×

Modeling the parasitic filariasis spread by mosquito in periodic environment. (English) Zbl 1369.92114

Summary: In this paper a mosquito-borne parasitic infection model in periodic environment is considered. Threshold parameter \(R_0\) is given by linear next infection operator, which determined the dynamic behaviors of system. We obtain that when \(R_0 < 1\), the disease-free periodic solution is globally asymptotically stable and when \(R_0 > 1\) by Poincaré map we obtain that disease is uniformly persistent. Numerical simulations support the results and sensitivity analysis shows effects of parameters on \(R_0\), which provided references to seek optimal measures to control the transmission of lymphatic filariasis.

MSC:

92D30 Epidemiology
34D20 Stability of solutions to ordinary differential equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] World Health Organization, WHO Global Programme to Eliminate Lymphatic Filariasis Progress Report for 2000-2009 and Strategic Plan 2010-2020 (2010), Geneva, Switzerland: World Health Organization, Geneva, Switzerland
[2] World Health Organization, Informal consultation on evaluation of morbidity in lymphatic filariasis, Document, WHO/TDR/FIL/923 (1992), Geneva, Switzerland: World Health Organization, Geneva, Switzerland
[3] Zeldenryk, L. M.; Gray, M.; Speare, R.; Gordon, S.; Melrose, W.; Brooker, S., The emerging story of disability associated with lymphatic filariasis: a critical review, PLoS Neglected Tropical Diseases, 5, 12 (2011) · doi:10.1371/journal.pntd.0001366
[4] Erickson, S. M.; Thomsen, E. K.; Keven, J. B.; Vincent, N.; Koimbu, G.; Siba, P. M.; Christensen, B. M.; Reimer, L. J., Mosquito-parasite interactions can shape filariasis transmission dynamics and impact elimination programs, PLoS Neglected Tropical Diseases, 7, 9 (2013) · doi:10.1371/journal.pntd.0002433
[5] Michael, E.; Spear, R. C., Modelling Parasite Transmission and Control. Modelling Parasite Transmission and Control, Advances in Experimental Medicine and Biology (2010), New York, NY, USA: Springer, New York, NY, USA
[6] Hayashi, S., A mathematical analysis on the epidemiology of Bancroftian and Malayan filariasis in Japan, Japanese Journal of Experimental Medicine, 32, 13-43 (1962)
[7] Hairston, N. G.; De Meillon, B., On the inefficiency of transmission of Wuchereria bancrofti from mosquito to human host, Bulletin of the World Health Organization, 38, 6, 935-941 (1967)
[8] Michael, E.; Bundy, D. A. P., Herd immunity to filarial infection is a function of vector biting rate, Proceedings of the Royal Society B: Biological Sciences, 265, 1399, 855-860 (1998) · doi:10.1098/rspb.1998.0370
[9] Michael, E.; Grenfell, B. T.; Isham, V. S.; Denham, D. A.; Bundy, D. A., Modelling variability in lymphatic filariasis: macrofilarial dynamics in the Brugia pahangi-cat model, Proceedings of the Royal Society B: Biological Sciences, 265, 1391, 155-165 (1998) · doi:10.1098/rspb.1998.0277
[10] Norman, R. A.; Chan, M. S.; Srividya, A.; Pani, S. P.; Ramaiah, K. D.; Vanamail, P.; Michael, E.; Das, P. K.; Bundy, D. A. P., EPIFIL: the development of an age-structured model for describing the transmission dynamics and control of lymphatic filariasis, Epidemiology and Infection, 124, 3, 529-541 (2000) · doi:10.1017/s0950268899003702
[11] Stolk, W. A.; De Vlas, S. J.; Borsboom, G. J. J. M.; Habbema, J. D. F., LYMFASIM, a simulation model for predicting the impact of lymphatic filariasis control: quantification for African villages, Parasitology, 135, 13, 1583-1598 (2008) · doi:10.1017/s0031182008000437
[12] Snow, L. C.; Bockarie, M. J.; Michael, E., Transmission dynamics of lymphatic filariasis: vector-specific density dependence in the development of wuchereria bancrofti infective larvae in mosquitoes, Medical and Veterinary Entomology, 20, 3, 261-272 (2006) · doi:10.1111/j.1365-2915.2006.00629.x
[13] Swaminathan, S.; Subash, P. P.; Rengachari, R.; Kaliannagounder, K.; Pradeep, D. K., Mathematical models for lymphatic filariasis transmission and control: challenges and prospects, Parasit & Vectors, 1, article 2 (2008) · doi:10.1186/1756-3305-1-2
[14] Michael, E.; Malecela-Lazaro, M. N.; Kazura, J. W.; Muller, R.; Rollinson, D.; Hay, S. I., Epidemiological modelling for monitoring and evaluation of lymphatic filariasis control, Advances in Parasitology, 65, 191-237 (2007)
[15] Gambhir, M.; Michael, E., Complex ecological dynamics and eradicability of the vector borne macroparasitic disease, lymphatic filariasis, PLoS ONE, 3, 8 (2008) · doi:10.1371/journal.pone.0002874
[16] Addiss, D. G., Global elimination of lymphatic filariasis: a ‘mass uprising of compassion’, PLoS Neglected Tropical Diseases, 7, 8 (2013) · doi:10.1371/journal.pntd.0002264
[17] Nutman, T. B., Insights into the pathogenesis of disease in human lymphatic filariasis, Lymphatic Research and Biology, 11, 3, 144-148 (2013) · doi:10.1089/lrb.2013.0021
[18] Liao, C. M.; Huang, T. L.; Lin, Y. J., Regional response of dengue fever epidemics to interannual variation and related climate variability, Stochastic Environmental Research and Risk Assessment, 29, 3, 947-958 (2015) · doi:10.1007/s00477-014-0948-6
[19] Liao, C. M.; You, S. H., Assessing risk perception and behavioral responses to influenza epidemics: linking information theory to probabilistic risk modeling, Stochastic Environmental Research and Risk Assessment, 28, 2, 189-200 (2014) · doi:10.1007/s00477-013-0739-5
[20] Slater, H. C.; Gambhir, M.; Parham, P. E.; Michael, E., Modelling co-infection with Malaria and Lymphatic filariasis, PLoS Computational Biology, 9, 6 (2013) · doi:10.1371/journal.pcbi.1003096
[21] Sabesan, S.; Raju, K. H. K.; Subramanian, S.; Srivastava, P. K.; Jambulingam, P., Lymphatic filariasis transmission risk map of India, based on a geo-environmental risk model, Vector-Borne and Zoonotic Diseases, 13, 9, 657-665 (2013) · doi:10.1089/vbz.2012.1238
[22] Wang, W.; Zhao, X.-Q., Threshold dynamics for compartmental epidemic models in periodic environments, Journal of Dynamics and Differential Equations, 20, 3, 699-717 (2008) · Zbl 1157.34041 · doi:10.1007/s10884-008-9111-8
[23] Zhidong, T.; Zhiming, L., Permanence and asymptotic behavior of the N-species nonautonomous LotkaVolterra competitive systems, Computers & Mathematics with Applications, 39, 7-8, 107-116 (2000) · Zbl 0959.34039
[24] Sun, G.; Bai, Z.; Zhang, Z.; Zhou, T.; Jin, Z., Positive periodic solutions of an epidemic model with seasonality, The Scientific World Journal, 2013 (2013) · doi:10.1155/2013/470646
[25] Muroya, Y., Global attractivity for discrete models of nonautonomous logistic equations, Computers & Mathematics with Applications, 53, 7, 1059-1073 (2007) · Zbl 1151.39007 · doi:10.1016/j.camwa.2006.12.010
[26] Invernizzi, S.; Terpin, K., A generalized logistic model for photosynthetic growth, Ecological Modelling, 94, 2-3, 231-242 (1997) · doi:10.1016/S0304-3800(96)00024-5
[27] Anita, L.-I.; Anita, S.; Arnautu, V., Global behavior for an age-dependent population model with logistic term and periodic vital rates, Applied Mathematics and Computation, 206, 1, 368-379 (2008) · Zbl 1152.92018 · doi:10.1016/j.amc.2008.09.016
[28] Wang, L.; Teng, Z.; Zhang, T., Threshold dynamics of a malaria transmission model in periodic environment, Communications in Nonlinear Science and Numerical Simulation, 18, 5, 1288-1303 (2013) · Zbl 1274.92050 · doi:10.1016/j.cnsns.2012.09.007
[29] Bai, Z.; Zhou, Y., Global dynamics of an SEIRS epidemic model with periodic vaccination and seasonal contact rate, Nonlinear Analysis. Real World Applications, 13, 3, 1060-1068 (2012) · Zbl 1239.34038 · doi:10.1016/j.nonrwa.2011.02.008
[30] Zhang, T.; Teng, Z., On a nonautonomous SEIRS model in epidemiology, Bulletin of Mathematical Biology, 69, 8, 2537-2559 (2007) · Zbl 1245.34040 · doi:10.1007/s11538-007-9231-z
[31] Teng, Z.; Liu, Y.; Zhang, L., Persistence and extinction of disease in non-autonomous SIRS epidemic models with disease-induced mortality, Nonlinear Analysis. Theory, Methods & Applications, 69, 8, 2599-2614 (2008) · Zbl 1162.34042 · doi:10.1016/j.na.2007.08.036
[32] Zhang, T.; Liu, J.; Teng, Z., Existence of positive periodic solutions of an SEIR model with periodic coefficients, Applications of Mathematics, 57, 6, 601-616 (2012) · Zbl 1274.34150 · doi:10.1007/s10492-012-0036-5
[33] Zhang, X.; Gao, S.; Cao, H., Threshold dynamics for a nonautonomous schistosomiasis model in a periodic environment, Journal of Applied Mathematics and Computing, 46, 1, 305-319 (2014) · Zbl 1302.34074 · doi:10.1007/s12190-013-0750-5
[34] Pereira, E.; Silva, C. M.; da Silva, J. A. L., A generalized nonautonomous SIRVS model, Mathematical Methods in the Applied Sciences, 36, 3, 275-289 (2013) · Zbl 1257.93013 · doi:10.1002/mma.2586
[35] Zhang, F.; Zhao, X.-Q., A periodic epidemic model in a patchy environment, Journal of Mathematical Analysis and Applications, 325, 1, 496-516 (2007) · Zbl 1101.92046 · doi:10.1016/j.jmaa.2006.01.085
[36] Bacaer, N.; Guernaoui, S., The epidemic threshold of vector-borne diseases with seasonality. The case of cutaneous leishmaniasis in Chichaoua, Morocco, Journal of Mathematical Biology, 53, 3, 421-436 (2006) · Zbl 1098.92056 · doi:10.1007/s00285-006-0015-0
[37] Bacaër, N., Approximation of the basic reproduction number R0 for vector-borne diseases with a periodic vector population, Bulletin of Mathematical Biology, 69, 3, 1067-1091 (2007) · Zbl 1298.92093 · doi:10.1007/s11538-006-9166-9
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.