title: On the time to reach a critical number of infections in epidemic models with infective and susceptible immigrants.
abstract: 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.
authors.affiliation: Department of Statistics and Operations Research I, Faculty of Mathematics, Complutense University of Madrid, 28040 Madrid, Spain. Electronic address: ealmarazluengo@mat.ucm.es.
authors.forename: E
authors.initials: E
authors.lastname: Almaraz
authors.affiliation: Instituto de Ciencias Matemáticas CSIC-UAM-UC3M-UCM, Calle Nicolás Cabrera 13-15, Campus de Cantoblanco UAM, 28049 Madrid, Spain. Electronic address: antonio.gomez@icmat.es.
authors.forename: A
authors.initials: A
authors.lastname: Gómez-Corral
authors.affiliation: Department of Statistics and Operations Research I, Faculty of Mathematics, Complutense University of Madrid, 28040 Madrid, Spain. Electronic address: mayter@mat.ucm.es.
authors.forename: M T
authors.initials: MT
authors.lastname: Rodríguez-Bernal
mesh_terms: D050936:Basic Reproduction Number / Q000706:statistics & numerical data*; D003141:Communicable Diseases / Q000453:epidemiology*; D004196:Disease Outbreaks / Q000706:statistics & numerical data*; D054242:Emigrants and Immigrants / Q000706:statistics & numerical data*; D058872:Epidemics / Q000706:statistics & numerical data*; D006801:Humans; D008962:Models, Theoretical; D013269:Stochastic Processes
keywords: Critical threshold; Immigration; Maximum number of infectives; Stochastic epidemic