Global analysis of a periodic epidemic model on cholera in presence of bacteriophage

We propose and analyze a recurrent epidemic model of cholera in the presence of bacteriophage. The model is extended by general periodic incidence functions for low-infectious bacterium and high-infectious bacterium, respectively. A general periodic shedding function for two infected class (phage-positive and phage-negative) and a generalized contact and intrinsic growth function for susceptible class are also considered. Under certain biological assumptions, we derive the basic reproduction number (R0) in a periodic environment for the proposed model. We also observe the global stability of the disease-free equilibrium, existence, permanence, and global stability of the positive endemic periodic solution of our proposed model. Finally, we verify our results with specific functional form. Copyright © 2016 John Wiley & Sons, Ltd.

[1]  C. Codeço.  Endemic and epidemic dynamics of cholera: the role of the aquatic reservoir , 2001, BMC infectious diseases.

[2]  B. Levin,et al.  Modeling the role of bacteriophage in the control of cholera outbreaks. , 2006, Proceedings of the National Academy of Sciences of the United States of America.

[3]  David L. Smith,et al.  Estimating the reproductive numbers for the 2008–2009 cholera outbreaks in Zimbabwe , 2011, Proceedings of the National Academy of Sciences.

[4]  David Greenhalgh,et al.  SIRS epidemic model and simulations using different types of seasonal contact rate , 2003 .

[5]  S. Ellner,et al.  Cholera dynamics and El Niño-Southern Oscillation. , 2000, Science.

[6]  Horst R. Thieme,et al.  Convergence results and a Poincaré-Bendixson trichotomy for asymptotically autonomous differential equations , 1992 .

[7]  Xiao-Qiang Zhao,et al.  A periodic epidemic model in a patchy environment , 2007 .

[8]  Ira M Longini,et al.  Epidemic and endemic cholera trends over a 33-year period in Bangladesh. , 2002, The Journal of infectious diseases.

[9]  Nicolas Bacaër,et al.  The epidemic threshold of vector-borne diseases with seasonality , 2006, Journal of mathematical biology.

[10]  David A. Sack,et al.  Self-limiting nature of seasonal cholera epidemics: Role of host-mediated amplification of phage , 2005, Proceedings of the National Academy of Sciences of the United States of America.

[11]  Xiao-Qiang Zhao,et al.  Threshold Dynamics for Compartmental Epidemic Models in Periodic Environments , 2008 .

[12]  A. Rinaldo,et al.  Reassessment of the 2010–2011 Haiti cholera outbreak and rainfall-driven multiseason projections , 2012, Proceedings of the National Academy of Sciences.

[13]  M. Pascual,et al.  Refractory periods and climate forcing in cholera dynamics , 2005, Nature.

[14]  Jianjun Paul Tian,et al.  Global stability for cholera epidemic models. , 2011, Mathematical biosciences.

[15]  A. Faruque,et al.  Effect of Phage on the Infectivity of Vibrio cholerae and Emergence of Genetic Variants , 2008, Infection and Immunity.

[16]  Amiya Ranjan Bhowmick,et al.  An Optimal Cost Effectiveness Study on Zimbabwe Cholera Seasonal Data from 2008–2011 , 2013, PloS one.

[17]  Yanling Zhu,et al.  Existence and global attractivity of positive periodic solutions for a predator–prey model with modified Leslie–Gower Holling-type II schemes , 2011 .

[18]  J. Cui,et al.  Threshold dynamics for a cholera epidemic model with periodic transmission rate , 2013 .

[19]  H. Endtz,et al.  Effects of local climate variability on transmission dynamics of cholera in Matlab, Bangladesh. , 2009, Transactions of the Royal Society of Tropical Medicine and Hygiene.

[20]  Drew Posny,et al.  Modelling cholera in periodic environments , 2014, Journal of biological dynamics.

[21]  Shu Liao,et al.  A generalized cholera model and epidemic–endemic analysis , 2012, Journal of biological dynamics.