# JumpProcesses.jl API

## Core Types

JumpProcesses.JumpProblemType
mutable struct JumpProblem{iip, P, A, C, J<:Union{Nothing, JumpProcesses.AbstractJumpAggregator}, J2, J3, J4, R} <: SciMLBase.AbstractJumpProblem{P, J<:Union{Nothing, JumpProcesses.AbstractJumpAggregator}}

Defines a collection of jump processes to associate with another problem type.

Constructors

JumpProblems can be constructed by first building another problem type to which the jumps will be associated. For example, to simulate a collection of jump processes for which the transition rates are constant between jumps (called ConstantRateJumps or MassActionJumps), we must first construct a DiscreteProblem

prob = DiscreteProblem(u0, p, tspan)

where u0 is the initial condition, p the parameters and tspan the time span. If we wanted to have the jumps coupled with a system of ODEs, or have transition rates with explicit time dependence, we would use an ODEProblem instead that defines the ODE portion of the dynamics.

Given prob we define the jumps via

• JumpProblem(prob, aggregator::AbstractAggregatorAlgorithm, jumps::JumpSet ; kwargs...)
• JumpProblem(prob, aggregator::AbstractAggregatorAlgorithm, jumps...; kwargs...)

Here aggregator specifies the underlying algorithm for calculating next jump times and types, for example Direct. The collection of different AbstractJump types can then be passed within a single JumpSet or as subsequent sequential arguments.

Fields

• prob

The type of problem to couple the jumps to. For a pure jump process use DiscreteProblem, to couple to ODEs, ODEProblem, etc.

• aggregator

The aggregator algorithm that determines the next jump times and types for ConstantRateJumps and MassActionJumps. Examples include Direct.

• discrete_jump_aggregation

The underlying state data associated with the chosen aggregator.

• jump_callback

CallBackSet with the underlying ConstantRate and VariableRate jumps.

• variable_jumps

The VariableRateJumps.

• regular_jump

The RegularJumps.

• massaction_jump

The MassActionJumps.

• rng

The random number generator to use.

Keyword Arguments

• rng, the random number generator to use. On 1.7 and up defaults to Julia's builtin generator, below 1.7 uses RandomNumbers.jl's Xorshifts.Xoroshiro128Star(rand(UInt64)).
• save_positions=(true,true), specifies whether to save the system's state (before,after) the jump occurs.
• spatial_system, for spatial problems the underlying spatial structure.
• hopping_constants, for spatial problems the spatial transition rate coefficients.

Please see the tutorial page in the DifferentialEquations.jl docs for usage examples and commonly asked questions.

JumpProcesses.SSAStepperType
struct SSAStepper <: SciMLBase.AbstractDEAlgorithm

Highly efficient integrator for pure jump problems that involve only ConstantRateJumps and/or MassActionJumps.

Notes

• Only works with JumProblems defined from DiscreteProblems.
• Only works with collections of ConstantRateJumps and MassActionJumps.
• Only supports DiscreteCallbacks for events.

Examples

SIR model:

using JumpProcesses
β = 0.1 / 1000.0; ν = .01;
p = (β,ν)
rate1(u,p,t) = p[1]*u[1]*u[2]  # β*S*I
function affect1!(integrator)
integrator.u[1] -= 1         # S -> S - 1
integrator.u[2] += 1         # I -> I + 1
end
jump = ConstantRateJump(rate1,affect1!)

rate2(u,p,t) = p[2]*u[2]      # ν*I
function affect2!(integrator)
integrator.u[2] -= 1        # I -> I - 1
integrator.u[3] += 1        # R -> R + 1
end
jump2 = ConstantRateJump(rate2,affect2!)
u₀    = [999,1,0]
tspan = (0.0,250.0)
prob = DiscreteProblem(u₀, tspan, p)
jump_prob = JumpProblem(prob, Direct(), jump, jump2)
sol = solve(jump_prob, SSAStepper())

see the tutorial for details.

JumpProcesses.reset_aggregated_jumps!Function
reset_aggregated_jumps!(integrator, uprev = nothing; update_jump_params=true)

Reset the state of jump processes and associated solvers following a change in parameters or such.

Notes - update_jump_params=true will recalculate the rates stored within any MassActionJump that was built from the parameter vector. If the parameter vector is unchanged this can safely be set to false to improve performance.

## Types of Jumps

JumpProcesses.ConstantRateJumpType
struct ConstantRateJump{F1, F2} <: JumpProcesses.AbstractJump

Defines a jump process with a rate (i.e. hazard, intensity, or propensity) that does not explicitly depend on time. More precisely, one where the rate function is constant between the occurrence of jumps. For detailed examples and usage information see the

Fields

• rate

Function rate(u,p,t) that returns the jump's current rate.

• affect!

Function affect(integrator) that updates the state for one occurrence of the jump.

Examples

Suppose u[1] gives the amount of particles and p[1] the probability per time each particle can decay away. A corresponding ConstantRateJump for this jump process is

rate(u,p,t) = p[1]*u[1]
affect!(integrator) = integrator.u[1] -= 1
crj = ConstantRateJump(rate, affect!)

Notice, here that rate changes in time, but is constant between the occurrence of jumps (when u[1] will decrease).

JumpProcesses.MassActionJumpType
struct MassActionJump{T, S, U, V} <: JumpProcesses.AbstractMassActionJump

Optimized representation for ConstantRateJumps that can be represented in mass action form, offering improved performance within jump algorithms compared to ConstantRateJump. For detailed examples and usage information see the

Constructors

• MassActionJump(reactant_stoich, net_stoich; scale_rates=true, param_idxs=nothing)

Here reactant_stoich denotes the reactant stoichiometry for each reaction and net_stoich the net stoichiometry for each reaction.

Fields

• scaled_rates

The (scaled) reaction rate constants.

• reactant_stoch

The reactant stoichiometry vectors.

• net_stoch

The net stoichiometry vectors.

• param_mapper

Parameter mapping functor to identify reaction rate constants with parameters in p vectors.

Keyword Arguments

• scale_rates=true, whether to rescale the reaction rate constants according to the stoichiometry.
• nocopy=false, whether the MassActionJump can alias the scaled_rates and reactant_stoch from the input. Note, if scale_rates=true this will potentially modify both of these.
• param_idxs=nothing, indexes in the parameter vector, JumpProblem.prob.p, that correspond to each reaction's rate.

See the tutorial and main docs for details.

Examples

An SIR model with S + I --> 2I at rate β as the first reaction and I --> R at rate ν as the second reaction can be encoded by

p        = (β=1e-4, ν=.01)
u0       = [999, 1, 0]       # (S,I,R)
tspan    = (0.0, 250.0)
rateidxs = [1, 2]           # i.e. [β,ν]
reactant_stoich =
[
[1 => 1, 2 => 1],         # 1*S and 1*I
[2 => 1]                  # 1*I
]
net_stoich =
[
[1 => -1, 2 => 1],        # -1*S and 1*I
[2 => -1, 3 => 1]         # -1*I and 1*R
]
maj = MassActionJump(reactant_stoich, net_stoich; param_idxs=rateidxs)
prob = DiscreteProblem(u0, tspan, p)
jprob = JumpProblem(prob, Direct(), maj)

Notes

• By default reaction rates are rescaled when constructing the MassActionJump as explained in the main docs. Disable this with the kwarg scale_rates=false.
• Also see the main docs for how to specify reactions with no products or no reactants.
JumpProcesses.VariableRateJumpType
struct VariableRateJump{R, F, I, T, T2} <: JumpProcesses.AbstractJump

Defines a jump process with a rate (i.e. hazard, intensity, or propensity) that may explicitly depend on time. More precisely, one where the rate function is allowed to change between the occurrence of jumps. For detailed examples and usage information see the

Fields

• rate

Function rate(u,p,t) that returns the jump's current rate.

• affect!

Function affect(integrator) that updates the state for one occurrence of the jump.

• idxs

• rootfind

• interp_points

• save_positions

• abstol

• reltol

Examples

Suppose u[1] gives the amount of particles and t*p[1] the probability per time each particle can decay away. A corresponding VariableRateJump for this jump process is

rate(u,p,t) = t*p[1]*u[1]
affect!(integrator) = integrator.u[1] -= 1
crj = VariableRateJump(rate, affect!)

Notes

• VariableRateJumps result in integrators storing an effective state type that wraps the main state vector. See ExtendedJumpArray for details on using this object. Note that the presence of any VariableRateJumps will result in all ConstantRateJump, VariableRateJump and callback affect! functions receiving an integrator with integrator.u an ExtendedJumpArray.
• Must be used with ODEProblems or SDEProblems to be correctly simulated (i.e. can not currently be used with DiscreteProblems).
• Salis H., Kaznessis Y., Accurate hybrid stochastic simulation of a system of coupled chemical or biochemical reactions, Journal of Chemical Physics, 122 (5), DOI:10.1063/1.1835951 is used for calculating jump times with VariableRateJumps within ODE/SDE integrators.
JumpProcesses.JumpSetType
struct JumpSet{T1, T2, T3, T4} <: JumpProcesses.AbstractJump

Defines a collection of jumps that should collectively be included in a simulation.

Fields

Examples

Here we construct two jumps, store them in a JumpSet, and then simulate the resulting process.

using JumpProcesses, OrdinaryDiffEq

rate1(u,p,t) = p[1]
affect1!(integrator) = (integrator.u[1] += 1)
crj = ConstantRateJump(rate1, affect1!)

rate2(u,p,t) = (t/(1+t))*p[2]*u[1]
affect2!(integrator) = (integrator.u[1] -= 1)
vrj = VariableRateJump(rate2, affect2!)

jset = JumpSet(crj, vrj)

f!(du,u,p,t) = (du .= 0)
u0 = [0.0]
p = (20.0, 2.0)
tspan = (0.0, 200.0)
oprob = ODEProblem(f!, u0, tspan, p)
jprob = JumpProblem(oprob, Direct(), jset)
sol = solve(jprob, Tsit5())

## Aggregators

Aggregators are the underlying algorithms used for sampling MassActionJumps and ConstantRateJumps.

JumpProcesses.DirectType

Gillespie's Direct method. ConstantRateJump rates and affects are stored in tuples. Fastest for a small (total) number of ConstantRateJumps or MassActionJumps (~10). For larger numbers of possible jumps use other methods.

Gillespie, Daniel T. (1976). A General Method for Numerically Simulating the Stochastic Time Evolution of Coupled Chemical Reactions. Journal of Computational Physics. 22 (4): 403–434. doi:10.1016/0021-9991(76)90041-3.

JumpProcesses.DirectCRType

The Composition-Rejection Direct method. Performs best relative to other methods for systems with large numbers of jumps with special structure (for example a linear chain of reactions, or jumps corresponding to particles hopping on a grid or graph).

• A. Slepoy, A.P. Thompson and S.J. Plimpton, A constant-time kinetic Monte Carlo algorithm for simulation of large biochemical reaction networks, Journal of Chemical Physics, 128 (20), 205101 (2008). doi:10.1063/1.2919546

• S. Mauch and M. Stalzer, Efficient formulations for exact stochastic simulation of chemical systems, ACM Transactions on Computational Biology and Bioinformatics, 8 (1), 27-35 (2010). doi:10.1109/TCBB.2009.47

JumpProcesses.FRMType

Gillespie's First Reaction Method. Should not be used for practical applications due to slow performance relative to all other methods.

Gillespie, Daniel T. (1976). A General Method for Numerically Simulating the Stochastic Time Evolution of Coupled Chemical Reactions. Journal of Computational Physics. 22 (4): 403–434. doi:10.1016/0021-9991(76)90041-3.

JumpProcesses.NRMType

The Next Reaction Method. Can significantly outperform Direct for systems with large numbers of jumps and sparse dependency graphs, but is usually slower than one of DirectCR, RSSA, or RSSACR for such systems.

M. A. Gibson and J. Bruck, Efficient exact stochastic simulation of chemical systems with many species and many channels, Journal of Physical Chemistry A, 104 (9), 1876-1889 (2000). doi:10.1021/jp993732q

JumpProcesses.RSSAType

The Rejection SSA method. One of the best methods for systems with hundreds to many thousands of jumps (along with RSSACR) and sparse dependency graphs.

• V. H. Thanh, C. Priami and R. Zunino, Efficient rejection-based simulation of biochemical reactions with stochastic noise and delays, Journal of Chemical Physics, 141 (13), 134116 (2014). doi:10.1063/1.4896985

• V. H. Thanh, R. Zunino and C. Priami, On the rejection-based algorithm for simulation and analysis of large-scale reaction networks, Journal of Chemical Physics, 142 (24), 244106 (2015). doi:10.1063/1.4922923

JumpProcesses.RSSACRType

The Rejection SSA Composition-Rejection method. Often the best performer for systems with tens of thousands of jumps and sparse depedency graphs.

V. H. Thanh, R. Zunino, and C. Priami, Efficient Constant-Time Complexity Algorithm for Stochastic Simulation of Large Reaction Networks, IEEE/ACM Transactions on Computational Biology and Bioinformatics, Vol. 14, No. 3, 657-667 (2017).

JumpProcesses.SortingDirectType

The Sorting Direct method. Often the fastest algorithm for smaller to moderate sized systems (tens of jumps), or systems where a few jumps occur much more frequently than others.

J. M. McCollum, G. D. Peterson, C. D. Cox, M. L. Simpson and N. F. Samatova, The sorting direct method for stochastic simulation of biochemical systems with varying reaction execution behavior, Computational Biology and Chemistry, 30 (1), 39049 (2006). doi:10.1016/j.compbiolchem.2005.10.007

# Private API Functions

JumpProcesses.ExtendedJumpArrayType
struct ExtendedJumpArray{T3<:Number, T1, T<:AbstractArray{T3<:Number, T1}, T2} <: AbstractArray{T3<:Number, 1}

Extended state definition used within integrators when there are VariableRateJumps in a system. For detailed examples and usage information see the

Fields

• u

The current state.

• jump_u

The current rate (i.e. hazard, intensity, or propensity) values for the VariableRateJumps.

Examples

using JumpProcesses, OrdinaryDiffEq
f(du,u,p,t) = du .= 0
rate(u,p,t) = (1+t)*u[1]*u[2]

# suppose we wish to decrease each of the two variables by one
# when a jump occurs
function affect!(integrator)
# Method 1, direct indexing works like normal
integrator.u[1] -= 1
integrator.u[2] -= 1

# Method 2, if we want to broadcast or use array operations we need
# to access integrator.u.u which is the actual state object.
# So equivalently to above we could have said:
# integrator.u.u .-= 1
end

u0 = [10.0, 10.0]
vrj = VariableRateJump(rate, affect!)
oprob = ODEProblem(f, u0, (0.0,2.0))
jprob = JumpProblem(oprob, Direct(), vrj)
sol = solve(jprob,Tsit5())

Notes

• If ueja isa ExtendedJumpArray with ueja.u of size N and ueja.jump_u of size num_variableratejumps then

# for 1 <= i <= N
ueja[i] == ueja.u[i]

# for N < i <= (N+num_variableratejumps)
ueja[i] == ueja.jump_u[i]
• In a system with VariableRateJumps all callback, ConstantRateJump, and VariableRateJump affect! functions will receive integrators with integrator.u an ExtendedJumpArray.

• As such, affect! functions that wish to modify the state via vector operations should use ueja.u.u to obtain the aliased state object.

JumpProcesses.SSAIntegratorType
mutable struct SSAIntegrator{F, uType, tType, tdirType, P, S, CB, SA, OPT, TS} <: SciMLBase.DEIntegrator{SSAStepper, Nothing, uType, tType}

Solution objects for pure jump problems solved via SSAStepper.

Fields

• f

The underlying prob.f function. Not currently used.

• u

The current solution values.

• t

The current solution time.

• tprev

The previous time a jump occured.

• tdir

The direction time is changing in (must be positive indicating time is increasing)

• p

The current parameters.

• sol

The current solution object.

• i

• tstop

The next jump time.

• cb

The jump aggregator callback.

• saveat

Times to save the solution at.

• save_everystep

Whether to save everytime a jump occurs.

• save_end

Whether to save at the final step.

• cur_saveat

Index of the next saveat time.

• opts

Tuple storing callbacks.

• tstops

User supplied times to step to, useful with callbacks.

• tstops_idx

• u_modified

• keep_stepping