Nonlinear Analysis: Real World Applications 13 (2012) 794–816 Contents lists available at SciVerse ScienceDirect Nonlinear Analysis: Real World Applications journal homepage: www.elsevier.com/locate/nonrwa Global properties of a two-scale network stochastic delayed human epidemic dynamic model Divine Wanduku ∗ , G.S. Ladde Department of Mathematics and Statistics, University of South Florida, 4202 East Fowler Avenue, PHY Tampa, FL 33620-5700, USA article info Article history: Received 20 July 2011 Accepted 17 August 2011 Keywords: Disease-free steady state Stochastic asymptotic stability Threshold value Positively invariant set Lyapunov functional abstract Complex population structure and the large-scale inter-patch connection human transportation underlie the recent rapid spread of infectious diseases of humans. Furthermore, the fluctuations in the endemicity of the diseases within patch dwelling populations are closely related with the hereditary features of the infectious agent. We present an SIR delayed stochastic dynamic epidemic process in a two-scale dynamic structured population. The disease confers temporary natural or infection- acquired immunity to recovered individuals. The time delay accounts for the time- lag during which naturally immune individuals become susceptible. We investigate the stochastic asymptotic stability of the disease free equilibrium of the scale structured mobile population, under environmental fluctuations and the impact on the emergence, propagation and resurgence of the disease. The presented results are demonstrated by numerical simulation results. © 2011 Elsevier Ltd. All rights reserved. 1. Introduction The recent high rates of globalization of human infectious diseases [1] are closely interrelated with the many inter- patch connections facilitated human transportation. Attempts to understand and study the dynamics of human mobility and infectious diseases in complex human meta-population structures are made [2–14]. The inclusion of the effects of disease latency or immunity into the epidemic dynamic modeling process leads to more realistic epidemic dynamic models. Several studies [15–21] incorporating temporary delays describing the effects of disease latency or immunity have been done. Stochastic models also offer a better representation of the reality. Several stochastic models describing single and multi-group disease dynamics have been investigated [22–27]. Assuming random perturbation about the endemic equilibrium of a two-group SIR model, the stochastic asymptotic stability of the endemic equilibrium via constructing a Lyapunov function according to the structure of the system is established in [26]. In [23,22], the random environmental perturbations manifest as fluctuations in the disease transmission rate, and are represented in the epidemic model by a white noise process. Furthermore, the stability of the competitive equilibrium [28], disease free equilibrium for SIRS [23,29] and SIR [27] single-group epidemic models are studied. In addition, by showing the existence of nonnegative solution for a stochastic model, the stochastic asymptotic stability behavior of the equilibria is proved in [22,24,25,28,30]. In [16], the stochastic and temporary immunity effects of the epidemic are both incorporated into the epidemic dynamic model. Furthermore, the global solution existence is exhibited in [16] by applying a Lyapunov energy function method. The understanding of the dynamics of human infectious diseases in complex population structures is still in the infancy level. This is due to the high degree of heterogeneities and complexity of spatial human population structures. Recently, Divine and Ladde [31] have introduced and studied the global properties of a deterministic two-scale network ∗ Corresponding author. Tel.: +1 8135626200. E-mail addresses: wandukudivine@yahoo.com, dwanduku@mail.usf.edu (D. Wanduku), gladde@cas.usf.edu (G.S. Ladde). 1468-1218/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.nonrwa.2011.08.017