## The standard SIS model

Recall the differential equations of the standard SIS model. \[
\frac{dS}{dt} = -\beta S I \\
\frac{dI}{dt} = \beta S I - \gamma I
\] Where \(S\) and \(I\) represent the fraction of the population of size \(N\) that are susceptible or infected, respectively. \(\beta\) is the transmission rate and \(\gamma\) is the recovery rate.

We can write this with just one equation if we only want to simulate change in the fraction of infected.

\[
S+I = 1
\\
\implies
\frac{dI}{dt} = \beta I (1-I) - \gamma I
\]

## One-dimensional habitat

As mentioned, we incorporate SIS dynamics in a Stepping Stone model framework. In the Stepping Stone model, we have \(L\) many discrete demes indexed by position \(r\). A symmetric migration rate \(m\) allows for spread of infection between nearest neighbor demes.

We can track changes in infection frequency at each deme \(r\) between time generations as \(I_r(t+1) - I_r(t)\). The Stepping Stone model incorporates changes in infection across the lattice as follows:

\[
I_r(t+1) - I_r(t) = m(I_{r-1}+I_{r+1}-2I_r)
\] Where each generation \(I\) fraction of infected individuals from nearest neighbors \(r+1\) and \(r-1\) move into deme \(r\). There are also two directions for individuals in \(r\) to leave from, so we subtract \(2I_r\). This discrete form is the main contribution of the Stepping Stone model. We will incorporate this into simulations, but will focus on our addition of SIS dynamics.

The infection spread over space and time is represented by the following difference equation:

\[
I_r(t+1) - I_r(t) = \beta I_r(1-I_r) - \gamma I_r + (m-d)(I_{r-1}+I_{r+1}-2I_r)
\] Where \(d\) is the “social distancing” parameter that decreases the migration rate.

We also want to consider stochastic changes to infection frequency. We model this by considering the conditional probability of having \(Y\) number of infected individuals in deme \(r\) at time \(t+1\). This only depends on the current number of infected in that location. Therefore:

\[
Y_r(t+1)|\vec{I}(t) \sim Binom(N,I_r(t+1)) \\
\, \\
I_r(t+1)|\vec{I}(t) \sim \frac{1}{N}Binom(N,I_r(t+1))
\] Where \(\vec{I}(t)\) is a vector of infection frequencies across the spatial lattice and \(I_r(t+1)\) is given by: \[
I_r(t+1) = I(t) + \beta I_r (1-I_r) - \gamma I_r + (m-d)(I_{r-1}+I_{r+1}-2I_r)
\] This allows for random changes due to finite population size. Also known as “drift.”

All together, this equation captures the four main forces of interest:

**Infectivity** with rate \(\beta\)

**Recovery** with rate \(\gamma\)

**Migration** with rate \(m\)

**Drift** with rate \(\frac{1}{N}\)

We consider each of these forces in our discrete simulations.