1
|
Amir-Ud-Din R, Khan M, Murad Z, Mussio I. Clear Waters, Bright Futures: Do Low-Cost Information Interventions Increase Health Preventive Behaviors. HEALTH ECONOMICS 2025. [PMID: 40394904 DOI: 10.1002/hec.4977] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/21/2024] [Revised: 05/06/2025] [Accepted: 05/12/2025] [Indexed: 05/22/2025]
Abstract
Contaminated drinking water poses a significant, long-term health challenge in developing countries. With the aim of shedding light on the most effective presentation of this information in awareness campaigns, we run a randomized control trial involving 1388 households in Punjab, Pakistan. We provide information about fecal matter (E.Coli) presence in drinking water and on ways to treat water to make it potable. This intervention increases the likelihood of adopting in-home water purification for those households who were provided with information about water contamination results. Those informed of both water contamination and potential water treatment methods exhibit an even higher likelihood of behavior change. This study is evidence of the potential efficacy of low-cost information-based interventions, offering valuable insights for health policy in resource-constrained settings.
Collapse
Affiliation(s)
- Rafi Amir-Ud-Din
- Department of Economics, COMSATS University Islamabad, Lahore, Pakistan
| | - Muhammad Khan
- Department of Economics, COMSATS University Islamabad, Lahore, Pakistan
| | - Zahra Murad
- Department of Accounting, Economics and Finance, University of Portsmouth, Portsmouth, UK
- UNEC Cognitive Economics Centre, Azerbaijan State University of Economics, Baku, Azerbaijan
| | - Irene Mussio
- Department of Economics, Leeds University Business School, Leeds, UK
| |
Collapse
|
2
|
Changes in water treatment, hygiene practices, household floors, and child health in times of Covid-19: A longitudinal cross-sectional survey in Surkhet District, Nepal. Int J Hyg Environ Health 2023; 249:114138. [PMID: 36821912 PMCID: PMC9925420 DOI: 10.1016/j.ijheh.2023.114138] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/30/2022] [Revised: 01/25/2023] [Accepted: 02/12/2023] [Indexed: 02/16/2023]
Abstract
INTRODUCTION Consistent and effective practice of water treatment, sanitation, and hygiene (WASH) behaviour is an indispensable requisite for realizing health improvements among children living in low-income areas with challenging hygienic conditions. Sustainably achieving such a behaviour change is challenging but more likely to be realized during epidemics, when health threats are high and the dissemination of information on preventative measures is intense. Our study conducted cross-sectional surveys in Surkhet District Nepal, before and during the Covid-19 pandemic to assess the impact of water safety interventions and hygiene training implemented before and during the pandemic on WASH conditions and practices and to assess the association of these changes with child health. METHODS Information on WASH infrastructure, WASH behaviour, nutrition, and child health, including on parasitic infections, was obtained before and during the Covid-19 pandemic in spring 2018 and spring 2021, from 589 children aged between 6 months and 10 years and their caregivers. Data was collected through quantitative, structured face-to-face interviews, observations, health examinations of children including anthropometric measurements, analysis of children's stool, and water quality analysis. The association of changes in WASH factors with changes in child health was analysed using multivariate generalized estimating equations for repeated measures. RESULTS Water safety management was significantly improved by the introduction of chlorination to piped water supply systems, which served 40% of households. In addition, the percentage of households using a ceramic water filter increased from 12.2% to 34.8%. Large and significant changes were observed in handwashing behaviour (frequency, use of soap and washing at critical times) and infrastructure: 35% of households constructed a new handwashing station. Kitchen and household hygiene also improved. An additional 22% of households improved the cleanliness of the toilet. The number of houses with a cemented floor increased by 20%. WASH changes were significantly associated with improved child health: the chlorination of piped water supply reduced odds ratios for diarrhoea (OR = 0.36, 95% CI = 0.15-0.88, p = 0.025), respiratory difficulties (OR = 0.39, 95% CI = 0.16-0.92, p = 0.033), fever (OR = 0.42, 95% CI = 0.26-0.71, p = 0.001) and cough (OR = 0.58, 95% CI = 0.36-0.93, p = 0.024), and. The frequency of handwashing with soap was associated with significantly reduced odds ratios for infections with Giardia lamblia (OR = 0.68, 95% CI = 0.50-0.91, p = 0.011), stunting and wasting (OR = 0.75, 95% CI = 0.66-0.92, p = 0.003) and fever (OR = 0.85, 95% CI = 0.75-0.96, p = 0.008),. The presence of a handwashing station at baseline was associated with significantly reduced odds ratios for respiratory difficulties (OR = 0.45, 95% CI = 0.26-0.78, p = 0.004). The construction of a handwashing station between baseline and endline was significantly associated with reduced odds ratios for pale conjunctiva (OR = 0.32, 95% CI = 0.17-0.60, p < 0.001), which is a clinical sign of iron deficiency and anaemia, respiratory difficulties (OR = 0.39, 95% CI = 0.17-0.89, p = 0.026) and cough (OR = 0.44, 95% CI = 0.26-0.76, p = 0.003). Using a clean container for the transport of drinking water was significantly associated with reduced odds ratios for infections with Giardia lamblia (OR = 0.39, 95% CI = 0.16-0.93, p = 0.033) and diarrhoea (OR = 0.48, 95% CI = 0.24-0.96, p = 0.038). Similarly, a cemented floor in the household was significantly associated with reduced odd ratios for diarrhoea (OR = 0.38, 95% CI = 0.16-0.87, p = 0.022) and infections with Giardia lamblia (OR = 0.44, 95% CI = 0.19-1.02, p = 0.056). CONCLUSION WASH training and the promotion of preventative measures during the Covid-19 pandemic supported improved water safety management and hygiene behaviour, which resulted in a reduction in infectious diseases among children in the study area.
Collapse
|
3
|
Crider YS, Tsuchiya M, Mukundwa M, Ray I, Pickering AJ. Adoption of Point-of-Use Chlorination for Household Drinking Water Treatment: A Systematic Review. ENVIRONMENTAL HEALTH PERSPECTIVES 2023; 131:16001. [PMID: 36715546 PMCID: PMC9885856 DOI: 10.1289/ehp10839] [Citation(s) in RCA: 2] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 12/21/2021] [Revised: 12/20/2022] [Accepted: 12/23/2022] [Indexed: 06/18/2023]
Abstract
BACKGROUND Centralized chlorination of urban piped water supplies has historically contributed to major reductions in waterborne illness. In locations without effective centralized water treatment, point-of-use (POU) chlorination for households is widely promoted to improve drinking water quality and health. Realizing these health benefits requires correct, consistent, and sustained product use, but real-world evaluations have often observed low levels of use. To our knowledge, no prior reviews exist on adoption of chlorine POU products. OBJECTIVES Our objectives were to identify which indicators of adoption are most often used in chlorine POU studies, summarize levels of adoption observed, understand how adoption changes over time, and determine how adoption is affected by frequency of contact between participants and study staff. METHODS We conducted a systematic review of household POU chlorination interventions or programs from 1990 through 2021 that reported a quantitative measure of adoption, were conducted in low- and middle-income countries, included data collection at households, and reported the intervention start date. RESULTS We identified 36 studies of household drinking water chlorination products that met prespecified eligibility criteria and extracted data from 46 chlorine intervention groups with a variety of chlorine POU products and locations. There was no consensus definition of adoption of household water treatment; the most common indicator was the proportion of household stored water samples with free chlorine residual > 0.1 or 0.2 mg / L . Among studies that reported either free or total chlorine-confirmed adoption of chlorine POU products, use was highly variable (across all chlorine intervention groups at the last time point measured in each study; range: 1.5%-100%; sample size-weighted median = 47 % ; unweighted median = 58 % ). The median follow-up duration among intervention groups was 3 months. On average, adoption declined over time and was positively associated with frequency of contact between respondents and study staff. DISCUSSION Although prior research has shown that POU chlorine products improve health when correctly and consistently used, a reliance on individual adoption for effective treatment is unlikely to lead to the widespread public health benefits historically associated with pressurized, centralized treatment of piped water supplies. https://doi.org/10.1289/EHP10839.
Collapse
Affiliation(s)
- Yoshika S. Crider
- Energy and Resources Group, University of California, Berkeley (UC Berkeley), Berkeley, California, USA
- Division of Epidemiology and Biostatistics, UC Berkeley, Berkeley, California, USA
- King Center on Global Development, Stanford University, Stanford, California, USA
| | - Miki Tsuchiya
- Master of Development Practice Program, UC Berkeley, Berkeley, California, USA
| | - Magnifique Mukundwa
- Department of Civil and Environmental Engineering, Tufts University, Medford, Massachusetts, USA
| | - Isha Ray
- Energy and Resources Group, University of California, Berkeley (UC Berkeley), Berkeley, California, USA
| | - Amy J. Pickering
- Department of Civil and Environmental Engineering, UC Berkeley, Berkeley, California, USA
| |
Collapse
|
4
|
Chirgwin H, Cairncross S, Zehra D, Sharma Waddington H. Interventions promoting uptake of water, sanitation and hygiene (WASH) technologies in low- and middle-income countries: An evidence and gap map of effectiveness studies. CAMPBELL SYSTEMATIC REVIEWS 2021; 17:e1194. [PMID: 36951806 PMCID: PMC8988822 DOI: 10.1002/cl2.1194] [Citation(s) in RCA: 12] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/25/2023]
Abstract
Background Lack of access to and use of water, sanitation and hygiene (WASH) cause 1.6 million deaths every year, of which 1.2 million are due to gastrointestinal illnesses like diarrhoea and acute respiratory infections like pneumonia. Poor WASH access and use also diminish nutrition and educational attainment, and cause danger and stress for vulnerable populations, especially for women and girls. The hardest hit regions are sub-Saharan Africa and South Asia. Sustainable Development Goal (SDG) 6 calls for the end of open defecation, and universal access to safely managed water and sanitation facilities, and basic hand hygiene, by 2030. WASH access and use also underpin progress in other areas such as SDG1 poverty targets, SDG3 health and SDG4 education targets. Meeting the SDG equity agenda to "leave none behind" will require WASH providers prioritise the hardest to reach including those living remotely and people who are disadvantaged. Objectives Decision makers need access to high-quality evidence on what works in WASH promotion in different contexts, and for different groups of people, to reach the most disadvantaged populations and thereby achieve universal targets. The WASH evidence map is envisioned as a tool for commissioners and researchers to identify existing studies to fill synthesis gaps, as well as helping to prioritise new studies where there are gaps in knowledge. It also supports policymakers and practitioners to navigate the evidence base, including presenting critically appraised findings from existing systematic reviews. Methods This evidence map presents impact evaluations and systematic reviews from the WASH sector, organised according to the types of intervention mechanisms, WASH technologies promoted, and outcomes measured. It is based on a framework of intervention mechanisms (e.g., behaviour change triggering or microloans) and outcomes along the causal pathway, specifically behavioural outcomes (e.g., handwashing and food hygiene practices), ill-health outcomes (e.g., diarrhoeal morbidity and mortality), nutrition and socioeconomic outcomes (e.g., school absenteeism and household income). The map also provides filters to examine the evidence for a particular WASH technology (e.g., latrines), place of use (e.g., home, school or health facility), location (e.g., global region, country, rural and urban) and group (e.g., people living with disability). Systematic searches for published and unpublished literature and trial registries were conducted of studies in low- and middle-income countries (LMICs). Searches were conducted in March 2018, and searches for completed trials were done in May 2020. Coding of information for the map was done by two authors working independently. Impact evaluations were critically appraised according to methods of conduct and reporting. Systematic reviews were critically appraised using a new approach to assess theory-based, mixed-methods evidence synthesis. Results There has been an enormous growth in impact evaluations and systematic reviews of WASH interventions since the International Year of Sanitation, 2008. There are now at least 367 completed or ongoing rigorous impact evaluations in LMICs, nearly three-quarters of which have been conducted since 2008, plus 43 systematic reviews. Studies have been done in 83 LMICs, with a high concentration in Bangladesh, India, and Kenya. WASH sector programming has increasingly shifted in focus from what technology to supply (e.g., a handwashing station or child's potty), to the best way in which to do so to promote demand. Research also covers a broader set of intervention mechanisms. For example, there has been increased interest in behaviour change communication using psychosocial "triggering", such as social marketing and community-led total sanitation. These studies report primarily on behavioural outcomes. With the advent of large-scale funding, in particular by the Bill & Melinda Gates Foundation, there has been a substantial increase in the number of studies on sanitation technologies, particularly latrines. Sustaining behaviour is fundamental for sustaining health and other quality of life improvements. However, few studies have been done of intervention mechanisms for, or measuring outcomes on sustained adoption of latrines to stop open defaecation. There has also been some increase in the number of studies looking at outcomes and interventions that disproportionately affect women and girls, who quite literally carry most of the burden of poor water and sanitation access. However, most studies do not report sex disaggregated outcomes, let alone integrate gender analysis into their framework. Other vulnerable populations are even less addressed; no studies eligible for inclusion in the map were done of interventions targeting, or reporting on outcomes for, people living with disabilities. We were only able to find a single controlled evaluation of WASH interventions in a health care facility, in spite of the importance of WASH in health facilities in global policy debates. The quality of impact evaluations has improved, such as the use of controlled designs as standard, attention to addressing reporting biases, and adequate cluster sample size. However, there remain important concerns about quality of reporting. The quality and usefulness of systematic reviews for policy is also improving, which draw clearer distinctions between intervention mechanisms and synthesise the evidence on outcomes along the causal pathway. Adopting mixed-methods approaches also provides information for programmes on barriers and enablers affecting implementation. Conclusion Ensuring everyone has access to appropriate water, sanitation, and hygiene facilities is one of the most fundamental of challenges for poverty elimination. Researchers and funders need to consider carefully where there is the need for new primary evidence, and new syntheses of that evidence. This study suggests the following priority areas:Impact evaluations incorporating understudied outcomes, such as sustainability and slippage, of WASH provision in understudied places of use, such as health care facilities, and of interventions targeting, or presenting disaggregated data for, vulnerable populations, particularly over the life-course and for people living with a disability;Improved reporting in impact evaluations, including presentation of participant flow diagrams; andSynthesis studies and updates in areas with sufficient existing and planned impact evaluations, such as for diarrhoea mortality, ARIs, WASH in schools and decentralisation. These studies will preferably be conducted as mixed-methods systematic reviews that are able to answer questions about programme targeting, implementation, effectiveness and cost-effectiveness, and compare alternative intervention mechanisms to achieve and sustain outcomes in particular contexts, preferably using network meta-analysis.
Collapse
Affiliation(s)
- Hannah Chirgwin
- International Initiative for Impact Evaluation (3ie)London International Development CentreLondonUK
| | | | | | - Hugh Sharma Waddington
- London School of Hygiene and Tropical Medicine and International Initiative for Impact Evaluation (3ie)London International Development CentreLondonUK
| |
Collapse
|
5
|
Wang Y, Hou T, Yang Z, Zhao L, Wu W, Yang W, Graham NJD. Nitrogen-Free Cationic Starch Flocculants: Flocculation Performance, Antibacterial Ability, and UF Membrane Fouling Control. ACS APPLIED BIO MATERIALS 2020; 3:2910-2919. [PMID: 35025338 DOI: 10.1021/acsabm.9b01129] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
Abstract
In light of growing concerns about the formation of nitrogen-based disinfection byproducts (N-DBP) and the possible contribution from the use of quaternary-ammonium-containing flocculants, there is growing interest in the alternative use of quaternary phosphonium salts, which have been reported to have a lower DBP formation potential, stronger cationic properties, lower cytotoxicity, and greater stability. In this study, the performance of N-free quaternary-phosphonium-modified starch flocculants (S-BTP), synthesized through a facile one-step method using commercially available raw materials, in the treatment of bacteria-laden waters (E. coli as the model bacteria) was assessed in both jar tests and a bench-scale continuous-flow flocculation-sedimentation-ultrafiltration process. In jar tests, the effects of the cationic degree of substitution (DS) and dosage of the flocculant, solution pH, and presence of model contaminants on treatment performance were studied. One particular flocculant (S-BTP3), with a DS of 19.3%, displayed high removal efficiencies of E. coli, turbidity, and UV254 from water, comparable with those of ammonium-based analogues and conventional alum, via a combination of charge attraction, polymer bridging, and antibacterial effects. S-BTP3 also possessed better bactericidal properties (99.4% of E. coli killed) than alum (41.4% killed) and did not cause the release of intracellular substances into the treated water. In the continuous-flow flocculation-sedimentation-UF tests, S-BTP3 was superior to alum in the flocculation and antibacterial performance, and in mitigating UF membrane fouling. The results have clearly demonstrated the multiple benefits of the use of N-free cationic starch flocculants in water treatment as an alternative to conventional chemicals.
Collapse
Affiliation(s)
- Yadong Wang
- School of Chemistry and Materials Science, Jiangsu Provincial Key Laboratory of Materials Cycling and Pollution Control, Nanjing Normal University, Nanjing 210023, P.R. China
| | - Tianyang Hou
- School of Chemistry and Materials Science, Jiangsu Provincial Key Laboratory of Materials Cycling and Pollution Control, Nanjing Normal University, Nanjing 210023, P.R. China
| | - Zhen Yang
- School of Chemistry and Materials Science, Jiangsu Provincial Key Laboratory of Materials Cycling and Pollution Control, Nanjing Normal University, Nanjing 210023, P.R. China.,Department of Civil and Environmental Engineering, Imperial College London, South Kensington Campus, London SW72AZ, U.K
| | - Lina Zhao
- School of Chemistry and Materials Science, Jiangsu Provincial Key Laboratory of Materials Cycling and Pollution Control, Nanjing Normal University, Nanjing 210023, P.R. China
| | - Wei Wu
- Jiangsu Academy of Environmental Industry and Technology Co. Ltd, Nanjing 210036, P.R. China
| | - Weiben Yang
- School of Chemistry and Materials Science, Jiangsu Provincial Key Laboratory of Materials Cycling and Pollution Control, Nanjing Normal University, Nanjing 210023, P.R. China
| | - Nigel J D Graham
- Department of Civil and Environmental Engineering, Imperial College London, South Kensington Campus, London SW72AZ, U.K
| |
Collapse
|
6
|
Brown J, Hayashi MAL, Eisenberg JNS. The Critical Role of Compliance in Delivering Health Gains from Environmental Health Interventions. Am J Trop Med Hyg 2020; 100:777-779. [PMID: 30675842 DOI: 10.4269/ajtmh.18-0624] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/07/2022] Open
Abstract
Gains in reducing childhood disease burden rely heavily on effective means of preventing environmental exposures. For many environmental health interventions, such as point-of-use water treatment, sanitation, or cookstoves, exposures are strongly influenced by user behaviors and the degree to which participants adhere to the prescribed preventive measures. In this commentary, we articulate the need for increased attention on user behaviors-critically, the careful measurement and inclusion of compliance-to strengthen exposure assessment and health impact trials in environmental health intervention research. We focus here on water, sanitation, and hygiene interventions to illustrate the problem with the understanding that this issue extends to other environmental health interventions.
Collapse
Affiliation(s)
- Joe Brown
- Georgia Institute of Technology, Atlanta, Georgia
| | | | | |
Collapse
|
7
|
Geremew A, Mengistie B, Mellor J, Lantagne DS, Alemayehu E, Sahilu G. Consistent point-of-use water chlorination among households using unimproved water sources and treatment preference in Eastern Ethiopia. INTERNATIONAL JOURNAL OF ENVIRONMENTAL HEALTH RESEARCH 2019; 29:686-701. [PMID: 30694087 DOI: 10.1080/09603123.2019.1569210] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/01/2018] [Accepted: 01/09/2019] [Indexed: 06/09/2023]
Abstract
A consistent and correct use of water treatment options is fundamental for health gain from point-of-use water chlorination. In Ethiopia, empirical evidence about the consistent use and preference for by end users is scarce. The current study is intended to explore these issues using Waterguard and Bishan Gari in Kersa Health and Demographic Surveillance Site, Eastern Ethiopia. Data were collected every 2 week for 4 months and analyzed using mixed effects logistic regression. Of 377 households, 31% consistently used Waterguard and Bishan Gari in all the study waves. Product use over time within a household varied based on the perception about the products and the type of products each household received. From the exit assessment, 56% of households stated a preference for Waterguard significantly different from 5.6% of households that stated a preference for Bishan Gari. An intervention for consistent use beyond their availability and accessibility to end users is needed.
Collapse
Affiliation(s)
- Abraham Geremew
- Ethiopian Institute of Water Resources, Water and Health, Addis Ababa University , Addis Ababa , Ethiopia
| | - Bezatu Mengistie
- College of Health and Medical Sciences, Haramaya University , Haramaya , Ethiopia
| | - Jonathan Mellor
- Department of Civil and Environmental Engineering, University of Connecticut , Hartford , CT , USA
| | - Daniele Susan Lantagne
- Department of Civil and Environmental Engineering, Tufts University , Medford , MA , USA
| | - Esayas Alemayehu
- Department of Civil and Environmental Engineering, Jimma University Institute of Technology , Jimma , Ethiopia
| | - Geremew Sahilu
- Ethiopian Institute of Water Resources, Addis Ababa University , Addis Ababa , Ethiopia
| |
Collapse
|
8
|
Hayashi MA, Eisenberg MC, Eisenberg JN. Linking Decision Theory and Quantitative Microbial Risk Assessment: Tradeoffs Between Compliance and Efficacy for Waterborne Disease Interventions. RISK ANALYSIS : AN OFFICIAL PUBLICATION OF THE SOCIETY FOR RISK ANALYSIS 2019; 39:2214-2226. [PMID: 31529800 PMCID: PMC8634750 DOI: 10.1111/risa.13381] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/04/2018] [Revised: 03/27/2019] [Accepted: 04/05/2019] [Indexed: 05/30/2023]
Abstract
Achieving health gains from the U.N. Sustainable Development Goals of universal coverage for water and sanitation will require interventions that can be widely adopted and maintained. Effectiveness-how an intervention performs based on actual use-as opposed to efficacy will therefore be central to evaluations of new and existing interventions. Incomplete compliance-when people do not always use the intervention and are therefore exposed to contamination-is thought to be responsible for the lower-than-expected risk reductions observed from water, sanitation, and hygiene interventions based on their efficacy at removing pathogens. We explicitly incorporated decision theory into a quantitative microbial risk assessment model. Specifically, we assume that the usability of household water treatment (HWT) devices (filters and chlorine) decreases as they become more efficacious due to issues such as taste or flow rates. Simulations were run to examine the tradeoff between device efficacy and usability. For most situations, HWT interventions that trade lower efficacy (i.e., remove less pathogens) for higher compliance (i.e., better usability) contribute substantial reductions in diarrheal disease risk compared to devices meeting current World Health Organization efficacy guidelines. Recommendations that take into account both the behavioral and microbiological properties of treatment devices are likely to be more effective at reducing the burden of diarrheal disease than current standards that only consider efficacy.
Collapse
Affiliation(s)
- Michael A.L. Hayashi
- Department of Epidemiology, University of Michigan School of Public Health, Ann Arbor, MI, USA
| | - Marisa C. Eisenberg
- Department of Epidemiology, University of Michigan School of Public Health, Ann Arbor, MI, USA
- Department of Mathematics, University of Michigan, Ann Arbor, MI, USA
- Center for the Study of Complex Systems, University of Michigan, Ann Arbor, MI, USA
| | - Joseph N.S. Eisenberg
- Department of Epidemiology, University of Michigan School of Public Health, Ann Arbor, MI, USA
| |
Collapse
|