On the asymptotic distribution of the maximum number of infectives in epidemic models with immigration. (English) Zbl 0817.92014
Summary: This paper considers the asymptotic distribution of the maximum number of infectives in an epidemic model by showing that, as the initial number of susceptibles converges to infinity, the process of infectives converges almost surely to a birth and death process. The model studied here is more general than usual in that it incorporates immigration and the limiting birth and death process is nonlinear. The main novelty of the present paper is the martingale approach used to prove the above- mentioned convergence.
MSC:
| 92D30 | Epidemiology |
| 60G42 | Martingales with discrete parameter |
| 60J20 | Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.) |
| 60F15 | Strong limit theorems |
| 60G40 | Stopping times; optimal stopping problems; gambling theory |
| 60J80 | Branching processes (Galton-Watson, birth-and-death, etc.) |
