We investigate biological processes, particularly the propagation of malaria. Both the continuous and the numerical models on some fixed mesh should preserve the basic qualitative properties of the original phenomenon. Our main goal is to give the conditions for the discrete (numerical) models of the malaria phenomena under which they possess some given qualitative property, namely, to be between zero and one. The conditions which guarantee this requirement are related to the time-discretization step-size. We give a sufficient condition for some explicit methods. For implicit methods we prove that the above property holds unconditionally.
AbstractEpidemiological models play an important role in the study of diseases. These models belong to population dynamics models and can be characterized with differential equations. In this paper we focus our attention on two epidemic models for malaria spreading, namely Ross-, and extended Ross model. As both the continous and the corresponding numerical models should preserve the basic qualitative properties of the phenomenon, we paid special attention to its examination, and proved their invariance with reference to the data set. Moreover, existence and uniqueness of equilibrium points for both models of malaria are considered. We demonstrate the theoritical results with numerical simulations.
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