On the time to reach a critical number of infections in epidemic models with infective and susceptible immigrants

• Published in: BioSystems Vol. 144; pp. 68 - 77
• Main Authors: Almaraz, E., Gómez-Corral, A., Rodríguez-Bernal, M.T.
• Format: Journal Article
• Language: English
• Published: Ireland Elsevier Ireland Ltd 01.06.2016
• Subjects: Basic Reproduction Number - statistics & numerical data; Communicable Diseases - epidemiology; Critical threshold; Disease Outbreaks - statistics & numerical data; Emigrants and Immigrants - statistics & numerical data; Epidemics - statistics & numerical data; Humans; Immigration; Maximum number of infectives; Models, Theoretical; Stochastic epidemic; Stochastic Processes; Stochastic epidemic; Immigration; 92D30; Critical threshold; Maximum number of infectives
• ISSN: 0303-2647; 1872-8324
• DOI: 10.1016/j.biosystems.2016.04.007

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.