zbMATH — the first resource for mathematics

Impact of predator on the host-vector disease model with stage structure for the vector. (English) Zbl 1448.92265
Summary: In this paper, we propose a host-vector-predator model with stage structure for the vector to explore the impact of biological control agents on host-vector dynamics and disease control. Here the total vector population is divided into two physiological subclasses which are immature and mature subclasses. Holling type II functional response is used to portray the interactions between vectors and predators. Stability analysis of the equilibria demonstrates that the basic reproduction number gives the threshold condition determining the persistence and extinction of the disease. Furthermore, the phenomenon of Hopf bifurcation occurs when predators are introduced. The stability of limit cycle arising from a Hopf bifurcation is rigorously investigated. Finally, numerical simulations are given to show the validity of analytical results, and the comparative results of disease dynamics with and without predators.
92D25 Population dynamics (general)
37N25 Dynamical systems in biology
Full Text: DOI
[1] Jones, K. E.; Patel, N. G.; Levy, M. A.; Storeygard, A.; Balk, D.; Gittleman, J. L.; Daszak, P., Global trends in emerging infectious diseases, Nature, 451, 990-993, (2008)
[2] Daszak, P.; Cunningham, A.; Hyatt, A. D., Anthropogenic environmental change and the infectious diseases in wildlife, Acta Trop., 78, 103-116, (2001)
[3] Yang, C. X.; Nie, L. F., Modelling the use of impulsive vaccination to control rift valley fever virus transmission, J. Differ. Equ., 2016, (2016)
[4] Nie, L. F.; Xue, Y. N., The roles of maturation delay and vaccination on the spread of dengue virus and optimal control, J. Differ. Equ., 2017, (2017) · Zbl 1422.92161
[5] Hemingway, J.; Ranson, H., Insecticide resistance in insect vectors of human disease, Annu. Rev. Entomol., 45, 371-391, (2000)
[6] Castro, M. C.; De Yamagata, Y.; Mtasiwa, D.; Tanner, M.; Utzinger, J.; Keiser, J.; Singer, B. H., Integrated urban malaria control: a case study in dares salaam, tanzania, Am. J. Trop. Med. Hyg., 71, 103-117, (2004)
[7] Benedict, M. Q.; Robinson, A. S., The first releases of transgenic mosquitoes: an argument for the sterile insect technique, Trends Parasitol., 19, 349-355, (2003)
[8] Kaaya, G. P.; Hassan, S., Entomogenous fungi as promising biopesticides for tick control, Exp. Appl. Acarol., 24, 913-926, (2000)
[9] Scholte, E. J.; Ng’Habi, K.; Kihonda, J.; Takken, W.; Paaijmans, K.; Abdulla, S.; Killeen, G. F.; Knols, B. G.J., An entomopathogenic fungus for control of adult african malaria mosquitoes, Science, 308, 1641-1642, (2005)
[10] Legner, E., Biological control of diptera of medical and veterinary importance, J. Vector Ecol., 20, 59-120, (1995)
[11] Stauffer, J. R.; Arnegard, M.; Cetron, M., Controlling vectors and hosts of parasitic diseases using fishes, Bioscience, 47, 41-49, (1997)
[12] Samish, M.; Rehacek, J., Pathogens and predators of ticks and their potential in biological control, Annu. Rev. Entomol., 44, 159-182, (1999)
[13] Ostfeld, R. S.; Price, A.; Hornbostel, V. L.; Benjamin, M. A.; Keesing, F., Controlling ticks and tick-borne zoonoses with biological and chemical agents, Bioscience, 56, 383-394, (2006)
[14] Walker, K.; Lynch, M., Contributions of anopheles larval control to malaria suppression in tropical africa: review of achievements and potential, Med. Vet. Entomol., 21, 2-21, (2007)
[15] Ghosh, S. K.; Tiwari, S. N.; Sathyanarayan, T. S.; Sampath, T. R.; Sharma, V. P.; Nanda, N.; Joshi, H.; Adak, T.; Subbarao, S. K., Larvivorous fish in wells target the malaria vector sibling species of the anopheles culicifacies complex in villages in karnataka, India, Trans. R. Soc. Trop. Med. Hyg., 99, 101-105, (2005)
[16] Ghosh, S. K.; Dash, A. P., Larvivorous fish against malaria vectors: a new outlook, Trans. R. Soc. Trop. Med. Hyg., 101, 1063-1064, (2007)
[17] Kay, B.; Nam, V. S., New strategy against aedes aegypti in Vietnam, Lancet, 365, 613-617, (2005)
[18] Kittayapong, P.; Yoksan, S.; Chansang, U.; Chansang, C.; Bhumiratana, A., Suppression of dengue transmission by application of integrated vector control strategies at sero-positive GIS-based foci, Am. J. Trop. Med. Hyg., 78, 70-76, (2008)
[19] Zhang, L. Q.; Liu, J.; Wu, H., The screening of a virulent strain of beauveria bassiana to monochamus alternatus, J. Nanjing For. Univ., 24, 33-37, (2000)
[20] Lai, Y. X.; Liu, J. D.; Xu, Q. Y.; Wang, Y. H.; Zhou, C. M., Trials on the parasitism of beauveria bassiana or verticillium lecanii on larvae of monochamus alternatus hope, J. Jiangsu. For. Sci. Technol., 30, 7-9, (2003)
[21] Jeger, M. J.; Holt, J.; Den Bosch, F.; Madden, L. V., Epidemiology of insect-transmitted plant viruses: modelling disease dynamics and control interventions, Physiol. Entomol., 29, 291-304, (2004)
[22] Otim, M.; Legg, D.; Kyamanywa, S.; Polaszek, A.; Gerling, D., Population dynamics of bemisia tabaci (homoptera: aleyrodidae) parasitoids on cassava mosaic disease-resistant and susceptible varieties, Biocontrol Sci. Technol., 16, 205-214, (2006)
[23] Luck, R. F.; Shepard, B. M.; Kenmore, P. E., Experimental methods for evaluating arthropod natural enemies, Annu. Rev. Entomol., 33, 367-389, (1988)
[24] Zehnder, G.; Gurr, G. M.; Kühne, S.; Wade, M. R.; Wratten, S. D.; Wyss, E., Arthropod pest management in organic crops, Annu. Rev. Entomol., 52, 57-80, (2007)
[25] Tian, B. D.; Yang, L.; Zhong, S. M., Global stability of a predator-prey model with allee effect, Int. J. Biomath., 8, 37-51, (2015)
[26] Zha, L. J.; Cui, J. A.; Zhou, X. Y., Ratio-dependent predator-prey model with stage structure and time delay, Int. J. Biomath., 5, 15-37, (2012) · Zbl 1280.92080
[27] Zhou, F. Y., Existence and global attractivity of a positive periodic solution for a non-autonomous predator-prey model under viral infection, Int. J. Biomath., 2, 419-442, (2009) · Zbl 1342.92220
[28] Schraiber, J. G.; Kaczmarczyk, A. N.; Kwok, R.; etal., Constraints on the use of lifespan-shortening wolbachia to control dengue fever, J. Theor. Biol., 297, 26-32, (2012) · Zbl 1336.92085
[29] Hughes, H.; Britton, N. F., Modelling the use of wolbachia to control dengue fever transmission, Bull. Math. Biol., 75, 796-818, (2013) · Zbl 1273.92034
[30] Moore, S. M.; Borer, E. T.; Hosseini, P. R., Predators indirectly control vector-borne disease: linking predator-prey and host-pathogen models, J. R. Soc. Interface, 7, 161-176, (2009)
[31] Zhou, F. Y.; Yao, H. Y., Dynamics and biocontrol: the indirect effects of a predator population on a host-vector disease model, Abstr. Appl. Anal., 2014, (2014)
[32] Okamoto, K. W.; Amarasekare, P., The biological control of disease vectors, J. Theor. Biol., 309, 47-57, (2012)
[33] Xiao, Y. N.; Chen, L. S., An SIS epidemic model with stage structure and a delay, Acta Math. Appl. Sin., 18, 607-618, (2002) · Zbl 1035.34054
[34] Cai, L.; Li, X.; Ghosh, M., Global stability of a stage-structured epidemic model with a nonlinear incidence, Appl. Math. Comput., 214, 73-82, (2009) · Zbl 1172.92027
[35] Hyman, J. M.; Li, J.; Stanley, E. A., The differential infectivity and staged progression models for the transmission of HIV, Math. Biosci., 155, 77-109, (1999) · Zbl 0942.92030
[36] Moghadas, S. M.; Gumel, A. B., Global stability of a two-stage epidemic model with generalized non-linear incidence, Math. Comput. Simul., 60, 107-118, (2002) · Zbl 1005.92031
[37] Martcheva, M.; Castillo-Chavez, C., Diseases with chronic stage in a population with varying size, Math. Biosci., 182, 1-25, (2003) · Zbl 1012.92024
[38] Tian, B. D.; Jin, Y. G.; Zhong, S. M.; Chen, N., Global stability of an epidemic model with stage structure and nonlinear incidence rates in a heterogeneous host population, J. Differ. Equ., 2015, (2015)
[39] World health organization: Frequently Asked Questions Scrub Typhus (2008)
[40] Zhao, B.G., Futai, K., Jack, R., Sutherland, J.R., Takeuchi, Y.: Pine Wilt Disease. Springer, New York (2008)
[41] Diekmann, O.; Heesterbeek, J. A.P.; Metz, J. A.J., On the definition and the computation of the basic reproduction ratio \(R_{0} \) in models for infectious diseases in heterogeneous populations, J. Math. Biol., 28, 365-382, (1990) · Zbl 0726.92018
[42] Driessche, V. P.; Watmough, J., Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180, 29-48, (2002) · Zbl 1015.92036
[43] Singh, B. K.; Chattopadhyay, J.; Sinha, S., The role of virus infection in a simple phytoplankton zooplankton system, J. Theor. Biol., 231, 153-166, (2004)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.