An Epidemic Model of Zoonotic Visceral Leishmaniasis with Time Delay
Keywords:ZVL, Stability, Hopf bifurcation, Time delay
This paper presents a mathematical model with time delay for the transmission dynamics of zoonotic visceral leishmaniasis (ZVL which is caused by protozoan parasite leishmania infantum and transmitted by female sandflies). Qualitative analysis of the ODE version of the model reveals that the disease-free equilibrium of the model is globally asymptotically stable when the basic reproduction number, R0, is less than unity. Using time delay as a bifurcation parameter in the delay-differential version of the model, it has been shown that the incubation period has a significant effect on the stability of the equilibria and we derived the condition for Hopf bifurcation to occur.
N. Hartemink, S. O. Vanwambeke, H. Heesterbeek, D. Rogers, D. Morley, B. Pesson, C. Davies, S. Mahamdallie & P. Ready, “Integrated Mapping of Establishment Risk for Emerging Vector-Borne Infections: ACase Study of Canine Leishmaniasis in Southwest France”, PLoS ONE 6 (2011) 20817.
H. Seifu, L. Torleif, H.M. Damen, W. Tassew & L. Bernt, “Climate change, crop production and child under nutrition in Ethiopia, a longitudinal panel study”, BMC Public Health 14 (2014) 884.
H.R. Thieme, “Convergence results and Poincare-Bendixson trichtomy for asymptotically autonomous differential equation”, Journal of Mathematical Biology 30 (1992) 755.
N. Hussaini, J. M-S Lubuma, K. Barley & A. B. Gumel, “Mathematical analysis of a model for AVL–HIV co-endemicity”, Mathematical Biosciences 271 (2016) 80.
Spickler, Anna Rovid, Technical factsheet (2009) @ http://www.cfsph.iastate.edu/DiseaseInfo/disease.php?name=leishmaniasis&lang=en (Accessed June 2017).
World Health Organization Post-kala-azar dermal leishmaniasis: a manual for case management and control: report of a WHO consultative meeting, Kolkata, India, 2-3 July 2012.
B. M. Younis , H. A. A. Mohammed, M. M. M. Dafalla, A. O. A. Adam, M. Y. Elamin, A. M. Musa, A. M. El-Hassan & E. A. G. Khalil, “Cure of post-Kala-azar dermal leishmaniasis with paromomycin/sodium stibogluconate combination: a proof of concept”, Int. J. Res. Med. Sci. 3 (2015) 16.
J. K.Evans and L. Kedzierski, “Development of vaccines against visceral leishmaniasis”, Journal of tropical medicine, Article ID 892817 (2012) 14.
E. Handman, “Development Leishmaniasis: Current Status of Vaccine”, Clin. Microbiol. Rev. 14 (2001) 229.
L. Ribas, V.L. Zaher, H.J. Shimozako & E. Massad, Estimating the Optimal Control of Zoonotic Visceral Leishmaniasis by the Use of a Mathematical Model, The Scientific World Journal, 2013.
A. Stauch, R. R. Sarkar, A. Picado, B. Ostyn, S. Sundar, S. Rijal, M. Boelaert, J. C. Dujardin & H. P. Duerr, “Visceral leishmaniasis in the indian subcontinent: modelling epidemiology and control”, PLoS Negl. Trop. Dis. 5 (2011) 1405.
C.B. Palatnik-de-sousa, “Vaccines for canine leishmaniasis, frontiers in immunology” 3 (2012) 69
A. P. Seva, F. G Ovallos, M. Amaku, E. Carrillo, J. Moreno, E. A. Galati, E. G. Lopes, R. M. Soares & F. Ferreira “Canine-Based Strategies for Prevention and Control of Visceral Leishmaniasis in Brazil”, PLoS ONE 11 (2016) 0160058.
F. Chappuis, S. Sundar, A. Hailu, H. Ghalib, S. Rijal, W. R. Peeling, J. Alvar & M. Boelaert, “Visceral leishmaniasis: what are the needs for diagnosis, treatment and control?”, Nature Reviews Microbiology 5 (2007) 7
E. M. Moore & D. N. Lockwood, Treatment of Visceral Leishmaniasis, Hospital for Tropical Diseases, University College London Hospital, London School of Hygiene and Tropical Medicine 2 (2010).
S. S. Menon, R. Rossi, L. Nshimyumukiza & K. Zinszer, “Decentralized control of human visceral leishmaniasis in endemic urban areas of Brazil: a literature review”, Tropical Medicine and Health 44 (2016) 9.
P. D. Ready, “Epidemiology of visceral leishmaniasis”, J. Clinical Epidemiology 6 (2014) 147.
I. M. ELmojtaba, J. Y. T. Mugisha & M. H. A. Hashim, “Mathematical analysis of the dynamics of visceral leishmaniasis in the Sudan”, Appl. Math. Comput. 217 (2010) 2567.
I. M. ELmojtaba, J. Y. T. Mugisha & M. H. A. Hashim, “Vaccination model for visceral leishmaniasis with infective immigrants”, Math. Meth. Appl. Sci. 36 (2013) 216.
Z. Muhammad, R. Ali, “Zoonotic Visceral Leishmania: Modeling and Control”, J. Appl. Comput. Math. 4 (2015) 4.
A. Subramanian, V. Singh, R.R. Sarkar, “Understanding Visceral Leishmaniasis Disease Transmission and its Control–A Study Based on Mathematical Modeling”, J. Mathematics 3 (2015) 913.
S. Zhao, Y. Kuang, C. Wu, D. Ben-Arieh, M. Ramalho-Ortigao & K. Bi, “Zoonotic visceral leishmaniasis transmission: modeling, backward bifurcation, and optimal control”, Journal of Mathematical Biology 73 (2016) 1525.
M. N. Burattini, F. A. B. Coutinho, L. F. Lopez & E. Massad, “Modelling the dynamics of leishmaniasis considering human, animal host and vector populations”, J. Biol. Syst. 6 (1998) 337.
H. J. Shimozako, J. Wu & E. Massad, “Mathematical modelling for Zoonotic Visceral Leishmaniasis.A new analysis considering updated parameters and notified human Brazilian data”, Infectious Disease Modelling 2 (2017) 143.
R. J. Quinnell, O. Couetenay, Transmission, reservoir hosts and control of zoonotic visceral leishmaniasis,Institute of Integrative and Comparative Biology”, University of Leeds, Leeds LS2 9JT, UK.
E. E. Zijlstra, A. M. Musa, E. A. G. Khalil, I. M. El-Hassan & A. M. El-Hassan, “Post-kala- azar dermal leishmaniasis”, Lancet Infect. Dis. 3 (2003) 87.
G. Baneth & S. E. Shaw, “Chemotherapy of canine leishmaniosis”, Veterinary Parasitology 106 (2002) 315.
L. A. Espejo, S. Costard & F. J. Zagmutt, “Modeling canine leishmaniasis spread to non-endemic areas of Europe”, Epidemiol. Infect. 143 (2015) 1936.
N. Hussaini, M. Winter & A. B. Gumel, “Qualitative assessment of the role of public health education program on HIV transmission dynamics”, Math. Med. Biol.: J. IMA 28 (2011) 245.
J. K. Hale, Ordinary Differential Equations, Wiley, NewYork, 1969.
Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, New York, 1993.
B. D. Hassard, N. D. K. Azarino & Y. H. Wan, Theory and Applications of Hopf Bifurcation, Cambridge University, Cambridge, 1981.
How to Cite
Copyright (c) 2019 Journal of the Nigerian Society of Physical Sciences
This work is licensed under a Creative Commons Attribution 4.0 International License.
The Journal of the Nigerian Society of Physical Sciences (JNSPS) is published under the Creative Commons Attribution 4.0 (CC BY-NC) license. This license was developed to facilitate open access, namely, it allows articles to be freely downloaded and to be re-used and re-distributed without restriction, as long as the original work is correctly cited. More specifically, anyone may copy, distribute or reuse these articles, create extracts, abstracts, and other revised versions, adaptations or derivative works of or from an article, mine the article even for commercial purposes, as long as they credit the author(s).
Most read articles by the same author(s)
- S. A. Abdullahi, N. Hussaini, A. G. Habib, Mathematical Model of In-host Dynamics of Snakebite Envenoming , Journal of the Nigerian Society of Physical Sciences: Volume 4, Issue 2, May 2022
- Adeyemi Olukayode Binuyo, Eigenvalue Elasticity and Sensitivity Analyses of the Transmission Dynamic Model of Corruption , Journal of the Nigerian Society of Physical Sciences: Volume 1, Issue 1, February 2019
You may also start an advanced similarity search for this article.