0.248,0.2485,0.249,0.2495,0.2499$ (from light blue to dark blue solid lines) with (a) $T^b=0.05$, so that $\beta_r=1$ and $\beta_R=1$, (b) $T^b=0.1$, so that $\beta_r=3/2$ and $\beta_R=1$, and (c) $T^b=0.2$, so that $\beta_r=2$ and $\beta_R=1$. Both internal links and bridge links are ER networks, with $\langle k \rangle=4$ and $\langle k^b \rangle=10$, respectively. All curves with different $T^I\lesssim T^I_c$ collapse under the scaling relation.}
\label{fig_crossovercollapsed}
\end{figure}

In Fig.~\ref{fig_crossovercollapsed} we show the plot of $R$ versus $r$ rescaled by $R^*\sim|T^I-T^I_c|^{\beta_R}$ and $r^*\sim|T^I-T^I_c|^{\beta_r}$ for three different values of $T^b$.
Since both communities and the bridge links are all ER networks, $\tau=\tau^b=5/2$, $\gamma=1$, $\sigma=1/2$, and $\beta=1$.
For all values of $T^b$, $\beta_R=1$, which is the same as the exponent $\beta$ in $R \propto |T-T_c|^\beta$ of an isolated network, as mentioned above.
When $T^b<1/(\kappa^b-1)$, e.g., $T^b=0.05$, we have $\beta_r=\gamma=1$.
When $T^b=1/(\kappa^b-1)$, e.g., $T^b=0.1$, we have $\beta_r=(\tau^b-2)\gamma+(3-\tau^b)/\sigma=3/2$.
When $T^b>1/(\kappa^b-1)$, e.g., $T^b=0.2$, we have $\beta_r=1/\sigma=2$. We can see that the curves of $R/R^*$ versus $r/r^*$ for different values of $T^I \lesssim T^I_c$ collapse.

Empirically, there are cases when we can not or do not want to further reduce internal transmissibility, for example due to the shortage of facial masks or to avoid severe economic consequences, so that $T^I$ is below but close to its critical value.
In those scenarios, depending on the value of $T^I$, if there are too many open international airports, we may go through a section where shutting them down does not show a huge effect on the total number of infected individuals. However, as long as the internal transmissibility is below its critical value so that the disease can not spread massively within a community, if we keep reducing $r$, then after a point, which is the crossover, the total number of infected individuals is going to drop dramatically.
A good understanding of this crossover is going to help us better estimate the impact of epidemic strategies.

\section{Conclusions}

In this paper, we study the effect of bridge nodes to the final state of the SIR model, by mapping it to link percolation.
We find power-law asymptotic behaviors between $R$ and $r$ in different regimes, depending on how $T^I$ and $T^b$ are compared to their critical values.
The different exponents are related to the different mechanisms of how finite clusters in each community are connected into the GC of the whole system.
Additionally, around but below the critical point of internal transmissibility (when $T^I \lesssim T^I_c$), we find the crossover points $r^*$ such that $R$ versus $r$ follows a different power-law behavior when $r^* \ll r\ll 1$ compared to its asymptotic one (when $r\ll r^*$).
The methodology and results in this paper can easily be generalized for NONs with multiple communities.

The results can provide the authorities with helpful guidance on making decisions about epidemic strategies. They enable us to better anticipate the impacts of epidemic strategies before adopting them, and
help us understand why strategies like shutting down international airports had better be combined with adequate social distancing strategies to be more effective.

\section{Acknowledgments}

The authors would like to thank Shlomo Havlin for useful discussions.
J.M., L.D.V., and L.A.B. acknowledge support from NSF Grant No. PHY-1505000, DTRA Grant No. HDTRA-1-14-1-0017, and DTRA Grant No. HDTRA-1-19-1-0016. L.A.B. and L.D.V. thank UNMdP (Grant EXA956/20) and CONICET (Grant No. PIP 00443/2014) for financial support.

\begin{appendices}

\section{Physical and Non-Physical Critical Values}
\label{app_sec_critical}

When $T^I \le 1/(\kappa-1)$, e.g., $T^I=0.2$, as in Fig.~\ref{app_fig_critical} (a), Eq.~(\ref{eq_tbc}) gives the physical critical value $T^b_c$, from which point the physical solution of $f$ and $f^b$ becomes nontrivial. However, when $T^I > 1/(\kappa-1)$, e.g., $T^I=0.4$, as in Fig.~\ref{app_fig_critical} (b), Eq.~(\ref{eq_tbc}) gives the value of $T^b$ where more nonphysical solutions show up, while the physical solution stays smooth, so there is no critical phenomenon.

\begin{figure}[htb]
\begin{minipage}[t]{0.49\linewidth}\centering
\includegraphics[width=5cm]{{./figures/theory_f_vs_T_b_T_i_0.2}.eps}
\medskip
\centerline{(a)}
\end{minipage}\hfill
\begin{minipage}[t]{0.49\linewidth}\centering
\includegraphics[width=5cm]{{./figures/theory_f_vs_T_b_T_i_0.4}.eps}
\medskip
\centerline{(b)}
\end{minipage}
\caption{Numerical solutions of $f$ (orange) and $f^b$ (blue) of Eqs.~(\ref{eq_f}) and (\ref{eq_fb}) with $r=0.1$, given (a) $T^I=0.2$ and (b) $T^I=0.4$.
The only physical solution in each case is plotted in solid lines, and pairs of nonphysical solutions are in dashed lines with different symbols.
Both internal links and bridge links are ER networks, with $\langle k \rangle=4$ and $\langle k^b \rangle=10$, respectively.
The vertical dashed line represents the value of $T^b_c$ predicted by Eq.~(\ref{eq_tbc}).}
\label{app_fig_critical}
\end{figure}

\section{Derivations of the Relations between $R$, $r$ and $R^b$ in Different Regimes}
\label{app_sec_exponents}

To derive how $R$ depends on $r R^b$, recall that $P(s) \sim s^{-\tau+1}\exp(-s/s_{\max})$, so Eq.~(\ref{eq_rsum}) becomes
\begin{equation}
\label{app_eq_r}
\begin{split}
R & = 1-\sum_{s=1}^\infty P(s)(1-r\,R^b)^s \\
    & \approx 1-\frac{\int_1^\infty s^{-\tau+1}e^{-s/s_{\max}}(1-r\,R^b)^s ds}{\int_1^\infty s^{-\tau+1}e^{-s/s_{\max}} ds} \\
    & = 1-\frac{\int_1^\infty s^{-\tau+1}e^{-s/s_{\max}}e^{s\ln(1-r\,R^b)} ds}{\int_1^\infty s^{-\tau+1}e^{-s/s_{\max}} ds} \\
    & \approx 1-\frac{\int_1^\infty s^{-\tau+1}e^{-s(1/s_{\max}+r\,R^b)} ds}{\int_1^\infty s^{-\tau+1}\exp(-s/s_{\max}) ds}.
\end{split}
\end{equation}

When $T^I=T^I_c=1/(\kappa-1)$, so $s_{\max}$ diverges and thus $P(s) \sim s^{-\tau+1}$,
or if $T^I\lesssim T^I_c=1/(\kappa-1)$, but $r$ is not too small,
so that $1/s_{\max}\ll r R^b \ll 1$ and thus $1/s_{\max}$ can be ignored,
\begin{equation}
\label{app_eq_rat}
\begin{split}
R & \approx 1-\frac{\int_1^\infty s^{-\tau+1}e^{-r\,R^b\,s} ds}{\int_1^\infty s^{-\tau+1} ds} \\
    & = 1-(\tau-2) \int_1^\infty s^{-\tau+1}e^{-r\,R^b\,s} ds \\
    & = 1-(\tau-2)(r\,R^b)^{\tau-2} \int_{r\,R^b}^\infty u^{-\tau+1}e^{-u} du, \text{ where }u=r\,R^bs \\
    & = 1-(\tau-2)(rR^b)^{\tau-2} \left[\frac{(rR^b)^{-\tau+2}}{\tau-2}e^{-rR^b} - \int_{rR^b}^\infty \frac{u^{-\tau+2}}{\tau-2}e^{-u} du \right] \\
    & \approx (r\,R^b)^{\tau-2} \int_{r\,R^b}^\infty u^{-\tau+2}e^{-u} du \\
    & \propto (r\,R^b)^{\tau-2}.
\end{split}
\end{equation}

When $T^I<T^I_c=1/(\kappa-1)$, so $s_{\max}$ is finite and thus $P(s) \sim s^{-\tau+1}\exp(-s/s_{\max})$,
or if $T^I\lesssim T^I_c=1/(\kappa-1)$, but $r$ is very small,
so that $1/s_{\max}$ can not be ignored compared with $r R^b$,
\begin{equation}
\label{app_eq_rbelow}
\begin{split}
R & \approx 1-\frac{\int_1^\infty s^{-\tau+1}e^{-s/s_{\max}}(1-r\,R^b)^s ds}{\int_1^\infty s^{-\tau+1}e^{-s/s_{\max}} ds} \\
    & \approx 1-\frac{\int_1^\infty s^{-\tau+1}e^{-s/s_{\max}}(1-r\,R^bs) ds}{\int_1^\infty s^{-\tau+1}e^{-s/s_{\max}} ds} \\
    & = r\,R^b \frac{\int_1^\infty s^{-\tau+2}e^{-s/s_{\max}} ds}{\int_1^\infty s^{-\tau+1}e^{-s/s_{\max}} ds} \\
    & \propto r\,R^b.
\end{split}
\end{equation}

When $T^I>T^I_c=1/(\kappa-1)$, it is in the epidemic phase and
$R$ does not approach $0$ as $r\rightarrow 0$,
so there is no power-law relation between $R$ and $r R^b$ in this regime.

On the other hand, to derive how $R^b$ depends on $R$, recall that $P^b(s) \sim s^{-\tau^b+1}\exp(-s/s^b_{\max})$.
When $T^b=T^b_c(r\rightarrow 0)=1/(\kappa^b-1)$,
so $s^b_{\max}$ diverges and $P^b(s) \sim s^{-\tau^b+1}$,
\begin{equation}
\label{app_eq_rbat}
\begin{split}
R^b & \approx 1-\sum_{s=1}^\infty P^b(s)(1-R)^s \\
    & \approx 1-\frac{\int_1^\infty s^{-\tau^b+1}(1-R)^s ds}{\int_1^\infty s^{-\tau^b+1} ds} \\
    & = 1-(\tau^b-2) \int_1^\infty s^{-\tau^b+1}e^{s\ln(1-R)} ds \\
    & \approx 1-(\tau^b-2) \int_1^\infty s^{-\tau^b+1}e^{-Rs} ds \\
    & = 1-(\tau^b-2)R^{\tau^b-2} \int_R^\infty u^{-\tau^b+1}e^{-u} du, \text{ where }u=Rs \\
    & = 1-(\tau^b-2)R^{\tau^b-2} \left[\frac{R^{-\tau^b+2}}{\tau^b-2}e^{-R} - \int_R^\infty \frac{u^{-\tau^b+2}}{\tau^b-2}e^{-u} du \right] \\
    & \approx R^{\tau^b-2} \int_R^\infty u^{-\tau^b+2}e^{-u} du \\
    & \propto R^{\tau^b-2}.
\end{split}
\end{equation}

When $T^b<T^b_c(r\rightarrow 0)=1/(\kappa^b-1)$, and thus $s^b_{\max}<\infty$,
\begin{equation}
\label{app_eq_rbbelow}
\begin{split}
R^b & \approx 1-\sum_{s=1}^\infty P^b(s)(1-R)^s \\
    & \approx 1-\frac{\int_1^\infty s^{-\tau^b+1}e^{-s/s^b_{\max}}(1-R)^s ds}{\int_1^\infty s^{-\tau^b+1}e^{-s/s^b_{\max}} ds} \\
    & \approx 1-\frac{\int_1^\infty s^{-\tau^b+1}e^{-s/s^b_{\max}}(1-Rs) ds}{\int_1^\infty s^{-\tau^b+1}e^{-s/s^b_{\max}} ds} \\
    & = R \frac{\int_1^\infty s^{-\tau^b+2}e^{-s/s^b_{\max}} ds}{\int_1^\infty s^{-\tau^b+1}e^{-s/s^b_{\max}} ds} \\
    & \propto R.
\end{split}
\end{equation}

When $T^b>T^b_c(r\rightarrow 0)=1/(\kappa^b-1)$,
most bridge nodes are connected into one big cluster through bridge links, so Eqs.~(\ref{eq_rbsum})~and~(\ref{eq_rbsumapprox}) do not hold and
$R^b$ is not a power law of $R$.

\section{Extension: Regimes when $P^A(k)\ne P^B(k)$}
\label{app_sec_panepb}

When the two communities of the system have different degree distributions,
i.e., $P^A(k)\ne P^B(k)$,
a similar methodology can be applied to make predictions about the asymptotic power-law relations between $R$ and $r$ in different regimes.

In this case,
instead of Eqs.~(\ref{eq_f}) and (\ref{eq_fb}),
we are going to have the following theoretical equations:
\begin{eqnarray}
\label{app_eq_fa}
f^A &=& (1-r)\left[1-G_1^A(1-T^I f^A)\right] + r\left[1-G_1^A(1-T^I f^A)G_0^b(1-T^bf^{A,b})\right],\\
\label{app_eq_fb}
f^B &=& (1-r)\left[1-G_1^B(1-T^I f^B)\right] + r\left[1-G_1^B(1-T^I f^B)G_0^b(1-T^bf^{B,b})\right],\\
\label{app_eq_fab}
f^{A,b} &=& 1-G_1^b(1-T^bf^{B,b})G_0^B(1-T^I f^B),\\
\label{app_eq_fbb}
f^{B,b} &=& 1-G_1^b(1-T^bf^{A,b})G_0^A(1-T^I f^A),
\end{eqnarray}
where $f^A$ (or $f^B$) is the probability to expand a branch to the infinity through an internal link in community A (or B), $f^{A,b}$ (or $f^{B,b}$) is the probability to expand a branch to the infinity through a bridge link,
which starts from a bridge node in community A (or B);
instead of Eqs.~(\ref{eq_r}) and (\ref{eq_rb}),
we are going to have
\begin{eqnarray}
\label{app_eq_ra} R^A & = &(1-r)\left[1-G_0^A(1-T^I f^A)\right] + r\left[1-G_0^A(1-T^I f^A)G_0^b(1-T^bf^{A,b})\right], \\
\label{app_eq_rb} R^B & = &(1-r)\left[1-G_0^B(1-T^I f^B)\right] + r\left[1-G_0^B(1-T^I f^B)G_0^b(1-T^bf^{B,b})\right], \\
\label{app_eq_r_avg}
R & = & (R^A+R^B)/2, \\
\label{app_eq_rab}
R^{A,b} &=& 1-G_0^b(1-T^bf^{A,b})G_0^A(1-T^I f^A), \\
\label{app_eq_rbb}
R^{B,b} &=& 1-G_0^b(1-T^bf^{B,b})G_0^B(1-T^I f^B), \\
\label{app_eq_rb_avg}
R^b & = & (R^{A,b}+R^{B,b})/2.
\end{eqnarray}
It is easy to see that the equations above will be reduced to Eqs.~(\ref{eq_f})-(\ref{eq_rb}) if $P^A(k)=P^B(k)=P(k)$.

To predict the relation between $R^A$, $R^B$, and $R$ as a function of $r$,
we assume $\kappa^A<\kappa^B$,
without loss of generality.
Then there will be 15 regimes,
namely, the five regimes for $T^I$ ($T^I<\frac{1}{\kappa^B-1}<\frac{1}{\kappa^A-1},T^I=\frac{1}{\kappa^B-1}<\frac{1}{\kappa^A-1},\frac{1}{\kappa^B-1}<T^I<\frac{1}{\kappa^A-1},\frac{1}{\kappa^B-1}<T^I=\frac{1}{\kappa^A-1},\frac{1}{\kappa^B-1}<\frac{1}{\kappa^A-1}<T^I$),
combined with the three regimes for $T^b$ ($T^b<\frac{1}{\kappa^b-1},T^b=\frac{1}{\kappa^b-1},T^b>\frac{1}{\kappa^b-1}$).

Here, we select two regimes as examples: (a) $T^I<\frac{1}{\kappa^B-1}<\frac{1}{\kappa^A-1},T^b=\frac{1}{\kappa^b-1}$, and (b) $T^I=\frac{1}{\kappa^B-1}<\frac{1}{\kappa^A-1},T^b=\frac{1}{\kappa^b-1}$.

In case (a),
where $T^I<\frac{1}{\kappa^A-1}$ and $T^I<\frac{1}{\kappa^B-1}$,
using a similar methodology as in Eq.~(\ref{eq_rvsrrb}),
we will get
$R^A \propto rR^{A,b}$ and $R^B \propto rR^{B,b}$.
Since $T^b=\frac{1}{\kappa^b-1}$,
using a similar methodology as in Eq.~(\ref{eq_rbvsr}),
but substituting $(1-R)$ by $\sqrt{(1-R^A)(1-R^B)}$
\footnote[2]{For a finite cluster of bridge nodes, especially if it is large enough to belong to the GC, there are approximately the same number of nodes that belong to each community, so instead of $R^b\approx 1-\sum_s^\infty P^b(s)(1-R)^s$ as in Eq.~(\ref{eq_rbsumapprox}), we will have $R^b\approx 1-\sum_s^\infty P^b(s)(1-R^A)^{s/2}(1-R^B)^{s/2}$ in this case.},
we will get $R^{A,b}\sim R^{B,b}\sim R^b\propto \left[1-\sqrt{(1-R^A)(1-R^B)}\right]^{\tau^b-2}$.
In the case of $r\rightarrow 0$ so that $R^A,R^B\rightarrow 0$,
as well as $\kappa^A<\kappa^B$ so that $R^B\gg R^A$,
we will have $1-\sqrt{(1-R^A)(1-R^B)}\approx \frac{R^A+R^B}{2}=R\propto R^B$.
That is to say,
$R$ is dominated by the community with a larger $\kappa$.
As a result,
we will have $R^b\propto (R^B)^{\tau^b-2}$.
Combining $R^A \propto rR^b,R^B \propto rR^b$ with $R^b\propto (R^B)^{\tau^b-2}$,
we will get
$R^A\propto r^{1/\epsilon^A}$,
where $\epsilon^A=1-(\tau^b-2)$, and $R^B\propto r^{1/\epsilon^B}$,
where $\epsilon^B=1-(\tau^b-2)$.
We have $\epsilon^A=\epsilon^B$,
as expected,
since $T^I$ is below critical in both communities,
and the $\epsilon$ as in $R\propto r^{1/\epsilon}$ has the same value $\epsilon=1-(\tau^b-2)$ as well.

In case (b),
where $T^I<\frac{1}{\kappa^A-1}$ and $T^I=\frac{1}{\kappa^B-1}$,
similarly,
we will get
$R^A \propto rR^b$ and $R^B \propto (rR^b)^{\tau-2}$.
Since $T^b=\frac{1}{\kappa^b-1}$,
we still get $R^b\propto \left[1-\sqrt{(1-R^A)(1-R^B)}\right]^{\tau^b-2}\propto \left(\frac{R^A+R^B}{2}\right)^{\tau^b-2}\propto (R^B)^{\tau^b-2}$.
Combining them,
we will get
$R^A\propto r^{1/\epsilon^A}$,
where $\epsilon^A=1-(\tau-2)(\tau^b-2)$, and $R^B\propto r^{1/\epsilon^B}$,
where $\epsilon^B=\frac{1}{\tau-2}-(\tau^b-2)$.
The $\epsilon$ as in $R\propto r^{1/\epsilon}$ is dominated by $\epsilon^B$,
i.e., $\epsilon=\epsilon^B=\frac{1}{\tau-2}-(\tau^b-2)$.

\begin{figure}[t] \begin{minipage}[t]{0.45\linewidth}\centering
\includegraphics[width=6cm]{{./figures/theory_R_vs_r_T_A_0.0625_T_B_0.0625_T_b_0.1}.eps}
\medskip
\centerline{(a)}
\end{minipage}\hfill
\begin{minipage}[t]{0.45\linewidth}\centering
\includegraphics[width=6cm]{{./figures/theory_R_vs_r_T_A_0.125_T_B_0.125_T_b_0.1}.eps}
\medskip
\centerline{(b)}
\end{minipage}
\caption{$R^A$ (blue), $R^B$ (orange), and $R=(R^A+R^B)/2$ (black) as a function of $r$, for two example regimes when $P^A(k)\ne P^B(k)$: (a) $T^I=0.0625$, $T^b=0.1$; (b) $T^I=0.125$, $T^b=0.1$. Both internal links and bridge links are ER networks, with $\langle k^A\rangle=4$, $\langle k^B\rangle=8$,
and $\langle k^b\rangle=10$.
In each regime, numerical solutions of Eqs.~(\ref{app_eq_fa})-(\ref{app_eq_r_avg}) are plotted in solid lines, and dashed lines are drawn with predicted slopes.}
\label{app_fig_exponents_ab}
\end{figure}

As in Fig.~\ref{app_fig_exponents_ab},
we consider a system where both internal and external links are ER networks,
with $\langle k^A\rangle=4$,
$\langle k^B\rangle=8$,
and $\langle k^b\rangle=10$.
As a result,
$\tau^I=\tau^b=5/2$,
$\kappa^A=5$ such that $T^A_c=\frac{1}{\kappa^A-1}=0.25$,
$\kappa^B=9$ such that $T^B_c=\frac{1}{\kappa^B-1}=0.125$,
and $\kappa^b=11$ such that $T^b_c(r\rightarrow 0)=\frac{1}{\kappa^b-1}=0.1$.
We can see from Fig.~\ref{app_fig_exponents_ab}
that the numerical solutions of $R^A,R^B$, and $R$ from Eqs.~(\ref{app_eq_fa})-(\ref{app_eq_r_avg}) (solid lines) agree well with dashed lines, whose slopes are predicted as above, as $r\rightarrow 0$.
It can be verified that a similar methodology can be used to give correct predictions for all 15 regimes.

\section{Simulation Results Compared with Numerical Solutions}
\label{app_sec_compare}

\begin{figure}[t]
\begin{minipage}[t]{0.49\linewidth}\centering
\includegraphics[width=5cm]{{./figures/compare_R_vs_r_T_i_0.25_T_b_0.1}.eps}
\medskip
\centerline{(a)}
\end{minipage}\hfill
\begin{minipage}[t]{0.49\linewidth}\centering
\includegraphics[width=5cm]{{./figures/compare_R_vs_r_T_i_0.25_T_b_0.05}.eps}
\medskip
\centerline{(b)}
\end{minipage}
\caption{
Theoretical solutions (black solid lines) compared with simulation results (from link percolation mapping, dashed lines with diamond symbols) with (a) $T^I=0.25$, $T^b=0.1$, and system size $N_A=N_B=10^7$ (blue), and (b) $T^I=0.25$, $T^b=0.05$, with system sizes $N_A=N_B=10^5,10^6,10^7,10^8$ (from light blue to dark blue). Both internal links and bridge links are ER networks, with $\langle k \rangle=4$ and $\langle k^b \rangle=10$, respectively. For the simulations, $k_{\min}=0$, $k_{\max}=100$, and are averaged over $100$ realizations.}
\label{app_fig_rvsrmultiple}
\end{figure}

\begin{figure}[htb]
\centering
\includegraphics[width=10cm]{./figures/boxplot.eps}
\caption{Box plots of simulation results (from link percolation mapping) compared with the theoretical solution (dashed horizontal line), for different system sizes with $r=0.003$, $T^I=0.25$, and $T^b=0.05$.
Both internal and bridge links are ER networks, with $\langle k \rangle=4$, and $\langle k^b \rangle=10$, respectively.
For the simulations, $k_{\min}=0$, $k_{\max}=100$, and are plotted with $200$ realizations.}
\label{app_fig_boxplot}
\end{figure}

In Fig.~\ref{app_fig_rvsrmultiple},
we show the simulation results of the link percolation mapping and numerical solutions of Eqs.~(\ref{eq_f})-(\ref{eq_r}) when both internal links and bridge links are ER networks, with $\langle k \rangle=4$, and $\langle k^b \rangle=10$. When $T^I=1/\langle k \rangle=0.25$, $T^b=1/\langle k^b \rangle=0.1$, the simulation agrees well with theoretical solutions [see Fig.~\ref{app_fig_rvsrmultiple} (a)]. When $T^I=0.25$, $T^b=0.05$, a finite-size effect shows up and a much larger system size is required in order to obtain the theoretical results. From Fig.~\ref{app_fig_rvsrmultiple} (b) we can see that as the system size increases, the simulation results converge to the theoretical solution.
This is further verified in Fig.~\ref{app_fig_boxplot}, in which we show the box plots of the simulation results of $R$ for different system sizes. We can see that as system size increases, the distribution of $R$ narrows and converges to the theoretical solution (horizontal dashed line).

\end{appendices}

redpajama_set_name: RedPajamaArXiv


"And the Lord did so. Thick swarms of flies came into the house of Pharaoh, into his servants' houses, and into all the land of Egypt. The land was corrupted because of the swarms of flies" (Exodus 8:24 NKJV).
In the beginning of John's Revelation Jesus addressed seven letters to seven congregations in the Roman province of Asia. The first of these is addressed to the Christians in Ephesus.
For the appeal we make does not come from error or impurity or with deceit, but just as we have been approved by God to be entrusted with the gospel, so we declare it, not to please people but God, who examines our hearts.
The Romans taxed nearly everything. They taxed personal income and the use of roads and harbors. They taxed carts according the number of wheels, and they taxed for the animal that pulled the cart. If a person was walking with a bundle on his back, a tax collector could require the bundle opened and taxes could be collected on any item therein.
In 17 years of ministry (part and full-time), I have met with numerous married, and to-be-married couples. One thing I've found by this experience is that to-be-married couples generally understand the concept of adultery, and see it as a legitimate – if not the only – reason for divorce. Even if their Bible knowledge is somewhat limited, or they are not Christians, they will often say that this is the only legitimate grounds for a biblical divorce. I have this in writing from almost every couple I've married.
Are some of Jesus' sayings too harsh?
"Unless you repent, you will all…perish" (Luke 13:3).
We read the biblical passage straight through, without pause, as if the dialogue were a race, without feeling the dynamic between two strangers, a man and a woman, a Jew and a Samaritan, he on the road and she taking care of home responsibilities.
It is said that a pessimist decries the fact that roses have thorns, while an optimist rejoices that thorn bushes yield beautiful roses.


text:
Agenția Națională pentru Pescuit și Acvacultură (ANPA) este o instituție publică din România, înființată în anul 2004, finanțată integral de la bugetul de stat, aflată în subordinea Ministerului Agriculturii și Dezvoltării Rurale, ca autoritate publică centrală, care răspunde de pescuit și acvacultură.

ANPA a fost organizată prin preluarea Inspecției Piscicole și atribuțiilor în domeniul pescuitului de la Autoritatea de Management Program Operațional Sectorial pentru Agricultură, Dezvoltare Rurală și Pescuit, direcție generală în structura Ministerului Agriculturii și Dezvoltării Rurale care se ocupa cu coordonarea, implementarea și gestionarea asistenței financiare comunitare prin instrumentele structurale.

ANPA elaborează strategia și cadrul legal privind dezvoltarea sectorului pescăresc în România, fiind responsabilă de implementarea Politicii Comune a Pescuitului în România și de implementarea tehnică a măsurilor și de controlul reglementărilor în domeniul pescuitului și acvaculturii.

Agenția acoperă întreg teritoriul țării prin cele 10 filiale: Iași, Galați, Tulcea, Constanța, Călărași, Mehedinți, Ilfov, Brașov, Timiș și Cluj.

În anul 2007, ANPA avea 150 de angajați.

Vezi și
 Ministerul Agriculturii și Dezvoltării Rurale (MADR)
 Fondul European pentru Pescuit (FEP)

Note

Legături externe
www.anpa.ro  - Site web oficial
 Se înființează Agenția Națională pentru Pescuit și Acvacultură, 5 octombrie 2004, Amos News
 Executivul a aprobat funcționarea Agenției Naționale pentru Pescuit și Acvacultură, 25 septembrie 2008, Claudia Dumbrava, Amos News

Organizații guvernamentale din România
Ministerul Agriculturii și Dezvoltării Rurale

redpajama_set_name: RedPajamaWikipedia


We've already heard that Samsung will launch the Note10 alongside a smaller version of the stylus-toting phablet, and today a new report from its home country of South Korea comes to elaborate further.
According to this new rumor, there will be no less than four different Galaxy Note 10 versions, two of them with 5G connectivity, and two topping out at 4G.
So, we'll apparently get the 6.75-inch Note10 along with a 6.28" version. The bigger model gets four rear cameras, while the smaller one has to make do with three. Then we have two almost identical phones in terms of specs, but with 5G added in for good measure.
Samsung is said to be making this unprecedented move, of launching four Note flagships in a year, because of how successful the same diversification strategy has been for the S10 line. Remember, there are four of those too: the S10, S10+, S10e, and S10 5G. Obviously the smaller Note 10 will try to replicate the S10e's success in the stylus-toting flagship line, while the two 5G variations will be there because it's 2019, the year in which marketing departments everywhere in the mobile world can't wait to utter that buzzword.
Okay, it still stands that I don't think it's disgraceful nor dishonorable.
Samsung remind me what Nokia did in the past.


Helping you to cook like it's 1829.
How did it manage to stay untouched for so long?
What is Lunch with the Curator?
It's another way to experience The Regency Town House. It's a tour with a lunch. Not just any tour, it's a tour of the Upstairs and Downstairs life in an 1830s house. Not just any lunch, it's a lunch held in the basement kitchen of the Town House, with dishes based on authentic recipes of the time.
What is the Regency Town House project?
The Regency Town House project started in 1984 and it is a labour of love for the curator, Nick Tyson. Together with Phil Blume they are the only two paid members of staff. Everyone else on the project gives their time for free. Upstairs there are teams of researchers busy with the project My House My Street. This is a project which enables people to search multiple historical records about one address, online with one website. With them, working in the old bedrooms which are now the offices, are students on placements from the UK and abroad. Downstairs there are teams of volunteers busy with painting & decorating, construction, cooking, sewing and event management. The Town House is a hive of working volunteers who join in with the project due to their love of history and the community that goes with the Town House.
The tour is given by curator Nick Tyson. He has given tours since the beginning and is a remarkable source of knowledge about every aspect of the house. He gives instruction on how the house was built, how Brighton and Hove developed and the sort of people that lived there. He is knowledgeable about both the upstairs gentry and the downstairs servants. Often visitors enjoy the stories about the servants, in the downstairs offices, the best.
The basement annexe is the highlight of the tour. Pat Nixon lived there from the 1920s to the late 1990s, that is when the Trust bought the basement. She was an elderly woman when Nick, still a young man, came and did her shopping for her. He saw the beauty of the basement she was living in. She couldn't see it herself. She was, in fact, living in a time capsule. The basement hadn't been modernised like much of the basement in the Square. It retains its meat safe, the flagstone floor, the rooms are in the original positions and even the well in the courtyard still has water in it.
The lunch, after the tour, gives house visitors a chance to put questions to Nick directly but also to speak to each other. The food menu is created by volunteer Paul Couchman who is beginning his own food business, giving cooking courses and catering for the many artists' openings that take place in the Town House. The menu is based on historic recipes, all of the food that he creates would have been made in the very kitchen the visitors are eating in. The menu consists of two courses.
Pickled fennel, pickled grapes and piccalilli.
How do you Dine Like A Servant?
The first thing visitor notice, as they enter the housekeeper's room at 10 Brunswick Square are the cupboards. Cupboards everywhere and all of them have locks. Locked away in the housekeeper's room were many things that servants could steal and sell. Porcelain, the finest table napkins, china and candles. And there were pickles. Hannah Glasse, the eighteenth-century cook I often turn to, had these wise words to say about pickles.
I went for glass, stone jars are the next thing to find for the kitchen. I decided also to go for pickled grapes and pickled fennel. There would also be pickled cucumber. Cucumber with onion and ginger. The recipes appear almost modern.
I was unsure about the fennel. But I knew that it would be one of many pickles on a plate, so I thought I could get away with it. I also adore fennel, I love the taste. In the recipe there are also fennel seeds and pepper.
The guests for Lunch with the Curator loved it. I will cook the fennel slightly longer this time and I have added additional spices, more pepper. The recipe will follow below. Please adapt it as you will. Hannah Glasse, at the end of many a recipe has these last words of wisdom.
Shrove Tuesday – 3 unusual things to add to a pancake.
Paula Wrightson on How did it manage to stay untouched for so long?
Paul Couchman on How did it manage to stay untouched for so long?
Martin Carrolchick on How did it manage to stay untouched for so long?


text:
With Chicks & Chickens, Keep Your Lips To Yourself
To safeguard your own health, love your flock—from a distance. Here's why and what you can do.
by Ana HotalingMay 8, 2017
They're soft. They're fuzzy. They're the epitome of innocence. Who can resist cuddling up to an adorable baby chick? The feel of that silky down caressing your skin may encourage you to plant a kiss on that tiny fluff ball, but resist the temptation: As cute as those chicks (and, later, adult chickens) may be, they might be hazardous to your health. Here are some of the risks associated with raising chicks and chickens, and how to keep yourself safe.
The Story On Salmonella
A group of bacteria that affect that the human digestive system, causing diarrhea, vomiting, fever and abdominal cramps, salmonella is carried by most warm-blooded animals. Due to reported outbreaks triggered by the consumption of infected eggs and meat, salmonella is most closely associated with poultry. Eating contaminated chicken meat and eggs is not the only way salmonellosis spreads, however: Transmission is far more common through direct contact with live chicks and adults. In fact, the Centers for Disease Control and Prevention has reported salmonella outbreaks resulting from the handling of infant and mature poultry every single year since 2010.
"Salmonella is shed from the bird's body through its manure," explains R.M. Fulton, an avian pathologist at the Diagnostic Center for Population and Animal Health at Michigan State University. "Of course, they come into contact with their own manure, and the bacteria gets on the birds' feathers and feet. When people kiss baby poultry or let their children have long periods of contact with them, the chance of being exposed and becoming sick increases dramatically."
Salmonella infection may be on the rise due to the growing trend of urban and suburban chicken keeping, which puts live poultry literally right in people's backyards or even inside their home.
"I am aware of people putting baby chicks in a playpen with their toddler," Fulton says. "There are people raising poultry in kitchen sinks, bathtubs and basements. It's very hard to house-train poultry, so all of these are dangerous situations."
Crash Course On Campylobacter
One of the leading causes of diarrheal disease in the U.S., campylobacter affects more than 1.3 million people every year, according to the CDC. Campylo-bacteriosis is frequently shrugged off as a stomach bug and often goes undiagnosed or untreated, despite its symptoms of abdominal pain, cramping, fever and blood-tinged diarrhea.
Like salmonella, campylobacter is associated with contaminated poultry and food that becomes cross-contaminated from using the same cutting board or utensils that came into contact with the raw poultry. According to the CDC, even one drop of liquid from raw chicken meat has enough campylobacter to cause infection. Like salmonella, ingesting the campylobacter is not the only way to become infected; campylobacteriosis is also spread by live contact.
Campylobacter naturally occurs in a chicken's digestive tract, where it lives harmlessly until it is shed via the animal's feces onto the ground or through its saliva into a common farmyard water source, spreading the bacteria to the rest of the flock. Once the microbe is picked up by a chicken's feathers or feet, it can easily transmit itself to a human. All it takes is one hug or cuddle.
Facts About The Flu
Much has been made of avian influenza, aka bird flu, over the past decade. Unlike salmonella and campylobacter, which can reside in birds without causing illness, avian influenza can acutely affect poultry, causing severe illness and death. Infection of backyard flocks so far has been minimal, but authorities have issued prevention guidelines and safety measures for what to do in case that happens.
The avian influenza virus can pass from bird to human not just by physical contact but also by breathing in the discharge from an infected animal's coughs and sneezes. Fortunately, the CDC reports that no human cases of avian influenza have yet been recorded in the U.S.
"News agencies are always concerned about a new strain of influenza arising from poultry and making people sick," Fulton says. "So far, we have not seen those strains in the U.S. They are, however, found in Asia."
Reducing Your Risk
While chances of infection do exist when raising poultry, these can be easily minimized by following a few preventative practices in your home and out in the chicken run.
Clean Hands
Wash your hands thoroughly with soap and water after coming in contact with your birds and their equipment. This includes feeders, waterers, perches, nest boxes and litter, all of which come into contact with chicken feet, feathers, saliva and droppings. Be sure to also wash your hands after handling and collecting eggs, which most definitely have been in close contact with a bird's bottom.
"My mom always told me to wash my hands before I eat," Fulton says. "She also taught me to cut up fruits and vegetables before cutting up whole chickens or raw meat. "If people would just follow that advice, they could prevent infection."
Clean Clothes
Designate a pair of work boots and gloves and some clothes—a coverall, a pair of jeans, a sweatshirt, a jacket—as your chicken gear. Wear only these when working in your runs and coops and, if you free-range your birds, anywhere in your yard. Store your chicken gear safely in a location well away from your personal living space and be sure to disinfect them frequently.
Clean Cartons
Don't reuse egg cartons. Egg cartons are porous and readily absorb bacteria from unwashed or broken eggs. Don't donate previously used egg cartons or trays to schools for use in craft projects.
Clean Disposal
Take extra precautions when handling a dead bird. Follow your local ordinances for the disposal of animal corpses. If you are uncertain why your bird died, contact your local extension office to arrange for a professional necropsy. Do not dissect the bird yourself. "Accidental puncture wounds from cutting up dead poultry can result in other, rarer diseases that people can get from poultry," Fulton says.
Above all, keep your kisser away from your feathered friends. It's perfectly fine to lavish a little love on your birds but just make sure you keep yourself healthy by giving your relationship a little space.
Instead of putting your smacker up to your birds' beaks, show you love them in other, more healthy ways.
Put some sticks and branches in the coop or run for your chickens to exercise on.
Hang a head of cabbage up for your birds to play with.
Give them some mealworms to snack on.
Spend some extra time with them in the coop, just watching and observing.
Create a dust-bathing area for them to preen their feathers.
This story appeared in the May/June 2017 issue of Chickens.
Ana Hotaling and her husband, Jae, own FMA Farms, a heritage-poultry farm in Southeast Michigan. When not writing or chasing after chickens and children, Ana teaches martial arts and yoga and is a competitive triathlete and runner. Follow Ana's poultry adventures at www.facebook.com/FMAFarms.

redpajama_set_name: RedPajamaCommonCrawl


LOOKING FOR EXPERIENCED SERVER STAFF WHO ARE LIKE MINDED AND CAN PROVIDE SOMETHING DIFFERENT TO BENEFIT THIS NEW SERVER.
It will be a 24/7 factions server but I want to set it all up properly before launching. Need to set up permissions properly, setting safe and war zones and banners for website ect.


text:
Unconventional personal diplomacy between US and Russia
Kilic Bugra Kanat Posted On January 10, 2017
With the recent intelligence briefing by the intelligence community for President-elect Donald Trump, once more relations between the U.S. and Russia top the agenda of U.S. foreign policy. In a report leaked to the press, the intelligence community asserted, "Russian efforts to influence the 2016 U.S. presidential election represent the most recent expression of Moscow's longstanding desire to undermine the U.S.-led liberal democratic order, but these activities demonstrated a significant escalation in directness, level of activity and scope of effort compared to previous operations." Accordingly, this operation was initiated from the very top of the Russian government. The report stated: "We assess Russian President Vladimir Putin ordered an influence campaign in 2016 aimed at the U.S. presidential election.
Russia's goals were to undermine public faith in the U.S. democratic process, denigrate Secretary Clinton, and harm her electability and potential presidency. We further assess Putin and the Russian government developed a clear preference for President-elect Trump. We have high confidence in these judgments." Everybody was curious how this long expected briefing would influence the approach of the president-elect and his policy toward Russia. However, a day after this briefing, Trump returned to tweet and harshly criticized those who have criticized his outreach efforts to Russia. He wrote in a series of tweets: "Having a good relationship with Russia is a good thing, not a bad thing. Only "stupid" people or fools would think that it is bad! We have enough problems around the world without yet another one. When I am president, Russia will respect us far more than they do now, and both countries will perhaps work together to solve some of the many great and pressing problems and issues around the WORLD!"
These recent developments launched a major debate in Washington, D.C. in regards to the future of U.S.-Russian relations. There are a lot of different scenarios provided by observers of bilateral relations between the two countries. Especially personal relations between Trump and Putin are the most frequently asked issue nowadays. Most recently, after the deportation of 35 Russian diplomats by the U.S. administration when Putin did not retaliate in a similar way President-elect Trump tweeted his reaction to this by writing, "Great move on delay (by V. Putin)- I always knew he was very smart!" Thus, the chemistry between the two leaders after Trump assumes office will be very important for the future of bilateral relations.
The next U.S. administration is more willing to work with Russia and work on eliminating Daesh threats. However, there is still some ambiguity about the extent of cooperation and whether there will be any joint effort to resolve the crisis in Syria
However it is important to remember that personal diplomacy will not be the only factor shaping U.S.-Russian relations. In the next administration, there will be serious challenges, risks as well as opportunities for both countries. As mentioned in the previous piece, there will be many variables involved in this process. Of course, foreign policy makers and their different approaches to Russia will be one of the most determining factors of bilateral relations. There are hardliners in the administration, such as Secretary of Defense James Mattis, whose approach to Russia differs from that of President Trump. Secretary of State Tillerson, on the other hand, is well known for his close ties with Russia because of the energy deals that he brokered with Rosneft. He is a recipient of the Russian order of Friendship by President Putin. In the immediate aftermath of his appointment, the opinions of National Security Advisor Mike Flynn on Russia have also become a hotly debated issue.
There are several issue areas that will be discussed in the coming months in relation to Russia. The crisis in Ukraine and sanctions against Russia due to this crisis will be one of them. The members of the administration may have differing views from the administration on how to handle this crisis, and the position of the U.S. will have some important repercussions on its relations with Central and Eastern European states as well as NATO. While the Trump administration tries to make its mind about Ukranian crises, there will also be external determinants that aim to influence decision making in Washington, D.C. For instance, shortly after the elections the Russian foreign ministry made a statement accusing the Ukranian government of trying to influence the elections in the U.S. by sabotaging the Trump campaign. There was a widespread belief that Ukranians were disappointed by the election of Trump due to his views about Russia. Former ambassador of the U.S. in Moscow Michael McFaul even tweeted about this and said, "Biggest loser in the world tonight is Ukraine."
In regards to Syria, some members of the incoming administration are more willing to work with Russia to, in particular, deal with Daesh threats. The president-elect also signaled a similar proposal in the aftermath of his election. However, there is still some ambiguity about the extent of cooperation, and if there will be any joint effort to resolve the crisis in Syria. The safe zone issue for instance was mentioned during the campaign; however, since then there has been no clarification about if and how to adopt such a strategy.
Other issues between the two countries will involve cybersecurity, which looks like one of the fault lines in relations between the two countries. The debate was triggered particularly due to Edward Snowden's residing in Russia, and since then in multiple instances the U.S. administration accused the Russian government of organizing cyber attacks on the American people. After the aforementioned intelligence briefing, it will be important to see precautions by the U.S. and possible retaliation that President Obama ordered.
Another significant issue, of course, will be energy politics. Given the resume of several high-level members of the administration, it is not hard to expect that energy will be an issue that will be frequently discussed by the Trump administration. Although the sanctions, oil prices and carbon fuels became the most frequently mentioned issues in these debates, recently the U.S. is also being more attentive to nuclear security and its dependence on foreign uranium sources for its reactors. Around 20 percent of uranium comes from Russia, and this has recently been reported as a possible significant national security risk factor.
In fact, there will definitely be critical choke points in bilateral relations between the two countries. The main challenges may be between those who want to resolve geopolitical issues with Russia before engaging in further collaboration and cooperation, and those who wants to extend the existing level of economic and, especially, energy-related cooperation between the two countries before handling other disputes. Recent probes and allegations in regards to the hacking of the Democratic National Convention by Russian hackers and deportation of 35 Russian diplomats will definitely be on the agenda, while the administration tries to make its mind about how to approach Russia. It even looks like a search for a new Russia policy may generate serious friction between the Congress and the White House and among different federal institutions in the country or, even more specifically, by members of the administration. This will likely be reflected in the media through different leaks and stories. Competition between different actors may lead to a dysfunctional Russia policy if it is not contained by the administration.
This article was first published in Daily Sabah on January 10, 2017.
Kilic Bugra Kanat
kanat@setadc.org
Challenges of the transition in the US


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Seoul E-Land Vs. Bucheon 1995: Can Bucheon Break the Streak?
Michael Redmond March 30, 2021 read ( words)
Last week saw a break from the already intense 2021 K League 2 Season. The magic of the FA Cup came to Jamsil when Seoul E-Land put five goals past Songwol FC. setting up the first-ever Seoul Derby in their existence. Attention now diverts back to the league, as the leaders welcome Bucheon 1995 to town. Seoul had a very successful head to head with Bucheon last year, winning all three of their meetings.
Bucheon will look to show that they have improved from the season prior. Revenge will be in the air for the travelling side. For Seoul, a win will keep them top of the table. Three points are also much needed after two draws in the previous two rounds. Michael Redmond previews this one here.
Rose Tinted Glasses for Seoul E-Land
A rule of three applied to Seoul when these two met in the 2020 K League 2 season. Seoul won all three of their games by three goals, claiming three points against Bucheon. The first game will easily go down as the most memorable one, as Seoul went 2-0 down in the game. Only to claw back and win in the dying moments of the match.
The second and third meetings were simple 3-0 wins for Seoul. Bucheon just did not look the same calibre as Seoul last year. It truly was a season to forget for the side. Leandro was the man Bucheon still have nightmares about. Scoring in all three fixtures and having four goals to his name against the side. Seoul only conceded two goals against Bucheon in the whole of 2020.
A Familiar Face
A familiar face that Bucheon fans will remember will be the one of Brazilian, William Barbio. He scored three goals last year for Bucheon, including a penalty against Seoul E-Land. At Seoul, he plays in a midfield role and uses his pace to push the ball up the pitch. Fans will only need to remember the goal against Jeonnam in round three to confirm this. The link-up with Leandro had the ball in the back of the Dragon's net within nine seconds of a Jeonnam corner.
Barbio wore the colours of Bucheon in the 2020 K League 2 Season.
Barbio has not yet scored for Seoul this year, but his character in midfield has really helped Seoul in breaking away on an attack. Bucheon will have eyes firmly on him this weekend. It won't be Barbio they will worry about scoring, but more that Barbio will be the player who creates the chances.
Tight at the Top
Of course, it is way too early to presume any league position in the K League 2. As only four rounds of the season has been played. However, Seoul have been 1st since round one and they have remained there ever since. Only Seoul and Jeonnam have yet to lose this year and it would be fair to say that Seoul had the hardest start. An opening day away game against Busan IPark followed by welcoming Gimcheon Sangmu and Jeonnam Dragons to Leoul Park. The last round was a trip to Changwon to take on playoff contenders Gyeongnam.
Ansan Greeners are the team closest to Seoul at the moment, with one point separating the two. Fans maybe quite shocked to see the two teams who, in the past were fighting for the wooden spoon are both glaring at the top of the table. With that being said, it takes one loss for either side and they could drop down to 5th. Seoul can't take teams like Bucheon and Chungnam Asan for granted now, simply because they may not have been seen as playoff contenders. If you may recall Seoul E-Land last season had an excellent record against the 'big five', yet failed miserably against the teams at the bottom.
Rotation, Rotation, Rotation
Last week brought FA Cup action to Leoul Park. This gave rest to a lot, if not most first-team players. It is still early on in the season. However, a break after their first four games would have been welcoming. Looking at Seoul, it appears that Kim Kyeong-min is the man of choice between the sticks. A Goalkeeper who has only conceded one goal so far in an E-Land shirt has commanded the line well.
Seoul E-Land manager Chung Jung-yong is not afraid to change and experiment with his team.
The partnership with Lee Sang-min and Lee In-jae is also working well in the capital. The two defenders have formed a partnership over the last few rounds. Kim Sun-min is once again taking the role in the middle of the field and excelling in the side. Captain Kim Min-kyun is still sidelined and his return is still not yet known.
The last few games saw boss Chung Jung-yong play all his foreign signings in the starting line up. Taking them off throughout the game. Barbio and Leandro bring the pace for Benegas upfront to get into positions to finish and send balls directly to him. However, It will be interesting to see if the boss changes things this week. The side have now drawn two games in a row and a win is becoming much needed.
Bucheon Versatility?
Bucheon plays in a 3-4-3 formation with normally Yun Ji-hyuk, Park Tae-hong, and Kim Jung-ho at the back. Park Jun-hui showed his versatility in the last game filling in for Park Tae-hong in defence.
They are playing three at the back but have only conceded three goals so far, two of them being in the 2-1 loss to Daejeon. So defensive wise they are not doing too bad. A team that has had somewhat of a difficult start, playing both Daejeon and Jeonnam in the first four games and getting 5 points.These points came in a win against Chungnam Asan and draws against Anyang and Jeonnam.
Park Chang-jun has been Bucheon's goal man as far in the 2021 K League 2 Season.
Their midfield normally has Cho Hyeon-taek Oh Jae-hyuk and Sung Hong-min in the midfield. But manager Lee Yeong-min does sometimes change it around. Benching or rotating only slightly. Bucheon's all three goals have come from Park Chang-jun. Fans might question are they relying on the 24-year-old too much to be their frontman. In Seoul's case, it will asked how will Lee Sang-min and Lee In-jae deal with a poacher like Park Chang-jun constantly appearing in the box. He seems to get himself in the right areas at the right time.
Bucheon have actually started or played better than some may have thought this year. They had a poor season last year and they could not come close to the level that Seoul played at throughout 2020. Bucheon will look to get some revenge against Seoul for embarrassing them in most of the games. Especially going 2-0 down to losing 3-2.
Can Bucheon Break the Streak?
Seoul flew out of the traps at the start of the season and they are still having an exceptional time in the K League 2. With that being said, they need another win on the board to cement their intentions of promotion. Two draws are better than two defeats, especially who they played to share the spoils. Although teams who win leagues, win games week in and week out. Teams will line up now to try and take Seoul down. This will also be true for Bucheon who are now looking to actually break their losing streak against the Leopards.
Seoul needs to once again bring on the goal flair that we saw against Busan, Gimcheon, and even Songwol in the cup. We all know that Seoul can score goals. They have the highest amount of goals scored in the league. They need to be firing goals into the back of the net as soon as possible.
TAGS K League 2 Seoul E-Land


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Crystallized is a series of spaces, which I will decorate with my crystals in different ways to enhance the visual outlook as well as the vibrational energies.
This time a have been spreading my crystal vibrations all over the hair dresser ESHK in Neukölln, to enhance the energy of the salon during Christmas.
I'm in LOVE with the Berlin hair salon! The hairdresser salons originally established in London. Not only do they cut great hair, however they combine the salon with an in house record store. A regularly updated selection of vinyl curated by our Director Miguel along with listening stations are available. You'll often find DJs & producers in the salon doing a set or presenting their latest work. So whether you're after great looking hair or an inspirational new selection of music, pop into the salon!