I’d like to get some advice on the best abstraction for dynamical systems where the parameters and/or states of the system change at discrete time intervals. As a motivating example, I’ll consider a simple compartmental model of a disease spreading through a population, in which there are susceptible, infectious, and recovered individuals. The model is described by the following system of ordinary differential equations:
where S, I, and R are the numbers of susceptible, infectious, and recovered individuals, respectively, and \beta and \gamma are parameters of the model. The system is initialized with initial values for S, I, and R, at time t_1 and the parameters \beta and \gamma are fixed throughout the simulation.
Now I want to consider vaccination of susceptible individuals that begins at time t_2, modeled in different ways. Firstly, I could consider a vaccination rate that is constant for t \geq t_2, such that the above system now becomes:
This would involve composing my original model with this new one, involving adding a new transition at time t_2.
Alternatively, I could consider a vaccination rate throughout the time interval [t_1, t_3], such that the above system becomes:
Now, to model the time-varying vaccination, I need to change \nu(t) from 0 to a constant value, (say \nu_0) at time t_2. I appreciate that I don’t need to compose systems here, if I’m prepared to have models with time-varying parameters, but’s let assume I’d like to think about implementing the above by composing one model with \nu=0 with another where \nu=\nu_0 at time t_2.
Alternatively, I could consider a different type of vaccination scenario where a fixed proportion of susceptible individuals are vaccinated at time t_2, then the model proceeds as before. My system remains the same, but I have a discontinuity in my states at time t_2, where S(t_2) \rightarrow S(t_2) - \delta S(t_2) and R(t_2) \rightarrow R(t_2) + \delta S(t_2), where \delta is the proportion of susceptible individuals vaccinated at time t_2.
What’s a reasonable categorical abstraction that captures both discontinuous changes in parameters and/or states for the above system?
I’m assuming that it helps if we assume that each period we consider is non-overlapping. What if my infectivity rate also changes e.g. from \beta_1 to \beta_2 at time t_4? The number of intervals where the parameters are contant will now depend on the value of t_4.