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I have a system of differential equation to solve, but it's a mixed system of ODE and SDE. I'm not sure whether there is any way to solve this kind of system or not. My equations are:

s'[t] == -a s[t] i[t]
di[t] == (a s[t] i[t] - µ i[t] + c (1 - s[t] -i[t]) i[t]) dt + σ dB
i[0]=.5
s[0]=.5

with the parameters a,c,σ,µ greater than 0. Is there any known way to solve this numerically?

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  • $\begingroup$ @rhermans Yes, It is, I fixed the post. $\endgroup$ – Herr Schrödinger May 31 '18 at 18:41
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    $\begingroup$ Si is this your system system? { s'[t] == -a s[t] i[t], i'[t] == (a s[t] i[t] - \[Micro] i[t] + c (1 - s[t] - i[t]) i[t]) + \[Sigma] B'[t], i[0] == .5, s[0] == .5 } $\endgroup$ – rhermans May 31 '18 at 18:42
  • $\begingroup$ Yeah, that's my system! $\endgroup$ – Herr Schrödinger May 31 '18 at 18:43
  • $\begingroup$ What about dB/dt i.e B'[t]? $\endgroup$ – rhermans May 31 '18 at 18:43
  • $\begingroup$ dB is a noise, a random noise. $\endgroup$ – Herr Schrödinger May 31 '18 at 18:44
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You want to use RandomFunction and ItoProcess to solve these stochastic differential equations.

a = 1;
μ = 0.1;
c = 1;
σ = 0.1;

sol = RandomFunction[ItoProcess[{
  \[DifferentialD]s[t] == -a s[t] i[t] \[DifferentialD]t,
  \[DifferentialD]i[t] == (a s[t] i[t] - μ i[t] + c (1 - s[t] - i[t]) i[t]) \[DifferentialD]t
  + σ \[DifferentialD]W[t]},
  {s[t], i[t]}, {{s, i}, {0.5, 0.5}}, t, W \[Distributed] WienerProcess[0, 1]],
  {0, 20, 0.01}];

ListLinePlot[sol, PlotRange -> All]

Mathematica graphics

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  • $\begingroup$ @Cris K, Is there any way to take the mean curve after 100 interations of this routine? $\endgroup$ – Herr Schrödinger Jul 18 '18 at 4:46

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