At any given time in the real world, a person may be sick or healthy. Sick people tend to make sick the others nearby, but sometimes a sickness seems to emerge on its own. Eventually, sick people become healthy again.

To model this phenomenon, we reduce time to discrete steps, fix who has contact with whom, and fix the probabilities of causing another’s sickness ($p$), becoming healthy again ($q$), and spontaneous emergence of sickness ($s$).

A graph, $𝔊=(V,E)$ may represent the situation. The vertices and edges respectively represent the persons and who are contactual neighbors.

The collective status of the population is a binary coloration of $V$ determining sickness and healthfulness. Thus, the $2^{|V|}$ binary colorations of $V$ form a state space, $𝒮$.

It makes sense that the probability of one neighbor causing sickness is independent of the probability of another. Let $I_t:V\to \{0,1\}$ indicate whether a vertex is sick (indicated by $1$) at time $t$. Then

$$\mathbb{P}(I_{t+1}(v)=0|I_t)=\underset{\begin{array}{c}\text{sick }\to \\ \text{healthy}\end{array}}{\underbrace{I_t(v)\cdot q}}+\underset{\begin{array}{c}\text{healthy }\to \text{ remains healthy}\end{array}}{\underbrace{(1-I_t(v))\cdot {(1-p)}^{\left[\sum _{u:E(v,u)}{I}_t(u)\right]}\cdot (1-s)}}.$$

Conditioning on the state of the system at time $t$, the events that people are healthy at time $t+1$ are independent because all the events of healing and events of becoming sick are conditionally independent.

For example, the probability of the above transition is

$$q\cdot (1-q)\cdot \left[(1-p)\cdot (1-s)\right]\cdot (1-q)\cdot \left[1-{(1-p)}^2\cdot (1-s)\right],$$

the factors given clockwise from the bottom.

Notice how the probability of transitioning to some state depends only on the current coloring of the vertices. That is to say, knowing which states were assigned to the previous time steps is irrelevant if we know the current state. This means that this model is a Markov chain. This is a discrete version of the contact process.

The following program simulates the contact process on a grid, where neighbors are exactly those squares adjoined in one of the four cardinal directions. The program displays the proportion of time that each vertex spends sick.

The contact process near stationary

Notice how the vertices at the edges and corners of the grid spend less time sick after the process has been running long enough. This is because they have fewer neighbors and less probability of becoming sick. So, it makes sense that the process converges to a stationary distribution that gives less mass to the states with more edge vertices sick.