Malaria is a major public health issue in many parts of the world, and the anopheles mosquitoes which drive transmission are key targets for interventions. Consequently, a best understanding of mosquito populations dynamics is necessary in the fight against the disease. Hence, in this paper we propose a delayed mathematical model of the life cycle of anopheles mosquitoes by using delayed-logistic population growth. The model is formulated by inserting the time delay into the logistic population growth rate, that accounts for the period of growth from eggs to the last aquatic stage, which is pupae. Depending on the system parameters, we establish a threshold for survival and extinction of the anopheles mosquitoes population. Moreover, by choosing the time delay as a bifurcation parameter, we prove that the system loses its stability and a Hopf bifurcation occurs when time delay passes through some critical values. Finally, we perform some numerical simulations and the results are well in keeping with the analytical analysis.
In this paper, we investigate a mathematical model of malaria transmission dynamics with maturation delay of a vector population in a periodic environment. The incidence rate between vector and human hosts is modeled by a general nonlinear incidence function which satisfies a set of conditions. Thus, the model is formulated as a system of retarded functional differential equations. Furthermore, through dynamical systems theory, we rigorously analyze the global behavior of the model. Therefore, we prove that the basic reproduction number of the model denoted by 0 is the threshold between the uniform persistence and the extinction of malaria virus transmission. More precisely, we show that if 0 is less than unity, then the disease-free periodic solution is globally asymptotically stable. Otherwise, the system exhibits at least one positive periodic solution if 0 is greater than unity. Finally, we perform some numerical simulations to illustrate our mathematical results and to analyze the impact of the delay on the disease transmission.
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