Jump Problems
Mathematical Specification of an problem with jumps
Jumps are defined as a Poisson process which changes states at some rate
. When there are multiple possible jumps, the process is a compound Poisson process. On its own, a jump equation is a continuous-time Markov Chain where the time to the next jump is exponentially distributed as calculated by the rate. This type of process, known in biology as "Gillespie discrete stochastic simulations" and modeled by the Chemical Master Equation (CME), is the same thing as adding jumps to a DiscreteProblem
. However, any differential equation can be extended by jumps as well. For example, we have an ODE with jumps, denoted by
\[\frac{du}{dt} = f(u,p,t) + \sum_{i}c_i(u,p,t)p_i(t)\]
where $p_i$ is a Poisson counter of rate $\lambda_i(u,p,t)$. Extending a stochastic differential equation to have jumps is commonly known as a Jump Diffusion, and is denoted by
\[du = f(u,p,t)dt + \sum_{j}g_j(u,t)dW_j(t) + \sum_{i}c_i(u,p,t)dp_i(t)\]
Types of Jumps: Regular, Variable, Constant Rate and Mass Action
A RegularJump
is a set of jumps that do not make structural changes to the underlying equation. These kinds of jumps only change values of the dependent variable (u
) and thus can be treated in an inexact manner. Other jumps, such as those which change the size of u
, require exact handling which is also known as time-adaptive jumping. These can only be specified as a ConstantRateJump
, MassActionJump
, or a VariableRateJump
.
We denote a jump as variable rate if its rate function is dependent on values which may change between constant rate jumps. For example, if there are multiple jumps whose rates only change when one of them occur, than that set of jumps is a constant rate jump. If a jump's rate depends on the differential equation, time, or by some value which changes outside of any constant rate jump, then it is denoted as variable.
A MassActionJump
is a specialized representation for a collection of constant rate jumps that can each be interpreted as a standard mass action reaction. For systems comprised of many mass action reactions, using the MassActionJump
type will offer improved performance. Note, only one MassActionJump
should be defined per JumpProblem
; it is then responsible for handling all mass action reaction type jumps. For systems with both mass action jumps and non-mass action jumps, one can create one MassActionJump
to handle the mass action jumps, and create a number of ConstantRateJumps
to handle the non-mass action jumps.
RegularJump
s are optimized for regular jumping algorithms like tau-leaping and hybrid algorithms. ConstantRateJump
s and MassActionJump
s are optimized for SSA algorithms. ConstantRateJump
s, MassActionJump
s and VariableRateJump
s can be added to standard DiffEq algorithms since they are simply callbacks, while RegularJump
s require special algorithms.
Defining a Regular Jump
The constructor for a RegularJump
is:
RegularJump(rate,c,numjumps;mark_dist = nothing)
rate(out,u,p,t)
is the function which computes the rate for every regular jump processc(du,u,p,t,counts,mark)
is calculates the update givencounts
number of jumps for each jump process in the interval.numjumps
is the number of jump processes, i.e. the number ofrate
equations and the number ofcounts
mark_dist
is the distribution for the mark.
Defining a Constant Rate Jump
The constructor for a ConstantRateJump
is:
ConstantRateJump(rate,affect!)
rate(u,p,t)
is a function which calculates the rate given the time and the state.affect!(integrator)
is the effect on the equation, using the integrator interface.
Defining a Mass Action Jump
The constructor for a MassActionJump
is:
MassActionJump(reactant_stoich, net_stoich; scale_rates = true, param_idxs=nothing)
reactant_stoich
is a vector whosek
th entry is the reactant stoichiometry of thek
th reaction. The reactant stoichiometry for an individual reaction is assumed to be represented as a vector ofPair
s, mapping species id to stoichiometric coefficient.net_stoich
is assumed to have the same type asreactant_stoich
; a vector whosek
th entry is the net stoichiometry of thek
th reaction. The net stoichiometry for an individual reaction is again represented as a vector ofPair
s, mapping species id to the net change in the species when the reaction occurs.scale_rates
is an optional parameter that specifies whether the rate constants correspond to stochastic rate constants in the sense used by Gillespie, and hence need to be rescaled. The default,scale_rates=true
, corresponds to rescaling the passed in rate constants. See below.param_idxs
is a vector of the indices within the parameter vector,p
, that correspond to the rate constant for each jump.
Notes for Mass Action Jumps
- When using
MassActionJump
the default behavior is to assume rate constants correspond to stochastic rate constants in the sense used by Gillespie (J. Comp. Phys., 1976, 22 (4)). This means that for a reaction such as $2A \overset{k}{\rightarrow} B$, the jump rate function constructed byMassActionJump
would bek*A*(A-1)/2!
. For a trimolecular reaction like $3A \overset{k}{\rightarrow} B$ the rate function would bek*A*(A-1)*(A-2)/3!
. To avoid having the reaction rates rescaled (by1/2
and1/6
for these two examples), one can pass theMassActionJump
constructor the optional named parameterscale_rates=false
, i.e. useMassActionJump(reactant_stoich, net_stoich; scale_rates = false, param_idxs)
- Zero order reactions can be passed as
reactant_stoich
s in one of two ways. Consider the $\varnothing \overset{k}{\rightarrow} A$ reaction with ratek=1
:
Alternatively one can create an empty vector of pairs to represent the reaction:p = [1.] reactant_stoich = [[0 => 1]] net_stoich = [[1 => 1]] jump = MassActionJump(reactant_stoich, net_stoich; param_idxs=[1])
p = [1.] reactant_stoich = [Vector{Pair{Int,Int}}()] net_stoich = [[1 => 1]] jump = MassActionJump(reactant_stoich, net_stoich; param_idxs=[1])
- For performance reasons, it is recommended to order species indices in stoichiometry vectors from smallest to largest. That is
is preferred overreactant_stoich = [[1 => 2, 3 => 1, 4 => 2], [2 => 2, 3 => 2]]
reactant_stoich = [[3 => 1, 1 => 2, 4 = > 2], [3 => 2, 2 => 2]]
Defining a Variable Rate Jump
The constructor for a VariableRateJump
is:
VariableRateJump(rate,affect!;
idxs = nothing,
rootfind=true,
save_positions=(true,true),
interp_points=10,
abstol=1e-12,reltol=0)
Note that this is the same as defining a ContinuousCallback
, except that instead of the condition
function, you provide a rate(u,p,t)
function for the rate
at a given time and state.
Defining a Jump Problem
To define a JumpProblem
, you must first define the basic problem. This can be a DiscreteProblem
if there is no differential equation, or an ODE/SDE/DDE/DAE if you would like to augment a differential equation with jumps. Denote this previously defined problem as prob
. Then the constructor for the jump problem is:
JumpProblem(prob,aggregator::Direct,jumps::JumpSet;
save_positions = typeof(prob) <: AbstractDiscreteProblem ? (false,true) : (true,true))
The aggregator is the method for aggregating the constant jumps. These are defined below. jumps
is a JumpSet
which is just a gathering of jumps. Instead of passing a JumpSet
, one may just pass a list of jumps themselves. For example:
JumpProblem(prob,aggregator,jump1,jump2)
and the internals will automatically build the JumpSet
. save_positions
is the save_positions
argument built by the aggregation of the constant rate jumps.
Note that a JumpProblem
/JumpSet
can only have 1 RegularJump
(since a RegularJump
itself describes multiple processes together). Similarly, it can only have one MassActionJump
(since it also describes multiple processes together).
Constant Rate Jump Aggregators
Constant rate jump aggregators are the methods by which constant rate jumps, including MassActionJump
s, are lumped together. This is required in all algorithms for both speed and accuracy. The current methods are:
Direct
: the Gillespie Direct method SSA.RDirect
: A variant of Gillespie's Direct method that uses rejection to sample the next reaction.DirectCR
: The Composition-Rejection Direct method of Slepoy et al. For large networks and linear chain-type networks it will often give better performance thanDirect
. (Requires dependency graph, see below.)DirectFW
: the Gillespie Direct method SSA withFunctionWrappers
. This aggregator uses a different internal storage format for collections ofConstantRateJumps
.FRM
: the Gillespie first reaction method SSA.Direct
should generally offer better performance and be preferred toFRM
.FRMFW
: the Gillespie first reaction method SSA withFunctionWrappers
.NRM
: The Gibson-Bruck Next Reaction Method. For some reaction network structures this may offer better performance thanDirect
(for example, large, linear chains of reactions). (Requires dependency graph, see below.)RSSA
: The Rejection SSA (RSSA) method of Thanh et al. WithRSSACR
, for very large reaction networks it often offers the best performance of all methods. (Requires dependency graph, see below.)RSSACR
: The Rejection SSA (RSSA) with Composition-Rejection method of Thanh et al. WithRSSA
, for very large reaction networks it often offers the best performance of all methods. (Requires dependency graph, see below.)SortingDirect
: The Sorting Direct Method of McCollum et al. It will usually offer performance as good asDirect
, and for some systems can offer substantially better performance. (Requires dependency graph, see below.)
To pass the aggregator, pass the instantiation of the type. For example:
JumpProblem(prob,Direct(),jump1,jump2)
will build a problem where the constant rate jumps are solved using Gillespie's Direct SSA method.
Constant Rate Jump Aggregators Requiring Dependency Graphs
Italicized constant rate jump aggregators require the user to pass a dependency graph to JumpProblem
. DirectCR
, NRM
and SortingDirect
require a jump-jump dependency graph, passed through the named parameter dep_graph
. i.e.
JumpProblem(prob,DirectCR(),jump1,jump2; dep_graph=your_dependency_graph)
For systems with only MassActionJump
s, or those generated from a Catalyst reaction_network
, this graph will be auto-generated. Otherwise you must construct the dependency graph manually. Dependency graphs are represented as a Vector{Vector{Int}}
, with the i
th vector containing the indices of the jumps for which rates must be recalculated when the i
th jump occurs. Internally, all MassActionJump
s are ordered before ConstantRateJump
s (with the latter internally ordered in the same order they were passed in).
RSSA
and RSSACR
require two different types of dependency graphs, passed through the following JumpProblem
kwargs:
vartojumps_map
- AVector{Vector{Int}}
mapping each variable index,i
, to a set of jump indices. The jump indices correspond to jumps with rate functions that depend on the value ofu[i]
.jumptovars_map
- AVector{Vector{Int}}
mapping each jump index to a set of variable indices. The corresponding variables are those that have their value,u[i]
, altered when the jump occurs.
For systems generated from a Catalyst reaction_network
these will be auto-generated. Otherwise you must explicitly construct and pass in these mappings.
Recommendations for Constant Rate Jumps
For representing and aggregating constant rate jumps
- Use a
MassActionJump
to handle all jumps that can be represented as mass action reactions. This will generally offer the fastest performance. - Use
ConstantRateJump
s for any remaining jumps. - For a small number of jumps, < ~10,
Direct
will often perform as well as the other aggregators. - For > ~10 jumps
SortingDirect
will often offer better performance thanDirect
. - For large numbers of jumps with sparse chain like structures and similar jump rates, for example continuous time random walks,
RSSACR
,DirectCR
and thenNRM
often have the best performance. - For very large networks, with many updates per jump,
RSSA
andRSSACR
will often substantially outperform the other methods.
In general, for systems with sparse dependency graphs if Direct
is slow, one of SortingDirect
, RSSA
or RSSACR
will usually offer substantially better performance. See DiffEqBenchmarks.jl for benchmarks on several example networks.
Remaking JumpProblem
s
When running many simulations, it can often be convenient to update the initial condition or simulation parameters without having to create and initialize a new JumpProblem
. In such situations remake
can be used to change the initial condition, time span, and the parameter vector. Note, the new JumpProblem
will alias internal data structures from the old problem, including core components of the SSA aggregators. As such, only the new problem generated by remake
should be used for subsequent simulations.
As an example, consider the following SIR model:
rate1(u,p,t) = (0.1/1000.0)*u[1]*u[2]
function affect1!(integrator)
integrator.u[1] -= 1
integrator.u[2] += 1
end
jump = ConstantRateJump(rate1,affect1!)
rate2(u,p,t) = 0.01u[2]
function affect2!(integrator)
integrator.u[2] -= 1
integrator.u[3] += 1
end
jump2 = ConstantRateJump(rate2,affect2!)
u0 = [999,1,0]
p = (0.1/1000,0.01)
tspan = (0.0,250.0)
dprob = DiscreteProblem(u0, tspan, p)
jprob = JumpProblem(dprob, Direct(), jump, jump2)
sol = solve(jprob, SSAStepper())
We can change any of u0
, p
and tspan
by either making a new DiscreteProblem
u02 = [10,1,0]
p2 = (.1/1000, 0.0)
tspan2 = (0.0,2500.0)
dprob2 = DiscreteProblem(u02, tspan2, p2)
jprob2 = remake(jprob, prob=dprob2)
sol2 = solve(jprob2, SSAStepper())
or by directly remaking with the new parameters
jprob2 = remake(jprob, u0=u02, p=p2, tspan=tspan2)
sol2 = solve(jprob2, SSAStepper())
To avoid ambiguities, the following will give an error
jprob2 = remake(jprob, prob=dprob2, u0=u02)
as will trying to update either p
or tspan
while passing a new DiscreteProblem
using the prob
kwarg.