The Effective Reproduction Number for a Class of Mixing Matrices in Meta-population Models in Epidemiology
Meta-populations models that account for population heterogeneities due to age, gender, spatial location, etc., and incorporate realistic mixing matrices are important for identifying optimal intervention strategies that can mitigate outbreaks of infectious diseases. Problems of finding optimal vaccination strategies for infectious diseases (including measles, mumps and rubella, HIV, and many others) based on such meta-population models are considered in this thesis. The problems are stated in the form of constrained minimization problems for the effective reproduction number, which is a threshold quantity that determines whether or not the disease can spread in the population. One of the most important findings in this thesis is the convexity of the effective reproduction number as a function of vaccine allocations among sub-groups of the population, which allows to prove bounds for the effective reproduction number and obtain estimates for the minimal number of vaccine doses needed to prevent the outbreak of a vaccine-preventable disease, as well as to bound the critical efficacy of the vaccine that would be able to do so. In the case of two sub-populations, explicit formulae are obtained for the optimal solutions. The results are applicable to a large class of mixing matrices, including commonly used Jacquez mixing matrices, as well as their generalizations introduced in this thesis.
Feng, Purdue University.
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