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A host-vector model for malaria with infective immigrants. (English) Zbl 1176.92045
Summary: This paper considers a host-vector mathematical model for the spread of malaria that incorporates recruitment of the human population through a constant immigration, with a fraction of infective immigrants. The model analysis is carried out to find the steady states and their stability. It is found that in the presence of infective immigrant humans, there is no disease-free equilibrium point. However, the model exhibits a unique endemic equilibrium state if the fraction of the infective immigrants \(\phi\) is positive. When the fraction of infective immigrants approaches a small value, there is a sharp threshold for which the disease can be reduced in the community. The unique endemic equilibrium for which there is a fraction of infective immigrants is globally asymptotically stable.

92D30 Epidemiology
34D23 Global stability of solutions to ordinary differential equations
34D05 Asymptotic properties of solutions to ordinary differential equations
92C60 Medical epidemiology
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[1] Brauer, F.; van den Driessche, P., Models of transmission of diseases with immigration of infectives, Math. biosci., 171, 143-154, (2001) · Zbl 0995.92041
[2] Chitnis, N.; Cushing, J.M.; Hyman, J.M., Bifurcation analysis for a mathematical model for malaria transmission, SIAM J. appl. math., 67, 1, 24-45, (2006) · Zbl 1107.92047
[3] Crowe, S., Malaria outbreak hit refugees in tanzania, Lancet, 350, 41, (1997)
[4] Fan, M.; Li, M.Y.; Wang, K., Global stability of an SEIS epidemic model with recruitment and a varying population size, Math. biosci., 171, 143-154, (2001)
[5] Franceschetti, A.; Pugliese, M., Threshold behaviour of a SIR epidemic model with age structure and immigration, J. math. biol., 171, 143-154, (2007)
[6] Hale, J.K., Ordinary differential equations, (1969), John Wiley New York · Zbl 0186.40901
[7] Hsu, B., A survey of constructing Lyapunov functions for mathematical models in population biology, Taiwanese J. math., 9, 2, 151-175, (2005) · Zbl 1087.34031
[8] Hastings, I.M.; Watkins, W.M., Intensity of malaria transmission and the evolution of drug resistance, Acta tropica, 94, 218-229, (2005)
[9] Iannelli, M.; Manfredi, P., Demographic change and immigration in epidemic models, Math. popul. stud., 14, 3, 169-191, (2007) · Zbl 1134.92356
[10] Khalil, H.K., Nonlinear systems, (2002), Prentice Hall New York · Zbl 0626.34052
[11] Li, G.; Wang, W.; Jin, Z., Global stability of an SEIR epidemic model with constant immigration, Chaos solitons fractals, 30, 4, 1012-1019, (2006) · Zbl 1142.34352
[12] Li, M.Y.; Muldowney, J.S., A geometric approach to global stability problems, SIAM J. math. anal. appl., 27, 4, 1070-1083, (1996) · Zbl 0873.34041
[13] López-vélez, R.; Huerga, H.; Turrientes, M.C., Infectious diseases in immigrants from the perspective of a tropical medicine referral unit, Am. J. trop. med. hyg., 69, 1, 115-121, (2003)
[14] Martens, P.; Hall, L., Malaria on the move: human population movement and malaria transmission, Emerging infect. dis., 6, 2, 103-109, (2000)
[15] Muldowney, J.S., Compound matrices and ordinary differential equations, Rocky mountain J. math., 20, 857-872, (1990) · Zbl 0725.34049
[16] McCluskey, C.C.; van den Driessche, P., Global analysis of tuberculosis models, J. differential equations, 16, 139-166, (2004) · Zbl 1056.92052
[17] Newman, R.D.; Parise, M.E.D.; Barber, A.M.; Steketee, R.W., Malaria-related deaths among U.S. travelers, 1963-2001, Ann. intern. med., 141, 7, 547-555, (2004)
[18] Prothero, R.M., Migration and malaria risk, Health risk society, 3, 1, 19-38, (2001)
[19] Ramlogan, R., Environmental refugees: A review, Environ. conservation, 23, 81-88, (1996)
[20] Singh, K.; Wester, W.C.; Gordon, M.; Trenholme, G.M., Problems in the therapy for imported malaria in the united states, Arch. intern. med., 163, 17, 2027-2030, (2003)
[21] Singh, S.; Shukla, J.B.; Chandra, P., Modelling and analysis the spread of malaria: environmental and ecological effects, J. biol. syst., 13, 1, 1-11, (2005) · Zbl 1073.92049
[22] Tatem, A.J.; Rogers, D.J.; Hay, S.I., Global transport networks and infectious disease spread, Adv. parasitology, 62, 294-333, (2006)
[23] Tumwiine, J.; Mugisha, J.Y.T.; Luboobi, L.S., A mathematical model for the dynamics of malaria in a human host and mosquito vector with temporary immunity, Appl. math. comput., 189, 2, 1953-1965, (2007) · Zbl 1117.92039
[24] Van den Driessche, P.; Watmough, J., Reproduction numbers and the sub-threshold endemic equilibria for compartmental models of disease transmission, Math. biosci., 180, 29-48, (2002) · Zbl 1015.92036
[25] Vandermeer, J.H.; Goldberg, D.E., Population ecology: first principles, (2003), Princeton University Press New Jersey
[26] Wilson, M.E., Infectious diseases: an ecological perspective, Br. med. J., 311, 1681-1684, (1998)
[27] Zhang, J.; Li, J.; Ma, Z., Global dynamics of an SEIR epidemic model with immigration of different compartments, Acta math. sci. ser. B, 26, 3, 551-567, (2006) · Zbl 1096.92039
[28] Zucker, J.R., Changing patterns of autochthonous malaria transmission in the united states: A review of recent outbreaks, Emerging infect. dis., 2, 1, 37-43, (1996)
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