Threshold behaviour for a chain-binomial S-I-S infectious disease
The paper is concerned with a discrete time model for the spread of an infection. Like in the classical Reed-Frost model an infectious individual can transmit the disease to any susceptible with a constant probability p (the infectious period is assumed to be very small). If an individual was infected, it itself becomes infectious after a certain period of fixed length, the latent period, which is taken as the time unit. After another time unit the individual returns to the susceptible state (SIS-epidemic). During the infectious period each susceptible makes a random number of contacts with other individuals; it is assumed that the j-th contact of a not yet infected individual with an infective will result in an infection with probability \(\beta_ j\). The process of infection is formulated stochastically and then the associated deterministic model is derived. For this model, which is essentially a first-order difference equation, it is studied under what conditions the disease becomes endemic. Besides some general results a detailed discussion is presented for the specific case where at least \(K+1\) (K\(\geq 1)\) contacts with infectives are needed for an infection to occur.
scientific article; zbMATH DE number 3989159