Somewhat. Let me explain the non-DAE form first, since that’s probably the right way to go here, but I’ll also do the other options. Notice that if you do two Brownian motions, you can associate the randomness with reactions instead of states.

```
function f(du,u,p,t)
du[1] = -ß*u[1]*u[2]
du[2] = ß*u[1]*u[2] -µ*u[2]
du[3] = µ*u[2]
end
function g(du,u,p,t)
du[1,1] = sqrt(abs(ß*u[1]*u[2]))
du[2,1] = -sqrt(abs(ß*u[1]*u[2]))
du[3,1] = 0
du[1,2] = 0
du[2,2] = sqrt(abs(µ*u[2]))
du[3,2] = -sqrt(abs(µ*u[2]))
end
prob = SDEProblem(f,g,[0.8, 0.1,0.1],(0.0,1.0),noise_rate_prototype=zeros(3,2))
```

This model is naturally conservative because you have randomness in the reaction, i.e. 1->2 is a stochastic reaction, and 2->3 is a stochastic reaction, and so the randomness is either increasing or decreasing those rates at any instantaneous time, but it’s still mass action kinetics. It’s just:

```
u1' = -b*u1*u2*dt + sqrt(b*u1*u2)*dW1
u2' = b*u1*u2*dt - m*u2*dt - sqrt(b*u1*u2)*dW1 + sqrt(m*u2)*dW2
u3' = m*u2*dt - sqrt(m*u2)*dW2
```

and so from reading the equations you can very easily see the conservative-ness property. This is actually known as the Chemical Langevin Equations and is and intermediate result where the noise is the approximate of the Poisson process of a Gillespie process, and with higher copy numbers it converges to the mass action kinetic ODE models. And BTW, Catalyst is a help library which generates this stuff directly from reaction descriptions:

https://catalyst.sciml.ai/dev/tutorials/using_catalyst/

But if you’re willing to drop mass action kinetics, you can use independent Brownian processes and keep the constraint. You just need to choose an equation to drop so that it’s not overdetermined, i.e.:

```
function f(du,u,p,t)
du[1] = -ß*u[1]*u[2]
du[2] = ß*u[1]*u[2] -µ*u[2]
du[3] = u[1] + u[2] + u[3] -1
end
mass_matrix = [1 0 0; 0 1 0; 0 0 0]
```

or

```
function f(du,u,p,t)
du[1] = -ß*u[1]*u[2]
du[2] = u[1] + u[2] + u[3] -1
du[3] = µ*u[2]
end
mass_matrix = [1 0 0; 0 0 0; 0 0 1]
```

or

```
function f(du,u,p,t)
du[1] = u[1] + u[2] + u[3] -1
du[2] = ß*u[1]*u[2] -µ*u[2]
du[3] = µ*u[2]
end
mass_matrix = [0 0 0; 0 1 0; 0 0 1]
```

All 3 of those are fine models, not necessarily MAK but an extension of sorts which could be what you want. But notice that in each case you’re eliminating one of the equations, because the dynamics of the other two fully determine the dynamics of the third.

However, as noted by @stephans3, this is a case where the constraint is linear, so you can just do the elimination by hand and then end up with a non-DAE SDE in any of those three cases. This is actually preferred since then it would be numerically easier to solve.

##
ModelingToolkit digression

But if you’re lazy, you can use ModelingToolkit.jl to eliminate the equations for you. for example:

```
using ModelingToolkit
@variables t u[1:3](t)
@parameters ß µ
D = Differential(t)
eqs = [D(u[1]) ~ -ß*u[1]*u[2],
D(u[2]) ~ ß*u[1]*u[2] - µ*u[2],
0 ~ u[1] + u[2] + u[3] - 1]
@named sys = ODESystem(eqs,t)
simpsys = structural_simplify(sys)
neqs = [
D(u[1]) ~ -1/2*u[1]*u[2]*u[3]
D(u[2]) ~ u[1]*u[2]*u[3]
]
@named stochsys = SDESystem(simpsys,neqs)
julia> equations(stochsys)
2-element Vector{Equation}:
Differential(t)(u[1](t)) ~ -ß*u[1](t)*u[2](t)
Differential(t)(u[2](t)) ~ ß*u[1](t)*u[2](t) - μ*u[2](t)
julia> ModelingToolkit.get_noiseeqs(stochsys)
2-element Vector{Equation}:
Differential(t)(u[1](t)) ~ -0.5u[1](t)*u[2](t)*u[3](t)
Differential(t)(u[2](t)) ~ u[1](t)*u[2](t)*u[3](t)
```

which will almost work. @yingboma how do I force the substitution of the `u[3]`

out of the noise equations? The constructor that’s required for this is:

```
function SDESystem(sys::ODESystem, neqs; name=nothing, kwargs...)
name === nothing && throw(ArgumentError("The `name` keyword must be provided. Please consider using the `@named` macro"))
SDESystem(
equations(sys), neqs, get_iv(sys), states(sys), parameters(sys),
get_var_to_name(sys), get_ctrls(sys), get_observed(sys), get_tgrad(sys),
get_jac(sys), get_ctrl_jac(sys), get_Wfact(sys), get_Wfact_t(sys),
name, get_systems(sys), get_defaults(sys),
get_connector_type(sys); kwargs...)
end
```

If we fix that up, this is automated. Of course, we should finish up `structural_simplify`

on SDESystem which would then make it much simpler.