In this paper we examine the time T to reach a critical number K0 of infections during an outbreak in an epidemic model with infective and susceptible immigrants. The underlying process X, which was first introduced by Ridler-Rowe (1967), is related to recurrent diseases and it appears to be analytically intractable. We present an approximating model inspired from the use of extreme values, and we derive formulae for the Laplace-Stieltjes transform of T and its moments, which are evaluated by using an iterative procedure. Numerical examples are presented to illustrate the effects of the contact and removal rates on the expected values of T and the threshold K0, when the initial time instant corresponds to an invasion time. We also study the exact reproduction number Rexact,0 and the population transmission number Rp, which are random versions of the basic reproduction number R0. © 2016 Elsevier Ireland Ltd.

Critical threshold Immigration Stochastic epidemic Maximum number of infectives

immigrant quantitative study ecological modeling human Communicable Diseases statistics and numerical data recurrent disease mathematical model Disease infectivity Epidemics Humans migrant Emigrants and Immigrants critical analysis mathematical analysis theoretical model Models, Theoretical Disease Outbreaks Article epidemic threshold numerical method infection mathematical computing stochasticity susceptible population Basic Reproduction Number coding algorithm Stochastic Processes Markov chain removal experiment immigrant population