Solving ODE with Exogenous Variable

Modeling
1

I’m currently working through Bob Carpenter’s fantastic Lotka-Volterra code found here (https://mc-stan.org/users/documentation/case-studies/lotka-volterra-predator-prey.html) to solve a set of ordinary differential equations. The main parameter transformation is as follows:

      transformed parameters {

      real z[N, 2]
          = integrate_ode_rk45(dz_dt, z_init, 0, ts, theta,
            rep_array(0.0, 0), rep_array(0, 0),
            1e-6, 1e-5, 1e3);
      }

My question, in general, is whether it is possible to use the integrate_ode_rk45 function (or some other ode solver) to solve a set of differential equations that include exogenous variables? Theta is a scalar; however, I need each individual parameter (alpha, beta, gamma, and delta) to be the result of a linear model with the form: \alpha = \alpha_0 + \alpha_1*x where x is the value of the exogenous variable at each time step.

Thank you in advance for any help or insight you can provide!

2

Nothing wrong with that. You’d just need to make sure that your \alpha doesn’t violate any constraints. The easiest way to do that would be throw in some exps/inv_logits.

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