BACKGROUND:Simulation models can be used to quantify the projected health impact of interventions. Quantifying heterogeneity in these impacts, for example by socioeconomic status, is important to understand impacts on health inequalities. We aim to disaggregate one type of Markov macro-simulation model, the proportional multistate lifetable, ensuring that under business-as-usual (BAU) the sum of deaths across disaggregated strata in each time step returns the same as the initial non-disaggregated model. We then demonstrate the application by deprivation quintiles for New Zealand (NZ), for: hypothetical interventions (50% lower all-cause mortality, 50% lower coronary heart disease mortality) and a dietary intervention to substitute 59% of sodium with potassium chloride in the food supply.METHODS:We developed a disaggregation algorithm that iteratively rescales mortality, incidence and case-fatality rates by time-step of the model to ensure correct total population counts were retained at each step. To demonstrate the algorithm on deprivation quintiles in NZ, we used the following inputs: overall (non-disaggregated) all-cause mortality & morbidity rates, coronary heart disease incidence & case fatality rates; stroke incidence & case fatality rates. We also obtained rate ratios by deprivation for these same measures. Given all-cause and cause-specific mortality rates by deprivation quintile, we derived values for the incidence, case fatality and mortality rates for each quintile, ensuring rate ratios across quintiles and the total population mortality and morbidity rates were returned when averaged across groups. The three interventions were then run on top of these scaled BAU scenarios.RESULTS:The algorithm exactly disaggregated populations by strata in BAU. The intervention scenario life years and health adjusted life years (HALYs) gained differed slightly when summed over the deprivation quintile compared to the aggregated model, due to the stratified model (appropriately) allowing for differential background mortality rates by strata. Modest differences in health gains (HALYs) resulted from rescaling of sub-population mortality and incidence rates to ensure consistency with the aggregate population.CONCLUSION:Policy makers ideally need to know the effect of population interventions estimated both overall, and by socioeconomic and other strata. We demonstrate a method and provide code to do this routinely within proportional multistate lifetable simulation models and similar Markov models.

Background We compared the health and economic consequences for the State of Victoria, Australia, of four COVID19 strategies: aggressive and moderate elimination, tight suppression (aiming for 1 to 5 cases per million per day) and loose suppression (5 to 25 cases per million per day). The strategies shifted up and down through five levels of policy stringency based on the number of cases per day, for one year. Methods An agent-based model (ABM) generated 100 runs of daily SARS-CoV-2 case numbers, that then fed into a proportional multistate lifetable to estimate health adjusted life years (HALYs) and costs. We used a net monetary benefit approach to estimate the optimal strategy. Findings Aggressive elimination resulted in the highest percentage of days with the lowest level of restrictions (median 31.7%, 90% simulation interval 6.6% to 64.4%). However, days in hard lockdown were similar across all four strategies (medians 27.5% to 36.1%). HALY losses (compared to a no-COVID-19 scenario) were similar for moderate elimination (286, 219 to 389) and moderate elimination (314, 228 to 413), and nearly eight and 40-times higher for tight and loose suppression. The median GDP loss was least for moderate elimination ($US41.7 billion, $29.0 to $63.6 billion), but there was substantial overlap in simulation intervals between the four strategies. From a health system perspective aggressive elimination was optimal in 64% of simulations above a willingness to pay of $15,000 per HALY, followed by moderate elimination in 35% of simulations. Moderate elimination was optimal from a partial societal perspective in half the simulations followed by aggressive elimination in a quarter. Shortening the pandemic duration to 6 months saw loose suppression become preferable under a partial societal perspective. Interpretation For this single high-income jurisdiction, elimination strategies were preferable over a 1-year pandemic duration. Funding Anonymous philanthropic donation to the University of Melbourne.

This economic evaluation determines the optimal policy response to the COVID-19 pandemic in Victoria, Australia, using a net monetary benefit approach for policies ranging from aggressive elimination and moderate elimination to tight suppression and loose suppression. Importance Countries have varied enormously in how they have responded to the COVID-19 pandemic, ranging from elimination strategies (eg, Australia, New Zealand, Taiwan) to tight suppression (not aiming for elimination but rather to keep infection rates low [eg, South Korea]) to loose suppression (eg, Europe, United States) to virtually unmitigated (eg, Brazil, India). Weighing the best option, based on health and economic consequences due to lockdowns, is necessary. Objective To determine the optimal policy response, using a net monetary benefit (NMB) approach, for policies ranging from aggressive elimination and moderate elimination to tight suppression (aiming for 1-5 cases per million per day) and loose suppression (5-25 cases per million per day). Design, Setting, and Participants Using governmental data from the state of Victoria, Australia, and other collected data, 2 simulation models in series were conducted of all residents (population, 6.4 million) for SARS-CoV-2 infections for 1 year from September 1, 2020. An agent-based model (ABM) was used to estimate daily SARS-CoV-2 infection rates and time in 5 stages of social restrictions (stages 1, 1b, 2, 3, and 4) for 4 policy response settings (aggressive elimination, moderate elimination, tight suppression, and loose suppression), and a proportional multistate life table (PMSLT) model was used to estimate health-adjusted life-years (HALYs) associated with COVID-19 and costs (health systems and health system plus gross domestic product [GDP]). The ABM is a generic COVID-19 model of 2500 agents, or simulants, that was scaled up to the population of interest. Models were specified with data from 2019 (eg, epidemiological data in the PMSLT model) and 2020 (eg, epidemiological and cost consequences of COVID-19). The NMB of each policy option at varying willingness to pay (WTP) per HALY was calculated: NMB = HALYs x WTP - cost. The estimated most cost-effective (optimal) policy response was that with the highest NMB. Main Outcome and Measures Estimated SARS-CoV-2 infection rates, time under 5 stages of restrictions, HALYs, health expenditure, and GDP losses. Results In 100 runs of both the ABM and PMSLT models for each of the 4 policy responses, 31.0% of SARS-CoV-2 infections, 56.5% of hospitalizations, and 84.6% of deaths occurred among those 60 years and older. Aggressive elimination was associated with the highest percentage of days with the lowest level of restrictions (median, 31.7%; 90% simulation interval [SI], 6.6%-64.4%). However, days in hard lockdown were similar across all 4 strategies. The HALY losses (compared with a scenario without COVID-19) were similar for aggressive elimination (median, 286 HALYs; 90% SI, 219-389 HALYs) and moderate elimination (median, 314 HALYs; 90% SI, 228-413 HALYs), and nearly 8 and 40 times higher for tight suppression and loose suppression, respectively. The median GDP loss was least for moderate elimination (median, $41.7 billion; 90% SI, $29.0-$63.6 billion), but there was substantial overlap in simulation intervals between the 4 strategies. From a health system perspective, aggressive elimination was optimal in 64% of simulations above a WTP of $15 000 per HALY, followed by moderate elimination in 35% of simulations. Moderate elimination was optimal from a GDP perspective in half of the simulations, followed by aggressive elimination in a quarter. Conclusions and Relevance In this simulation modeling economic evaluation of estimated SARS-CoV-infection rates, time under 5 stages of restrictions, HALYs, health expenditure, and GDP losses in Victoria, Australia, an elimination strategy was associated with the least health losses and usually the fewest GDP losses. Question What has the least health losses and is the most cost-effective of 4 policy responses to the COVID-19 pandemic (aggressive elimination, moderate elimination, tight suppression, and loose suppression) in the state of Victoria, Australia? Findings In this simulation modeling economic evaluation of health losses and costs from COVID-19 policy responses, aggressive elimination was the most cost-effective from a health system perspective in 64% of simulations above a willingness to pay of $15 000 per health-adjusted life-years, followed by moderate elimination in 35% of simulations. Moderate elimination was most cost-effective from a gross domestic product (GDP) perspective (ie, including GDP losses in addition to health expenditure) in half of the simulations, followed by aggressive elimination in a quarter. Meaning While there is considerable uncertainty in outcomes for all 4 policy responses, the 2 elimination options appear to be the most optimal from both health system and health plus GDP perspectives.

The geometric $\delta$-minimum spanning tree problem ($\delta$-MST) is the problem of finding a minimum spanning tree for a set of points in a normed vector space, such that no vertex in the tree has a degree which exceeds $\delta$, and the sum of the lengths of the edges in the tree is minimum. The similarly defined geometric $\delta$-minimum bottleneck spanning tree problem ($\delta$-MBST), is the problem of finding a degree bounded spanning tree such that the length of the longest edge is minimum. For point sets that lie in the Euclidean plane, both of these problems have been shown to be NP-hard for certain specific values of $\delta$. In this paper, we investigate the $\delta$-MBST problem in $3$-dimensional Euclidean space and $3$-dimensional rectilinear space. We show that the problems are NP-hard for certain values of $\delta$, and we provide inapproximability results for these cases. We also describe new approximation algorithms for solving these $3$-dimensional variants, and then analyse their worst-case performance.

Background: Countries can decide between one of three COVID-19 control strategies: 1) elimination (e.g., some island countries); 2) suppression, to low infection rates; 3) or mitigation, as per pandemic influenza strategies with ensuing herd immunity. This paper quantifies the health (direct COVID-19 impact, and indirect through unemployment onto self-harm and road traffic crash) and cost (health system and societal) consequences for these strategies across Australia, New Zealand (NZ) and Sweden. Methods: We used proportional multistate lifetable (PMSLT) models for each country, with mortality and morbidity data from the Global Burden of Disease Study and health system expenditure from country-specific sources. Feeding into the PMSLT were monthly SARS-CoV-2 infection rates (0.1%/2.5%/60% for elimination/suppression/mitigation by 18 months), and anticipated changes in unemployment rates that generated changes in suicide/self-harm and RTC injury rates. For Australia and NZ, we also estimated fixed health system costs by strategy and GDP loss for societal costing. We used a 3% per annum discount rate, over a 20-year time horizon. FINDINGS Compared to the pre-pandemic baseline, health adjusted life year (HALY) losses were 15.0/ 11.0/ 23.0 per million population under elimination for Australia/ NZ/ Sweden, 1,540/ 1,500/ 1,820 per million for suppression, and 19,800/ 19,500/ 22,400 for mitigation. For Australia and NZ, the optimal policy from a health system perspective was mitigation up to a US$20,000 willingness-to-pay (WTP) per HALY gained. For higher WTP, elimination (or suppression if elimination is not feasible) was favoured. From a societal perspective (health system costs plus GDP losses), mitigation was optimal up to US$240,000 per HALY, then elimination (or suppression). Interpretation: This modelling analysis suggests that elimination or suppression is optimal across the usual range of WTP from a health system perspective in high-income countries (HICs). But from a societal perspective, mitigation is favoured unless a HALY is valued at over US$240,000. Funding Statement: Health Research Council of New Zealand (16/443) for core model development). Strategic and COVID-19 specific funding for this research was provided by the Melbourne School of Population and Global Health, University of Melbourne. Carvalho is supported by the University of Melbourne McKenzie Postdoctoral Fellowship. Declaration of Interest: None.

Given a set of points in the Euclidean plane, the Euclidean [Formula: see text]-minimum spanning tree ([Formula: see text]-MST) problem is the problem of finding a spanning tree with maximum degree no more than [Formula: see text] for the set of points such the sum of the total length of its edges is minimum. Similarly, the Euclidean [Formula: see text]-minimum bottleneck spanning tree ([Formula: see text]-MBST) problem, is the problem of finding a degree-bounded spanning tree for a set of points in the plane such that the length of the longest edge is minimum. When [Formula: see text], these two problems may yield disjoint sets of optimal solutions for the same set of points. In this paper, we perform computational experiments to compare the accuracies of a variety of heuristic and approximation algorithms for both these problems. We develop heuristics for these problems and compare them with existing algorithms. We also describe a new type of edge swap algorithm for these problems that outperforms all the algorithms we tested.

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Given a graph G with edge lengths, the minimum bottleneck spanning tree (MBST) problem is to find a spanning tree where the length of the longest edge in tree is minimum. It is a well‐known fact that every minimum spanning tree (MST) is a minimum bottleneck spanning tree. In this article, we introduce the δ ‐MBST problem, which is the problem of finding an MBST such that every vertex in the tree has degree at most δ . We show that optimal solutions to the similarly defined δ ‐MST problem are not necessarily optimal solutions to the δ ‐MBST, and we establish that the δ ‐MBST problem is NP‐complete for any . We show that when edge lengths of the graph are Euclidean distances between points in the plane, the problem is NP‐hard for δ = 2 and 3, and tractable for . We give a dual approximation scheme for the general graph version of the problem which is the best possible with respect to feasibility. For the Euclidean version, we give a ‐factor approximation algorithm for the 4‐MBST. We also give a 2‐factor algorithm for the Euclidean 3‐MBST and a 3‐factor approximation algorithm for the general Euclidean δ ‐MBST, both of which can be generalized to metric spaces. © 2016 Wiley Periodicals, Inc. NETWORKS, Vol. 68(4), 302–314 2016

For a simple graph [Formula: see text] and for a pair of vertices [Formula: see text], we say that a vertex [Formula: see text] resolves [Formula: see text] and [Formula: see text] if the shortest path from [Formula: see text] to [Formula: see text] is of a different length than the shortest path from [Formula: see text] to [Formula: see text]. A set of vertices [Formula: see text] is a resolving set if for every pair of vertices [Formula: see text] and [Formula: see text] in [Formula: see text], there exists a vertex [Formula: see text] that resolves [Formula: see text] and [Formula: see text]. The minimum weight resolving set problem is to find a resolving set [Formula: see text] for a weighted graph [Formula: see text] such that [Formula: see text] is minimum, where [Formula: see text] is the weight of vertex [Formula: see text]. In this paper, we explore the possible solutions of this problem for grid graphs [Formula: see text] where [Formula: see text]. We give a complete characterization of solutions whose cardinalities are 2 or 3, and show that the maximum cardinality of a solution is [Formula: see text]. We show that the grid has the property that given a landmark set, we only need to investigate whether or not all pairs of vertices that share common neighbors are resolved to determine if the whole graph is resolved. We use this result to provide a characterization of a class of minimals whose cardinalities range from [Formula: see text] to [Formula: see text] and show that the number of such minimals is [Formula: see text].