# Gillespie's SSA¶

At several points, epipack allows to run stochastic simulations which are all based on Gillespie's stochastic simulation algorithm (SSA). This algorithm comes with the advantage of working in continuous time, i.e. with rates instead of probabilities, which preserves neighboring state correlations in network simulations and therefore correctly samples trajectories from the original underlying Markov process.

There's a mountain of literature describing how this algorithm works, so we will be satisfied with a short description of its underlying principles in the following.

## Homogeneous Poisson Processes¶

Gillespie's SSA samples events $$e$$ from an event set $$E$$. Each event is associated with a constant rate $$\lambda_e$$. The total rate of any event happening is given as

$\Lambda = \sum_{e\in E} \lambda_e.$

Now, suppose we're currently at time $$t$$ and we want to know a time leap $$\tau$$. Derived from first principles, this time leap is distributed according to an exponential distribution

$\tau \sim \mathcal E (\Lambda).$

The event that takes place at this time can be chosen from the event set with probability

$\pi_e = \lambda_e / \Lambda.$

After the event takes place, the event set and the event rates can be updated according to the new situation and time is advanced as

$t \leftarrow t + \tau.$

## Inhomogeneous Poisson Processes¶

When rates are time-dependent the math becomes a little more tedious but works out as follows.

Each event is associated with a time-varying rate $$\lambda_e(t)$$. The total rate of any event happening is given as

$\Lambda(t) = \sum_{e\in E} \lambda_e(t).$

Now, suppose we're currently at time $$t$$ and we want to know a time leap $$\tau$$. We find that the quantity

$\Theta(\tau) = \int\limits_t^{t+\tau} dt' \sum_{e\in E}\Lambda(t')$

is exponentially distributed as $$\Theta \sim \mathcal E(1)$$. Hence, we draw a value for $$\tilde\Theta$$ from the default exponential distribution and solve the integral $$\Theta(\tau)$$ for $$\tau$$.

The event that takes place at this time can be chosen from the event set with probability

$\pi_e(t+\tau) = \lambda_e(t+\tau) / \Lambda(t+\tau).$

In epipack, two methods exist to solve this integral numerically. The first method solves the integral $$\Theta(\tau)$$ with a quadrature method from the scipy.integrate module and applies a Newton-Raphson root finding method to this numerical integral function.

The second (and default) method redefines the problem as an initial value problem

$\frac{d}{d\tau}\Theta(\tau) = \Lambda(t+\tau), \qquad \Theta(0) = 0$

And applies a Runge-Kutta 2(3) method until $$\tilde\Theta = \Theta(\tau)$$. This method is faster and has therefore been chosen as the default method.

Both methods yield values of $$\tau$$ that lie within 0.1% of each other.

## Temporal Networks¶

In principle, temporal networks lead to rate functions that change as step functions. Vestergaard and Génois have used this fact to come up with a fast simulation algorithm for temporal networks:

"Temporal Gillespie algorithm: Fast simulation of contagion processes on time-varying networks". C.L. Vestergaard & M. Génois. PLoS Computational Biology (2015) 11, e1004579 (http://arxiv.org/abs/1504.01298).

We do not use this algorithm here though. We simply make use of the fact that we're dealing with Poisson processes. I.e. we treat a network as static first and simulate model processes until we overshoot the time at which the network changes next. As soon as we overshoot a time value at which the network changes, we simply stop at this time value, reset the model with the new network structure and restart the simulation again. Doing so does not lead to erroneous results. Since we're dealing with Poisson processes, we can restart the simulation at this time point, because we've already decided that the next event will happen after this time point. If the rates would not have changed, restarting the simulation is equal to resampling from the tail of the original distribution that was cut off after the network change time.

Resetting the entire model/event set every time the network changes is quite inefficient. Yet, our primary goal here is to have a framework where we can prototype epidemiological models in a fast manner while simulation efficacy is left aside for now.

If you want to simulate canonical models on temporal networks efficiently, check out tacoma where Vestergaard's and Génois's method is implemented (it's really fast).