In this paper we present a susceptible–infectious–susceptible (SIS) model that describes the transmission dynamics of cutaneous Leishmaniasis. The model treats a vector population and several populations of different mammals. Members of the human population serve as the incidental hosts, and members of the various animals populations serve as reservoir hosts. We establish the basic reproduction number and the equilibrium conditions of the system. We use a generalization of the Lyapunov function approach to show that when the basic reproduction number is less than or equal to one, the diseases-free equilibrium is a global attractor, and that when it is greater than one the endemic equilibrium is a global attractor. We present numerical simulations that demonstrate the dynamics of the model for a system containing a human population and a single animal population.

This paper presents a deterministic SIS model for the transmission dynamics of malaria, a life-threatening disease transmitted by mosquitos. Four species of the parasite genus Plasmodium are known to cause human malaria. Some species of the parasite have evolved into strains that are resistant to treatment. Although proportions of Plasmodium species vary considerably between geographic regions, multiple species and strains do coexist within some communities. The mathematical model derived here includes all available species and strains for a given community. The model has a disease-free equilibrium, which is a global attractor when the reproduction number of each species or strain is less than one. The model possesses quasiendemic equilibria; local asymptotic stability is established for two species, and numerical simulations suggest that the species or strain with the highest reproduction number exhibits competitive exclusion.

Abstract. Epidermal wound healing is a complex process that repairs injured tissue. The complexity of this process increases when bacteria are present in a wound; the bacteria interaction determines whether infection sets in. Because of underlying physiological problems infected wounds do not follow the normal healing pattern. In this paper we present a mathematical model of the healing of both infected and uninfected wounds. At the core of our model is an account of the initiation of angiogenesis by macrophage-derived growth factors. We express the model as a system of reaction-diffusion equations, and we present results of computations for a version of the model with one spatial dimension.

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This paper presents a generalized high order block hybridk-step backward differentiation formula (HBDF) for solving stiff systems, including large systems resulting from the semidiscretization parabolic partial differential equations (PDEs). A block scheme in which two off-grid points are specified by the zeros of the second degree Chebyshev polynomial of the first kind is examined for convergence,LandAstabilities. Numerical simulations that illustrate the accuracy of a Chebyshev based method are given for selected stiff systems and partial differential equations.