Rates via linear time-invariant systems
Don't be intimidated by the title of this section! There are a lot of big words, but the concepts are actually quite simple and important building blocks for understanding disease dynamics.
What is a rate?
A rate is a measure of how quickly something happens. In the context of disease modeling, rates can describe the speed at which individuals transition between different states (like from exposed to infectious). Rates can also characterize arrival frequencies.
Rates have units that have time in the denominator, like "per day" or "per year."
A hazard rate, or simply hazard, is a type of rate that describes how quickly individuals transition from one state to another. You’ll often encounter hazard rates in survival analysis, where they represent the risk of an event (like disease progression or death) occurring at any given time. Most of the rates discussed in this guide fall into this category.
An intensity is another type of rate, often used in the context of counting processes. It describes the expected number of events per unit time, like the number of new infections per day. Therefore, the units are "events per time", like "infections per day."
What is a linear time-invariant system?
In the world of dynamics, a system is a mathematical model that describes how the state changes over time. The system state is a scalar number or vector of numbers that fully defines the current configuration of the system. The evolution of the system state, how it changes over time, is described by one or more ordinary differential equations (ODEs).
Linear time-invariant systems are a special case that will help us to understand rates, transition probabilities, and much more. Let's break this down.
A linear system allows us to describe the rate of change in the state as a linear function of the current state. In differential equations, the rate of change of a variable is denoted by \(\frac{dx}{dt}\), where \(x\) is the state variable and \(t\) is time. A linear system has the form:
is linear because the rate of change of the state, \(x\), denoted \(\frac{dx}{dt}\), is linearly proportional to the current value, \(x\).
The negative sign in the above equation is to allow us to work with rates \(\lambda\) that are positive. With this convention, positive rates result in a system that will stabilize to zero over time.
We can also see that the units work out. The left side, \(\frac{dx}{dt}\), has units of "state per time," like infections per day. The right side has \(x\) with state units (infections) multiplied by the rate \(\lambda\), which has "per time" units.
In contrast, a nonlinear system might instead have some nonlinear function to describe the rate of change of \(x\) with respect to time. Most disease models are actually non-linear as the number of new infections depends on the non-linear product of the number of susceptible and infectious individuals. Non-linear systems require special handling, but you don't need to worry about that for the purposes of this guide.
A time-invariant system does not change over time. In the example above, the rate \(\lambda\) is just a constant number. In contrast, a time-varying system may have a rate that changes over time. We'll focus on time-invariant systems in this guide.
So how does this help us understand rates?
In the examples above, \(\lambda\) is a rate! It describes the "rate" at which the state \(x\) changes over time. Intuitively, larger rates mean that the system state, \(x\) here, changes more quickly.
If the right hand side of the equation is positive, then the state \(x\) is increasing over time. If it is negative, then the state \(x\) is decreasing over time. Here, we are assuming that the rate \(\lambda\) is positive, so the the state will decrease if \(x\) is positive and increase if \(x\) is negative. Thus over time, the state will approach zero.
The amazing thing about linear time-invariant systems is that they are mathematically tractable. We can write down the solution in closed form! This means we can directly calculate what the state will be at any future time \(t\) if we know the initial state -- all without needing to actually run the simulation!
The solution to the equation \(dx/dt = -\lambda x\) is:
where \(x(0)\) is the value of the state \(x\) at time \(t=0\). This solution tells us that the state \(x\) decays exponentially over time. The rate \(\lambda\) determines the speed of this decay.