A mathematical model of malaria transmission with structured vector population and seasonality. (English) Zbl 1437.92081
Summary: In this paper, we formulate a mathematical model of nonautonomous ordinary differential equations describing the dynamics of malaria transmission with age structure for the vector population. The biting rate of mosquitoes is considered as a positive periodic function which depends on climatic factors. The basic reproduction ratio of the model is obtained and we show that it is the threshold parameter between the extinction and the persistence of the disease. Thus, by applying the theorem of comparison and the theory of uniform persistence, we prove that if the basic reproduction ratio is less than 1, then the disease-free equilibrium is globally asymptotically stable and if it is greater than 1, then there exists at least one positive periodic solution. Finally, numerical simulations are carried out to illustrate our analytical results.
MSC:
| 92C60 | Medical epidemiology |
| 92-10 | Mathematical modeling or simulation for problems pertaining to biology |
References:
| [1] | Ross, R., The prevention of malaria (1911), John Murray: London, John Murray |
| [2] | Macdonald, G., The epidemiology and control of malaria (1957), London: Oxford University press, London |
| [3] | Chiyaka, C.; Tchuenche, J. M.; Garira, W.; Dube, S., A mathematical analysis of the effects of control strategies on the transmission dynamics of malaria, Applied Mathematics and Computation, 195, 2, 641-662 (2008) · Zbl 1128.92022 · doi:10.1016/j.amc.2007.05.016 |
| [4] | Ngwa, G. A.; Shu, W. S., A mathematical model for endemic malaria with variable human and mosquito populations, Mathematical and Computer Modelling, 32, 7-8, 747-763 (2000) · Zbl 0998.92035 · doi:10.1016/S0895-7177(00)00169-2 |
| [5] | Beck-Johnson, L. M.; Nelson, W. A.; Paaijmans, K. P.; Read, A. F.; Thomas, M. B.; Bjørnstad, O. N., The effect of temperature on Anopheles mosquito population dynamics and the potential for malaria transmission, PLoS ONE, 8, 11 (2013) · doi:10.1371/journal.pone.0079276 |
| [6] | Moulay, D.; Aziz-Alaoui, M. A.; Cadivel, M., The chikungunya disease: modeling, vector and transmission global dynamics, Mathematical Biosciences, 229, 1, 50-63 (2011) · Zbl 1208.92044 · doi:10.1016/j.mbs.2010.10.008 |
| [7] | Chukwu, E. N., On the boundedness and stability properties of solutions of some differential equations of the fifth order, Annali di Matematica Pura ed Applicata. Serie Quarta, 106, 245-258 (1975) · Zbl 0341.34038 · doi:10.1007/BF02415032 |
| [8] | Sinha, A. S., Stability result of a sixth order non-linear system, 7, 641-643 (1971) · Zbl 0228.34031 · doi:10.1016/0005-1098(71)90029-X |
| [9] | Tunç, C., On the stability and boundedness of solutions in a class of nonlinear differential equations of fourth order with constant delay, Vietnam Journal of Mathematics, 38, 4, 453-466 (2010) · Zbl 1226.34072 |
| [10] | Tunç, C., New results on the stability and boundedness of nonlinear differential equations of fifth order with multiple deviating arguments, Bulletin of the Malaysian Mathematical Sciences Society, 36, 3, 671-682 (2013) · Zbl 1275.34095 |
| [11] | Lou, Y.; Zhao, X.-Q., A climate-based malaria transmission model with structured vector population, SIAM Journal on Applied Mathematics, 70, 6, 2023-2044 (2010) · Zbl 1221.34224 · doi:10.1137/080744438 |
| [12] | Wang, W.; Zhao, X., 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 |
| [13] | Wang, J.; Gao, S.; Luo, Y.; Xie, D., Threshold dynamics of a huanglongbing model with logistic growth in periodic environments, Abstract and Applied Analysis, 2014 (2014) · Zbl 1474.34360 · doi:10.1155/2014/841367 |
| [14] | Xiao-Qiang, Z., CMS Books in mathematics/Ouvrages de mathématiques de la SMC (2003), New York, NY, USA: Springer Verlag, New York, NY, USA · Zbl 1023.37047 |
| [15] | Magal, P.; Zhao, X.-Q., Global attractors and steady states for uniformly persistent dynamical systems, SIAM Journal on Mathematical Analysis, 37, 1, 251-275 (2005) · Zbl 1128.37016 · doi:10.1137/S0036141003439173 |
| [16] | Ouedraogo, W.; Sangaré, B.; Traoré, S., Some mathematical problems arising in biological models: a predator-prey model fish-plankton, Journal of Applied Mathematics and Bioinformatics, 5, 4, 1-27 (2015) · Zbl 1377.92075 |
| [17] | Chitnis, N.; Hyman, J. M.; Cushing, J. M., Determining important parameters in the spread of malaria through the sensitivity analysis of a mathematical model, Bulletin of Mathematical Biology, 70, 5, 1272-1296 (2008) · Zbl 1142.92025 · doi:10.1007/s11538-008-9299-0 |
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