Journal of Statistical Planning and Inference 71 (1998) 229–244 Statistical inference for a multitype epidemic model Paul S.F. Yip ∗ , Qizhi Chen Department of Statistics, The University of Hong Kong, Hong Kong Received 18 February 1997; received in revised form 4 February 1998; accepted 4 February 1998 Abstract A continuous-time model which allows for k types of susceptible individuals is constructed to study the spread of an infectious disease. Estimation of parameters is considered for the purpose in determining the mechanism of spread and in assessing the potential of a major epidemic. The inference procedure makes use of martingale estimating equations and explicit expressions for the estimates are obtained. Some of the estimators do not require complete observation of the epidemic. Only information at the end of the epidemic is needed. Asymptotic properties of the estimators and eciencies among the estimators are determined. Simulation studies were done to assess the performance of the proposed estimators. The inference procedures were applied to epidemics of respiratory disease on the Island of Tristan de Cunha in the South Atlantic. c  1998 Elsevier Science B.V. All rights reserved. AMS classication: 60G45; 62F20; 62M99; 62P10 Keywords: Counting process models; Maximum likelihood estimation; Potential; Relative infection rate; Zero mean martingale (ZMM) 1. Introduction Consider an infectious disease model with k types of susceptibles whose number ini- tially at t = 0 are n 1 ;n 2 ;:::;n k and n = ∑ k j=1 n j . The number of infectives introduced into the population at the start of the outbreak is a. Suppose that at time t there are S j (t ) susceptibles, I j (t ) infective persons, and R j (t ) removals of type j for j =1; 2;:::;k . With regard to the spread of the disease we assume that the probability of one type-j susceptible being infected in a small time interval (t; t +dt ), given the val- ues of {S j (t );I j (t );R j (t ); j =1; 2;:::;k } is n −1  j S j (t ) I (t )dt + o(dt ); j =1; 2;:::;k; where I (t )= ∑ k j=1 I j (t ), I (0) = a. The corresponding probability for two or more sus- ceptibles of any type being infected is assumed to be o(dt ), and so negligible for small * Corresponding author. Fax: 00852 28589041; e-mail: SFPYIP@HKUCC.HKU.HK. 0378-3758/98/$19.00 c  1998 Elsevier Science B.V. All rights reserved. PII: S0378-3758(98)00087-1