Let Γ be a rooted (and directed) tree, and let t be a positive integer. The path ideal I t (Γ ) is generated by monomials that correspond to directed paths of length (t − 1) in Γ . In this paper, we study algebraic properties and invariants of I t (Γ ). We give a recursive formula to compute the graded Betti numbers of I t (Γ ) in terms of path ideals of subtrees. We also give a general bound for the regularity, explicitly compute the linear strand, and investigate when I t (Γ ) has a linear resolution.

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A path ideal of a tree is an ideal whose minimal generating set corresponds to paths of a specified length in a tree. We provide a description of a collection of induced subtrees whose vertex sets correspond to the multi-graded Betti numbers on the linear strand in the corresponding minimal free resolution of the path ideal. For two classes of path ideals, we give an explicit description of a collection of induced subforests whose vertex sets correspond to the multi-graded Betti numbers in the corresponding minimal free resolutions. Lastly, in both classes of path ideals considered, the graded Betti numbers are explicitly computed for [Formula: see text]-ary trees.

Onchocerciasis is an endemic disease in parts of sub-Saharan Africa. Complex mathematical models are being used to assess the likely efficacy of efforts to eradicate the disease; however, their predictions have not always been borne out in practice. In this paper, we represent the immunological aspects of the disease with a single empirical parameter in order to reduce the model complexity. Asymptotic approximation allows us to reduce the vector-borne epidemiological model to a model of an infectious disease with nonlinear incidence. We then consider two versions, one with continuous treatment and a more realistic one where treatment occurs only at intervals. Thorough mathematical analysis of these models yields equilibrium solutions for the continuous case, periodic solutions for the pulsed case, and conditions for the existence of endemic disease equilibria in both cases, thereby leading to simple model criteria for eradication. The analytical results and numerical experiments show that the continuous treatment version is an excellent approximation for the pulsed version and that the current onchocerciasis eradication strategy is inadequate for regions where the incidence is highest and unacceptably slow even when the long-term behavior is the disease-free state.