Representation of Phenomena by Probability Models

If you toss a coin and move right for heads and left for tails, this movement becomes a random walk, which is the most typical stochastic process. The behavior of the random walk over a long period of time differs depending on the ratio of heads to tails. For example, if the probability of getting heads is higher, there is a positive probability that it will go to infinity without returning to the starting point. Various probability models are created by generalizing the random walk. Here, let us introduce one of them, which is called an epidemic model.

Initially, we assume that there is at least one healthy person and one sick person. In an unoccupied space, healthy people are born at a rate of λ1 times the number of adjacent healthy people. A healthy person dies at a rate of δ1 regardless of the state of their neighbors and becomes sick at a rate of λ2 times the number of adjacent sick people. A sick person dies at a rate of δ2 (> δ1) regardless of the state of their neighbors and recovers to become healthy at a rate of γ (≥ 0). In this model, the behavior over a long period of time differs depending on the relationship between the parameters λ1, λ2, δ1, δ2, and γ1. Furthermore, it is greatly influenced by the dimension of the space in which people exist. In two or more dimensions, if the recovery rate γ1 is positive, it is known that by appropriately setting the parameters, there is a positive probability that healthy and sick people will coexist forever. When the rate γ is zero, that is, when recovery is not possible, in the one-dimensional case, the final state is either that all people become healthy or that all people die and the space becomes unoccupied. In other words, healthy and sick people do not coexist. The case of two or more dimensions is still unresolved, but it is expected that healthy and sick people can coexist. This is because the number of clusters of sick people increases, and coexistence can occur where the number of contacts between healthy and sick people is large compared to the total number of sick people.   

When considering this in relation to the actual spread of a disease, a one-dimensional space corresponds to a situation where the number of contacts is small and everyone is staying at home, while a high-dimensional space corresponds to a situation where there is contact with a large number of people in a crowded place. In this model, each person does not move, and the number of neighbors is fixed, but if the model were one where people move, for example, in a random walk, coexistence would become even easier. Regarding clusters, the fact that large clusters with simple shapes are likely to disappear is related to the fact that the infection ends when herd immunity is achieved (which in this model means death).