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Given the following text description, write Python code to implement the functionality described below step by step
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Description:
Step1:  Inferring from long versus short-time regimes
How well can we infer parameters of choice from data that is at different stages? For example, if we are dealing with a system that has equilibrated versus a system that is still in the transition regime. 
The system of choice shall be a simple birth-death process, described by the relations 
Step2:  Above we can see the inference from STAN. Remember that $a=1$ and $\mu=10$, which gives $\gamma=a/\mu=0.1$. Looking at the values above, we can see that STAN didn't quite hit the mark with $a$, whereas it did better at $\mu$ and $\gamma$. 
We shall now try again, but instead of using a single path, we shall use multiple paths.
Step3:  Now that we have data that are mostly from the transition regime, we run our inference again.
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Python Code:
N = 1000
alpha = 1.0
mu = 10.0
x, t = SSA(100,N,a=alpha,mu=mu)
x = x.astype(int) # path data supposed to be integers.
path_data = {
    'N' : N,
    't' : t,
    'x' : x
}
print 'Simulated up to time T =', round(t[N-1]) 
# Setup STAN :
model_description = 
data{
      int<lower=0> N; ## number of time steps
      vector[N] t; ## time value at each time step
      int<lower=0> x[N]; ## population value at each time step
    }
    transformed data{
      int<lower=0> x0; ## starting population value
      x0 <- x[1];
    }
    parameters{
      real<lower=0> alpha; ## birth rate parameter 
      real<lower=0> mu; ## death rate parameter
    }
    
    transformed parameters{
      real<lower=0> gamma;
      gamma <- alpha/mu;
    }
 model {
      x ~ poisson(gamma+(x0-gamma)*exp(-mu*t));
    }
fit = stan(model_code=model_description,
           data=path_data, chains=1,iter=1000)
print fit
Explanation: Inferring from long versus short-time regimes
How well can we infer parameters of choice from data that is at different stages? For example, if we are dealing with a system that has equilibrated versus a system that is still in the transition regime. 
The system of choice shall be a simple birth-death process, described by the relations : 
$$
\begin{align}
\emptyset \stackrel{a}{\to} X,\
X \stackrel{\mu X}{\to} \emptyset.
\end{align}
$$
$X$ here is the population number.
Throughout, we shall use $a=10$ and $\mu=1.0$. 
For some intuition on what the system is doing, we shall use the corresponding reaction rate equation. The reaction rate equation for this system is : 
$$
\begin{align}
\dot{x}=a-\mu\cdot x,\
x(0)=x_0.
\end{align}
$$
As this is a linear equation, we can solve it exactly, with solution
$$
x(t)= a/\mu+(x_0-a/\mu) e^{-t\mu}
$$
Let us set $\gamma = a/\mu$. Then, the solution above can be written as 
$$
x(t)= \gamma+(x_0-\gamma) e^{-t\mu}.
$$
But now note that, since $\mu>0$, we have that $$\lim_{t\to\infty} x(t)=\gamma.$$
That is, for long times, the relevant quantity that drives the system is $\gamma$, and by that time, we can no longer recover the exact values of $a$ and $\mu$.
End of explanation
N = 300
Npath = 10
alpha = 1.0
mu = 10.0
x = np.zeros([Npath,N])
t = np.zeros([Npath,N])
for i in xrange(Npath):
    x[i,], t[i,] = SSA(100,N,a=alpha,mu=mu)
    
x = x.astype(int) # path data supposed to be integers.
x = x.flatten()
t = t.flatten()
Ndata = N*Npath
path_data = {
    'N' : Ndata,
    't' : t,
    'x' : x
}
Explanation: Above we can see the inference from STAN. Remember that $a=1$ and $\mu=10$, which gives $\gamma=a/\mu=0.1$. Looking at the values above, we can see that STAN didn't quite hit the mark with $a$, whereas it did better at $\mu$ and $\gamma$. 
We shall now try again, but instead of using a single path, we shall use multiple paths.
End of explanation
fit_trans = stan(model_code=model_description,
           data=path_data, chains=1,iter=2000)
print fit_trans
Explanation: Now that we have data that are mostly from the transition regime, we run our inference again.
End of explanation
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