On the spread of a disease with gamma distributed latent and infectious periods

Abstract

SUMMARY In this paper we investigate a model with assumptions as for the so-called general epidemic model, except that the latent and infectious periods are independently gamma distributed. For this model, we obtain a number of results concerning the progress of the epidemic. These results allow some assessment of the effects of the latent period and the infectious period on the behaviour of the epidemic model. Further we note that many of the results can be generalized. k large, and the degenerate distribution at /-1, as k -? oo. The latent period has density f10s and the infectious period density fm,. The model go, is in fact the standard epidemic model. For the model glm' we obtain results concerning the behaviour of the path of the process by means of the deterministic approximation and a martingale central limit theorem result. Equations for the distribution of the size of the outbreak are specified and approximations are derived for the probability of a minor outbreak, the distribution of the size of a minor outbreak and the distribution of the size of a major outbreak. Further, we obtain an approxi- mation for the initial rate of increase of the number of infected individuals and hence the mean time for the epidemic to reach its peak. These results give a reasonably detailed description of the behaviour of the model glm' allowing an assessment of the effects of the parameters I and m, and also enabling a comparison of ?pAm with the standard epidemic model 9Yi, to be made.