Steady-State Models

Background

Suppose one derives some differential equation governed by Equation 1 below:

The time rate of change in X is given by some function f(X). Given our interest in steady-state behavior, we have assumed that the differential Equation 1 does not explicitly depend on time. However, Equation 1 has greater generality than is first apparent.

For example, when X is the vector [x(t), dx/dt, d2x/dt2, K], one can model higher-order equations. In this framework, the equation

can be represented as

Here, we use the expressions x1(t) and x2(t) to indicate two different functions that depend on time t and that evolve over time in accordance with Equation 3.

Steady-state values satisfy the relationship f(X) = 0. When some X satisfies this relationship in a deterministic model, once the system reaches this value, it will stay there. In the case of Equation 3, x(t) = dx/dt = 0 is the unique steady-state solution. A system that starts at this value will stay at this value indefinitely.

Many physical systems converge on a steady-state solution if one waits a sufficiently long period of time. Yet as the above example indicates, there is no reason any physical or economic system necessarily tends to the steady state. Readers may recognize Equation 2 as embodying Hooke's law. A mass attached to a spring experiences a force proportional to its displacement from its rest position. It is readily demonstrated that any function of the form

will satisfy Equation 2, with A and B chosen to match the mass's initial position and velocity. [Page 1070]Unless A = B = 0, this system will oscillate indefinitely, without converging to any steady-state position. Equation 5 below is even worse:

Unless x = dx/dt = 0, Equation 5 describes a system that diverges to ±∞. Undergraduate texts in differential equations provide further information about the existence, uniqueness, and stability of steady-state solutions.

Strengths and Weaknesses: Infectious Disease Transmission among Injection Drug Users

The remainder of this entry illustrates the value of steady-state analysis by exploring the “random mixing” model of infectious disease spread. It then applies this framework to a particular application: HIV and hepatitis C transmission among IDUs. This critical public health challenge provides a simple framework to exhibit both the strengths and the limitations of steady-state analysis.

More elaborate discussion is provided elsewhere (e.g., Pollack, 2001, 2002). This model seeks to capture two interrelated processes: entry and exit from the population of active IDUs and the process of mixing and infection spread among active injectors. It also seeks to explore how the introduction of substance abuse treatment can alter both the overall population of drug users and infectious disease prevalence.

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