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Abstract
The role of animal reservoir in the disease dynamics is not yet properly studied. In the present investigation a mathematical model of a vector-host-reservoir is proposed and analyzed to observe the global dynamics of the disease. We observe that the disease free equilibrium is globally asymptotically stable if the basic reproduction number ( ) is less than unity whereas unique positive equilibrium is globally asymptotically stable if and transcritical bifurcation occurs at . Our numerical result suggests that the biting rate plays an important role for the propagation of the disease and the recovery rate has not such important contribution towards eradication of the disease. We also perform sensitivity analysis of the model parameters and the results suggest that the death rate of reservoir may be used as a control parameter to eradicate the disease.
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References
- Ashford, R.W. When Is a Reservoir Not a Reservoir? Emerg. Infect. Dis., 2003, 9(11), 1495-1496.
- Aguirre, A.A., Richard, O., and Peter, D. New Directions in Conservation Medicine: Applied Cases of Ecological Health. Oxford University Press. ISBN 9780199731473. pp. 196.
- Salkeld, D.J., Leonhard, S., Girard, Y.A., Hahn, N., Mun, J., Padgett, K.A. and Lane, R.S. Identifying the Reservoir Hosts of the Lyme Disease Spirochete Borrelia burgdorferi in California: The Role of the Western Gray Squirrel (Sciurus griseus). Am. J. Trop. Med. Hyg., 2008, 79(4), 535-540.
- Craine, N.G., Nuttall. P.A., Marriott, A.C. and Randolph, S.E. Role of grey squirrels and pheasants in the transmission of Borrelia burgdorferi sensulato, the Lyme disease spirochaete, in the U.K. Folia. Parasitol. (Praha), 1997, 44, 155-160.
- Richter, D., Spielman, A., Komar, N. and Matuschka, F. Competence of American Robins as Reservoir Hosts for Lyme Disease Spirochetes. Emerging Infec. Dis., 2000, 6(2), 133-138.
- Diniz, S.A., Silva, F.L., Neta, A.V.C., Bueno, R., Guerra, R., Abreu-Silva, A.L., and Santos, R.L. Animal reservoirs for visceral leishmaniasis in densely populated urban areas. J. Infect. Developing Countries, 2008, 2(1), 24-33.
- Dantas-Torres, F. The role of dogs as reservoirs of Leishmania parasites, with emphasis on Leishmania (Leishmania) infantum and Leishmania (Viannia) braziliensis. Vet. Parasitol., 2007, 149, 139-146.
- Faiman, R., Abbasi, I., Jaffe, C., Motro, Y., Nasereddin, A., Schnur, L.F., Torem, M., Pratlong, F., Dedet, J. and Warburg, A. A Newly Emerged Cutaneous Leishmaniasis Focus in Northern Israel and Two New Reservoir Hosts of Leishmania major. PLOS Neglected Tropical Diseases, 2013, 7(2), 1-10. e2058.
- Quinnell, R.J. and Courtenay, O. Transmission, reservoir hosts and control of zoonotic visceral leishmaniasis. Parasitology, 2009, 136(14), 1915-1934.
- Melaun, C., Werblow, A., Busch, M.W., Liston, A. and Klimpel, S. Bats as Potential Reservoir Hosts for Vector-Borne Diseases. in: S. Klimpel and H. Mehlhorn (eds.), Bats (Chiroptera) as Vectors of Diseases and Parasites, Parasitology Research Monographs. (Springer-Verlag Berlin Heidelberg, 2014).
- Ashrafur, S.M., Rahman, M. and Xingfu Zou. Modelling the impact of vaccination on infectious diseases dynamics. J. Biological Dynamics, 2015, 9(sup1), 307-320.
- Paulhus, C. and Xiang-Sheng Wang. Global stability analysis of a delayed susceptible-infected-susceptible epidemic model, J. Biological Dynamics, 2015, 9(sup1), 45-50.
- Razvan, M.R. and Yasaman, S. Global analysis of an SAIS model. J. Biological Dynamics, 2012, 6(2), 457-474.
- Elmojtaba, I.M., Mugisha, J.Y.T. and Hashim, M.H.A. Mathematical analysis of the dynamics of visceral leishmaniasis in the Sudan. Applied Mathematics and Computation, 2010. 217, 2567-2578.
- Elmojtaba, I.M., Mugisha, J.Y.T. and Hashim, M.H.A. Modeling the role of crossimmunity between two different strains of leishmania. Nonlinear Analysis: Real World Applications, 2010, 11(3), 2175-2189.
- Elmojtaba, I.M., Mugisha, J.Y.T. and Hashim, M.H.A. Vaccination Model for Visceral Leishmaniasis with infective immigrants. Mathematical Methods in the Applied Sciences, 2012, 36(2), 216-226.
- Macdonald, G. The Epidemiology and Control of Malaria (Oxford University Press, London, 1957).
- Ross, R. The Prevention of Malaria. (John Murray, Oxford, London, (1911).
- Aneke, S.J. Mathematical modelling of drug resistant malaria parasites and vector population. Mathematical Methods in the Applied Sciences, 2002, 90, 385-396.
- Hale, J.K. Ordinary Differential Equations (John Wiley, New York, 1969).
- Vandermeer, J.H. and Goldberg, D.E. Population Ecology: First Principles (Princeton University Press, New Jersey, 2003).
- Diekman, O. Heesterbeek, J.A.P and Metz, J.A.J. On the definition and computation of the basic reproduction ratio in models for infectious diseases in heterogeneous populations. J. Math. Biol., 1990, 28,365-382.
- van den Driessche, P. and Watmough, J. Reproduction numbers and the sub-threshold endemic equilibria for compartmental models of disease transmission. Mathematical Biosciences, 2002, 180, 29-48.
- Castillo-Chavez, C. and Song, B. Dynamical models of tuberculosis and their applications. Mathematical Biosciences and Engineering, 2004, 1(2), 361-404.
- Zhang, F. and Zhao, X-Q. A periodic epidemic model in patchy environment. J. Math. Anal. Appl., 2007, 325, 496-516.
- Zhao, X-Q. Dynamical Systems in Population Biology (Springer-Verlag, New York, 2003).
- Smith, H.L. and Waltman, P. The Theory of the Chemostat: Dynamics of Microbial Competition. (Cambridge Univ. Press, Cambridge, England, 1995).
- Heesterbeek, J.A.P and Dietz, K. The concept of in epidemic theory. Stat. Neerl., 1996, 50(1), 89-110.
- Hamby, D.M. A review of techniques for parameter sensitivity analysis of environmental models. Environmental Monitoring and Assessment, 1994, 32, 135-154.
- Helton, J.C., Iman, R.L. and Brown, J.B. Sensitivity analysis of the asymptotic behavior of a model for the environmental movement of radionuclides. Ecol. Modeling, 1985, 28, 243-278.
- Abdulrahman, S., Akinwande, N.I., Awojoyogbe, O.B. and Abubakar, U.Y. Sensitivity Analysis of the Parameters of a Mathematical Model of Hepatitis B Virus Transmission. Universal J. Applied Mathematics, 2013, 1(4), 230-241.
- Chitnis, N., Hyman, J.M. and Manore, C.A. Modelling vertical transmission in vector-borne diseases with applications to Rift Valley fever. J. Biological Dynamics, 2013, 7(1), 11-40.
- Coutinho, F.A.B., Burattini, M.N., Lopez, L.F. and Massad, E. An approximation threshold condition for non-autonomous system: an application to a vector-borne infection. Mathematics and Computers in Simulation, 2005, 70, 149-158.
- Kasap, O.E. and Alten, B. Comparative demography of the sandfly Phlebotomus papatasi (Diptera: Psychodidae) at constant temperatures. J. Vector Ecology, 2006, 31(2), 378-385.
- Fialho, R.F. and Schall, J.J. Thermal ecology of malarial parasite and it is insect vector: consequence for parasite's transmission success. J. Anim. Ecol., 1995, 46(5), 553-562.
- Lainson, R., Ryan, L. and Shaw, J.J. Infective stages of Leishmania in the sandfly vector and some observations on the mechanism of transmission. Mem. Inst. Oswaldo Cruz, Rio de Janeiro, 1987, 82(3), 421-424.
References
Ashford, R.W. When Is a Reservoir Not a Reservoir? Emerg. Infect. Dis., 2003, 9(11), 1495-1496.
Aguirre, A.A., Richard, O., and Peter, D. New Directions in Conservation Medicine: Applied Cases of Ecological Health. Oxford University Press. ISBN 9780199731473. pp. 196.
Salkeld, D.J., Leonhard, S., Girard, Y.A., Hahn, N., Mun, J., Padgett, K.A. and Lane, R.S. Identifying the Reservoir Hosts of the Lyme Disease Spirochete Borrelia burgdorferi in California: The Role of the Western Gray Squirrel (Sciurus griseus). Am. J. Trop. Med. Hyg., 2008, 79(4), 535-540.
Craine, N.G., Nuttall. P.A., Marriott, A.C. and Randolph, S.E. Role of grey squirrels and pheasants in the transmission of Borrelia burgdorferi sensulato, the Lyme disease spirochaete, in the U.K. Folia. Parasitol. (Praha), 1997, 44, 155-160.
Richter, D., Spielman, A., Komar, N. and Matuschka, F. Competence of American Robins as Reservoir Hosts for Lyme Disease Spirochetes. Emerging Infec. Dis., 2000, 6(2), 133-138.
Diniz, S.A., Silva, F.L., Neta, A.V.C., Bueno, R., Guerra, R., Abreu-Silva, A.L., and Santos, R.L. Animal reservoirs for visceral leishmaniasis in densely populated urban areas. J. Infect. Developing Countries, 2008, 2(1), 24-33.
Dantas-Torres, F. The role of dogs as reservoirs of Leishmania parasites, with emphasis on Leishmania (Leishmania) infantum and Leishmania (Viannia) braziliensis. Vet. Parasitol., 2007, 149, 139-146.
Faiman, R., Abbasi, I., Jaffe, C., Motro, Y., Nasereddin, A., Schnur, L.F., Torem, M., Pratlong, F., Dedet, J. and Warburg, A. A Newly Emerged Cutaneous Leishmaniasis Focus in Northern Israel and Two New Reservoir Hosts of Leishmania major. PLOS Neglected Tropical Diseases, 2013, 7(2), 1-10. e2058.
Quinnell, R.J. and Courtenay, O. Transmission, reservoir hosts and control of zoonotic visceral leishmaniasis. Parasitology, 2009, 136(14), 1915-1934.
Melaun, C., Werblow, A., Busch, M.W., Liston, A. and Klimpel, S. Bats as Potential Reservoir Hosts for Vector-Borne Diseases. in: S. Klimpel and H. Mehlhorn (eds.), Bats (Chiroptera) as Vectors of Diseases and Parasites, Parasitology Research Monographs. (Springer-Verlag Berlin Heidelberg, 2014).
Ashrafur, S.M., Rahman, M. and Xingfu Zou. Modelling the impact of vaccination on infectious diseases dynamics. J. Biological Dynamics, 2015, 9(sup1), 307-320.
Paulhus, C. and Xiang-Sheng Wang. Global stability analysis of a delayed susceptible-infected-susceptible epidemic model, J. Biological Dynamics, 2015, 9(sup1), 45-50.
Razvan, M.R. and Yasaman, S. Global analysis of an SAIS model. J. Biological Dynamics, 2012, 6(2), 457-474.
Elmojtaba, I.M., Mugisha, J.Y.T. and Hashim, M.H.A. Mathematical analysis of the dynamics of visceral leishmaniasis in the Sudan. Applied Mathematics and Computation, 2010. 217, 2567-2578.
Elmojtaba, I.M., Mugisha, J.Y.T. and Hashim, M.H.A. Modeling the role of crossimmunity between two different strains of leishmania. Nonlinear Analysis: Real World Applications, 2010, 11(3), 2175-2189.
Elmojtaba, I.M., Mugisha, J.Y.T. and Hashim, M.H.A. Vaccination Model for Visceral Leishmaniasis with infective immigrants. Mathematical Methods in the Applied Sciences, 2012, 36(2), 216-226.
Macdonald, G. The Epidemiology and Control of Malaria (Oxford University Press, London, 1957).
Ross, R. The Prevention of Malaria. (John Murray, Oxford, London, (1911).
Aneke, S.J. Mathematical modelling of drug resistant malaria parasites and vector population. Mathematical Methods in the Applied Sciences, 2002, 90, 385-396.
Hale, J.K. Ordinary Differential Equations (John Wiley, New York, 1969).
Vandermeer, J.H. and Goldberg, D.E. Population Ecology: First Principles (Princeton University Press, New Jersey, 2003).
Diekman, O. Heesterbeek, J.A.P and Metz, J.A.J. On the definition and computation of the basic reproduction ratio in models for infectious diseases in heterogeneous populations. J. Math. Biol., 1990, 28,365-382.
van den Driessche, P. and Watmough, J. Reproduction numbers and the sub-threshold endemic equilibria for compartmental models of disease transmission. Mathematical Biosciences, 2002, 180, 29-48.
Castillo-Chavez, C. and Song, B. Dynamical models of tuberculosis and their applications. Mathematical Biosciences and Engineering, 2004, 1(2), 361-404.
Zhang, F. and Zhao, X-Q. A periodic epidemic model in patchy environment. J. Math. Anal. Appl., 2007, 325, 496-516.
Zhao, X-Q. Dynamical Systems in Population Biology (Springer-Verlag, New York, 2003).
Smith, H.L. and Waltman, P. The Theory of the Chemostat: Dynamics of Microbial Competition. (Cambridge Univ. Press, Cambridge, England, 1995).
Heesterbeek, J.A.P and Dietz, K. The concept of in epidemic theory. Stat. Neerl., 1996, 50(1), 89-110.
Hamby, D.M. A review of techniques for parameter sensitivity analysis of environmental models. Environmental Monitoring and Assessment, 1994, 32, 135-154.
Helton, J.C., Iman, R.L. and Brown, J.B. Sensitivity analysis of the asymptotic behavior of a model for the environmental movement of radionuclides. Ecol. Modeling, 1985, 28, 243-278.
Abdulrahman, S., Akinwande, N.I., Awojoyogbe, O.B. and Abubakar, U.Y. Sensitivity Analysis of the Parameters of a Mathematical Model of Hepatitis B Virus Transmission. Universal J. Applied Mathematics, 2013, 1(4), 230-241.
Chitnis, N., Hyman, J.M. and Manore, C.A. Modelling vertical transmission in vector-borne diseases with applications to Rift Valley fever. J. Biological Dynamics, 2013, 7(1), 11-40.
Coutinho, F.A.B., Burattini, M.N., Lopez, L.F. and Massad, E. An approximation threshold condition for non-autonomous system: an application to a vector-borne infection. Mathematics and Computers in Simulation, 2005, 70, 149-158.
Kasap, O.E. and Alten, B. Comparative demography of the sandfly Phlebotomus papatasi (Diptera: Psychodidae) at constant temperatures. J. Vector Ecology, 2006, 31(2), 378-385.
Fialho, R.F. and Schall, J.J. Thermal ecology of malarial parasite and it is insect vector: consequence for parasite's transmission success. J. Anim. Ecol., 1995, 46(5), 553-562.
Lainson, R., Ryan, L. and Shaw, J.J. Infective stages of Leishmania in the sandfly vector and some observations on the mechanism of transmission. Mem. Inst. Oswaldo Cruz, Rio de Janeiro, 1987, 82(3), 421-424.