title: Dynamics of SI models with both horizontal and vertical transmissions as well as Allee effects.
abstract: A general SI (Susceptible-Infected) epidemic system of host-parasite interactions operating under Allee effects, horizontal and/or vertical transmission, and where infected individuals experience pathogen-induced reductions in reproductive ability, is introduced. The initial focus of this study is on the analyses of the dynamics of density-dependent and frequency-dependent effects on SI models (SI-DD and SI-FD). The analyses identify conditions involving horizontal and vertical transmitted reproductive numbers, namely those used to characterize and contrast SI-FD and SI-DD dynamics. Conditions that lead to disease-driven extinction, or disease-free dynamics, or susceptible-free dynamics, or endemic disease patterns are identified. The SI-DD system supports richer dynamics including limit cycles while the SI-FD model only supports equilibrium dynamics. SI models under "small" horizontal transmission rates may result in disease-free dynamics. SI models under with and inefficient reproductive infectious class may lead to disease-driven extinction scenarios. The SI-DD model supports stable periodic solutions that emerge from an unstable equilibrium provided that either the Allee threshold and/or the disease transmission rate is large; or when the disease has limited influence on the infectives growth rate; and/or when disease-induced mortality is low. Host-parasite systems where diffusion or migration of local populations manage to destabilize them are examples of what is known as diffusive instability. The exploration of SI-dynamics in the presence of dispersal brings up the question of whether or not diffusive instability is a possible outcome. Here, we briefly look at such possibility within two-patch coupled SI-DD and SI-FD systems. It is shown that relative high levels of asymmetry, two modes of transmission, frequency dependence, and Allee effects are capable of supporting diffusive instability.
authors.affiliation: Science and Mathematics Faculty, School of Letters and Sciences, Arizona State University, Mesa, AZ 85212, USA. Electronic address: yun.kang@asu.edu.
authors.forename: Yun
authors.initials: Y
authors.lastname: Kang
authors.affiliation: Mathematical, Computational and Modeling Sciences Center, Arizona State University, Tempe, AZ 85287-1904, USA; School of Human Evolution and Social Changes, Santa Fe Institute, Santa Fe, NM 87501, USA; School of Sustainability, Santa Fe Institute, Santa Fe, NM 87501, USA; Cornell University, Biological Statistics and Computational Biology, Ithaca, NY 14853-2601, USA; Department of Aeronautics and Astronautics, Massachusetts Institute of Technology, 77 MASS Ave. 33-404, Cambridge, MA 02139-4307, USA. Electronic address: ccchavez@asu.edu.
authors.forename: Carlos
authors.initials: C
authors.lastname: Castillo-Chavez
mesh_terms: D050936:Basic Reproduction Number / Q000706:statistics & numerical data; D005075:Biological Evolution; D018562:Disease Transmission, Infectious / Q000706:statistics & numerical data; D006790:Host-Parasite Interactions; D006801:Humans; D018445:Infectious Disease Transmission, Vertical / Q000706:statistics & numerical data; D055641:Mathematical Concepts; D008954:Models, Biological; D010272:Parasitic Diseases / Q000453:epidemiology / Q000635:transmission*; D011156:Population Density
keywords: Allee effects; Diffusive instability; Disease-driven extinction; Disease-free dynamics; Horizontal transmission; Vertical transmission