Steady States & Stable Orbits

ramp.xds computes steady states and stable orbits as part of the qualitative analysis of dynamical systems for mosquito-transmitted pathogens.

Steady States

Steady states are computed in two ways.

1. If there is a closed form expression for the steady state of one of the dynamical modules, it can be computed as a function of the inputs. There are 5 generics that compute the steady states:

   • steady_state_L – given a static egg deposition rate, compute the steady states, including the emergence rate of adult mosquitoes

   • steady_state_M – given a static emergence rate of adult mosquitoes from aquatic habitats, compute the steady state mosquito population density

   • steady_state_MY – given a static emergence rate of adult mosquitoes from aquatic habitats, and given a static net infectiousness (\(\kappa\)), compute the steady states of an adult population model

   • steady_state_X – given a static human population size, compute the steady states of a model for infection and immunity in the human population

   • steady_state_XH – compute the steady state for a model of demography, infection and immunity

2. The function xds_steady computes the steady state numerically. The function has two optional arguments, a runtime, Y (in years, default Y=10) and a tolerance argument (default tol=1e-5). The system is repeatedly solved, and the sum of squared errors of the final states are checked. The system stops after the difference is less than tol or after iterating 10 times.

The algorithms assume that the equilibrium is globally, asymptotically stable. This may not be true for some models.

Steady Orbits