On complexity of algorithms for modeling disease transmission and optimal vaccination strategy

We consider simple deterministic models of disease transmission. Given a set of individuals I, we assign a hypergraph Hi = (I \ {i}, Ei) to each i ∈ I and assume that i will be infected whenever there is a fully infected edge e ∈ Ei. Along with this general model MH we also study two special cases MG and MD when for all i ∈ I the hypergraphs Hi are specified implicitly by a (directed) graph G = (I, E) and integral positive thresholds k(i) for all i ∈ I. Then we assume that i will be infected whenever at least k(i) of his neighbors (predecessors) are infected. Given a set S of the originally infected individuals (a source) we generate the closure T (S) = cl(S), that is, the set of all individuals that will be infected if the above transmission rules are applied iteratively sufficiently many times. We study all minimal sources such that (i) T (S) = I, or (ii) T (S) contains a given individual q ∈ I, or (iii) T (S) contains an edge of a given “target” hypergraph H. We denote these three types of “targets” by TI , Tq, and TH respectively. We show that, given a threshold t, it is NP-complete to decide whether there is a source S of size at most t. The problem remains NP-complete for each of the three models MR, MG or MD and targets TI , Tq or TH . We also consider enumeration problems and show that if the transmission rule is given explicitly, MR, then all inclusion minimal sources can be generated in incremental polynomial time for all targets TI , Tq, or TH . On the other hand, generating minimal sources is hard for all targets if the transmission model is given by a (directed) graph, MG or MD, since for these two cases the input size may be logarithmic in the input size of MR. Indeed, given G = (I, E) and k(i) for all i ∈ I, a corresponding hypergraph Hi for some i ∈ I may be exponential in |I| unless k(i) is bounded by a constant.