%0 Journal Article %1 Kang201497 %A Kang, Y. %A Castillo-Chavez, C. %D 2014 %J Mathematical Biosciences %K Allee Basic Biological Biological; Concepts; Density Disease Disease-induced Diseases; Dynamics, Equilibrium Evolution; Finite Horizontal Host-Parasite Host-parasite Humans; Infectious Infectious; Interactions; Mathematical Models, Number; Parasitic Population Reproduction Stability, Transmission, Unstable Vertical Vertical; analysis; and article; basic biological controlled data; density density; dependent difference diffusive disease driven dynamics; effect; effects; endemic equilibriums; evolution; extinction; free frequency growth horizontal host host-parasite human; infected instability; interaction; interactions; mathematical method; model; mortality; number; numerical parasite pattern; phenomena; population rate; reproduction statistics study; susceptible transmission, transmission; transmissions; vertical %N 1 %P 97-116 %R http://dx.doi.org/10.1016/j.mbs.2013.12.006 %T Dynamics of SI models with both horizontal and vertical transmissions as well as Allee effects %U http://dx.doi.org/10.1016/j.mbs.2013.12.006 %V 248 %X 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. Â\copyright 2013.

@article{Kang201497, 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. {\^A}{\copyright} 2013.}, added-at = {2017-11-10T22:48:29.000+0100}, affiliation = {Science and Mathematics Faculty, School of Letters and Sciences, Arizona State University, Mesa, AZ 85212, United States; Mathematical, Computational and Modeling Sciences Center, Arizona State University, Tempe, AZ 85287-1904, United States; School of Human Evolution and Social Changes, Santa Fe Institute, Santa Fe, NM 87501, United States; School of Sustainability, Santa Fe Institute, Santa Fe, NM 87501, United States; Cornell University, Biological Statistics and Computational Biology, Ithaca, NY 14853-2601, United States; Department of Aeronautics and Astronautics, Massachusetts Institute of Technology, 77 MASS Ave. 33-404, Cambridge, MA 02139-4307, United States}, author = {Kang, Y. and Castillo-Chavez, C.}, author_keywords = {Allee effects; Diffusive instability; Disease-driven extinction; Disease-free dynamics; Horizontal transmission; Vertical transmission}, biburl = {https://www.bibsonomy.org/bibtex/2944178abc75ed27628a7a7c4ff4650af/ccchavez}, coden = {MABIA}, correspondence_address1 = {Kang, Y.; Science and Mathematics Faculty, School of Letters and Sciences, Arizona State University, Mesa, AZ 85212, United States; email: yun.kang@asu.edu}, date-added = {2017-11-10 21:45:26 +0000}, date-modified = {2017-11-10 21:45:26 +0000}, document_type = {Article}, doi = {http://dx.doi.org/10.1016/j.mbs.2013.12.006}, interhash = {1cd274addd3419f7e0d3e258a2221b26}, intrahash = {944178abc75ed27628a7a7c4ff4650af}, issn = {00255564}, journal = {Mathematical Biosciences}, keywords = {Allee Basic Biological Biological; Concepts; Density Disease Disease-induced Diseases; Dynamics, Equilibrium Evolution; Finite Horizontal Host-Parasite Host-parasite Humans; Infectious Infectious; Interactions; Mathematical Models, Number; Parasitic Population Reproduction Stability, Transmission, Unstable Vertical Vertical; analysis; and article; basic biological controlled data; density density; dependent difference diffusive disease driven dynamics; effect; effects; endemic equilibriums; evolution; extinction; free frequency growth horizontal host host-parasite human; infected instability; interaction; interactions; mathematical method; model; mortality; number; numerical parasite pattern; phenomena; population rate; reproduction statistics study; susceptible transmission, transmission; transmissions; vertical}, language = {English}, number = 1, pages = {97-116}, pubmed_id = {24389426}, timestamp = {2017-11-10T22:48:29.000+0100}, title = {Dynamics of SI models with both horizontal and vertical transmissions as well as Allee effects}, url = {http://dx.doi.org/10.1016/j.mbs.2013.12.006}, volume = 248, year = 2014 }