| 1. |
- Britton, Tom, 1965-, et al.
(författare)
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Optimal Intervention Strategies for Minimizing Total Incidence During an Epidemic
- 2023
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Ingår i: SIAM Journal on Applied Mathematics. - 0036-1399 .- 1095-712X. ; 83:2, s. 354-373
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Tidskriftsartikel (refereegranskat)abstract
- This article considers the minimization of the total number of infected individuals over the course of an epidemic in which the rate of infectious contacts can be reduced by time-dependent nonpharmaceutical interventions. The societal and economic costs of interventions are taken into account using a linear budget constraint which imposes a trade-off between short-term heavy interventions and long-term light interventions. We search for an optimal intervention strategy in an infinite-dimensional space of controls containing multiple consecutive lockdowns, gradually imposed and lifted restrictions, and various heuristic controls based, for example, on tracking the effective reproduction number. Mathematical analysis shows that among all such strategies, the global optimum is achieved by a single constant-level lockdown of maximum possible magnitude. Numerical simulations highlight the need for careful timing of such interventions and illustrate their benefits and disadvantages compared to strategies designed for minimizing peak prevalence. Rather counterintuitively, adding restrictions prior to the start of a well-planned intervention strategy may even increase the total incidence.
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| 2. |
- Wall, David J.N., et al.
(författare)
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On an inverse problem from magnetic resonance elastic imaging
- 2011
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Ingår i: SIAM Journal on Applied Mathematics. - : Society for Industrial & Applied Mathematics (SIAM). - 0036-1399 .- 1095-712X. ; 71:5, s. 1578-1605
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Tidskriftsartikel (refereegranskat)abstract
- The imaging problem of elastography is an inverse problem. The nature of an inverse problem is that it is ill-conditioned. We consider properties of the mathematical map which describes how the elastic properties of the tissue being reconstructed vary with the field measured by magnetic resonance imaging (MRI). This map is a nonlinear mapping, and our interest is in proving certain conditioning and regularity results for this operator which occurs implicitly in this problem of imaging in elastography. In this treatment we consider the tissue to be linearly elastic, isotropic, and spatially heterogeneous. We determine the conditioning of this problem of function reconstruction, in particular for the stiffness function. We further examine the conditioning when determining both stiffness and density. We examine the Frechet derivative of the nonlinear mapping, which enables us to describe the properties of how the field affects the individual maps to the stiffness and density functions. We illustrate how use of the implicit function theorem can considerably simplify the analysis of Frechet differentiability and regularity properties of this underlying operator. We present new results which show that the stiffness map is mildly ill-posed, whereas the density map suffers from medium ill-conditioning. Computational work has been done previously to study the sensitivity of these maps, but our work here is analytical. The validity of the Newton-Kantorovich and optimization methods for the computational solution of this inverse problem is directly linked to the Frechet differentiability of the appropriate nonlinear operator, which we justify.
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