Exercises part I: SIS dynamics

Exercises part I: SIS dynamics#

SIS dynamics with spontaneous infections#

Master equations are powerful because they capture the stochasticity, discreteness, finiteness, and absorbing states, important to many complex systems. Let’s revisit one of the most classic and simple model: the Susceptible-Infectious-Susceptible dynamics. The compartmental version of this model goes as follows. We study a population of \(N\) individuals which can all be in one of two states, such that \(S(t)\) are susceptible and \(I(t)\) are infectious at time \(t\). Infectious individuals infect susceptible individuals at rate \(\beta\) and recover to become susceptible again at rate \(\alpha\).

sis_schematic

Importantly, we now add an extra mechanism and assume that susceptible individuals can also get infected by their environment at some rate \(\epsilon\). We can call those spontaneous infections or self-infections. What are the exact dynamics of this process?

  1. First, construct the master equation.

Hint

Remember that master equations allow us to capture the finiteness of \(N\)

  1. Integrate the master equation over time to look at the behavior of the system for different values of the parameters. What happens if \(N\) goes to one? What happens if \(\beta\) or \(\epsilon\) go to zero?

  2. Solve the steady-state of the master equation analytically using any of the methods studied in this tutorial. Does the solution capture the intuition you built by integrating the system for different values of \(\epsilon\)?

  3. Can you derive the standard SIS equation using the method of moments? Do you have to make any approximation to recover the standard solution?

Hint

When \(\epsilon = 0\), the mean-field SIS model is described by a single equation: \(\dot{S} = \alpha (N-S) - \beta S (N-S)\)