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Saldaña F, Kebir A, Camacho-Gutiérrez JA, Aguiar M. Optimal vaccination strategies for a heterogeneous population using multiple objectives: The case of L 1- and L 2-formulations. Math Biosci 2023; 366:109103. [PMID: 37918477 DOI: 10.1016/j.mbs.2023.109103] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/12/2023] [Revised: 09/29/2023] [Accepted: 10/28/2023] [Indexed: 11/04/2023]
Abstract
The choice of the objective functional in optimization problems coming from biomedical and epidemiological applications plays a key role in optimal control outcomes. In this study, we investigate the role of the objective functional on the structure of the optimal control solution for an epidemic model for sexually transmitted infections that includes a core group with higher sexual activity levels than the rest of the population. An optimal control problem is formulated to find a targeted vaccination program able to control the spread of the infection with minimum vaccine deployment. Both L1- and L2-objectives are considered as an attempt to explore the trade-offs between control dynamics and the functional form characterizing optimality. The results show that the optimal vaccination policies for both the L1- and the L2-formulation share one important qualitative property, that is, immunization of the core group should be prioritized by policymakers to achieve a fast reduction of the epidemic. However, quantitative aspects of this result can be significantly affected depending on the choice of the control weights between formulations. Overall, the results suggest that with appropriate weight constants, the optimal control outcomes are reasonably robust with respect to the L1- or L2-formulation. This is particularly true when the monetary cost of the control policy is substantially lower than the cost associated with the disease burden. Under these conditions, even if the L1-formulation is more realistic from a modeling perspective, the L2-formulation can be used as an approximation and yield qualitatively comparable outcomes.
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Affiliation(s)
| | - Amira Kebir
- Basque Center for Applied Mathematics (BCAM), Bilbao, Spain; IPEIT, Tunis University, Tunis, Tunisia; BIMS-IPT, Tunis El Manar University, Tunis, Tunisia
| | | | - Maíra Aguiar
- Basque Center for Applied Mathematics (BCAM), Bilbao, Spain; Ikerbasque, Basque Foundation for Science, Bilbao, Spain; Dipartimento di Matematica, Università degli Studi di Trento, Trento, Italy
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2
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Hill AN, Glasser JW, Feng Z. Implications for infectious disease models of heterogeneous mixing on control thresholds. J Math Biol 2023; 86:53. [PMID: 36884154 PMCID: PMC9993378 DOI: 10.1007/s00285-023-01886-9] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/14/2022] [Revised: 02/07/2023] [Accepted: 02/10/2023] [Indexed: 03/09/2023]
Abstract
Mixing among sub-populations, as well as heterogeneity in characteristics affecting their reproduction numbers, must be considered when evaluating public health interventions to prevent or control infectious disease outbreaks. In this overview, we apply a linear algebraic approach to re-derive some well-known results pertaining to preferential within- and proportionate among-group contacts in compartmental models of pathogen transmission. We give results for the meta-population effective reproduction number ([Formula: see text]) assuming different levels of vaccination in the sub-populations. Specifically, we unpack the dependency of [Formula: see text] on the fractions of contacts reserved for individuals within one's own subgroup and, by obtaining implicit expressions for the partial derivatives of [Formula: see text], we show that these increase as this preferential-mixing fraction increases in any sub-population.
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Affiliation(s)
- Andrew N Hill
- Global Health Center, U.S. Centers for Disease Control and Prevention, Atlanta, GA, USA.
| | - John W Glasser
- National Center for Immunization and Respiratory Diseases, U.S. Centers for Disease Control and Prevention, Atlanta, GA, USA
| | - Zhilan Feng
- Division of Mathematical Sciences, National Science Foundation, Alexandria, VA, USA.,Department of Mathematics, Purdue University, West Lafayette, IN, USA
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3
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Renardy M, Kirschner D, Eisenberg M. Structural identifiability analysis of age-structured PDE epidemic models. J Math Biol 2022; 84:9. [PMID: 34982260 PMCID: PMC8724244 DOI: 10.1007/s00285-021-01711-1] [Citation(s) in RCA: 9] [Impact Index Per Article: 4.5] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/12/2021] [Revised: 10/21/2021] [Accepted: 12/22/2021] [Indexed: 11/24/2022]
Abstract
Computational and mathematical models rely heavily on estimated parameter values for model development. Identifiability analysis determines how well the parameters of a model can be estimated from experimental data. Identifiability analysis is crucial for interpreting and determining confidence in model parameter values and to provide biologically relevant predictions. Structural identifiability analysis, in which one assumes data to be noiseless and arbitrarily fine-grained, has been extensively studied in the context of ordinary differential equation (ODE) models, but has not yet been widely explored for age-structured partial differential equation (PDE) models. These models present additional difficulties due to increased number of variables and partial derivatives as well as the presence of boundary conditions. In this work, we establish a pipeline for structural identifiability analysis of age-structured PDE models using a differential algebra framework and derive identifiability results for specific age-structured models. We use epidemic models to demonstrate this framework because of their wide-spread use in many different diseases and for the corresponding parallel work previously done for ODEs. In our application of the identifiability analysis pipeline, we focus on a Susceptible-Exposed-Infected model for which we compare identifiability results for a PDE and corresponding ODE system and explore effects of age-dependent parameters on identifiability. We also show how practical identifiability analysis can be applied in this example.
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Affiliation(s)
- Marissa Renardy
- Department of Microbiology and Immunology, University of Michigan Medical School, Ann Arbor, USA
| | - Denise Kirschner
- Department of Microbiology and Immunology, University of Michigan Medical School, Ann Arbor, USA
| | - Marisa Eisenberg
- Department of Epidemiology, University of Michigan, Ann Arbor, USA
- Department of Mathematics, University of Michigan, Ann Arbor, USA
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4
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Liu Q, Jiang D. Global dynamical behavior of a multigroup SVIR epidemic model with Markovian switching. INT J BIOMATH 2021. [DOI: 10.1142/s1793524521500807] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022]
Abstract
In this paper, we are concerned with the global dynamical behavior of a multigroup SVIR epidemic model, which is formulated as a piecewise-deterministic Markov process. We first obtain sufficient criteria for extinction of the diseases. Then we establish sufficient criteria for persistence in the mean of the diseases. Moreover, in the case of persistence, we find a domain which is positive recurrence for the solution of the stochastic system by constructing an appropriate Lyapunov function with regime switching.
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Affiliation(s)
- Qun Liu
- Key Laboratory of Applied Statistics of MOE, School of Mathematics and Statistics, Northeast Normal University, Changchun 130024, Jilin Province, P. R. China
| | - Daqing Jiang
- Key Laboratory of Unconventional Oil and Gas Development, China University of Petroleum (East China), Ministry of Education, Qingdao 266580, P. R. China
- College of Science, China University of Petroleum, Qingdao 266580, Shandong Province, P. R. China
- Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, King Abdulaziz University, Jeddah, Saudi Arabia
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5
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Feng Z, Glasser JW. Infectious Disease Modeling. SYSTEMS MEDICINE 2021. [DOI: 10.1016/b978-0-12-801238-3.11471-0] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/25/2022] Open
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6
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Chen Z, Liu K, Liu X, Lou Y. Modelling epidemics with fractional-dose vaccination in response to limited vaccine supply. J Theor Biol 2019; 486:110085. [PMID: 31758966 DOI: 10.1016/j.jtbi.2019.110085] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/18/2019] [Revised: 10/10/2019] [Accepted: 11/16/2019] [Indexed: 11/26/2022]
Abstract
The control strategies of emergency infectious diseases are constrained by limited medical resources. The fractional dose vaccination strategy as one of feasible strategies was proposed in response to global shortages of vaccine stockpiles. Although a variety of epidemic models have been developed under the circumstances of limited resources in treatment, few models particularly investigated vaccination strategies in resource-limited settings. In this paper, we develop a two-group SIR model with incorporation of proportionate mixing patterns and n-fold fractional dose vaccination related parameters to evaluate the efficiency of fractional dose vaccination on disease control at the population level. The existence and uniqueness of the final size of the two-group SIR epidemic model, the formulation of the basic reproduction number and the relationship between them are established. Moreover, numerical simulations are performed based on this two-group vector-free model to investigate the effectiveness of n-fold fractional dose vaccination by using the emergency outbreaks of yellow fever in Angola in 2016. By employing linear and nonlinear dose-response relationships, we compare the resulting fluctuations of four characteristics of the epidemics, which are the outbreak size, the peak time of the outbreak, the basic reproduction number and the infection attack rate (IAR). For both types of dose-response relationships, dose-fractionation takes positive effects in lowering the outbreak size, delay the peak time of the outbreak, reducing the basic reproduction number and the IAR of yellow fever only when the vaccine efficacy is high enough. Moreover, five-fold fractional dose vaccination strategy may not be the optimal vaccination strategy as proposed by the World Health Organization if the dose-response relationship is nonlinear.
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Affiliation(s)
- Zhimin Chen
- School of Mathematical Sciences, South China Normal University, Guangzhou 510631, PR China.
| | - Kaihui Liu
- Faculty of Science, Jiangsu University, Zhenjiang, Jiangsu 212013, PR China.
| | - Xiuxiang Liu
- School of Mathematical Sciences, South China Normal University, Guangzhou 510631, PR China.
| | - Yijun Lou
- Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong.
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A general theory for target reproduction numbers with applications to ecology and epidemiology. J Math Biol 2019; 78:2317-2339. [PMID: 30854577 DOI: 10.1007/s00285-019-01345-4] [Citation(s) in RCA: 16] [Impact Index Per Article: 3.2] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/02/2018] [Revised: 02/16/2019] [Indexed: 10/27/2022]
Abstract
A general framework for threshold parameters in population dynamics is developed using the concept of target reproduction numbers. This framework identifies reproduction numbers and other threshold parameters in the literature in terms of their roles in population control. The framework is applied to the analysis of single and multiple control strategies in ecology and epidemiology, and this provides new biological insights.
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Dénes A, Székely L. Global dynamics of a mathematical model for the possible re-emergence of polio. Math Biosci 2017; 293:64-74. [PMID: 28859911 DOI: 10.1016/j.mbs.2017.08.010] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/26/2016] [Revised: 08/03/2017] [Accepted: 08/25/2017] [Indexed: 11/26/2022]
Abstract
Motivated by studies warning about a possible re-emergence of poliomyelitis in Europe, we analyse a compartmental model for the transmission of polio describing the possible effect of unvaccinated people arriving to a region with low vaccination coverage. We calculate the basic reproduction number, and determine the global dynamics of the system: we show that, depending on the parameters, one of the two equilibria is globally asymptotically stable. The main tools applied are Lyapunov functions and persistence theory. We illustrate the analytic results by numerical examples, which also suggest that in order to avoid the risk of polio re-emergence, vaccinating the immigrant population might result insufficient, and also the vaccination coverage of countries with low rates should be increased.
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Affiliation(s)
- Attila Dénes
- Bolyai Institute, University of Szeged, Aradi vértanúk tere 1., Szeged H-6720, Hungary.
| | - László Székely
- Institute for Environmental Systems, Szent István University, Páter Károly utca 1., Gödöllő H-2103, Hungary
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9
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Li S, Fan M, Rong X. Global threshold dynamics of SIQS epidemic model in time fluctuating environment. INT J BIOMATH 2017. [DOI: 10.1142/s1793524517500607] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022]
Abstract
The paper characterizes the global threshold dynamics of an epidemic model of SIQS type in environments with fluctuations, where the quarantine class is explicitly involved. Criteria are established for the permanence and extinction of the infective in environments with time oscillations. In particular, we further consider an environment which varies periodically in time. The global threshold dynamic scenarios i.e. the existence and global asymptotic stability of the disease-free periodic solution, the existence of the endemic periodic solution and the permanence of the infective are completely characterized by the basic reproduction number defined by the spectral radius of an associated linear integral operator.
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Affiliation(s)
- Shang Li
- School of Mathematics and Statistics, Northeast Normal University, 5268 Renmin Street, Changchun, Jilin 130024, P. R. China
| | - Meng Fan
- School of Mathematics and Statistics, Northeast Normal University, 5268 Renmin Street, Changchun, Jilin 130024, P. R. China
| | - Xinmiao Rong
- School of Mathematics and Statistics, Northeast Normal University, 5268 Renmin Street, Changchun, Jilin 130024, P. R. China
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Assessing the Potential Impact of Hormonal-Based Contraceptives on HIV Transmission Dynamics Among Heterosexuals. Bull Math Biol 2017; 79:738-771. [PMID: 28258539 DOI: 10.1007/s11538-017-0252-y] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/18/2015] [Accepted: 02/02/2017] [Indexed: 10/20/2022]
Abstract
HIV susceptibility linked to hormonal contraception (HC) has been studied before, but with mixed results. Reports from some of the recent findings have prompted the World Health Organisation to encourage women who use HC to concurrently use condoms in order to prevent HIV infection in the light of possible increased HIV risk of infection associated with hormone-based contraceptives. A two-sex HIV model classifying women into three risk groups consisting of individuals who use condoms, natural methods, and hormone-based contraceptives is formulated and analysed to assess the possible effects of various birth control strategies on the transmission dynamics of the disease. Our model results showed that women who use HC could be key drivers of the epidemic and that their increased infectivity may be critical in driving the epidemic. Women who use hormone-based contraceptives potentially act as a core group from which men get infected and in turn transmit the disease to other population groups. We fitted the model to HIV prevalence data for Zimbabwe reported by UNAIDS and Zimbabwe Ministry of Health and Child Care and used the model fit to project HIV prevalence. Predictions using HIV data for Zimbabwe suggest that a hypothesised increase in susceptibility and infectivity of two-, three-, and fourfold would result in a 25, 50, and 100% increase in baseline HIV prevalence projection, respectively, thus suggesting possible increased disease burden even in countries reporting plausible HIV prevalence declines. Although a possible causal relationship between HIV susceptibility and HC use remains subject of continuing scientific probe, its inclusion as part of birth control strategy has been shown in this study, to possibly increase HIV transmission. If proven, HC use may potentially explain the inordinate spread of HIV within the sub-Saharan Africa region and therefore compel for urgent assessment with a view to reorienting birth control methods in use in settings with generalised epidemics.
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Feng Z, Hill AN, Smith PJ, Glasser JW. An elaboration of theory about preventing outbreaks in homogeneous populations to include heterogeneity or preferential mixing. J Theor Biol 2015; 386:177-87. [PMID: 26375548 DOI: 10.1016/j.jtbi.2015.09.006] [Citation(s) in RCA: 32] [Impact Index Per Article: 3.6] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/02/2014] [Revised: 09/03/2015] [Accepted: 09/04/2015] [Indexed: 11/30/2022]
Abstract
The goal of many vaccination programs is to attain the population immunity above which pathogens introduced by infectious people (e.g., travelers from endemic areas) will not cause outbreaks. Using a simple meta-population model, we demonstrate that, if sub-populations either differ in characteristics affecting their basic reproduction numbers or if their members mix preferentially, weighted average sub-population immunities cannot be compared with the proportionally-mixing homogeneous population-immunity threshold, as public health practitioners are wont to do. Then we review the effect of heterogeneity in average per capita contact rates on the basic meta-population reproduction number. To the extent that population density affects contacts, for example, rates might differ in urban and rural sub-populations. Other differences among sub-populations in characteristics affecting their basic reproduction numbers would contribute similarly. In agreement with more recent results, we show that heterogeneous preferential mixing among sub-populations increases the basic meta-population reproduction number more than homogeneous preferential mixing does. Next we refine earlier results on the effects of heterogeneity in sub-population immunities and preferential mixing on the effective meta-population reproduction number. Finally, we propose the vector of partial derivatives of this reproduction number with respect to the sub-population immunities as a fundamentally new tool for targeting vaccination efforts.
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Affiliation(s)
- Zhilan Feng
- Department of Mathematics, Purdue University, West Lafayette, IN, USA
| | - Andrew N Hill
- National Center for HIV/AIDS, Viral Hepatitis, STD, and TB Prevention, CDC, Atlanta, GA, USA
| | - Philip J Smith
- National Center for Immunization and Respiratory Diseases, CDC, 1600 Clifton Road, NE, Mail Stop A-34, Atlanta, GA 30333, USA
| | - John W Glasser
- National Center for Immunization and Respiratory Diseases, CDC, 1600 Clifton Road, NE, Mail Stop A-34, Atlanta, GA 30333, USA
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12
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De Vos AS, Kretzschmar MEE. The efficiency of targeted intervention in limiting the spread of HIV and Hepatitis C Virus among injecting drug users. J Theor Biol 2013; 333:126-34. [PMID: 23733004 DOI: 10.1016/j.jtbi.2013.05.017] [Citation(s) in RCA: 12] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/21/2013] [Revised: 04/10/2013] [Accepted: 05/21/2013] [Indexed: 11/29/2022]
Abstract
BACKGROUND Interventions aimed at minimizing the spread of blood borne infections among Injecting Drug Users (IDU) are impeded by limitations in resources. To enhance their efficiency, it may be beneficial to target specific behavioural subpopulations, distinguished by syringe sharing tendencies. METHODS We used mathematical modelling to explore the effects of two types of intervention: removal of individuals from the injecting population and risk decrease at group-level (e.g. distribution of syringes). We computed the direct effects of intervention on the probability of obtaining and spreading infection as a function of baseline risk behaviour. Population level effects of (targeted) intervention were explored using a differential equations model, which incorporated two levels of risk. RESULTS Within most scenarios of risk distribution considered, HIV could be substantially reduced or eliminated by targeting high risk IDU only. Conversely, higher incidence reductions for HCV were reached in many scenarios when targeting low risk IDU. The potential for preventing infections by removal of uninfected IDU increases with baseline risk, but so does the probability that an IDU is already infected before being reached by intervention. Decreasing risk is likely to only delay rather than prevent infection for IDU borrowing many syringes, especially for a very infectious disease such as HCV. CONCLUSIONS The efficiency of intervention on injecting drug users may be much enhanced by targeting specific risk subgroups. However, the optimal targeting policy depends strongly on the infection under consideration.
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Affiliation(s)
- Anneke S De Vos
- Julius Center, University Medical Center Utrecht, Stratenum 6.131, Postbus 85500, 3508GA Utrecht, The Netherlands.
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Shuai Z, Heesterbeek JAP, van den Driessche P. Extending the type reproduction number to infectious disease control targeting contacts between types. J Math Biol 2012; 67:1067-82. [PMID: 22941454 DOI: 10.1007/s00285-012-0579-9] [Citation(s) in RCA: 49] [Impact Index Per Article: 4.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/02/2012] [Revised: 07/20/2012] [Indexed: 11/25/2022]
Abstract
A new quantity called the target reproduction number is defined to measure control strategies for infectious diseases with multiple host types such as waterborne, vector-borne and zoonotic diseases. The target reproduction number includes as a special case and extends the type reproduction number to allow disease control targeting contacts between types. Relationships among the basic, type and target reproduction numbers are established. Examples of infectious disease models from the literature are given to illustrate the use of the target reproduction number.
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Affiliation(s)
- Zhisheng Shuai
- Department of Mathematics and Statistics, University of Victoria, Victoria, BC, V8W 3R4, Canada,
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Ward K, Chow MYK, King C, Leask J. Strategies to improve vaccination uptake in Australia, a systematic review of types and effectiveness. Aust N Z J Public Health 2012. [DOI: 10.1111/j.1753-6405.2012.00897.x] [Citation(s) in RCA: 37] [Impact Index Per Article: 3.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022] Open
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15
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Malunguza NJ, Hove-Musekwa SD, Musuka G, Mukandavire Z. Investigating Alcohol Consumption as a Risk Factor for HIV Transmission in Heterosexual Settings in Sub-Saharan African Communities. Bull Math Biol 2012; 74:2094-124. [DOI: 10.1007/s11538-012-9747-8] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/25/2010] [Accepted: 06/26/2012] [Indexed: 11/24/2022]
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