# 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 their own, a jump equation is 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

$\frac{du}{dt} = f(u,p,t) + Σ c_i(u,p,t)dp_i$

where $dp_i$ 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

$\frac{du}{dt} = f(u,p,t) + Σgᵢ(u,t)dWⁱ + Σ c_i(u,p,t)dp_i$

## 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.

RegularJumps are optimized for regular jumping algorithms like tau-leaping and hybrid algorithms. ConstantRateJumps and MassActionJumps are optimized for SSA algorithms. ConstantRateJumps, MassActionJumps and VariableRateJumps can be added to standard DiffEq algorithms since they are simply callbacks, while RegularJumps 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 process
• c(du,u,p,t,counts,mark) is calculates the update given counts number of jumps for each jump process in the interval.
• numjumps is the number of jump processes, i.e. the number of rate equations and the number of counts
• 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(rate_consts, reactant_stoich, net_stoich; scale_rates = true)
• rate_consts is a vector of the rate constants for each reaction.
• reactant_stoich is a vector whose kth entry is the reactant stoichiometry of the kth reaction. The reactant stoichiometry for an individual reaction is assumed to be represented as a vector of Pairs, mapping species id to stoichiometric coefficient.
• net_stoich is assumed to have the same type as reactant_stoich; a vector whose kth entry is the net stoichiometry of the kth reaction. The net stoichiometry for an individual reaction is again represented as a vector of Pairs, 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.

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 by MassActionJump would be k*A*(A-1)/2!. For a trimolecular reaction like $3A \overset{k}{\rightarrow} B$ the rate function would be k*A*(A-1)*(A-2)/3!. To avoid having the reaction rates rescaled (by 1/2 and 1/6 for these two examples), one can pass the MassActionJump constructor the optional named parameter scale_rates=false, i.e. use
MassActionJump(rates, reactant_stoich, net_stoich; scale_rates = false)
• Zero order reactions can be passed as reactant_stoichs in one of two ways. Consider the $\varnothing \overset{k}{\rightarrow} A$ reaction with rate k=1:
k = [1.]
reactant_stoich = [[0 => 1]]
net_stoich = [[1 => 1]]
jump = MassActionJump(k, reactant_stoich, net_stoich)
Alternatively one can create an empty vector of pairs to represent the reaction:
k = [1.]
reactant_stoich = [Vector{Pair{Int,Int}}()]
net_stoich = [[1 => 1]]
jump = MassActionJump(k, reactant_stoich, net_stoich)
• For performance reasons, it is recommended to order species indices in stoichiometry vectors from smallest to largest. That is
reactant_stoich = [[1 => 2, 3 => 1, 4 => 2], [2 => 2, 3 => 2]]
is preferred over
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 MassActionJumps, are lumped together. This is required in all algorithms for both speed and accuracy. The current methods are:

• Direct: the Gillespie Direct method SSA.
• DirectCR: The Composition-Rejection Direct method of Slepoy et al. For large networks and linear chain-type networks it will often give better performance than Direct. (Requires dependency graph, see below.)
• DirectFW: the Gillespie Direct method SSA with FunctionWrappers. This aggregator uses a different internal storage format for collections of ConstantRateJumps.
• FRM: the Gillespie first reaction method SSA. Direct should generally offer better performance and be preferred to FRM.
• FRMFW: the Gillespie first reaction method SSA with FunctionWrappers.
• NRM: The Gibson-Bruck Next Reaction Method. For some reaction network structures this may offer better performance than Direct (for example, large, linear chains of reactions). (Requires dependency graph, see below.)
• RSSA: The Rejection SSA (RSSA) method of Thanh et al. 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 as Direct, 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 MassActionJumps, or those generated from a DiffEqBiological 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 ith vector containing the indices of the jumps for which rates must be recalculated when the ith jump occurs.

RSSA requires two different types of dependency graphs, passed through the following JumpProblem kwargs:

• vartojumps_map - A Vector{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 of u[i].
• jumptovars_map - A Vector{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 DiffEqBiological 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 ConstantRateJumps 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 than Direct.
• For large number of jumps with sparse chain like structures and similar jump rates, for example continuous time random walks, DirectCR and then NRM often have the best performance.
• For very large networks, with many updates per jump, RSSA will often substantially outperform the other methods.

In general, for systems with sparse dependency graphs if Direct is slow, one of SortingDirect, DirectCR or RSSA will usually offer substantially better performance. See DiffEqBenchmarks.jl for benchmarks on several example networks.