Asymptotic distribution of martingale estimators for a class of epidemic models
This article is a contribution to the asymptotic inference on the parameters of a quite general class of stochastic models for the spread of epidemics developing in closed populations. Various epidemic models are contained within our framework, for instance, a stochastic version of the Kermack and McKendrick model and the SIS epidemic model. Each model belonging to this class, which consists in a family of discrete-time stochastic process, contains certain parameters to be estimated by means of martingale estimators. Some particular cases defined by means of Markov chains are included in our setting. The main aim of this work is to prove consistency and asymptotic normality of these estimators. Some hypothesis tests based on the main results are also shown.
KeywordsStochastic epidemic models Hypothesis testing Asymptotic normality Basic reproduction number
Unable to display preview. Download preview PDF.
- Buckley FM, Pollet PK (2009) Analytical methods for a stochastic mainland-island metapopulation model. In: Anderssen R, Braddock R, Newham L (eds) Proceedings of the 18th World IMACS Congress and MODSIM09 International congress on modelling and simulation. Modelling and Simulation Society of Australia and New Zealand and International Association for Mathematics and Computers in Simulation, Camberra, Australia, pp 1767–1773Google Scholar
- Kendall DG (1956) Deterministic and stochastic epidemic in closed populations. In: Proceedings of third berkeley symposium in mathematical statistics and probability, vol 4, pp 149–165Google Scholar
- Kurtz TG (1981) Approximation of population processes. SIAM Applied Mathematical Series, 36Google Scholar
- Lefèvre C (1990) Stochastic epidemic models for S-I-R infectious diseases: a brief survey of the recent general theory. In: Gabriel JP, Lefèvre C, Picard P (eds) Stochastic processes in epidemic theory. Lecture notes in biomathematics, vol 86. Springer, New York, pp 1–12Google Scholar
- Mcvinish R, Pollett PK (2009) Limits of large metapopulations with patch dependent extintion probabilities. Appl Probab Trust:1–15Google Scholar
- Rebolledo R La méthode des martingales appliquée á l’etude de la convergence en loi de processus. Mémoires de la Société Mathématique de France 62:1–125Google Scholar